LMFDB - Embedded newform 8001.2.a.v.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-0.782325$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.782325 q^{2} -1.38797 q^{4} +2.85652 q^{5} +1.00000 q^{7} -2.65049 q^{8} +O(q^{10})$	\(q+0.782325 q^{2} -1.38797 q^{4} +2.85652 q^{5} +1.00000 q^{7} -2.65049 q^{8} +2.23472 q^{10} +5.95291 q^{11} -2.24308 q^{13} +0.782325 q^{14} +0.702387 q^{16} -0.355250 q^{17} +4.58452 q^{19} -3.96475 q^{20} +4.65711 q^{22} -1.77438 q^{23} +3.15968 q^{25} -1.75482 q^{26} -1.38797 q^{28} +9.18000 q^{29} -2.61888 q^{31} +5.85048 q^{32} -0.277921 q^{34} +2.85652 q^{35} +0.446439 q^{37} +3.58659 q^{38} -7.57117 q^{40} +5.55748 q^{41} -3.81437 q^{43} -8.26245 q^{44} -1.38814 q^{46} +4.89679 q^{47} +1.00000 q^{49} +2.47190 q^{50} +3.11332 q^{52} -12.9711 q^{53} +17.0046 q^{55} -2.65049 q^{56} +7.18175 q^{58} -5.34918 q^{59} +3.71025 q^{61} -2.04882 q^{62} +3.17221 q^{64} -6.40739 q^{65} +10.3248 q^{67} +0.493075 q^{68} +2.23472 q^{70} -1.31960 q^{71} -8.85675 q^{73} +0.349261 q^{74} -6.36316 q^{76} +5.95291 q^{77} +13.1696 q^{79} +2.00638 q^{80} +4.34776 q^{82} -15.2753 q^{83} -1.01478 q^{85} -2.98408 q^{86} -15.7782 q^{88} +2.60509 q^{89} -2.24308 q^{91} +2.46278 q^{92} +3.83088 q^{94} +13.0958 q^{95} -3.10938 q^{97} +0.782325 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.782325	0.553188	0.276594	−	0.960987i	$-0.410794\pi$
$2$	0.782325	0.553188	0.276594	+	0.960987i	$0.410794\pi$
$3$	0	0
$3$	0	0
$4$	−1.38797	−0.693984
$5$	2.85652	1.27747	0.638736	−	0.769426i	$-0.279457\pi$
$5$	2.85652	1.27747	0.638736	+	0.769426i	$0.279457\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.65049	−0.937091
$9$	0	0
$10$	2.23472	0.706682
$11$	5.95291	1.79487	0.897435	−	0.441146i	$-0.145428\pi$
$11$	5.95291	1.79487	0.897435	+	0.441146i	$0.145428\pi$
$12$	0	0
$13$	−2.24308	−0.622118	−0.311059	−	0.950391i	$-0.600684\pi$
$13$	−2.24308	−0.622118	−0.311059	+	0.950391i	$0.600684\pi$
$14$	0.782325	0.209085
$15$	0	0
$16$	0.702387	0.175597
$17$	−0.355250	−0.0861607	−0.0430803	−	0.999072i	$-0.513717\pi$
$17$	−0.355250	−0.0861607	−0.0430803	+	0.999072i	$0.513717\pi$
$18$	0	0
$19$	4.58452	1.05176	0.525881	−	0.850558i	$-0.323736\pi$
$19$	4.58452	1.05176	0.525881	+	0.850558i	$0.323736\pi$
$20$	−3.96475	−0.886545
$21$	0	0
$22$	4.65711	0.992900
$23$	−1.77438	−0.369984	−0.184992	−	0.982740i	$-0.559226\pi$
$23$	−1.77438	−0.369984	−0.184992	+	0.982740i	$0.559226\pi$
$24$	0	0
$25$	3.15968	0.631936
$26$	−1.75482	−0.344148
$27$	0	0
$28$	−1.38797	−0.262301
$29$	9.18000	1.70468	0.852342	−	0.522985i	$-0.175182\pi$
$29$	9.18000	1.70468	0.852342	+	0.522985i	$0.175182\pi$
$30$	0	0
$31$	−2.61888	−0.470365	−0.235183	−	0.971951i	$-0.575569\pi$
$31$	−2.61888	−0.470365	−0.235183	+	0.971951i	$0.575569\pi$
$32$	5.85048	1.03423
$33$	0	0
$34$	−0.277921	−0.0476630
$35$	2.85652	0.482839
$36$	0	0
$37$	0.446439	0.0733942	0.0366971	−	0.999326i	$-0.488316\pi$
$37$	0.446439	0.0733942	0.0366971	+	0.999326i	$0.488316\pi$
$38$	3.58659	0.581821
$39$	0	0
$40$	−7.57117	−1.19711
$41$	5.55748	0.867932	0.433966	−	0.900929i	$-0.357114\pi$
$41$	5.55748	0.867932	0.433966	+	0.900929i	$0.357114\pi$
$42$	0	0
$43$	−3.81437	−0.581685	−0.290843	−	0.956771i	$-0.593936\pi$
$43$	−3.81437	−0.581685	−0.290843	+	0.956771i	$0.593936\pi$
$44$	−8.26245	−1.24561
$45$	0	0
$46$	−1.38814	−0.204671
$47$	4.89679	0.714271	0.357135	−	0.934053i	$-0.383754\pi$
$47$	4.89679	0.714271	0.357135	+	0.934053i	$0.383754\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.47190	0.349579
$51$	0	0
$52$	3.11332	0.431740
$53$	−12.9711	−1.78171	−0.890856	−	0.454286i	$-0.849894\pi$
$53$	−12.9711	−1.78171	−0.890856	+	0.454286i	$0.849894\pi$
$54$	0	0
$55$	17.0046	2.29290
$56$	−2.65049	−0.354187
$57$	0	0
$58$	7.18175	0.943010
$59$	−5.34918	−0.696403	−0.348202	−	0.937420i	$-0.613208\pi$
$59$	−5.34918	−0.696403	−0.348202	+	0.937420i	$0.613208\pi$
$60$	0	0
$61$	3.71025	0.475049	0.237524	−	0.971382i	$-0.423664\pi$
$61$	3.71025	0.475049	0.237524	+	0.971382i	$0.423664\pi$
$62$	−2.04882	−0.260200
$63$	0	0
$64$	3.17221	0.396526
$65$	−6.40739	−0.794739
$66$	0	0
$67$	10.3248	1.26138	0.630689	−	0.776035i	$-0.282772\pi$
$67$	10.3248	1.26138	0.630689	+	0.776035i	$0.282772\pi$
$68$	0.493075	0.0597941
$69$	0	0
$70$	2.23472	0.267101
$71$	−1.31960	−0.156608	−0.0783041	−	0.996930i	$-0.524950\pi$
$71$	−1.31960	−0.156608	−0.0783041	+	0.996930i	$0.524950\pi$
$72$	0	0
$73$	−8.85675	−1.03660	−0.518302	−	0.855198i	$-0.673436\pi$
$73$	−8.85675	−1.03660	−0.518302	+	0.855198i	$0.673436\pi$
$74$	0.349261	0.0406007
$75$	0	0
$76$	−6.36316	−0.729905
$77$	5.95291	0.678397
$78$	0	0
$79$	13.1696	1.48169	0.740847	−	0.671673i	$-0.234424\pi$
$79$	13.1696	1.48169	0.740847	+	0.671673i	$0.234424\pi$
$80$	2.00638	0.224320
$81$	0	0
$82$	4.34776	0.480129
