LMFDB - Embedded newform 8001.2.a.p.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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 $ x^14 - 5x^13 - 9x^12 + 76x^11 - 12x^10 - 414x^9 + 331x^8 + 959x^7 - 1067x^6 - 875x^5 + 1134x^4 + 301x^3 - 418x^2 - 42*x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.16813$ 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)

$f(q)$	$=$	$q-2.16813 q^{2} +2.70077 q^{4} +1.98599 q^{5} -1.00000 q^{7} -1.51936 q^{8} +O(q^{10})$	\(q-2.16813 q^{2} +2.70077 q^{4} +1.98599 q^{5} -1.00000 q^{7} -1.51936 q^{8} -4.30589 q^{10} -0.510474 q^{11} +3.25111 q^{13} +2.16813 q^{14} -2.10738 q^{16} -0.751379 q^{17} +1.71057 q^{19} +5.36372 q^{20} +1.10677 q^{22} +6.15159 q^{23} -1.05582 q^{25} -7.04881 q^{26} -2.70077 q^{28} +9.05238 q^{29} +4.10900 q^{31} +7.60778 q^{32} +1.62908 q^{34} -1.98599 q^{35} -0.843790 q^{37} -3.70873 q^{38} -3.01744 q^{40} +0.431385 q^{41} -9.52893 q^{43} -1.37867 q^{44} -13.3374 q^{46} +2.32862 q^{47} +1.00000 q^{49} +2.28916 q^{50} +8.78049 q^{52} +10.0827 q^{53} -1.01380 q^{55} +1.51936 q^{56} -19.6267 q^{58} +10.9113 q^{59} +3.80129 q^{61} -8.90882 q^{62} -12.2799 q^{64} +6.45668 q^{65} +0.655991 q^{67} -2.02930 q^{68} +4.30589 q^{70} -7.96140 q^{71} -8.99944 q^{73} +1.82944 q^{74} +4.61986 q^{76} +0.510474 q^{77} -0.0468384 q^{79} -4.18524 q^{80} -0.935298 q^{82} +1.02880 q^{83} -1.49224 q^{85} +20.6599 q^{86} +0.775594 q^{88} -5.59309 q^{89} -3.25111 q^{91} +16.6140 q^{92} -5.04875 q^{94} +3.39718 q^{95} -12.4534 q^{97} -2.16813 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.16813	−1.53310	−0.766548	−	0.642187i	$-0.778027\pi$
$2$	−2.16813	−1.53310	−0.766548	+	0.642187i	$0.778027\pi$
$3$	0	0
$3$	0	0
$4$	2.70077	1.35039
$5$	1.98599	0.888164	0.444082	−	0.895986i	$-0.353530\pi$
$5$	1.98599	0.888164	0.444082	+	0.895986i	$0.353530\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.51936	−0.537175
$9$	0	0
$10$	−4.30589	−1.36164
$11$	−0.510474	−0.153914	−0.0769568	−	0.997034i	$-0.524520\pi$
$11$	−0.510474	−0.153914	−0.0769568	+	0.997034i	$0.524520\pi$
$12$	0	0
$13$	3.25111	0.901695	0.450847	−	0.892601i	$-0.351122\pi$
$13$	3.25111	0.901695	0.450847	+	0.892601i	$0.351122\pi$
$14$	2.16813	0.579456
$15$	0	0
$16$	−2.10738	−0.526844
$17$	−0.751379	−0.182236	−0.0911181	−	0.995840i	$-0.529044\pi$
$17$	−0.751379	−0.182236	−0.0911181	+	0.995840i	$0.529044\pi$
$18$	0	0
$19$	1.71057	0.392432	0.196216	−	0.980561i	$-0.437135\pi$
$19$	1.71057	0.392432	0.196216	+	0.980561i	$0.437135\pi$
$20$	5.36372	1.19936
$21$	0	0
$22$	1.10677	0.235964
$23$	6.15159	1.28269	0.641347	−	0.767251i	$-0.278376\pi$
$23$	6.15159	1.28269	0.641347	+	0.767251i	$0.278376\pi$
$24$	0	0
$25$	−1.05582	−0.211165
$26$	−7.04881	−1.38238
$27$	0	0
$28$	−2.70077	−0.510398
$29$	9.05238	1.68098	0.840492	−	0.541823i	$-0.182266\pi$
$29$	9.05238	1.68098	0.840492	+	0.541823i	$0.182266\pi$
$30$	0	0
$31$	4.10900	0.737997	0.368999	−	0.929430i	$-0.379701\pi$
$31$	4.10900	0.737997	0.368999	+	0.929430i	$0.379701\pi$
$32$	7.60778	1.34488
$33$	0	0
$34$	1.62908	0.279386
$35$	−1.98599	−0.335694
$36$	0	0
$37$	−0.843790	−0.138718	−0.0693591	−	0.997592i	$-0.522095\pi$
$37$	−0.843790	−0.138718	−0.0693591	+	0.997592i	$0.522095\pi$
$38$	−3.70873	−0.601636
$39$	0	0
$40$	−3.01744	−0.477100
$41$	0.431385	0.0673711	0.0336855	−	0.999432i	$-0.489276\pi$
$41$	0.431385	0.0673711	0.0336855	+	0.999432i	$0.489276\pi$
$42$	0	0
$43$	−9.52893	−1.45315	−0.726574	−	0.687088i	$-0.758888\pi$
$43$	−9.52893	−1.45315	−0.726574	+	0.687088i	$0.758888\pi$
$44$	−1.37867	−0.207843
$45$	0	0
$46$	−13.3374	−1.96650
$47$	2.32862	0.339664	0.169832	−	0.985473i	$-0.445677\pi$
$47$	2.32862	0.339664	0.169832	+	0.985473i	$0.445677\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.28916	0.323736
$51$	0	0
$52$	8.78049	1.21764
$53$	10.0827	1.38497	0.692484	−	0.721433i	$-0.256516\pi$
$53$	10.0827	1.38497	0.692484	+	0.721433i	$0.256516\pi$
$54$	0	0
$55$	−1.01380	−0.136701
$56$	1.51936	0.203033
$57$	0	0
$58$	−19.6267	−2.57711
$59$	10.9113	1.42053	0.710265	−	0.703934i	$-0.248575\pi$
$59$	10.9113	1.42053	0.710265	+	0.703934i	$0.248575\pi$
$60$	0	0
$61$	3.80129	0.486705	0.243353	−	0.969938i	$-0.421753\pi$
$61$	3.80129	0.486705	0.243353	+	0.969938i	$0.421753\pi$
$62$	−8.90882	−1.13142
$63$	0	0
$64$	−12.2799	−1.53498
$65$	6.45668	0.800853
$66$	0	0
$67$	0.655991	0.0801420	0.0400710	−	0.999197i	$-0.487242\pi$
$67$	0.655991	0.0801420	0.0400710	+	0.999197i	$0.487242\pi$
$68$	−2.02930	−0.246089
$69$	0	0
$70$	4.30589	0.514652
$71$	−7.96140	−0.944844	−0.472422	−	0.881372i	$-0.656620\pi$
$71$	−7.96140	−0.944844	−0.472422	+	0.881372i	$0.656620\pi$
$72$	0	0
$73$	−8.99944	−1.05330	−0.526652	−	0.850081i	$-0.676553\pi$
$73$	−8.99944	−1.05330	−0.526652	+	0.850081i	$0.676553\pi$
$74$	1.82944	0.212668
$75$	0	0
$76$	4.61986	0.529934
