LMFDB - Embedded newform 8001.2.a.l.1.4

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:	$7$
Coefficient field:	7.7.118870813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 $ x^7 - 9x^5 - 3x^4 + 20x^3 + 7x^2 - 13*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.52532$ 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)

$f(q)$	$=$	$q-0.246202 q^{2} -1.93938 q^{4} +0.318209 q^{5} +1.00000 q^{7} +0.969884 q^{8} +O(q^{10})$	\(q-0.246202 q^{2} -1.93938 q^{4} +0.318209 q^{5} +1.00000 q^{7} +0.969884 q^{8} -0.0783436 q^{10} -2.79943 q^{11} -6.49995 q^{13} -0.246202 q^{14} +3.63998 q^{16} +1.33136 q^{17} +4.36328 q^{19} -0.617130 q^{20} +0.689226 q^{22} +5.29356 q^{23} -4.89874 q^{25} +1.60030 q^{26} -1.93938 q^{28} +2.35110 q^{29} -4.76449 q^{31} -2.83594 q^{32} -0.327782 q^{34} +0.318209 q^{35} +1.06889 q^{37} -1.07425 q^{38} +0.308626 q^{40} +1.46345 q^{41} +8.04774 q^{43} +5.42918 q^{44} -1.30329 q^{46} +7.16642 q^{47} +1.00000 q^{49} +1.20608 q^{50} +12.6059 q^{52} -9.12591 q^{53} -0.890805 q^{55} +0.969884 q^{56} -0.578846 q^{58} +5.02542 q^{59} -2.48901 q^{61} +1.17303 q^{62} -6.58175 q^{64} -2.06834 q^{65} +0.549822 q^{67} -2.58201 q^{68} -0.0783436 q^{70} +11.9210 q^{71} -13.8588 q^{73} -0.263163 q^{74} -8.46208 q^{76} -2.79943 q^{77} -1.49518 q^{79} +1.15827 q^{80} -0.360305 q^{82} +17.1283 q^{83} +0.423650 q^{85} -1.98137 q^{86} -2.71513 q^{88} -12.8927 q^{89} -6.49995 q^{91} -10.2663 q^{92} -1.76439 q^{94} +1.38844 q^{95} -7.20075 q^{97} -0.246202 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * 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 $

$2$	−0.246202	−0.174091	−0.0870455	−	0.996204i	$-0.527743\pi$
$2$	−0.246202	−0.174091	−0.0870455	+	0.996204i	$0.527743\pi$
$3$	0	0
$3$	0	0
$4$	−1.93938	−0.969692
$5$	0.318209	0.142307	0.0711537	−	0.997465i	$-0.477332\pi$
$5$	0.318209	0.142307	0.0711537	+	0.997465i	$0.477332\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.969884	0.342906
$9$	0	0
$10$	−0.0783436	−0.0247744
$11$	−2.79943	−0.844061	−0.422031	−	0.906582i	$-0.638683\pi$
$11$	−2.79943	−0.844061	−0.422031	+	0.906582i	$0.638683\pi$
$12$	0	0
$13$	−6.49995	−1.80276	−0.901381	−	0.433027i	$-0.857445\pi$
$13$	−6.49995	−1.80276	−0.901381	+	0.433027i	$0.857445\pi$
$14$	−0.246202	−0.0658002
$15$	0	0
$16$	3.63998	0.909996
$17$	1.33136	0.322901	0.161451	−	0.986881i	$-0.448383\pi$
$17$	1.33136	0.322901	0.161451	+	0.986881i	$0.448383\pi$