$83$	−15.2753	−1.67668	−0.838342	−	0.545145i	$-0.816475\pi$
$83$	−15.2753	−1.67668	−0.838342	+	0.545145i	$0.816475\pi$
$84$	0	0
$85$	−1.01478	−0.110068
$86$	−2.98408	−0.321781
$87$	0	0
$88$	−15.7782	−1.68196
$89$	2.60509	0.276139	0.138070	−	0.990423i	$-0.455910\pi$
$89$	2.60509	0.276139	0.138070	+	0.990423i	$0.455910\pi$
$90$	0	0
$91$	−2.24308	−0.235139
$92$	2.46278	0.256763
$93$	0	0
$94$	3.83088	0.395126
$95$	13.0958	1.34360
$96$	0	0
$97$	−3.10938	−0.315709	−0.157855	−	0.987462i	$-0.550458\pi$
$97$	−3.10938	−0.315709	−0.157855	+	0.987462i	$0.550458\pi$
$98$	0.782325	0.0790268
$99$	0	0
$100$	−4.38553	−0.438553
$101$	−2.36661	−0.235487	−0.117743	−	0.993044i	$-0.537566\pi$
$101$	−2.36661	−0.235487	−0.117743	+	0.993044i	$0.537566\pi$
$102$	0	0
$103$	3.24902	0.320135	0.160068	−	0.987106i	$-0.448829\pi$
$103$	3.24902	0.320135	0.160068	+	0.987106i	$0.448829\pi$
$104$	5.94526	0.582981
$105$	0	0
$106$	−10.1476	−0.985621
$107$	0.104034	0.0100573	0.00502866	−	0.999987i	$-0.498399\pi$
$107$	0.104034	0.0100573	0.00502866	+	0.999987i	$0.498399\pi$
$108$	0	0
$109$	19.8237	1.89877	0.949383	−	0.314120i	$-0.101710\pi$
$109$	19.8237	1.89877	0.949383	+	0.314120i	$0.101710\pi$
$110$	13.3031	1.26840
$111$	0	0
$112$	0.702387	0.0663693
$113$	13.2923	1.25043	0.625216	−	0.780452i	$-0.285011\pi$
$113$	13.2923	1.25043	0.625216	+	0.780452i	$0.285011\pi$
$114$	0	0
$115$	−5.06855	−0.472645
$116$	−12.7415	−1.18302
$117$	0	0
$118$	−4.18480	−0.385242
$119$	−0.355250	−0.0325657
$120$	0	0
$121$	24.4372	2.22156
$122$	2.90262	0.262791
$123$	0	0
$124$	3.63492	0.326426
$125$	−5.25690	−0.470191
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−9.21926	−0.814875
$129$	0	0
$130$	−5.01266	−0.439640
$131$	−1.48402	−0.129660	−0.0648298	−	0.997896i	$-0.520650\pi$
$131$	−1.48402	−0.129660	−0.0648298	+	0.997896i	$0.520650\pi$
$132$	0	0
$133$	4.58452	0.397528
$134$	8.07738	0.697779
$135$	0	0
$136$	0.941587	0.0807404
$137$	−3.92401	−0.335251	−0.167626	−	0.985851i	$-0.553610\pi$
$137$	−3.92401	−0.335251	−0.167626	+	0.985851i	$0.553610\pi$
$138$	0	0
$139$	−1.61387	−0.136887	−0.0684435	−	0.997655i	$-0.521803\pi$
$139$	−1.61387	−0.136887	−0.0684435	+	0.997655i	$0.521803\pi$
$140$	−3.96475	−0.335082
$141$	0	0
$142$	−1.03236	−0.0866337
$143$	−13.3529	−1.11662
$144$	0	0
$145$	26.2228	2.17769
$146$	−6.92886	−0.573437
$147$	0	0
$148$	−0.619643	−0.0509343
$149$	−9.10778	−0.746138	−0.373069	−	0.927804i	$-0.621695\pi$
$149$	−9.10778	−0.746138	−0.373069	+	0.927804i	$0.621695\pi$
$150$	0	0
$151$	−9.73474	−0.792202	−0.396101	−	0.918207i	$-0.629637\pi$
$151$	−9.73474	−0.792202	−0.396101	+	0.918207i	$0.629637\pi$
$152$	−12.1512	−0.985595
$153$	0	0
$154$	4.65711	0.375281
$155$	−7.48088	−0.600879
$156$	0	0
$157$	10.5542	0.842315	0.421158	−	0.906987i	$-0.361624\pi$
$157$	10.5542	0.842315	0.421158	+	0.906987i	$0.361624\pi$
$158$	10.3029	0.819655
$159$	0	0
$160$	16.7120	1.32120
$161$	−1.77438	−0.139841
$162$	0	0
$163$	6.03292	0.472534	0.236267	−	0.971688i	$-0.424076\pi$
$163$	6.03292	0.472534	0.236267	+	0.971688i	$0.424076\pi$
$164$	−7.71360	−0.602331
$165$	0	0
$166$	−11.9503	−0.927520
$167$	−1.24770	−0.0965497	−0.0482748	−	0.998834i	$-0.515372\pi$
$167$	−1.24770	−0.0965497	−0.0482748	+	0.998834i	$0.515372\pi$
$168$	0	0
$169$	−7.96860	−0.612969
$170$	−0.793885	−0.0608882
$171$	0	0
$172$	5.29421	0.403680
$173$	7.02419	0.534039	0.267020	−	0.963691i	$-0.413961\pi$
$173$	7.02419	0.534039	0.267020	+	0.963691i	$0.413961\pi$
$174$	0	0
$175$	3.15968	0.238849
$176$	4.18125	0.315173
$177$	0	0
$178$	2.03803	0.152757
$179$	16.0501	1.19964	0.599822	−	0.800134i	$-0.295238\pi$
$179$	16.0501	1.19964	0.599822	+	0.800134i	$0.295238\pi$
$180$	0	0
$181$	−10.0388	−0.746181	−0.373091	−	0.927795i	$-0.621702\pi$
$181$	−10.0388	−0.746181	−0.373091	+	0.927795i	$0.621702\pi$
$182$	−1.75482	−0.130076
$183$	0	0
$184$	4.70299	0.346709
$185$	1.27526	0.0937590
$186$	0	0
$187$	−2.11477	−0.154647
$188$	−6.79659	−0.495692
$189$	0	0
$190$	10.2451	0.743260
$191$	−12.8858	−0.932386	−0.466193	−	0.884683i	$-0.654375\pi$
$191$	−12.8858	−0.932386	−0.466193	+	0.884683i	$0.654375\pi$
$192$	0	0
$193$	15.5417	1.11871	0.559357	−	0.828927i	$-0.311048\pi$
$193$	15.5417	1.11871	0.559357	+	0.828927i	$0.311048\pi$
$194$	−2.43254	−0.174647
$195$	0	0
$196$	−1.38797	−0.0991405
$197$	9.10451	0.648669	0.324335	−	0.945942i	$-0.394860\pi$
$197$	9.10451	0.648669	0.324335	+	0.945942i	$0.394860\pi$
$198$	0	0
$199$	20.7799	1.47305	0.736524	−	0.676411i	$-0.236466\pi$
$199$	20.7799	1.47305	0.736524	+	0.676411i	$0.236466\pi$
$200$	−8.37471	−0.592181
$201$	0	0
$202$	−1.85146	−0.130268
$203$	9.18000	0.644310
$204$	0	0
$205$	15.8750	1.10876
$206$	2.54179	0.177095
$207$	0	0
$208$	−1.57551	−0.109242
$209$	27.2912	1.88778
$210$	0	0
$211$	−7.23948	−0.498387	−0.249193	−	0.968454i	$-0.580165\pi$