$77$	0.510474	0.0581739
$78$	0	0
$79$	−0.0468384	−0.00526973	−0.00263487	−	0.999997i	$-0.500839\pi$
$79$	−0.0468384	−0.00526973	−0.00263487	+	0.999997i	$0.500839\pi$
$80$	−4.18524	−0.467924
$81$	0	0
$82$	−0.935298	−0.103286
$83$	1.02880	0.112925	0.0564626	−	0.998405i	$-0.482018\pi$
$83$	1.02880	0.112925	0.0564626	+	0.998405i	$0.482018\pi$
$84$	0	0
$85$	−1.49224	−0.161856
$86$	20.6599	2.22782
$87$	0	0
$88$	0.775594	0.0826786
$89$	−5.59309	−0.592866	−0.296433	−	0.955054i	$-0.595797\pi$
$89$	−5.59309	−0.592866	−0.296433	+	0.955054i	$0.595797\pi$
$90$	0	0
$91$	−3.25111	−0.340808
$92$	16.6140	1.73213
$93$	0	0
$94$	−5.04875	−0.520738
$95$	3.39718	0.348544
$96$	0	0
$97$	−12.4534	−1.26445	−0.632225	−	0.774785i	$-0.717858\pi$
$97$	−12.4534	−1.26445	−0.632225	+	0.774785i	$0.717858\pi$
$98$	−2.16813	−0.219014
$99$	0	0
$100$	−2.85154	−0.285154
$101$	0.460231	0.0457947	0.0228973	−	0.999738i	$-0.492711\pi$
$101$	0.460231	0.0457947	0.0228973	+	0.999738i	$0.492711\pi$
$102$	0	0
$103$	5.66914	0.558597	0.279299	−	0.960204i	$-0.409898\pi$
$103$	5.66914	0.558597	0.279299	+	0.960204i	$0.409898\pi$
$104$	−4.93961	−0.484368
$105$	0	0
$106$	−21.8606	−2.12329
$107$	5.32050	0.514352	0.257176	−	0.966365i	$-0.417208\pi$
$107$	5.32050	0.514352	0.257176	+	0.966365i	$0.417208\pi$
$108$	0	0
$109$	7.45032	0.713612	0.356806	−	0.934179i	$-0.383866\pi$
$109$	7.45032	0.713612	0.356806	+	0.934179i	$0.383866\pi$
$110$	2.19804	0.209575
$111$	0	0
$112$	2.10738	0.199128
$113$	−9.00060	−0.846705	−0.423352	−	0.905965i	$-0.639147\pi$
$113$	−9.00060	−0.846705	−0.423352	+	0.905965i	$0.639147\pi$
$114$	0	0
$115$	12.2170	1.13924
$116$	24.4484	2.26998
$117$	0	0
$118$	−23.6571	−2.17781
$119$	0.751379	0.0688788
$120$	0	0
$121$	−10.7394	−0.976311
$122$	−8.24167	−0.746166
$123$	0	0
$124$	11.0975	0.996581
$125$	−12.0268	−1.07571
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	11.4088	1.00840
$129$	0	0
$130$	−13.9989	−1.22778
$131$	−3.06170	−0.267502	−0.133751	−	0.991015i	$-0.542702\pi$
$131$	−3.06170	−0.267502	−0.133751	+	0.991015i	$0.542702\pi$
$132$	0	0
$133$	−1.71057	−0.148325
$134$	−1.42227	−0.122865
$135$	0	0
$136$	1.14162	0.0978928
$137$	17.9015	1.52943	0.764716	−	0.644368i	$-0.222879\pi$
$137$	17.9015	1.52943	0.764716	+	0.644368i	$0.222879\pi$
$138$	0	0
$139$	12.3891	1.05083	0.525413	−	0.850847i	$-0.323911\pi$
$139$	12.3891	1.05083	0.525413	+	0.850847i	$0.323911\pi$
$140$	−5.36372	−0.453317
$141$	0	0
$142$	17.2613	1.44854
$143$	−1.65960	−0.138783
$144$	0	0
$145$	17.9780	1.49299
$146$	19.5119	1.61482
$147$	0	0
$148$	−2.27888	−0.187323
$149$	5.85233	0.479441	0.239721	−	0.970842i	$-0.422944\pi$
$149$	5.85233	0.479441	0.239721	+	0.970842i	$0.422944\pi$
$150$	0	0
$151$	10.7951	0.878489	0.439245	−	0.898368i	$-0.355246\pi$
$151$	10.7951	0.878489	0.439245	+	0.898368i	$0.355246\pi$
$152$	−2.59898	−0.210805
$153$	0	0
$154$	−1.10677	−0.0891862
$155$	8.16044	0.655463
$156$	0	0
$157$	−2.80193	−0.223618	−0.111809	−	0.993730i	$-0.535665\pi$
$157$	−2.80193	−0.223618	−0.111809	+	0.993730i	$0.535665\pi$
$158$	0.101552	0.00807901
$159$	0	0
$160$	15.1090	1.19447
$161$	−6.15159	−0.484813
$162$	0	0
$163$	12.2769	0.961599	0.480800	−	0.876831i	$-0.340346\pi$
$163$	12.2769	0.961599	0.480800	+	0.876831i	$0.340346\pi$
$164$	1.16507	0.0909769
$165$	0	0
$166$	−2.23056	−0.173125
$167$	11.2074	0.867258	0.433629	−	0.901091i	$-0.357233\pi$
$167$	11.2074	0.867258	0.433629	+	0.901091i	$0.357233\pi$
$168$	0	0
$169$	−2.43031	−0.186947
$170$	3.23535	0.248140
$171$	0	0
$172$	−25.7355	−1.96231
$173$	−5.65042	−0.429593	−0.214797	−	0.976659i	$-0.568909\pi$
$173$	−5.65042	−0.429593	−0.214797	+	0.976659i	$0.568909\pi$
$174$	0	0
$175$	1.05582	0.0798128
$176$	1.07576	0.0810884
$177$	0	0
$178$	12.1265	0.908921
$179$	6.41623	0.479572	0.239786	−	0.970826i	$-0.422923\pi$
$179$	6.41623	0.479572	0.239786	+	0.970826i	$0.422923\pi$
$180$	0	0
$181$	4.05868	0.301679	0.150840	−	0.988558i	$-0.451802\pi$
$181$	4.05868	0.301679	0.150840	+	0.988558i	$0.451802\pi$
$182$	7.04881	0.522492
$183$	0	0
$184$	−9.34649	−0.689032
$185$	−1.67576	−0.123205
$186$	0	0
$187$	0.383559	0.0280486
$188$	6.28908	0.458678
$189$	0	0
$190$	−7.36552	−0.534351
$191$	−13.9617	−1.01023	−0.505115	−	0.863052i	$-0.668550\pi$
$191$	−13.9617	−1.01023	−0.505115	+	0.863052i	$0.668550\pi$
$192$	0	0
$193$	8.41589	0.605789	0.302895	−	0.953024i	$-0.402047\pi$
$193$	8.41589	0.605789	0.302895	+	0.953024i	$0.402047\pi$
$194$	27.0005	1.93853
$195$	0	0
$196$	2.70077	0.192912
$197$	−15.2959	−1.08979	−0.544895	−	0.838504i	$-0.683431\pi$
$197$	−15.2959	−1.08979	−0.544895	+	0.838504i	$0.683431\pi$
$198$	0	0
$199$	0.737303	0.0522660	0.0261330	−	0.999658i	$-0.491681\pi$
$199$	0.737303	0.0522660	0.0261330	+	0.999658i	$0.491681\pi$
$200$	1.60418	0.113433
$201$	0	0
$202$	−0.997839	−0.0702077
$203$	−9.05238	−0.635353