$18$	0	0
$19$	4.36328	1.00101	0.500503	−	0.865735i	$-0.333149\pi$
$19$	4.36328	1.00101	0.500503	+	0.865735i	$0.333149\pi$
$20$	−0.617130	−0.137994
$21$	0	0
$22$	0.689226	0.146943
$23$	5.29356	1.10378	0.551892	−	0.833915i	$-0.313906\pi$
$23$	5.29356	1.10378	0.551892	+	0.833915i	$0.313906\pi$
$24$	0	0
$25$	−4.89874	−0.979749
$26$	1.60030	0.313845
$27$	0	0
$28$	−1.93938	−0.366509
$29$	2.35110	0.436589	0.218295	−	0.975883i	$-0.429951\pi$
$29$	2.35110	0.436589	0.218295	+	0.975883i	$0.429951\pi$
$30$	0	0
$31$	−4.76449	−0.855729	−0.427864	−	0.903843i	$-0.640734\pi$
$31$	−4.76449	−0.855729	−0.427864	+	0.903843i	$0.640734\pi$
$32$	−2.83594	−0.501328
$33$	0	0
$34$	−0.327782	−0.0562142
$35$	0.318209	0.0537871
$36$	0	0
$37$	1.06889	0.175725	0.0878624	−	0.996133i	$-0.471996\pi$
$37$	1.06889	0.175725	0.0878624	+	0.996133i	$0.471996\pi$
$38$	−1.07425	−0.174266
$39$	0	0
$40$	0.308626	0.0487980
$41$	1.46345	0.228553	0.114277	−	0.993449i	$-0.463545\pi$
$41$	1.46345	0.228553	0.114277	+	0.993449i	$0.463545\pi$
$42$	0	0
$43$	8.04774	1.22727	0.613635	−	0.789590i	$-0.289707\pi$
$43$	8.04774	1.22727	0.613635	+	0.789590i	$0.289707\pi$
$44$	5.42918	0.818480
$45$	0	0
$46$	−1.30329	−0.192159
$47$	7.16642	1.04533	0.522665	−	0.852538i	$-0.324938\pi$
$47$	7.16642	1.04533	0.522665	+	0.852538i	$0.324938\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.20608	0.170565
$51$	0	0
$52$	12.6059	1.74812
$53$	−9.12591	−1.25354	−0.626770	−	0.779204i	$-0.715623\pi$
$53$	−9.12591	−1.25354	−0.626770	+	0.779204i	$0.715623\pi$
$54$	0	0
$55$	−0.890805	−0.120116
$56$	0.969884	0.129606
$57$	0	0
$58$	−0.578846	−0.0760063
$59$	5.02542	0.654255	0.327127	−	0.944980i	$-0.393919\pi$
$59$	5.02542	0.654255	0.327127	+	0.944980i	$0.393919\pi$
$60$	0	0
$61$	−2.48901	−0.318685	−0.159343	−	0.987223i	$-0.550937\pi$
$61$	−2.48901	−0.318685	−0.159343	+	0.987223i	$0.550937\pi$
$62$	1.17303	0.148975
$63$	0	0
$64$	−6.58175	−0.822719
$65$	−2.06834	−0.256546
$66$	0	0
$67$	0.549822	0.0671714	0.0335857	−	0.999436i	$-0.489307\pi$
$67$	0.549822	0.0671714	0.0335857	+	0.999436i	$0.489307\pi$
$68$	−2.58201	−0.313115
$69$	0	0
$70$	−0.0783436	−0.00936386
$71$	11.9210	1.41476	0.707380	−	0.706833i	$-0.249877\pi$
$71$	11.9210	1.41476	0.707380	+	0.706833i	$0.249877\pi$
$72$	0	0
$73$	−13.8588	−1.62205	−0.811027	−	0.585008i	$-0.801091\pi$
$73$	−13.8588	−1.62205	−0.811027	+	0.585008i	$0.801091\pi$
$74$	−0.263163	−0.0305921
$75$	0	0
$76$	−8.46208	−0.970667