$211$	−7.23948	−0.498387	−0.249193	+	0.968454i	$0.580165\pi$
$212$	18.0034	1.23648
$213$	0	0
$214$	0.0813882	0.00556358
$215$	−10.8958	−0.743087
$216$	0	0
$217$	−2.61888	−0.177781
$218$	15.5086	1.05037
$219$	0	0
$220$	−23.6018	−1.59123
$221$	0.796853	0.0536021
$222$	0	0
$223$	−16.9713	−1.13648	−0.568241	−	0.822862i	$-0.692376\pi$
$223$	−16.9713	−1.13648	−0.568241	+	0.822862i	$0.692376\pi$
$224$	5.85048	0.390902
$225$	0	0
$226$	10.3989	0.691723
$227$	−26.3750	−1.75057	−0.875285	−	0.483607i	$-0.839326\pi$
$227$	−26.3750	−1.75057	−0.875285	+	0.483607i	$0.839326\pi$
$228$	0	0
$229$	22.6839	1.49900	0.749498	−	0.662006i	$-0.230295\pi$
$229$	22.6839	1.49900	0.749498	+	0.662006i	$0.230295\pi$
$230$	−3.96525	−0.261461
$231$	0	0
$232$	−24.3315	−1.59744
$233$	−19.9035	−1.30392	−0.651960	−	0.758253i	$-0.726053\pi$
$233$	−19.9035	−1.30392	−0.651960	+	0.758253i	$0.726053\pi$
$234$	0	0
$235$	13.9878	0.912461
$236$	7.42448	0.483292
$237$	0	0
$238$	−0.277921	−0.0180149
$239$	−14.8701	−0.961870	−0.480935	−	0.876756i	$-0.659703\pi$
$239$	−14.8701	−0.961870	−0.480935	+	0.876756i	$0.659703\pi$
$240$	0	0
$241$	29.5656	1.90449	0.952245	−	0.305335i	$-0.0987685\pi$
$241$	29.5656	1.90449	0.952245	+	0.305335i	$0.0987685\pi$
$242$	19.1178	1.22894
$243$	0	0
$244$	−5.14970	−0.329676
$245$	2.85652	0.182496
$246$	0	0
$247$	−10.2834	−0.654320
$248$	6.94133	0.440775
$249$	0	0
$250$	−4.11261	−0.260104
$251$	−20.0199	−1.26364	−0.631821	−	0.775114i	$-0.717692\pi$
$251$	−20.0199	−1.26364	−0.631821	+	0.775114i	$0.717692\pi$
$252$	0	0
$253$	−10.5627	−0.664074
$254$	0.782325	0.0490875
$255$	0	0
$256$	−13.5569	−0.847305
$257$	26.4101	1.64742	0.823709	−	0.567012i	$-0.191901\pi$
$257$	26.4101	1.64742	0.823709	+	0.567012i	$0.191901\pi$
$258$	0	0
$259$	0.446439	0.0277404
$260$	8.89325	0.551536
$261$	0	0
$262$	−1.16099	−0.0717261
$263$	18.5090	1.14131	0.570656	−	0.821189i	$-0.306689\pi$
$263$	18.5090	1.14131	0.570656	+	0.821189i	$0.306689\pi$
$264$	0	0
$265$	−37.0520	−2.27609
$266$	3.58659	0.219908
$267$	0	0
$268$	−14.3305	−0.875376
$269$	−0.573334	−0.0349568	−0.0174784	−	0.999847i	$-0.505564\pi$
$269$	−0.573334	−0.0349568	−0.0174784	+	0.999847i	$0.505564\pi$
$270$	0	0
$271$	7.26958	0.441595	0.220798	−	0.975320i	$-0.429134\pi$
$271$	7.26958	0.441595	0.220798	+	0.975320i	$0.429134\pi$
$272$	−0.249523	−0.0151295
$273$	0	0
$274$	−3.06986	−0.185457
$275$	18.8093	1.13424
$276$	0	0
$277$	−16.1927	−0.972924	−0.486462	−	0.873702i	$-0.661713\pi$
$277$	−16.1927	−0.972924	−0.486462	+	0.873702i	$0.661713\pi$
$278$	−1.26257	−0.0757242
$279$	0	0
$280$	−7.57117	−0.452464
$281$	0.461522	0.0275321	0.0137661	−	0.999905i	$-0.495618\pi$
$281$	0.461522	0.0275321	0.0137661	+	0.999905i	$0.495618\pi$
$282$	0	0
$283$	3.00080	0.178379	0.0891895	−	0.996015i	$-0.471572\pi$
$283$	3.00080	0.178379	0.0891895	+	0.996015i	$0.471572\pi$
$284$	1.83157	0.108683
$285$	0	0
$286$	−10.4463	−0.617701
$287$	5.55748	0.328048
$288$	0	0
$289$	−16.8738	−0.992576
$290$	20.5148	1.20467
$291$	0	0
$292$	12.2929	0.719386
$293$	−14.8706	−0.868752	−0.434376	−	0.900732i	$-0.643031\pi$
$293$	−14.8706	−0.868752	−0.434376	+	0.900732i	$0.643031\pi$
$294$	0	0
$295$	−15.2800	−0.889636
$296$	−1.18328	−0.0687770
$297$	0	0
$298$	−7.12524	−0.412754
$299$	3.98008	0.230174
$300$	0	0
$301$	−3.81437	−0.219856
$302$	−7.61573	−0.438236
$303$	0	0
$304$	3.22011	0.184686
$305$	10.5984	0.606862
$306$	0	0
$307$	20.0380	1.14363	0.571815	−	0.820383i	$-0.306240\pi$
$307$	20.0380	1.14363	0.571815	+	0.820383i	$0.306240\pi$
$308$	−8.26245	−0.470797
$309$	0	0
$310$	−5.85248	−0.332399
$311$	1.42299	0.0806904	0.0403452	−	0.999186i	$-0.487154\pi$
$311$	1.42299	0.0806904	0.0403452	+	0.999186i	$0.487154\pi$
$312$	0	0
$313$	8.24182	0.465855	0.232928	−	0.972494i	$-0.425170\pi$
$313$	8.24182	0.465855	0.232928	+	0.972494i	$0.425170\pi$
$314$	8.25680	0.465958
$315$	0	0
$316$	−18.2790	−1.02827
$317$	−22.8267	−1.28208	−0.641038	−	0.767509i	$-0.721496\pi$
$317$	−22.8267	−1.28208	−0.641038	+	0.767509i	$0.721496\pi$
$318$	0	0
$319$	54.6477	3.05969
$320$	9.06145	0.506551
$321$	0	0
$322$	−1.38814	−0.0773582
$323$	−1.62865	−0.0906205
$324$	0	0
$325$	−7.08741	−0.393139
$326$	4.71970	0.261400
$327$	0	0
$328$	−14.7301	−0.813331
$329$	4.89679	0.269969
$330$	0	0
$331$	23.7844	1.30731	0.653656	−	0.756792i	$-0.273235\pi$
$331$	23.7844	1.30731	0.653656	+	0.756792i	$0.273235\pi$
$332$	21.2016	1.16359
$333$	0	0
$334$	−0.976105	−0.0534101
$335$	29.4930	1.61138
$336$	0	0
$337$	−6.16356	−0.335750	−0.167875	−	0.985808i	$-0.553691\pi$
$337$	−6.16356	−0.335750	−0.167875	+	0.985808i	$0.553691\pi$
$338$	−6.23403	−0.339087
$339$	0	0
$340$	1.40848	0.0763853
$341$	−15.5900	−0.844245
$342$	0	0
$343$	1.00000	0.0539949
$344$	10.1099	0.545092
$345$	0	0
$346$	5.49520	0.295424
$347$	10.9479	0.587716	0.293858	−	0.955849i	$-0.405061\pi$