$204$	0	0
$205$	0.856729	0.0598365
$206$	−12.2914	−0.856384
$207$	0	0
$208$	−6.85130	−0.475052
$209$	−0.873201	−0.0604006
$210$	0	0
$211$	12.1447	0.836076	0.418038	−	0.908430i	$-0.362718\pi$
$211$	12.1447	0.836076	0.418038	+	0.908430i	$0.362718\pi$
$212$	27.2311	1.87024
$213$	0	0
$214$	−11.5355	−0.788552
$215$	−18.9244	−1.29063
$216$	0	0
$217$	−4.10900	−0.278937
$218$	−16.1532	−1.09404
$219$	0	0
$220$	−2.73804	−0.184598
$221$	−2.44281	−0.164321
$222$	0	0
$223$	7.12459	0.477098	0.238549	−	0.971131i	$-0.423328\pi$
$223$	7.12459	0.477098	0.238549	+	0.971131i	$0.423328\pi$
$224$	−7.60778	−0.508316
$225$	0	0
$226$	19.5144	1.29808
$227$	7.12486	0.472894	0.236447	−	0.971644i	$-0.424017\pi$
$227$	7.12486	0.472894	0.236447	+	0.971644i	$0.424017\pi$
$228$	0	0
$229$	−16.1345	−1.06620	−0.533098	−	0.846054i	$-0.678972\pi$
$229$	−16.1345	−1.06620	−0.533098	+	0.846054i	$0.678972\pi$
$230$	−26.4880	−1.74657
$231$	0	0
$232$	−13.7538	−0.902984
$233$	−17.5265	−1.14820	−0.574098	−	0.818787i	$-0.694647\pi$
$233$	−17.5265	−1.14820	−0.574098	+	0.818787i	$0.694647\pi$
$234$	0	0
$235$	4.62463	0.301678
$236$	29.4689	1.91826
$237$	0	0
$238$	−1.62908	−0.105598
$239$	−7.72068	−0.499410	−0.249705	−	0.968322i	$-0.580334\pi$
$239$	−7.72068	−0.499410	−0.249705	+	0.968322i	$0.580334\pi$
$240$	0	0
$241$	8.53586	0.549843	0.274921	−	0.961467i	$-0.411348\pi$
$241$	8.53586	0.549843	0.274921	+	0.961467i	$0.411348\pi$
$242$	23.2844	1.49678
$243$	0	0
$244$	10.2664	0.657240
$245$	1.98599	0.126881
$246$	0	0
$247$	5.56125	0.353854
$248$	−6.24305	−0.396434
$249$	0	0
$250$	26.0757	1.64917
$251$	11.2396	0.709437	0.354718	−	0.934973i	$-0.384577\pi$
$251$	11.2396	0.709437	0.354718	+	0.934973i	$0.384577\pi$
$252$	0	0
$253$	−3.14022	−0.197424
$254$	2.16813	0.136040
$255$	0	0
$256$	−0.175889	−0.0109931
$257$	18.5101	1.15463	0.577314	−	0.816522i	$-0.304101\pi$
$257$	18.5101	1.15463	0.577314	+	0.816522i	$0.304101\pi$
$258$	0	0
$259$	0.843790	0.0524306
$260$	17.4380	1.08146
$261$	0	0
$262$	6.63815	0.410106
$263$	5.84531	0.360438	0.180219	−	0.983627i	$-0.442319\pi$
$263$	5.84531	0.360438	0.180219	+	0.983627i	$0.442319\pi$
$264$	0	0
$265$	20.0242	1.23008
$266$	3.70873	0.227397
$267$	0	0
$268$	1.77168	0.108223
$269$	8.89744	0.542487	0.271243	−	0.962511i	$-0.412565\pi$
$269$	8.89744	0.542487	0.271243	+	0.962511i	$0.412565\pi$
$270$	0	0
$271$	−22.8532	−1.38823	−0.694116	−	0.719863i	$-0.744204\pi$
$271$	−22.8532	−1.38823	−0.694116	+	0.719863i	$0.744204\pi$
$272$	1.58344	0.0960100
$273$	0	0
$274$	−38.8128	−2.34477
$275$	0.538971	0.0325011
$276$	0	0
$277$	−4.66394	−0.280229	−0.140115	−	0.990135i	$-0.544747\pi$
$277$	−4.66394	−0.280229	−0.140115	+	0.990135i	$0.544747\pi$
$278$	−26.8610	−1.61102
$279$	0	0
$280$	3.01744	0.180327
$281$	−31.0478	−1.85216	−0.926078	−	0.377332i	$-0.876842\pi$
$281$	−31.0478	−1.85216	−0.926078	+	0.377332i	$0.876842\pi$
$282$	0	0
$283$	−1.77414	−0.105461	−0.0527307	−	0.998609i	$-0.516793\pi$
$283$	−1.77414	−0.105461	−0.0527307	+	0.998609i	$0.516793\pi$
$284$	−21.5019	−1.27590
$285$	0	0
$286$	3.59823	0.212768
$287$	−0.431385	−0.0254639
$288$	0	0
$289$	−16.4354	−0.966790
$290$	−38.9785	−2.28890
$291$	0	0
$292$	−24.3054	−1.42237
$293$	13.4832	0.787696	0.393848	−	0.919176i	$-0.371144\pi$
$293$	13.4832	0.787696	0.393848	+	0.919176i	$0.371144\pi$
$294$	0	0
$295$	21.6698	1.26166
$296$	1.28202	0.0745160
$297$	0	0
$298$	−12.6886	−0.735030
$299$	19.9995	1.15660
$300$	0	0
$301$	9.52893	0.549238
$302$	−23.4050	−1.34681
$303$	0	0
$304$	−3.60481	−0.206750
$305$	7.54934	0.432274
$306$	0	0
$307$	30.0812	1.71682	0.858411	−	0.512962i	$-0.171452\pi$
$307$	30.0812	1.71682	0.858411	+	0.512962i	$0.171452\pi$
$308$	1.37867	0.0785572
$309$	0	0
$310$	−17.6929	−1.00489
$311$	0.109629	0.00621648	0.00310824	−	0.999995i	$-0.499011\pi$
$311$	0.109629	0.00621648	0.00310824	+	0.999995i	$0.499011\pi$
$312$	0	0
$313$	6.34521	0.358652	0.179326	−	0.983790i	$-0.442608\pi$
$313$	6.34521	0.358652	0.179326	+	0.983790i	$0.442608\pi$
$314$	6.07494	0.342829
$315$	0	0
$316$	−0.126500	−0.00711617
$317$	−0.777211	−0.0436525	−0.0218263	−	0.999762i	$-0.506948\pi$
$317$	−0.777211	−0.0436525	−0.0218263	+	0.999762i	$0.506948\pi$
$318$	0	0
$319$	−4.62100	−0.258726
$320$	−24.3878	−1.36332
$321$	0	0
$322$	13.3374	0.743265
$323$	−1.28529	−0.0715153
$324$	0	0
$325$	−3.43260	−0.190406
$326$	−26.6178	−1.47422
$327$	0	0
$328$	−0.655430	−0.0361901
$329$	−2.32862	−0.128381
$330$	0	0
$331$	16.1089	0.885424	0.442712	−	0.896664i	$-0.354016\pi$
$331$	16.1089	0.885424	0.442712	+	0.896664i	$0.354016\pi$
$332$	2.77855	0.152493
$333$	0	0
$334$	−24.2992	−1.32959
$335$	1.30279	0.0711792
$336$	0	0
$337$	26.6848	1.45362	0.726808	−	0.686841i	$-0.241003\pi$
$337$	26.6848	1.45362	0.726808	+	0.686841i	$0.241003\pi$
$338$	5.26922	0.286608
$339$	0	0
$340$	−4.03019	−0.218568