$77$	−2.79943	−0.319025
$78$	0	0
$79$	−1.49518	−0.168221	−0.0841106	−	0.996456i	$-0.526805\pi$
$79$	−1.49518	−0.168221	−0.0841106	+	0.996456i	$0.526805\pi$
$80$	1.15827	0.129499
$81$	0	0
$82$	−0.360305	−0.0397891
$83$	17.1283	1.88008	0.940039	−	0.341067i	$-0.110788\pi$
$83$	17.1283	1.88008	0.940039	+	0.341067i	$0.110788\pi$
$84$	0	0
$85$	0.423650	0.0459512
$86$	−1.98137	−0.213657
$87$	0	0
$88$	−2.71513	−0.289433
$89$	−12.8927	−1.36662	−0.683309	−	0.730129i	$-0.739460\pi$
$89$	−12.8927	−1.36662	−0.683309	+	0.730129i	$0.739460\pi$
$90$	0	0
$91$	−6.49995	−0.681380
$92$	−10.2663	−1.07033
$93$	0	0
$94$	−1.76439	−0.181983
$95$	1.38844	0.142450
$96$	0	0
$97$	−7.20075	−0.731125	−0.365563	−	0.930787i	$-0.619123\pi$
$97$	−7.20075	−0.731125	−0.365563	+	0.930787i	$0.619123\pi$
$98$	−0.246202	−0.0248701
$99$	0	0
$100$	9.50055	0.950055
$101$	0.894063	0.0889625	0.0444813	−	0.999010i	$-0.485836\pi$
$101$	0.894063	0.0889625	0.0444813	+	0.999010i	$0.485836\pi$
$102$	0	0
$103$	10.2130	1.00631	0.503157	−	0.864195i	$-0.332172\pi$
$103$	10.2130	1.00631	0.503157	+	0.864195i	$0.332172\pi$
$104$	−6.30420	−0.618177
$105$	0	0
$106$	2.24682	0.218230
$107$	−1.98724	−0.192114	−0.0960569	−	0.995376i	$-0.530623\pi$
$107$	−1.98724	−0.192114	−0.0960569	+	0.995376i	$0.530623\pi$
$108$	0	0
$109$	−18.2252	−1.74566	−0.872831	−	0.488022i	$-0.837718\pi$
$109$	−18.2252	−1.74566	−0.872831	+	0.488022i	$0.837718\pi$
$110$	0.219318	0.0209111
$111$	0	0
$112$	3.63998	0.343946
$113$	6.23590	0.586624	0.293312	−	0.956017i	$-0.405243\pi$
$113$	6.23590	0.586624	0.293312	+	0.956017i	$0.405243\pi$
$114$	0	0
$115$	1.68446	0.157077
$116$	−4.55970	−0.423357
$117$	0	0
$118$	−1.23727	−0.113900
$119$	1.33136	0.122045
$120$	0	0
$121$	−3.16317	−0.287561
$122$	0.612799	0.0554802
$123$	0	0
$124$	9.24019	0.829793
$125$	−3.14987	−0.281733
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	7.29232	0.644556
$129$	0	0
$130$	0.509230	0.0446624
$131$	18.7890	1.64160	0.820800	−	0.571215i	$-0.193528\pi$
$131$	18.7890	1.64160	0.820800	+	0.571215i	$0.193528\pi$
$132$	0	0
$133$	4.36328	0.378344
$134$	−0.135367	−0.0116939
$135$	0	0
$136$	1.29126	0.110725
$137$	−20.9329	−1.78842	−0.894209	−	0.447649i	$-0.852261\pi$
$137$	−20.9329	−1.78842	−0.894209	+	0.447649i	$0.852261\pi$
$138$	0	0
$139$	−5.77630	−0.489940	−0.244970	−	0.969531i	$-0.578778\pi$
$139$	−5.77630	−0.489940	−0.244970	+	0.969531i	$0.578778\pi$
$140$	−0.617130	−0.0521570