$347$	10.9479	0.587716	0.293858	+	0.955849i	$0.405061\pi$
$348$	0	0
$349$	−4.35089	−0.232898	−0.116449	−	0.993197i	$-0.537151\pi$
$349$	−4.35089	−0.232898	−0.116449	+	0.993197i	$0.537151\pi$
$350$	2.47190	0.132129
$351$	0	0
$352$	34.8274	1.85631
$353$	23.5716	1.25459	0.627294	−	0.778782i	$-0.284162\pi$
$353$	23.5716	1.25459	0.627294	+	0.778782i	$0.284162\pi$
$354$	0	0
$355$	−3.76947	−0.200063
$356$	−3.61578	−0.191636
$357$	0	0
$358$	12.5564	0.663628
$359$	10.7509	0.567412	0.283706	−	0.958911i	$-0.408436\pi$
$359$	10.7509	0.567412	0.283706	+	0.958911i	$0.408436\pi$
$360$	0	0
$361$	2.01782	0.106201
$362$	−7.85364	−0.412778
$363$	0	0
$364$	3.11332	0.163182
$365$	−25.2995	−1.32423
$366$	0	0
$367$	13.4042	0.699694	0.349847	−	0.936807i	$-0.386234\pi$
$367$	13.4042	0.699694	0.349847	+	0.936807i	$0.386234\pi$
$368$	−1.24630	−0.0649680
$369$	0	0
$370$	0.997669	0.0518663
$371$	−12.9711	−0.673424
$372$	0	0
$373$	2.43314	0.125983	0.0629915	−	0.998014i	$-0.479936\pi$
$373$	2.43314	0.125983	0.0629915	+	0.998014i	$0.479936\pi$
$374$	−1.65444	−0.0855490
$375$	0	0
$376$	−12.9789	−0.669336
$377$	−20.5915	−1.06051
$378$	0	0
$379$	1.12262	0.0576652	0.0288326	−	0.999584i	$-0.490821\pi$
$379$	1.12262	0.0576652	0.0288326	+	0.999584i	$0.490821\pi$
$380$	−18.1765	−0.932433
$381$	0	0
$382$	−10.0809	−0.515784
$383$	12.1235	0.619484	0.309742	−	0.950821i	$-0.399757\pi$
$383$	12.1235	0.619484	0.309742	+	0.950821i	$0.399757\pi$
$384$	0	0
$385$	17.0046	0.866634
$386$	12.1586	0.618858
$387$	0	0
$388$	4.31571	0.219097
$389$	7.49818	0.380173	0.190086	−	0.981767i	$-0.439123\pi$
$389$	7.49818	0.380173	0.190086	+	0.981767i	$0.439123\pi$
$390$	0	0
$391$	0.630349	0.0318781
$392$	−2.65049	−0.133870
$393$	0	0
$394$	7.12268	0.358836
$395$	37.6191	1.89282
$396$	0	0
$397$	2.60531	0.130757	0.0653785	−	0.997861i	$-0.479175\pi$
$397$	2.60531	0.130757	0.0653785	+	0.997861i	$0.479175\pi$
$398$	16.2566	0.814872
$399$	0	0
$400$	2.21932	0.110966
$401$	34.5142	1.72356	0.861778	−	0.507285i	$-0.169351\pi$
$401$	34.5142	1.72356	0.861778	+	0.507285i	$0.169351\pi$
$402$	0	0
$403$	5.87436	0.292623
$404$	3.28478	0.163424
$405$	0	0
$406$	7.18175	0.356424
$407$	2.65761	0.131733
$408$	0	0
$409$	−18.5829	−0.918864	−0.459432	−	0.888213i	$-0.651947\pi$
$409$	−18.5829	−0.918864	−0.459432	+	0.888213i	$0.651947\pi$
$410$	12.4194	0.613352
$411$	0	0
$412$	−4.50953	−0.222168
$413$	−5.34918	−0.263216
$414$	0	0
$415$	−43.6342	−2.14192
$416$	−13.1231	−0.643412
$417$	0	0
$418$	21.3506	1.04429
$419$	34.4281	1.68192	0.840962	−	0.541094i	$-0.181990\pi$
$419$	34.4281	1.68192	0.840962	+	0.541094i	$0.181990\pi$
$420$	0	0
$421$	10.5543	0.514383	0.257191	−	0.966360i	$-0.417203\pi$
$421$	10.5543	0.514383	0.257191	+	0.966360i	$0.417203\pi$
$422$	−5.66363	−0.275701
$423$	0	0
$424$	34.3797	1.66963
$425$	−1.12248	−0.0544481
$426$	0	0
$427$	3.71025	0.179552
$428$	−0.144395	−0.00697961
$429$	0	0
$430$	−8.52406	−0.411067
$431$	−23.7306	−1.14306	−0.571531	−	0.820581i	$-0.693650\pi$
$431$	−23.7306	−1.14306	−0.571531	+	0.820581i	$0.693650\pi$
$432$	0	0
$433$	3.32779	0.159923	0.0799617	−	0.996798i	$-0.474520\pi$
$433$	3.32779	0.159923	0.0799617	+	0.996798i	$0.474520\pi$
$434$	−2.04882	−0.0983464
$435$	0	0
$436$	−27.5146	−1.31771
$437$	−8.13469	−0.389135
$438$	0	0
$439$	20.6852	0.987252	0.493626	−	0.869674i	$-0.335671\pi$
$439$	20.6852	0.987252	0.493626	+	0.869674i	$0.335671\pi$
$440$	−45.0705	−2.14865
$441$	0	0
$442$	0.623398	0.0296520
$443$	40.0896	1.90472	0.952358	−	0.304981i	$-0.0986503\pi$
$443$	40.0896	1.90472	0.952358	+	0.304981i	$0.0986503\pi$
$444$	0	0
$445$	7.44149	0.352760
$446$	−13.2771	−0.628687
$447$	0	0
$448$	3.17221	0.149873
$449$	20.3346	0.959648	0.479824	−	0.877365i	$-0.340700\pi$
$449$	20.3346	0.959648	0.479824	+	0.877365i	$0.340700\pi$
$450$	0	0
$451$	33.0832	1.55783
$452$	−18.4492	−0.867779
$453$	0	0
$454$	−20.6338	−0.968394
$455$	−6.40739	−0.300383
$456$	0	0
$457$	−11.4154	−0.533991	−0.266996	−	0.963698i	$-0.586031\pi$
$457$	−11.4154	−0.533991	−0.266996	+	0.963698i	$0.586031\pi$
$458$	17.7462	0.829226
$459$	0	0
$460$	7.03498	0.328008
$461$	35.1107	1.63527	0.817634	−	0.575738i	$-0.195285\pi$
$461$	35.1107	1.63527	0.817634	+	0.575738i	$0.195285\pi$
$462$	0	0
$463$	−9.61602	−0.446894	−0.223447	−	0.974716i	$-0.571731\pi$
$463$	−9.61602	−0.446894	−0.223447	+	0.974716i	$0.571731\pi$
$464$	6.44791	0.299337
$465$	0	0
$466$	−15.5710	−0.721313
$467$	−32.7492	−1.51545	−0.757725	−	0.652574i	$-0.773689\pi$
$467$	−32.7492	−1.51545	−0.757725	+	0.652574i	$0.773689\pi$
$468$	0	0
$469$	10.3248	0.476756
$470$	10.9430	0.504762
$471$	0	0
$472$	14.1780	0.652593
$473$	−22.7066	−1.04405
$474$	0	0
$475$	14.4856	0.664646
$476$	0.493075	0.0226000
$477$	0	0
$478$	−11.6333	−0.532095
$479$	−4.11326	−0.187940	−0.0939698	−	0.995575i	$-0.529956\pi$