$341$	−2.09753	−0.113588
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	14.4779	0.780595
$345$	0	0
$346$	12.2508	0.658608
$347$	16.2059	0.869981	0.434990	−	0.900435i	$-0.356752\pi$
$347$	16.2059	0.869981	0.434990	+	0.900435i	$0.356752\pi$
$348$	0	0
$349$	−0.917312	−0.0491026	−0.0245513	−	0.999699i	$-0.507816\pi$
$349$	−0.917312	−0.0491026	−0.0245513	+	0.999699i	$0.507816\pi$
$350$	−2.28916	−0.122361
$351$	0	0
$352$	−3.88357	−0.206995
$353$	−14.9303	−0.794659	−0.397329	−	0.917676i	$-0.630063\pi$
$353$	−14.9303	−0.794659	−0.397329	+	0.917676i	$0.630063\pi$
$354$	0	0
$355$	−15.8113	−0.839177
$356$	−15.1057	−0.800598
$357$	0	0
$358$	−13.9112	−0.735230
$359$	8.96795	0.473310	0.236655	−	0.971594i	$-0.423949\pi$
$359$	8.96795	0.473310	0.236655	+	0.971594i	$0.423949\pi$
$360$	0	0
$361$	−16.0739	−0.845997
$362$	−8.79973	−0.462504
$363$	0	0
$364$	−8.78049	−0.460223
$365$	−17.8728	−0.935507
$366$	0	0
$367$	−28.2327	−1.47373	−0.736867	−	0.676037i	$-0.763696\pi$
$367$	−28.2327	−1.47373	−0.736867	+	0.676037i	$0.763696\pi$
$368$	−12.9637	−0.675780
$369$	0	0
$370$	3.63327	0.188884
$371$	−10.0827	−0.523469
$372$	0	0
$373$	−31.3000	−1.62065	−0.810325	−	0.585981i	$-0.800709\pi$
$373$	−31.3000	−1.62065	−0.810325	+	0.585981i	$0.800709\pi$
$374$	−0.831605	−0.0430013
$375$	0	0
$376$	−3.53802	−0.182459
$377$	29.4302	1.51573
$378$	0	0
$379$	28.7825	1.47846	0.739230	−	0.673453i	$-0.235190\pi$
$379$	28.7825	1.47846	0.739230	+	0.673453i	$0.235190\pi$
$380$	9.17502	0.470669
$381$	0	0
$382$	30.2707	1.54878
$383$	−29.7239	−1.51882	−0.759411	−	0.650612i	$-0.774513\pi$
$383$	−29.7239	−1.51882	−0.759411	+	0.650612i	$0.774513\pi$
$384$	0	0
$385$	1.01380	0.0516679
$386$	−18.2467	−0.928734
$387$	0	0
$388$	−33.6338	−1.70750
$389$	24.4038	1.23732	0.618660	−	0.785659i	$-0.287676\pi$
$389$	24.4038	1.23732	0.618660	+	0.785659i	$0.287676\pi$
$390$	0	0
$391$	−4.62217	−0.233753
$392$	−1.51936	−0.0767394
$393$	0	0
$394$	33.1635	1.67075
$395$	−0.0930208	−0.00468039
$396$	0	0
$397$	−20.9102	−1.04945	−0.524727	−	0.851271i	$-0.675832\pi$
$397$	−20.9102	−1.04945	−0.524727	+	0.851271i	$0.675832\pi$
$398$	−1.59857	−0.0801289
$399$	0	0
$400$	2.22502	0.111251
$401$	23.4289	1.16999	0.584993	−	0.811038i	$-0.301097\pi$
$401$	23.4289	1.16999	0.584993	+	0.811038i	$0.301097\pi$
$402$	0	0
$403$	13.3588	0.665448
$404$	1.24298	0.0618405
$405$	0	0
$406$	19.6267	0.974057
$407$	0.430733	0.0213506
$408$	0	0
$409$	−2.56825	−0.126992	−0.0634958	−	0.997982i	$-0.520225\pi$
$409$	−2.56825	−0.126992	−0.0634958	+	0.997982i	$0.520225\pi$
$410$	−1.85750	−0.0917352
$411$	0	0
$412$	15.3111	0.754322
$413$	−10.9113	−0.536910
$414$	0	0
$415$	2.04319	0.100296
$416$	24.7337	1.21267
$417$	0	0
$418$	1.89321	0.0926000
$419$	−29.6586	−1.44892	−0.724459	−	0.689318i	$-0.757910\pi$
$419$	−29.6586	−1.44892	−0.724459	+	0.689318i	$0.757910\pi$
$420$	0	0
$421$	−23.1837	−1.12990	−0.564952	−	0.825124i	$-0.691105\pi$
$421$	−23.1837	−1.12990	−0.564952	+	0.825124i	$0.691105\pi$
$422$	−26.3313	−1.28179
$423$	0	0
$424$	−15.3193	−0.743971
$425$	0.793324	0.0384819
$426$	0	0
$427$	−3.80129	−0.183957
$428$	14.3695	0.694574
$429$	0	0
$430$	41.0305	1.97867
$431$	15.7031	0.756393	0.378196	−	0.925725i	$-0.376544\pi$
$431$	15.7031	0.756393	0.378196	+	0.925725i	$0.376544\pi$
$432$	0	0
$433$	22.4586	1.07929	0.539645	−	0.841893i	$-0.318559\pi$
$433$	22.4586	1.07929	0.539645	+	0.841893i	$0.318559\pi$
$434$	8.90882	0.427637
$435$	0	0
$436$	20.1216	0.963651
$437$	10.5227	0.503370
$438$	0	0
$439$	−8.32414	−0.397290	−0.198645	−	0.980072i	$-0.563654\pi$
$439$	−8.32414	−0.397290	−0.198645	+	0.980072i	$0.563654\pi$
$440$	1.54033	0.0734322
$441$	0	0
$442$	5.29633	0.251921
$443$	33.0474	1.57013	0.785064	−	0.619415i	$-0.212630\pi$
$443$	33.0474	1.57013	0.785064	+	0.619415i	$0.212630\pi$
$444$	0	0
$445$	−11.1078	−0.526562
$446$	−15.4470	−0.731437
$447$	0	0
$448$	12.2799	0.580170
$449$	−8.80883	−0.415714	−0.207857	−	0.978159i	$-0.566649\pi$
$449$	−8.80883	−0.415714	−0.207857	+	0.978159i	$0.566649\pi$
$450$	0	0
$451$	−0.220211	−0.0103693
$452$	−24.3086	−1.14338
$453$	0	0
$454$	−15.4476	−0.724992
$455$	−6.45668	−0.302694
$456$	0	0
$457$	−3.98370	−0.186350	−0.0931749	−	0.995650i	$-0.529702\pi$
$457$	−3.98370	−0.186350	−0.0931749	+	0.995650i	$0.529702\pi$
$458$	34.9816	1.63458
$459$	0	0
$460$	32.9954	1.53842
$461$	12.0079	0.559263	0.279632	−	0.960107i	$-0.409788\pi$
$461$	12.0079	0.559263	0.279632	+	0.960107i	$0.409788\pi$
$462$	0	0
$463$	−4.98431	−0.231641	−0.115820	−	0.993270i	$-0.536950\pi$
$463$	−4.98431	−0.231641	−0.115820	+	0.993270i	$0.536950\pi$
$464$	−19.0768	−0.885617
$465$	0	0
$466$	37.9996	1.76030
$467$	−4.89870	−0.226685	−0.113342	−	0.993556i	$-0.536156\pi$
$467$	−4.89870	−0.226685	−0.113342	+	0.993556i	$0.536156\pi$
$468$	0	0
$469$	−0.655991	−0.0302908
$470$	−10.0268	−0.462501