$141$	0	0
$142$	−2.93497	−0.246297
$143$	18.1962	1.52164
$144$	0	0
$145$	0.748143	0.0621299
$146$	3.41207	0.282385
$147$	0	0
$148$	−2.07299	−0.170399
$149$	5.65638	0.463389	0.231694	−	0.972789i	$-0.425573\pi$
$149$	5.65638	0.463389	0.231694	+	0.972789i	$0.425573\pi$
$150$	0	0
$151$	−2.50255	−0.203655	−0.101827	−	0.994802i	$-0.532469\pi$
$151$	−2.50255	−0.203655	−0.101827	+	0.994802i	$0.532469\pi$
$152$	4.23188	0.343250
$153$	0	0
$154$	0.689226	0.0555394
$155$	−1.51610	−0.121776
$156$	0	0
$157$	−14.3393	−1.14440	−0.572201	−	0.820113i	$-0.693910\pi$
$157$	−14.3393	−1.14440	−0.572201	+	0.820113i	$0.693910\pi$
$158$	0.368117	0.0292858
$159$	0	0
$160$	−0.902421	−0.0713426
$161$	5.29356	0.417191
$162$	0	0
$163$	−16.7787	−1.31421	−0.657105	−	0.753799i	$-0.728219\pi$
$163$	−16.7787	−1.31421	−0.657105	+	0.753799i	$0.728219\pi$
$164$	−2.83820	−0.221626
$165$	0	0
$166$	−4.21703	−0.327305
$167$	10.9980	0.851048	0.425524	−	0.904947i	$-0.360090\pi$
$167$	10.9980	0.851048	0.425524	+	0.904947i	$0.360090\pi$
$168$	0	0
$169$	29.2493	2.24995
$170$	−0.104303	−0.00799970
$171$	0	0
$172$	−15.6077	−1.19007
$173$	−14.9307	−1.13516	−0.567579	−	0.823319i	$-0.692120\pi$
$173$	−14.9307	−1.13516	−0.567579	+	0.823319i	$0.692120\pi$
$174$	0	0
$175$	−4.89874	−0.370310
$176$	−10.1899	−0.768092
$177$	0	0
$178$	3.17419	0.237916
$179$	−10.5989	−0.792200	−0.396100	−	0.918207i	$-0.629637\pi$
$179$	−10.5989	−0.792200	−0.396100	+	0.918207i	$0.629637\pi$
$180$	0	0
$181$	14.6508	1.08898	0.544491	−	0.838766i	$-0.316723\pi$
$181$	14.6508	1.08898	0.544491	+	0.838766i	$0.316723\pi$
$182$	1.60030	0.118622
$183$	0	0
$184$	5.13414	0.378494
$185$	0.340131	0.0250069
$186$	0	0
$187$	−3.72704	−0.272548
$188$	−13.8985	−1.01365
$189$	0	0
$190$	−0.341835	−0.0247993
$191$	17.6738	1.27883	0.639415	−	0.768862i	$-0.279177\pi$
$191$	17.6738	1.27883	0.639415	+	0.768862i	$0.279177\pi$
$192$	0	0
$193$	−13.3112	−0.958158	−0.479079	−	0.877772i	$-0.659029\pi$
$193$	−13.3112	−0.958158	−0.479079	+	0.877772i	$0.659029\pi$
$194$	1.77284	0.127282
$195$	0	0
$196$	−1.93938	−0.138527
$197$	−3.10847	−0.221469	−0.110735	−	0.993850i	$-0.535320\pi$
$197$	−3.10847	−0.221469	−0.110735	+	0.993850i	$0.535320\pi$
$198$	0	0
$199$	−21.0921	−1.49518	−0.747590	−	0.664160i	$-0.768789\pi$
$199$	−21.0921	−1.49518	−0.747590	+	0.664160i	$0.768789\pi$
$200$	−4.75121	−0.335961
$201$	0	0
$202$	−0.220120	−0.0154876
$203$	2.35110	0.165015
$204$	0	0
$205$	0.465684	0.0325248