$479$	−4.11326	−0.187940	−0.0939698	+	0.995575i	$0.529956\pi$
$480$	0	0
$481$	−1.00140	−0.0456598
$482$	23.1299	1.05354
$483$	0	0
$484$	−33.9180	−1.54173
$485$	−8.88199	−0.403310
$486$	0	0
$487$	−10.3942	−0.471007	−0.235504	−	0.971873i	$-0.575674\pi$
$487$	−10.3942	−0.471007	−0.235504	+	0.971873i	$0.575674\pi$
$488$	−9.83399	−0.445164
$489$	0	0
$490$	2.23472	0.100955
$491$	−11.6309	−0.524895	−0.262447	−	0.964946i	$-0.584530\pi$
$491$	−11.6309	−0.524895	−0.262447	+	0.964946i	$0.584530\pi$
$492$	0	0
$493$	−3.26119	−0.146877
$494$	−8.04500	−0.361961
$495$	0	0
$496$	−1.83947	−0.0825946
$497$	−1.31960	−0.0591923
$498$	0	0
$499$	18.6834	0.836384	0.418192	−	0.908359i	$-0.362664\pi$
$499$	18.6834	0.836384	0.418192	+	0.908359i	$0.362664\pi$
$500$	7.29640	0.326305
$501$	0	0
$502$	−15.6620	−0.699031
$503$	37.2918	1.66276	0.831380	−	0.555705i	$-0.187551\pi$
$503$	37.2918	1.66276	0.831380	+	0.555705i	$0.187551\pi$
$504$	0	0
$505$	−6.76026	−0.300828
$506$	−8.26350	−0.367357
$507$	0	0
$508$	−1.38797	−0.0615811
$509$	−30.3885	−1.34695	−0.673474	−	0.739211i	$-0.735199\pi$
$509$	−30.3885	−1.34695	−0.673474	+	0.739211i	$0.735199\pi$
$510$	0	0
$511$	−8.85675	−0.391800
$512$	7.83264	0.346157
$513$	0	0
$514$	20.6613	0.911331
$515$	9.28086	0.408964
$516$	0	0
$517$	29.1502	1.28202
$518$	0.349261	0.0153456
$519$	0	0
$520$	16.9827	0.744742
$521$	−13.9497	−0.611146	−0.305573	−	0.952169i	$-0.598848\pi$
$521$	−13.9497	−0.611146	−0.305573	+	0.952169i	$0.598848\pi$
$522$	0	0
$523$	−22.2163	−0.971450	−0.485725	−	0.874112i	$-0.661444\pi$
$523$	−22.2163	−0.971450	−0.485725	+	0.874112i	$0.661444\pi$
$524$	2.05977	0.0899816
$525$	0	0
$526$	14.4800	0.631360
$527$	0.930357	0.0405270
$528$	0	0
$529$	−19.8516	−0.863112
$530$	−28.9867	−1.25910
$531$	0	0
$532$	−6.36316	−0.275878
$533$	−12.4659	−0.539957
$534$	0	0
$535$	0.297174	0.0128479
$536$	−27.3659	−1.18203
$537$	0	0
$538$	−0.448534	−0.0193377
$539$	5.95291	0.256410
$540$	0	0
$541$	25.9878	1.11730	0.558651	−	0.829403i	$-0.311319\pi$
$541$	25.9878	1.11730	0.558651	+	0.829403i	$0.311319\pi$
$542$	5.68718	0.244285
$543$	0	0
$544$	−2.07838	−0.0891098
$545$	56.6267	2.42562
$546$	0	0
$547$	−38.7425	−1.65651	−0.828254	−	0.560352i	$-0.810666\pi$
$547$	−38.7425	−1.65651	−0.828254	+	0.560352i	$0.810666\pi$
$548$	5.44640	0.232659
$549$	0	0
$550$	14.7150	0.627450
$551$	42.0859	1.79292
$552$	0	0
$553$	13.1696	0.560028
$554$	−12.6679	−0.538209
$555$	0	0
$556$	2.24000	0.0949973
$557$	13.2224	0.560252	0.280126	−	0.959963i	$-0.409624\pi$
$557$	13.2224	0.560252	0.280126	+	0.959963i	$0.409624\pi$
$558$	0	0
$559$	8.55592	0.361877
$560$	2.00638	0.0847850
$561$	0	0
$562$	0.361061	0.0152304
$563$	12.5799	0.530180	0.265090	−	0.964224i	$-0.414598\pi$
$563$	12.5799	0.530180	0.265090	+	0.964224i	$0.414598\pi$
$564$	0	0
$565$	37.9696	1.59739
$566$	2.34760	0.0986771
$567$	0	0
$568$	3.49760	0.146756
$569$	−22.4858	−0.942653	−0.471326	−	0.881959i	$-0.656225\pi$
$569$	−22.4858	−0.942653	−0.471326	+	0.881959i	$0.656225\pi$
$570$	0	0
$571$	10.9617	0.458733	0.229366	−	0.973340i	$-0.426335\pi$
$571$	10.9617	0.458733	0.229366	+	0.973340i	$0.426335\pi$
$572$	18.5333	0.774917
$573$	0	0
$574$	4.34776	0.181472
$575$	−5.60648	−0.233806
$576$	0	0
$577$	−7.86167	−0.327286	−0.163643	−	0.986520i	$-0.552324\pi$
$577$	−7.86167	−0.327286	−0.163643	+	0.986520i	$0.552324\pi$
$578$	−13.2008	−0.549081
$579$	0	0
$580$	−36.3964	−1.51128
$581$	−15.2753	−0.633727
$582$	0	0
$583$	−77.2156	−3.19794
$584$	23.4748	0.971392
$585$	0	0
$586$	−11.6337	−0.480583
$587$	−45.8628	−1.89296	−0.946479	−	0.322764i	$-0.895388\pi$
$587$	−45.8628	−1.89296	−0.946479	+	0.322764i	$0.895388\pi$
$588$	0	0
$589$	−12.0063	−0.494712
$590$	−11.9539	−0.492136
$591$	0	0
$592$	0.313573	0.0128878
$593$	−19.8056	−0.813319	−0.406660	−	0.913580i	$-0.633307\pi$
$593$	−19.8056	−0.813319	−0.406660	+	0.913580i	$0.633307\pi$
$594$	0	0
$595$	−1.01478	−0.0416018
$596$	12.6413	0.517808
$597$	0	0
$598$	3.11372	0.127329
$599$	−34.5952	−1.41352	−0.706761	−	0.707452i	$-0.749844\pi$
$599$	−34.5952	−1.41352	−0.706761	+	0.707452i	$0.749844\pi$
$600$	0	0
$601$	10.5867	0.431840	0.215920	−	0.976411i	$-0.430725\pi$
$601$	10.5867	0.431840	0.215920	+	0.976411i	$0.430725\pi$
$602$	−2.98408	−0.121622
$603$	0	0
$604$	13.5115	0.549775
$605$	69.8052	2.83798
$606$	0	0
$607$	−28.2830	−1.14797	−0.573986	−	0.818865i	$-0.694604\pi$
$607$	−28.2830	−1.14797	−0.573986	+	0.818865i	$0.694604\pi$
$608$	26.8216	1.08776
$609$	0	0
$610$	8.29138	0.335708
$611$	−10.9839	−0.444361
$612$	0	0
$613$	−8.39067	−0.338896	−0.169448	−	0.985539i	$-0.554198\pi$
$613$	−8.39067	−0.338896	−0.169448	+	0.985539i	$0.554198\pi$
$614$	15.6762	0.632642
$615$	0	0
$616$	−15.7782	−0.635720
$617$	−1.54113	−0.0620437	−0.0310219	−	0.999519i	$-0.509876\pi$