$471$	0	0
$472$	−16.5782	−0.763074
$473$	4.86427	0.223659
$474$	0	0
$475$	−1.80606	−0.0828678
$476$	2.02930	0.0930130
$477$	0	0
$478$	16.7394	0.765643
$479$	2.15827	0.0986140	0.0493070	−	0.998784i	$-0.484299\pi$
$479$	2.15827	0.0986140	0.0493070	+	0.998784i	$0.484299\pi$
$480$	0	0
$481$	−2.74325	−0.125081
$482$	−18.5068	−0.842962
$483$	0	0
$484$	−29.0047	−1.31840
$485$	−24.7324	−1.12304
$486$	0	0
$487$	3.73224	0.169124	0.0845619	−	0.996418i	$-0.473051\pi$
$487$	3.73224	0.169124	0.0845619	+	0.996418i	$0.473051\pi$
$488$	−5.77553	−0.261446
$489$	0	0
$490$	−4.30589	−0.194520
$491$	16.4764	0.743567	0.371784	−	0.928319i	$-0.378746\pi$
$491$	16.4764	0.743567	0.371784	+	0.928319i	$0.378746\pi$
$492$	0	0
$493$	−6.80177	−0.306336
$494$	−12.0575	−0.542492
$495$	0	0
$496$	−8.65920	−0.388809
$497$	7.96140	0.357118
$498$	0	0
$499$	−13.3331	−0.596871	−0.298435	−	0.954430i	$-0.596465\pi$
$499$	−13.3331	−0.596871	−0.298435	+	0.954430i	$0.596465\pi$
$500$	−32.4817	−1.45263
$501$	0	0
$502$	−24.3689	−1.08764
$503$	−14.4781	−0.645545	−0.322772	−	0.946477i	$-0.604615\pi$
$503$	−14.4781	−0.645545	−0.322772	+	0.946477i	$0.604615\pi$
$504$	0	0
$505$	0.914016	0.0406732
$506$	6.80840	0.302670
$507$	0	0
$508$	−2.70077	−0.119827
$509$	21.7425	0.963717	0.481859	−	0.876249i	$-0.339962\pi$
$509$	21.7425	0.963717	0.481859	+	0.876249i	$0.339962\pi$
$510$	0	0
$511$	8.99944	0.398112
$512$	−22.4362	−0.991548
$513$	0	0
$514$	−40.1322	−1.77016
$515$	11.2589	0.496126
$516$	0	0
$517$	−1.18870	−0.0522790
$518$	−1.82944	−0.0803811
$519$	0	0
$520$	−9.81003	−0.430198
$521$	−22.2530	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$521$	−22.2530	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$522$	0	0
$523$	−31.3171	−1.36940	−0.684700	−	0.728825i	$-0.740067\pi$
$523$	−31.3171	−1.36940	−0.684700	+	0.728825i	$0.740067\pi$
$524$	−8.26895	−0.361231
$525$	0	0
$526$	−12.6734	−0.552586
$527$	−3.08741	−0.134490
$528$	0	0
$529$	14.8420	0.645306
$530$	−43.4150	−1.88583
$531$	0	0
$532$	−4.61986	−0.200296
$533$	1.40248	0.0607481
$534$	0	0
$535$	10.5665	0.456829
$536$	−0.996687	−0.0430503
$537$	0	0
$538$	−19.2908	−0.831685
$539$	−0.510474	−0.0219877
$540$	0	0
$541$	7.39900	0.318108	0.159054	−	0.987270i	$-0.449156\pi$
$541$	7.39900	0.318108	0.159054	+	0.987270i	$0.449156\pi$
$542$	49.5486	2.12829
$543$	0	0
$544$	−5.71633	−0.245085
$545$	14.7963	0.633804
$546$	0	0
$547$	7.22154	0.308771	0.154385	−	0.988011i	$-0.450660\pi$
$547$	7.22154	0.308771	0.154385	+	0.988011i	$0.450660\pi$
$548$	48.3480	2.06532
$549$	0	0
$550$	−1.16856	−0.0498274
$551$	15.4847	0.659672
$552$	0	0
$553$	0.0468384	0.00199177
$554$	10.1120	0.429618
$555$	0	0
$556$	33.4600	1.41902
$557$	22.9512	0.972472	0.486236	−	0.873828i	$-0.338370\pi$
$557$	22.9512	0.972472	0.486236	+	0.873828i	$0.338370\pi$
$558$	0	0
$559$	−30.9795	−1.31030
$560$	4.18524	0.176859
$561$	0	0
$562$	67.3156	2.83954
$563$	−22.6246	−0.953515	−0.476757	−	0.879035i	$-0.658188\pi$
$563$	−22.6246	−0.953515	−0.476757	+	0.879035i	$0.658188\pi$
$564$	0	0
$565$	−17.8751	−0.752013
$566$	3.84655	0.161683
$567$	0	0
$568$	12.0962	0.507547
$569$	21.0376	0.881943	0.440972	−	0.897521i	$-0.354634\pi$
$569$	21.0376	0.881943	0.440972	+	0.897521i	$0.354634\pi$
$570$	0	0
$571$	7.94595	0.332528	0.166264	−	0.986081i	$-0.446830\pi$
$571$	7.94595	0.332528	0.166264	+	0.986081i	$0.446830\pi$
$572$	−4.48221	−0.187411
$573$	0	0
$574$	0.935298	0.0390386
$575$	−6.49500	−0.270860
$576$	0	0
$577$	−2.55198	−0.106240	−0.0531201	−	0.998588i	$-0.516917\pi$
$577$	−2.55198	−0.106240	−0.0531201	+	0.998588i	$0.516917\pi$
$578$	35.6341	1.48218
$579$	0	0
$580$	48.5544	2.01611
$581$	−1.02880	−0.0426817
$582$	0	0
$583$	−5.14696	−0.213165
$584$	13.6734	0.565810
$585$	0	0
$586$	−29.2333	−1.20761
$587$	−41.5275	−1.71402	−0.857012	−	0.515297i	$-0.827682\pi$
$587$	−41.5275	−1.71402	−0.857012	+	0.515297i	$0.827682\pi$
$588$	0	0
$589$	7.02873	0.289614
$590$	−46.9828	−1.93425
$591$	0	0
$592$	1.77818	0.0730828
$593$	−28.4913	−1.17000	−0.585000	−	0.811034i	$-0.698905\pi$
$593$	−28.4913	−1.17000	−0.585000	+	0.811034i	$0.698905\pi$
$594$	0	0
$595$	1.49224	0.0611757
$596$	15.8058	0.647431
$597$	0	0
$598$	−43.3614	−1.77318
$599$	−25.9964	−1.06218	−0.531091	−	0.847315i	$-0.678218\pi$
$599$	−25.9964	−1.06218	−0.531091	+	0.847315i	$0.678218\pi$
$600$	0	0
$601$	26.6602	1.08749	0.543747	−	0.839249i	$-0.317005\pi$
$601$	26.6602	1.08749	0.543747	+	0.839249i	$0.317005\pi$
$602$	−20.6599	−0.842035
$603$	0	0
$604$	29.1550	1.18630
$605$	−21.3284	−0.867124
$606$	0	0
$607$	−27.2012	−1.10406	−0.552032	−	0.833823i	$-0.686147\pi$
$607$	−27.2012	−1.10406	−0.552032	+	0.833823i	$0.686147\pi$
$608$	13.0136	0.527773
$609$	0	0
$610$	−16.3679	−0.662718
$611$	7.57060	0.306274
$612$	0	0
$613$	−1.09435	−0.0442004	−0.0221002	−	0.999756i	$-0.507035\pi$