$617$	−1.54113	−0.0620437	−0.0310219	+	0.999519i	$0.509876\pi$
$618$	0	0
$619$	−18.5681	−0.746313	−0.373157	−	0.927768i	$-0.621725\pi$
$619$	−18.5681	−0.746313	−0.373157	+	0.927768i	$0.621725\pi$
$620$	10.3832	0.417000
$621$	0	0
$622$	1.11324	0.0446369
$623$	2.60509	0.104371
$624$	0	0
$625$	−30.8148	−1.23259
$626$	6.44778	0.257705
$627$	0	0
$628$	−14.6489	−0.584553
$629$	−0.158597	−0.00632369
$630$	0	0
$631$	34.2361	1.36292	0.681459	−	0.731857i	$-0.261346\pi$
$631$	34.2361	1.36292	0.681459	+	0.731857i	$0.261346\pi$
$632$	−34.9059	−1.38848
$633$	0	0
$634$	−17.8579	−0.709229
$635$	2.85652	0.113357
$636$	0	0
$637$	−2.24308	−0.0888740
$638$	42.7523	1.69258
$639$	0	0
$640$	−26.3350	−1.04098
$641$	−0.674416	−0.0266378	−0.0133189	−	0.999911i	$-0.504240\pi$
$641$	−0.674416	−0.0266378	−0.0133189	+	0.999911i	$0.504240\pi$
$642$	0	0
$643$	29.7807	1.17444	0.587219	−	0.809428i	$-0.300223\pi$
$643$	29.7807	1.17444	0.587219	+	0.809428i	$0.300223\pi$
$644$	2.46278	0.0970473
$645$	0	0
$646$	−1.27413	−0.0501301
$647$	−19.3926	−0.762401	−0.381200	−	0.924492i	$-0.624489\pi$
$647$	−19.3926	−0.762401	−0.381200	+	0.924492i	$0.624489\pi$
$648$	0	0
$649$	−31.8432	−1.24995
$650$	−5.54466	−0.217480
$651$	0	0
$652$	−8.37349	−0.327931
$653$	7.50126	0.293547	0.146773	−	0.989170i	$-0.453111\pi$
$653$	7.50126	0.293547	0.146773	+	0.989170i	$0.453111\pi$
$654$	0	0
$655$	−4.23913	−0.165637
$656$	3.90350	0.152406
$657$	0	0
$658$	3.83088	0.149343
$659$	1.97123	0.0767882	0.0383941	−	0.999263i	$-0.487776\pi$
$659$	1.97123	0.0767882	0.0383941	+	0.999263i	$0.487776\pi$
$660$	0	0
$661$	−24.4180	−0.949750	−0.474875	−	0.880053i	$-0.657507\pi$
$661$	−24.4180	−0.949750	−0.474875	+	0.880053i	$0.657507\pi$
$662$	18.6072	0.723188
$663$	0	0
$664$	40.4871	1.57120
$665$	13.0958	0.507831
$666$	0	0
$667$	−16.2888	−0.630706
$668$	1.73176	0.0670039
$669$	0	0
$670$	23.0732	0.891394
$671$	22.0868	0.852651
$672$	0	0
$673$	−39.0533	−1.50539	−0.752697	−	0.658367i	$-0.771247\pi$
$673$	−39.0533	−1.50539	−0.752697	+	0.658367i	$0.771247\pi$
$674$	−4.82191	−0.185733
$675$	0	0
$676$	11.0601	0.425390
$677$	37.5231	1.44213	0.721065	−	0.692868i	$-0.243653\pi$
$677$	37.5231	1.44213	0.721065	+	0.692868i	$0.243653\pi$
$678$	0	0
$679$	−3.10938	−0.119327
$680$	2.68966	0.103144
$681$	0	0
$682$	−12.1964	−0.467026
$683$	42.5582	1.62845	0.814223	−	0.580552i	$-0.197163\pi$
$683$	42.5582	1.62845	0.814223	+	0.580552i	$0.197163\pi$
$684$	0	0
$685$	−11.2090	−0.428274
$686$	0.782325	0.0298693
$687$	0	0
$688$	−2.67916	−0.102142
$689$	29.0951	1.10844
$690$	0	0
$691$	8.50252	0.323451	0.161726	−	0.986836i	$-0.448294\pi$
$691$	8.50252	0.323451	0.161726	+	0.986836i	$0.448294\pi$
$692$	−9.74934	−0.370614
$693$	0	0
$694$	8.56485	0.325117
$695$	−4.61006	−0.174869
$696$	0	0
$697$	−1.97429	−0.0747817
$698$	−3.40381	−0.128836
$699$	0	0
$700$	−4.38553	−0.165758
$701$	−41.0282	−1.54962	−0.774808	−	0.632197i	$-0.782153\pi$
$701$	−41.0282	−1.54962	−0.774808	+	0.632197i	$0.782153\pi$
$702$	0	0
$703$	2.04671	0.0771931
$704$	18.8839	0.711712
$705$	0	0
$706$	18.4406	0.694023
$707$	−2.36661	−0.0890055
$708$	0	0
$709$	−27.8419	−1.04563	−0.522813	−	0.852448i	$-0.675117\pi$
$709$	−27.8419	−1.04563	−0.522813	+	0.852448i	$0.675117\pi$
$710$	−2.94895	−0.110672
$711$	0	0
$712$	−6.90478	−0.258767
$713$	4.64690	0.174028
$714$	0	0
$715$	−38.1426	−1.42645
$716$	−22.2771	−0.832533
$717$	0	0
$718$	8.41072	0.313885
$719$	−35.7992	−1.33509	−0.667543	−	0.744572i	$-0.732654\pi$
$719$	−35.7992	−1.33509	−0.667543	+	0.744572i	$0.732654\pi$
$720$	0	0
$721$	3.24902	0.121000
$722$	1.57859	0.0587492
$723$	0	0
$724$	13.9336	0.517838
$725$	29.0059	1.07725
$726$	0	0
$727$	28.1120	1.04262	0.521308	−	0.853369i	$-0.325444\pi$
$727$	28.1120	1.04262	0.521308	+	0.853369i	$0.325444\pi$
$728$	5.94526	0.220346
$729$	0	0
$730$	−19.7924	−0.732550
$731$	1.35505	0.0501184
$732$	0	0
$733$	−3.35859	−0.124053	−0.0620263	−	0.998075i	$-0.519756\pi$
$733$	−3.35859	−0.124053	−0.0620263	+	0.998075i	$0.519756\pi$
$734$	10.4865	0.387062
$735$	0	0
$736$	−10.3810	−0.382648
$737$	61.4628	2.26401
$738$	0	0
$739$	−8.81699	−0.324338	−0.162169	−	0.986763i	$-0.551849\pi$
$739$	−8.81699	−0.324338	−0.162169	+	0.986763i	$0.551849\pi$
$740$	−1.77002	−0.0650672
$741$	0	0
$742$	−10.1476	−0.372530
$743$	33.5929	1.23240	0.616201	−	0.787589i	$-0.288671\pi$
$743$	33.5929	1.23240	0.616201	+	0.787589i	$0.288671\pi$
$744$	0	0
$745$	−26.0165	−0.953171
$746$	1.90350	0.0696922
$747$	0	0
$748$	2.93523	0.107323
$749$	0.104034	0.00380131
$750$	0	0
$751$	38.7986	1.41578	0.707891	−	0.706321i	$-0.249647\pi$
$751$	38.7986	1.41578	0.707891	+	0.706321i	$0.249647\pi$
$752$	3.43944	0.125424
$753$	0	0
$754$	−16.1092	−0.586663
$755$	−27.8074	−1.01202
$756$	0	0
$757$	13.6304	0.495406	0.247703	−	0.968836i	$-0.420324\pi$