$613$	−1.09435	−0.0442004	−0.0221002	+	0.999756i	$0.507035\pi$
$614$	−65.2197	−2.63205
$615$	0	0
$616$	−0.775594	−0.0312496
$617$	−25.6593	−1.03300	−0.516502	−	0.856286i	$-0.672766\pi$
$617$	−25.6593	−1.03300	−0.516502	+	0.856286i	$0.672766\pi$
$618$	0	0
$619$	30.2920	1.21754	0.608770	−	0.793347i	$-0.291663\pi$
$619$	30.2920	1.21754	0.608770	+	0.793347i	$0.291663\pi$
$620$	22.0395	0.885128
$621$	0	0
$622$	−0.237689	−0.00953046
$623$	5.59309	0.224082
$624$	0	0
$625$	−18.6061	−0.744245
$626$	−13.7572	−0.549849
$627$	0	0
$628$	−7.56737	−0.301971
$629$	0.634006	0.0252795
$630$	0	0
$631$	−12.0709	−0.480534	−0.240267	−	0.970707i	$-0.577235\pi$
$631$	−12.0709	−0.480534	−0.240267	+	0.970707i	$0.577235\pi$
$632$	0.0711645	0.00283077
$633$	0	0
$634$	1.68509	0.0669236
$635$	−1.98599	−0.0788118
$636$	0	0
$637$	3.25111	0.128814
$638$	10.0189	0.396653
$639$	0	0
$640$	22.6577	0.895626
$641$	30.6573	1.21089	0.605445	−	0.795887i	$-0.292995\pi$
$641$	30.6573	1.21089	0.605445	+	0.795887i	$0.292995\pi$
$642$	0	0
$643$	38.9651	1.53663	0.768316	−	0.640070i	$-0.221095\pi$
$643$	38.9651	1.53663	0.768316	+	0.640070i	$0.221095\pi$
$644$	−16.6140	−0.654685
$645$	0	0
$646$	2.78666	0.109640
$647$	5.51264	0.216724	0.108362	−	0.994111i	$-0.465439\pi$
$647$	5.51264	0.216724	0.108362	+	0.994111i	$0.465439\pi$
$648$	0	0
$649$	−5.56993	−0.218639
$650$	7.44230	0.291911
$651$	0	0
$652$	33.1570	1.29853
$653$	−29.3988	−1.15046	−0.575231	−	0.817991i	$-0.695088\pi$
$653$	−29.3988	−1.15046	−0.575231	+	0.817991i	$0.695088\pi$
$654$	0	0
$655$	−6.08052	−0.237585
$656$	−0.909091	−0.0354940
$657$	0	0
$658$	5.04875	0.196821
$659$	36.6282	1.42683	0.713416	−	0.700741i	$-0.247147\pi$
$659$	36.6282	1.42683	0.713416	+	0.700741i	$0.247147\pi$
$660$	0	0
$661$	48.6973	1.89411	0.947053	−	0.321078i	$-0.104045\pi$
$661$	48.6973	1.89411	0.947053	+	0.321078i	$0.104045\pi$
$662$	−34.9261	−1.35744
$663$	0	0
$664$	−1.56312	−0.0606607
$665$	−3.39718	−0.131737
$666$	0	0
$667$	55.6865	2.15619
$668$	30.2688	1.17113
$669$	0	0
$670$	−2.82462	−0.109125
$671$	−1.94046	−0.0749106
$672$	0	0
$673$	9.51470	0.366765	0.183382	−	0.983042i	$-0.441295\pi$
$673$	9.51470	0.366765	0.183382	+	0.983042i	$0.441295\pi$
$674$	−57.8561	−2.22853
$675$	0	0
$676$	−6.56372	−0.252451
$677$	−6.20788	−0.238588	−0.119294	−	0.992859i	$-0.538063\pi$
$677$	−6.20788	−0.238588	−0.119294	+	0.992859i	$0.538063\pi$
$678$	0	0
$679$	12.4534	0.477917
$680$	2.26724	0.0869449
$681$	0	0
$682$	4.54772	0.174141
$683$	10.5024	0.401862	0.200931	−	0.979605i	$-0.435603\pi$
$683$	10.5024	0.401862	0.200931	+	0.979605i	$0.435603\pi$
$684$	0	0
$685$	35.5524	1.35839
$686$	2.16813	0.0827794
$687$	0	0
$688$	20.0810	0.765582
$689$	32.7800	1.24882
$690$	0	0
$691$	11.4489	0.435537	0.217769	−	0.976000i	$-0.430122\pi$
$691$	11.4489	0.435537	0.217769	+	0.976000i	$0.430122\pi$
$692$	−15.2605	−0.580117
$693$	0	0
$694$	−35.1365	−1.33376
$695$	24.6046	0.933306
$696$	0	0
$697$	−0.324134	−0.0122774
$698$	1.98885	0.0752790
$699$	0	0
$700$	2.85154	0.107778
$701$	43.0161	1.62470	0.812348	−	0.583173i	$-0.198189\pi$
$701$	43.0161	1.62470	0.812348	+	0.583173i	$0.198189\pi$
$702$	0	0
$703$	−1.44336	−0.0544374
$704$	6.26855	0.236255
$705$	0	0
$706$	32.3707	1.21829
$707$	−0.460231	−0.0173088
$708$	0	0
$709$	−46.7793	−1.75683	−0.878416	−	0.477897i	$-0.841399\pi$
$709$	−46.7793	−1.75683	−0.878416	+	0.477897i	$0.841399\pi$
$710$	34.2809	1.28654
$711$	0	0
$712$	8.49792	0.318473
$713$	25.2768	0.946625
$714$	0	0
$715$	−3.29597	−0.123262
$716$	17.3288	0.647607
$717$	0	0
$718$	−19.4436	−0.725631
$719$	38.3665	1.43083	0.715414	−	0.698701i	$-0.246238\pi$
$719$	38.3665	1.43083	0.715414	+	0.698701i	$0.246238\pi$
$720$	0	0
$721$	−5.66914	−0.211130
$722$	34.8504	1.29700
$723$	0	0
$724$	10.9616	0.407383
$725$	−9.55772	−0.354965
$726$	0	0
$727$	14.8939	0.552385	0.276193	−	0.961102i	$-0.410927\pi$
$727$	14.8939	0.552385	0.276193	+	0.961102i	$0.410927\pi$
$728$	4.93961	0.183074
$729$	0	0
$730$	38.7506	1.43422
$731$	7.15984	0.264816
$732$	0	0
$733$	27.5065	1.01597	0.507987	−	0.861365i	$-0.330390\pi$
$733$	27.5065	1.01597	0.507987	+	0.861365i	$0.330390\pi$
$734$	61.2120	2.25938
$735$	0	0
$736$	46.7999	1.72507
$737$	−0.334866	−0.0123349
$738$	0	0
$739$	4.92960	0.181338	0.0906692	−	0.995881i	$-0.471099\pi$
$739$	4.92960	0.181338	0.0906692	+	0.995881i	$0.471099\pi$
$740$	−4.52585	−0.166374
$741$	0	0
$742$	21.8606	0.802528
$743$	0.844507	0.0309820	0.0154910	−	0.999880i	$-0.495069\pi$
$743$	0.844507	0.0309820	0.0154910	+	0.999880i	$0.495069\pi$
$744$	0	0
$745$	11.6227	0.425823
$746$	67.8623	2.48461
$747$	0	0
$748$	1.03591	0.0378765
$749$	−5.32050	−0.194407
$750$	0	0
$751$	47.8293	1.74532	0.872658	−	0.488333i	$-0.162395\pi$
$751$	47.8293	1.74532	0.872658	+	0.488333i	$0.162395\pi$
$752$	−4.90728	−0.178950
$753$	0	0
$754$	−63.8085	−2.32377