$757$	13.6304	0.495406	0.247703	+	0.968836i	$0.420324\pi$
$758$	0.878256	0.0318997
$759$	0	0
$760$	−34.7102	−1.25907
$761$	7.00438	0.253909	0.126954	−	0.991909i	$-0.459480\pi$
$761$	7.00438	0.253909	0.126954	+	0.991909i	$0.459480\pi$
$762$	0	0
$763$	19.8237	0.717666
$764$	17.8851	0.647061
$765$	0	0
$766$	9.48456	0.342691
$767$	11.9986	0.433245
$768$	0	0
$769$	−25.3807	−0.915251	−0.457625	−	0.889145i	$-0.651300\pi$
$769$	−25.3807	−0.915251	−0.457625	+	0.889145i	$0.651300\pi$
$770$	13.3031	0.479411
$771$	0	0
$772$	−21.5713	−0.776368
$773$	−45.1789	−1.62497	−0.812485	−	0.582982i	$-0.801886\pi$
$773$	−45.1789	−1.62497	−0.812485	+	0.582982i	$0.801886\pi$
$774$	0	0
$775$	−8.27483	−0.297241
$776$	8.24138	0.295848
$777$	0	0
$778$	5.86601	0.210307
$779$	25.4784	0.912857
$780$	0	0
$781$	−7.85548	−0.281091
$782$	0.493138	0.0176346
$783$	0	0
$784$	0.702387	0.0250852
$785$	30.1482	1.07603
$786$	0	0
$787$	17.2345	0.614344	0.307172	−	0.951654i	$-0.400617\pi$
$787$	17.2345	0.614344	0.307172	+	0.951654i	$0.400617\pi$
$788$	−12.6368	−0.450166
$789$	0	0
$790$	29.4304	1.04709
$791$	13.2923	0.472619
$792$	0	0
$793$	−8.32238	−0.295536
$794$	2.03820	0.0723331
$795$	0	0
$796$	−28.8418	−1.02227
$797$	−11.1590	−0.395273	−0.197636	−	0.980275i	$-0.563327\pi$
$797$	−11.1590	−0.395273	−0.197636	+	0.980275i	$0.563327\pi$
$798$	0	0
$799$	−1.73958	−0.0615420
$800$	18.4856	0.653566
$801$	0	0
$802$	27.0013	0.953450
$803$	−52.7235	−1.86057
$804$	0	0
$805$	−5.06855	−0.178643
$806$	4.59566	0.161875
$807$	0	0
$808$	6.27268	0.220672
$809$	8.81764	0.310012	0.155006	−	0.987914i	$-0.450460\pi$
$809$	8.81764	0.310012	0.155006	+	0.987914i	$0.450460\pi$
$810$	0	0
$811$	23.6575	0.830726	0.415363	−	0.909656i	$-0.363655\pi$
$811$	23.6575	0.830726	0.415363	+	0.909656i	$0.363655\pi$
$812$	−12.7415	−0.447140
$813$	0	0
$814$	2.07912	0.0728731
$815$	17.2331	0.603650
$816$	0	0
$817$	−17.4870	−0.611794
$818$	−14.5379	−0.508304
$819$	0	0
$820$	−22.0340	−0.769461
$821$	10.4013	0.363006	0.181503	−	0.983390i	$-0.441904\pi$
$821$	10.4013	0.363006	0.181503	+	0.983390i	$0.441904\pi$
$822$	0	0
$823$	−55.3493	−1.92935	−0.964677	−	0.263434i	$-0.915145\pi$
$823$	−55.3493	−1.92935	−0.964677	+	0.263434i	$0.915145\pi$
$824$	−8.61149	−0.299996
$825$	0	0
$826$	−4.18480	−0.145608
$827$	−48.9737	−1.70298	−0.851491	−	0.524369i	$-0.824301\pi$
$827$	−48.9737	−1.70298	−0.851491	+	0.524369i	$0.824301\pi$
$828$	0	0
$829$	13.1541	0.456859	0.228430	−	0.973560i	$-0.426641\pi$
$829$	13.1541	0.456859	0.228430	+	0.973560i	$0.426641\pi$
$830$	−34.1361	−1.18488
$831$	0	0
$832$	−7.11551	−0.246686
$833$	−0.355250	−0.0123087
$834$	0	0
$835$	−3.56407	−0.123340
$836$	−37.8794	−1.31008
$837$	0	0
$838$	26.9340	0.930420
$839$	22.3972	0.773237	0.386618	−	0.922240i	$-0.373643\pi$
$839$	22.3972	0.773237	0.386618	+	0.922240i	$0.373643\pi$
$840$	0	0
$841$	55.2724	1.90594
$842$	8.25686	0.284550
$843$	0	0
$844$	10.0482	0.345872
$845$	−22.7624	−0.783051
$846$	0	0
$847$	24.4372	0.839671
$848$	−9.11070	−0.312863
$849$	0	0
$850$	−0.878141	−0.0301200
$851$	−0.792154	−0.0271547
$852$	0	0
$853$	6.45732	0.221094	0.110547	−	0.993871i	$-0.464740\pi$
$853$	6.45732	0.221094	0.110547	+	0.993871i	$0.464740\pi$
$854$	2.90262	0.0993257
$855$	0	0
$856$	−0.275741	−0.00942462
$857$	2.95220	0.100845	0.0504226	−	0.998728i	$-0.483943\pi$
$857$	2.95220	0.100845	0.0504226	+	0.998728i	$0.483943\pi$
$858$	0	0
$859$	−36.7317	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$859$	−36.7317	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$860$	15.1230	0.515690
$861$	0	0
$862$	−18.5650	−0.632327
$863$	−20.3620	−0.693131	−0.346565	−	0.938026i	$-0.612652\pi$
$863$	−20.3620	−0.693131	−0.346565	+	0.938026i	$0.612652\pi$
$864$	0	0
$865$	20.0647	0.682220
$866$	2.60342	0.0884677
$867$	0	0
$868$	3.63492	0.123377
$869$	78.3974	2.65945
$870$	0	0
$871$	−23.1594	−0.784727
$872$	−52.5426	−1.77932
$873$	0	0
$874$	−6.36397	−0.215265
$875$	−5.25690	−0.177716
$876$	0	0
$877$	24.0567	0.812338	0.406169	−	0.913798i	$-0.366864\pi$
$877$	24.0567	0.812338	0.406169	+	0.913798i	$0.366864\pi$
$878$	16.1826	0.546136
$879$	0	0
$880$	11.9438	0.402625
$881$	−46.6478	−1.57161	−0.785803	−	0.618477i	$-0.787750\pi$
$881$	−46.6478	−1.57161	−0.785803	+	0.618477i	$0.787750\pi$
$882$	0	0
$883$	25.3836	0.854225	0.427112	−	0.904198i	$-0.359531\pi$
$883$	25.3836	0.854225	0.427112	+	0.904198i	$0.359531\pi$
$884$	−1.10601	−0.0371990
$885$	0	0
$886$	31.3631	1.05367
$887$	−25.9636	−0.871771	−0.435886	−	0.900002i	$-0.643565\pi$
$887$	−25.9636	−0.871771	−0.435886	+	0.900002i	$0.643565\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	5.82166	0.195143
$891$	0	0
$892$	23.5556	0.788699
$893$	22.4494	0.751242
$894$	0	0
$895$	45.8475	1.53251
$896$	−9.21926	−0.307994
$897$	0	0
$898$	15.9083	0.530866
$899$	−24.0413	−0.801824
$900$	0	0
$901$	4.60796	0.153514
$902$	25.8818	0.861770
$903$	0	0
$904$	−35.2310	−1.17177
$905$	−28.6761	−0.953226
$906$	0	0
$907$	−16.6421	−0.552591	−0.276296	−	0.961073i	$-0.589107\pi$
$907$	−16.6421	−0.552591	−0.276296	+	0.961073i	$0.589107\pi$
$908$	36.6076	1.21487
$909$	0	0
$910$	−5.01266	−0.166168
$911$	−23.3831	−0.774717	−0.387358	−	0.921929i	$-0.626612\pi$
$911$	−23.3831	−0.774717	−0.387358	+	0.921929i	$0.626612\pi$
$912$	0	0
$913$	−90.9326	−3.00943
$914$	−8.93058	−0.295397
$915$	0	0
$916$	−31.4845	−1.04028
$917$	−1.48402	−0.0490067
$918$	0	0
$919$	24.1039	0.795114	0.397557	−	0.917578i	$-0.369858\pi$
$919$	24.1039	0.795114	0.397557	+	0.917578i	$0.369858\pi$
$920$	13.4342	0.442911
$921$	0	0
$922$	27.4680	0.904610
$923$	2.95997	0.0974288
$924$	0	0
$925$	1.41061	0.0463804
$926$	−7.52286	−0.247216
$927$	0	0
$928$	53.7074	1.76303
$929$	11.7991	0.387117	0.193559	−	0.981089i	$-0.437997\pi$
$929$	11.7991	0.387117	0.193559	+	0.981089i	$0.437997\pi$
$930$	0	0
$931$	4.58452	0.150252
$932$	27.6254	0.904899
$933$	0	0
$934$	−25.6205	−0.838328
$935$	−6.04087	−0.197558
$936$	0	0
$937$	−13.8473	−0.452371	−0.226186	−	0.974084i	$-0.572626\pi$
$937$	−13.8473	−0.452371	−0.226186	+	0.974084i	$0.572626\pi$
$938$	8.07738	0.263736
$939$	0	0
$940$	−19.4146	−0.633233
$941$	−44.6036	−1.45404	−0.727018	−	0.686618i	$-0.759095\pi$
$941$	−44.6036	−1.45404	−0.727018	+	0.686618i	$0.759095\pi$
$942$	0	0
$943$	−9.86109	−0.321121
$944$	−3.75719	−0.122286
$945$	0	0
$946$	−17.7639	−0.577555
$947$	−23.5909	−0.766601	−0.383301	−	0.923624i	$-0.625213\pi$
$947$	−23.5909	−0.766601	−0.383301	+	0.923624i	$0.625213\pi$
$948$	0	0
$949$	19.8664	0.644891
$950$	11.3325	0.367674
$951$	0	0
$952$	0.941587	0.0305170
$953$	−22.2076	−0.719374	−0.359687	−	0.933073i	$-0.617116\pi$
$953$	−22.2076	−0.719374	−0.359687	+	0.933073i	$0.617116\pi$
$954$	0	0
$955$	−36.8086	−1.19110
$956$	20.6393	0.667522
$957$	0	0
$958$	−3.21791	−0.103966
$959$	−3.92401	−0.126713
$960$	0	0
$961$	−24.1415	−0.778757
$962$	−0.783419	−0.0252585
$963$	0	0
$964$	−41.0361	−1.32168
$965$	44.3950	1.42913
$966$	0	0
$967$	−15.9265	−0.512161	−0.256081	−	0.966655i	$-0.582431\pi$
$967$	−15.9265	−0.512161	−0.256081	+	0.966655i	$0.582431\pi$
$968$	−64.7705	−2.08180
$969$	0	0
$970$	−6.94860	−0.223106
$971$	34.2446	1.09896	0.549480	−	0.835507i	$-0.314826\pi$
$971$	34.2446	1.09896	0.549480	+	0.835507i	$0.314826\pi$
$972$	0	0
$973$	−1.61387	−0.0517384
$974$	−8.13166	−0.260555
$975$	0	0
$976$	2.60603	0.0834170
$977$	−6.04555	−0.193414	−0.0967072	−	0.995313i	$-0.530831\pi$
$977$	−6.04555	−0.193414	−0.0967072	+	0.995313i	$0.530831\pi$
$978$	0	0
$979$	15.5079	0.495634
$980$	−3.96475	−0.126649
$981$	0	0
$982$	−9.09914	−0.290365
$983$	−50.1818	−1.60055	−0.800275	−	0.599633i	$-0.795313\pi$
$983$	−50.1818	−1.60055	−0.800275	+	0.599633i	$0.795313\pi$
$984$	0	0
$985$	26.0072	0.828657
$986$	−2.55131	−0.0812504
$987$	0	0
$988$	14.2731	0.454087
$989$	6.76814	0.215214
$990$	0	0
$991$	34.7358	1.10342	0.551710	−	0.834036i	$-0.313976\pi$
$991$	34.7358	1.10342	0.551710	+	0.834036i	$0.313976\pi$
$992$	−15.3217	−0.486465
$993$	0	0
$994$	−1.03236	−0.0327444
$995$	59.3581	1.88178
$996$	0	0
$997$	−28.1054	−0.890108	−0.445054	−	0.895504i	$-0.646815\pi$
$997$	−28.1054	−0.890108	−0.445054	+	0.895504i	$0.646815\pi$
$998$	14.6165	0.462677
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.13		19
3.2	odd	2		2667.2.a.q.1.7	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.7	✓	19	3.2	odd	2
8001.2.a.v.1.13		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.782325 q^{2} -1.38797 q^{4} +2.85652 q^{5} +1.00000 q^{7} -2.65049 q^{8} +O(q^{10})\)	\(q+0.782325 q^{2} -1.38797 q^{4} +2.85652 q^{5} +1.00000 q^{7} -2.65049 q^{8} +2.23472 q^{10} +5.95291 q^{11} -2.24308 q^{13} +0.782325 q^{14} +0.702387 q^{16} -0.355250 q^{17} +4.58452 q^{19} -3.96475 q^{20} +4.65711 q^{22} -1.77438 q^{23} +3.15968 q^{25} -1.75482 q^{26} -1.38797 q^{28} +9.18000 q^{29} -2.61888 q^{31} +5.85048 q^{32} -0.277921 q^{34} +2.85652 q^{35} +0.446439 q^{37} +3.58659 q^{38} -7.57117 q^{40} +5.55748 q^{41} -3.81437 q^{43} -8.26245 q^{44} -1.38814 q^{46} +4.89679 q^{47} +1.00000 q^{49} +2.47190 q^{50} +3.11332 q^{52} -12.9711 q^{53} +17.0046 q^{55} -2.65049 q^{56} +7.18175 q^{58} -5.34918 q^{59} +3.71025 q^{61} -2.04882 q^{62} +3.17221 q^{64} -6.40739 q^{65} +10.3248 q^{67} +0.493075 q^{68} +2.23472 q^{70} -1.31960 q^{71} -8.85675 q^{73} +0.349261 q^{74} -6.36316 q^{76} +5.95291 q^{77} +13.1696 q^{79} +2.00638 q^{80} +4.34776 q^{82} -15.2753 q^{83} -1.01478 q^{85} -2.98408 q^{86} -15.7782 q^{88} +2.60509 q^{89} -2.24308 q^{91} +2.46278 q^{92} +3.83088 q^{94} +13.0958 q^{95} -3.10938 q^{97} +0.782325 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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