$755$	21.4389	0.780242
$756$	0	0
$757$	7.18644	0.261196	0.130598	−	0.991435i	$-0.458310\pi$
$757$	7.18644	0.261196	0.130598	+	0.991435i	$0.458310\pi$
$758$	−62.4042	−2.26662
$759$	0	0
$760$	−5.16155	−0.187229
$761$	−7.40425	−0.268404	−0.134202	−	0.990954i	$-0.542847\pi$
$761$	−7.40425	−0.268404	−0.134202	+	0.990954i	$0.542847\pi$
$762$	0	0
$763$	−7.45032	−0.269720
$764$	−37.7073	−1.36420
$765$	0	0
$766$	64.4452	2.32850
$767$	35.4738	1.28088
$768$	0	0
$769$	38.7720	1.39815	0.699076	−	0.715047i	$-0.253595\pi$
$769$	38.7720	1.39815	0.699076	+	0.715047i	$0.253595\pi$
$770$	−2.19804	−0.0792120
$771$	0	0
$772$	22.7294	0.818049
$773$	39.7030	1.42802	0.714009	−	0.700136i	$-0.246878\pi$
$773$	39.7030	1.42802	0.714009	+	0.700136i	$0.246878\pi$
$774$	0	0
$775$	−4.33838	−0.155839
$776$	18.9212	0.679232
$777$	0	0
$778$	−52.9105	−1.89693
$779$	0.737915	0.0264385
$780$	0	0
$781$	4.06409	0.145424
$782$	10.0215	0.358367
$783$	0	0
$784$	−2.10738	−0.0752634
$785$	−5.56462	−0.198610
$786$	0	0
$787$	−13.5482	−0.482941	−0.241471	−	0.970408i	$-0.577630\pi$
$787$	−13.5482	−0.482941	−0.241471	+	0.970408i	$0.577630\pi$
$788$	−41.3108	−1.47164
$789$	0	0
$790$	0.201681	0.00717549
$791$	9.00060	0.320024
$792$	0	0
$793$	12.3584	0.438859
$794$	45.3360	1.60891
$795$	0	0
$796$	1.99129	0.0705793
$797$	11.2866	0.399793	0.199896	−	0.979817i	$-0.435939\pi$
$797$	11.2866	0.399793	0.199896	+	0.979817i	$0.435939\pi$
$798$	0	0
$799$	−1.74968	−0.0618992
$800$	−8.03248	−0.283991
$801$	0	0
$802$	−50.7969	−1.79370
$803$	4.59398	0.162118
$804$	0	0
$805$	−12.2170	−0.430593
$806$	−28.9635	−1.02020
$807$	0	0
$808$	−0.699257	−0.0245998
$809$	48.1332	1.69227	0.846137	−	0.532965i	$-0.178922\pi$
$809$	48.1332	1.69227	0.846137	+	0.532965i	$0.178922\pi$
$810$	0	0
$811$	42.9501	1.50818	0.754091	−	0.656770i	$-0.228078\pi$
$811$	42.9501	1.50818	0.754091	+	0.656770i	$0.228078\pi$
$812$	−24.4484	−0.857971
$813$	0	0
$814$	−0.933883	−0.0327326
$815$	24.3818	0.854058
$816$	0	0
$817$	−16.2999	−0.570261
$818$	5.56828	0.194690
$819$	0	0
$820$	2.31383	0.0808024
$821$	−44.6870	−1.55959	−0.779793	−	0.626037i	$-0.784676\pi$
$821$	−44.6870	−1.55959	−0.779793	+	0.626037i	$0.784676\pi$
$822$	0	0
$823$	−43.2608	−1.50798	−0.753989	−	0.656887i	$-0.771873\pi$
$823$	−43.2608	−1.50798	−0.753989	+	0.656887i	$0.771873\pi$
$824$	−8.61348	−0.300065
$825$	0	0
$826$	23.6571	0.823135
$827$	12.4723	0.433705	0.216852	−	0.976204i	$-0.430421\pi$
$827$	12.4723	0.433705	0.216852	+	0.976204i	$0.430421\pi$
$828$	0	0
$829$	7.46841	0.259389	0.129694	−	0.991554i	$-0.458600\pi$
$829$	7.46841	0.259389	0.129694	+	0.991554i	$0.458600\pi$
$830$	−4.42989	−0.153764
$831$	0	0
$832$	−39.9232	−1.38409
$833$	−0.751379	−0.0260337
$834$	0	0
$835$	22.2579	0.770268
$836$	−2.35832	−0.0815641
$837$	0	0
$838$	64.3036	2.22133
$839$	12.1870	0.420742	0.210371	−	0.977622i	$-0.432533\pi$
$839$	12.1870	0.420742	0.210371	+	0.977622i	$0.432533\pi$
$840$	0	0
$841$	52.9456	1.82571
$842$	50.2652	1.73225
$843$	0	0
$844$	32.8001	1.12903
$845$	−4.82659	−0.166040
$846$	0	0
$847$	10.7394	0.369011
$848$	−21.2481	−0.729662
$849$	0	0
$850$	−1.72003	−0.0589965
$851$	−5.19065	−0.177933
$852$	0	0
$853$	−22.3736	−0.766056	−0.383028	−	0.923737i	$-0.625119\pi$
$853$	−22.3736	−0.766056	−0.383028	+	0.923737i	$0.625119\pi$
$854$	8.24167	0.282024
$855$	0	0
$856$	−8.08376	−0.276297
$857$	45.1001	1.54059	0.770295	−	0.637688i	$-0.220109\pi$
$857$	45.1001	1.54059	0.770295	+	0.637688i	$0.220109\pi$
$858$	0	0
$859$	42.5744	1.45262	0.726310	−	0.687368i	$-0.241234\pi$
$859$	42.5744	1.45262	0.726310	+	0.687368i	$0.241234\pi$
$860$	−51.1105	−1.74285
$861$	0	0
$862$	−34.0464	−1.15962
$863$	24.2647	0.825978	0.412989	−	0.910736i	$-0.364485\pi$
$863$	24.2647	0.825978	0.412989	+	0.910736i	$0.364485\pi$
$864$	0	0
$865$	−11.2217	−0.381549
$866$	−48.6930	−1.65466
$867$	0	0
$868$	−11.0975	−0.376672
$869$	0.0239098	0.000811084 0
$870$	0	0
$871$	2.13269	0.0722636
$872$	−11.3197	−0.383335
$873$	0	0
$874$	−22.8146	−0.771715
$875$	12.0268	0.406581
$876$	0	0
$877$	3.40441	0.114959	0.0574793	−	0.998347i	$-0.481694\pi$
$877$	3.40441	0.114959	0.0574793	+	0.998347i	$0.481694\pi$
$878$	18.0478	0.609083
$879$	0	0
$880$	2.13645	0.0720198
$881$	−5.37939	−0.181236	−0.0906182	−	0.995886i	$-0.528884\pi$
$881$	−5.37939	−0.181236	−0.0906182	+	0.995886i	$0.528884\pi$
$882$	0	0
$883$	−40.3435	−1.35767	−0.678833	−	0.734292i	$-0.737514\pi$
$883$	−40.3435	−1.35767	−0.678833	+	0.734292i	$0.737514\pi$
$884$	−6.59748	−0.221897
$885$	0	0
$886$	−71.6509	−2.40716
$887$	−16.1014	−0.540631	−0.270316	−	0.962772i	$-0.587128\pi$
$887$	−16.1014	−0.540631	−0.270316	+	0.962772i	$0.587128\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	24.0832	0.807271
$891$	0	0
$892$	19.2419	0.644266
$893$	3.98327	0.133295
$894$	0	0
$895$	12.7426	0.425938
$896$	−11.4088	−0.381140
$897$	0	0
$898$	19.0987	0.637331
$899$	37.1962	1.24056
$900$	0	0
$901$	−7.57594	−0.252391
$902$	0.477445	0.0158972
$903$	0	0
$904$	13.6752	0.454829
$905$	8.06052	0.267941
$906$	0	0
$907$	13.2149	0.438795	0.219397	−	0.975636i	$-0.429591\pi$
$907$	13.2149	0.438795	0.219397	+	0.975636i	$0.429591\pi$
$908$	19.2426	0.638589
$909$	0	0
$910$	13.9989	0.464059
$911$	21.8075	0.722514	0.361257	−	0.932466i	$-0.382348\pi$
$911$	21.8075	0.722514	0.361257	+	0.932466i	$0.382348\pi$
$912$	0	0
$913$	−0.525174	−0.0173807
$914$	8.63717	0.285692
$915$	0	0
$916$	−43.5755	−1.43978
$917$	3.06170	0.101106
$918$	0	0
$919$	−22.0193	−0.726349	−0.363174	−	0.931721i	$-0.618307\pi$
$919$	−22.0193	−0.726349	−0.363174	+	0.931721i	$0.618307\pi$
$920$	−18.5621	−0.611973
$921$	0	0
$922$	−26.0346	−0.857405
$923$	−25.8834	−0.851961
$924$	0	0
$925$	0.890894	0.0292924
$926$	10.8066	0.355127
$927$	0	0
$928$	68.8685	2.26072
$929$	5.69507	0.186849	0.0934245	−	0.995626i	$-0.470219\pi$
$929$	5.69507	0.186849	0.0934245	+	0.995626i	$0.470219\pi$
$930$	0	0
$931$	1.71057	0.0560617
$932$	−47.3350	−1.55051
$933$	0	0
$934$	10.6210	0.347530
$935$	0.761747	0.0249118
$936$	0	0
$937$	−32.9534	−1.07654	−0.538270	−	0.842772i	$-0.680922\pi$
$937$	−32.9534	−1.07654	−0.538270	+	0.842772i	$0.680922\pi$
$938$	1.42227	0.0464388
$939$	0	0
$940$	12.4901	0.407381
$941$	37.8877	1.23510	0.617552	−	0.786530i	$-0.288124\pi$
$941$	37.8877	1.23510	0.617552	+	0.786530i	$0.288124\pi$
$942$	0	0
$943$	2.65370	0.0864165
$944$	−22.9942	−0.748398
$945$	0	0
$946$	−10.5463	−0.342891
$947$	39.9963	1.29970	0.649852	−	0.760061i	$-0.274831\pi$
$947$	39.9963	1.29970	0.649852	+	0.760061i	$0.274831\pi$
$948$	0	0
$949$	−29.2581	−0.949759
$950$	3.91577	0.127044
$951$	0	0
$952$	−1.14162	−0.0370000
$953$	−10.4927	−0.339893	−0.169947	−	0.985453i	$-0.554360\pi$
$953$	−10.4927	−0.339893	−0.169947	+	0.985453i	$0.554360\pi$
$954$	0	0
$955$	−27.7278	−0.897250
$956$	−20.8518	−0.674396
$957$	0	0
$958$	−4.67941	−0.151185
$959$	−17.9015	−0.578071
$960$	0	0
$961$	−14.1162	−0.455360
$962$	5.94771	0.191762
$963$	0	0
$964$	23.0534	0.742500
$965$	16.7139	0.538040
$966$	0	0
$967$	1.66714	0.0536116	0.0268058	−	0.999641i	$-0.491466\pi$
$967$	1.66714	0.0536116	0.0268058	+	0.999641i	$0.491466\pi$
$968$	16.3171	0.524450
$969$	0	0
$970$	53.6229	1.72173
$971$	45.1406	1.44863	0.724316	−	0.689468i	$-0.242156\pi$
$971$	45.1406	1.44863	0.724316	+	0.689468i	$0.242156\pi$
$972$	0	0
$973$	−12.3891	−0.397175
$974$	−8.09196	−0.259283
$975$	0	0
$976$	−8.01074	−0.256418
$977$	18.2447	0.583700	0.291850	−	0.956464i	$-0.405729\pi$
$977$	18.2447	0.583700	0.291850	+	0.956464i	$0.405729\pi$
$978$	0	0
$979$	2.85512	0.0912502
$980$	5.36372	0.171338
$981$	0	0
$982$	−35.7228	−1.13996
$983$	7.41897	0.236628	0.118314	−	0.992976i	$-0.462251\pi$
$983$	7.41897	0.236628	0.118314	+	0.992976i	$0.462251\pi$
$984$	0	0
$985$	−30.3776	−0.967912
$986$	14.7471	0.469643
$987$	0	0
$988$	15.0197	0.477839
$989$	−58.6180	−1.86394
$990$	0	0
$991$	36.6998	1.16581	0.582904	−	0.812541i	$-0.301916\pi$
$991$	36.6998	1.16581	0.582904	+	0.812541i	$0.301916\pi$
$992$	31.2603	0.992517
$993$	0	0
$994$	−17.2613	−0.547496
$995$	1.46428	0.0464208
$996$	0	0
$997$	−1.39485	−0.0441753	−0.0220877	−	0.999756i	$-0.507031\pi$
$997$	−1.39485	−0.0441753	−0.0220877	+	0.999756i	$0.507031\pi$
$998$	28.9078	0.915061
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.2		14
3.2	odd	2		2667.2.a.m.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.13	✓	14	3.2	odd	2
8001.2.a.p.1.2		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.16813 q^{2} +2.70077 q^{4} +1.98599 q^{5} -1.00000 q^{7} -1.51936 q^{8} +O(q^{10})\)	\(q-2.16813 q^{2} +2.70077 q^{4} +1.98599 q^{5} -1.00000 q^{7} -1.51936 q^{8} -4.30589 q^{10} -0.510474 q^{11} +3.25111 q^{13} +2.16813 q^{14} -2.10738 q^{16} -0.751379 q^{17} +1.71057 q^{19} +5.36372 q^{20} +1.10677 q^{22} +6.15159 q^{23} -1.05582 q^{25} -7.04881 q^{26} -2.70077 q^{28} +9.05238 q^{29} +4.10900 q^{31} +7.60778 q^{32} +1.62908 q^{34} -1.98599 q^{35} -0.843790 q^{37} -3.70873 q^{38} -3.01744 q^{40} +0.431385 q^{41} -9.52893 q^{43} -1.37867 q^{44} -13.3374 q^{46} +2.32862 q^{47} +1.00000 q^{49} +2.28916 q^{50} +8.78049 q^{52} +10.0827 q^{53} -1.01380 q^{55} +1.51936 q^{56} -19.6267 q^{58} +10.9113 q^{59} +3.80129 q^{61} -8.90882 q^{62} -12.2799 q^{64} +6.45668 q^{65} +0.655991 q^{67} -2.02930 q^{68} +4.30589 q^{70} -7.96140 q^{71} -8.99944 q^{73} +1.82944 q^{74} +4.61986 q^{76} +0.510474 q^{77} -0.0468384 q^{79} -4.18524 q^{80} -0.935298 q^{82} +1.02880 q^{83} -1.49224 q^{85} +20.6599 q^{86} +0.775594 q^{88} -5.59309 q^{89} -3.25111 q^{91} +16.6140 q^{92} -5.04875 q^{94} +3.39718 q^{95} -12.4534 q^{97} -2.16813 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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