$206$	−2.51445	−0.175190
$207$	0	0
$208$	−23.6597	−1.64051
$209$	−12.2147	−0.844910
$210$	0	0
$211$	3.11651	0.214549	0.107275	−	0.994229i	$-0.465788\pi$
$211$	3.11651	0.214549	0.107275	+	0.994229i	$0.465788\pi$
$212$	17.6986	1.21555
$213$	0	0
$214$	0.489262	0.0334453
$215$	2.56086	0.174649
$216$	0	0
$217$	−4.76449	−0.323435
$218$	4.48709	0.303904
$219$	0	0
$220$	1.72761	0.116476
$221$	−8.65375	−0.582114
$222$	0	0
$223$	−14.2513	−0.954337	−0.477169	−	0.878812i	$-0.658337\pi$
$223$	−14.2513	−0.954337	−0.477169	+	0.878812i	$0.658337\pi$
$224$	−2.83594	−0.189484
$225$	0	0
$226$	−1.53529	−0.102126
$227$	−14.3103	−0.949808	−0.474904	−	0.880038i	$-0.657517\pi$
$227$	−14.3103	−0.949808	−0.474904	+	0.880038i	$0.657517\pi$
$228$	0	0
$229$	19.1899	1.26810	0.634051	−	0.773291i	$-0.281391\pi$
$229$	19.1899	1.26810	0.634051	+	0.773291i	$0.281391\pi$
$230$	−0.414717	−0.0273456
$231$	0	0
$232$	2.28030	0.149709
$233$	11.6209	0.761310	0.380655	−	0.924717i	$-0.375699\pi$
$233$	11.6209	0.761310	0.380655	+	0.924717i	$0.375699\pi$
$234$	0	0
$235$	2.28042	0.148758
$236$	−9.74623	−0.634426
$237$	0	0
$238$	−0.327782	−0.0212470
$239$	−0.298510	−0.0193090	−0.00965449	−	0.999953i	$-0.503073\pi$
$239$	−0.298510	−0.0193090	−0.00965449	+	0.999953i	$0.503073\pi$
$240$	0	0
$241$	18.4344	1.18747	0.593734	−	0.804662i	$-0.297653\pi$
$241$	18.4344	1.18747	0.593734	+	0.804662i	$0.297653\pi$
$242$	0.778778	0.0500617
$243$	0	0
$244$	4.82715	0.309027
$245$	0.318209	0.0203296
$246$	0	0
$247$	−28.3611	−1.80457
$248$	−4.62101	−0.293434
$249$	0	0
$250$	0.775504	0.0490472
$251$	−27.6399	−1.74462	−0.872308	−	0.488956i	$-0.837378\pi$
$251$	−27.6399	−1.74462	−0.872308	+	0.488956i	$0.837378\pi$
$252$	0	0
$253$	−14.8190	−0.931662
$254$	0.246202	0.0154481
$255$	0	0
$256$	11.3681	0.710508
$257$	−2.39616	−0.149469	−0.0747343	−	0.997203i	$-0.523811\pi$
$257$	−2.39616	−0.149469	−0.0747343	+	0.997203i	$0.523811\pi$
$258$	0	0
$259$	1.06889	0.0664177
$260$	4.01131	0.248771
$261$	0	0
$262$	−4.62588	−0.285788
$263$	−19.2268	−1.18558	−0.592789	−	0.805358i	$-0.701973\pi$
$263$	−19.2268	−1.18558	−0.592789	+	0.805358i	$0.701973\pi$
$264$	0	0
$265$	−2.90395	−0.178388
$266$	−1.07425	−0.0658664
$267$	0	0
$268$	−1.06632	−0.0651356
$269$	−16.0800	−0.980414	−0.490207	−	0.871606i	$-0.663079\pi$
$269$	−16.0800	−0.980414	−0.490207	+	0.871606i	$0.663079\pi$
$270$	0	0
$271$	−24.7242	−1.50189	−0.750945	−	0.660365i	$-0.770402\pi$
$271$	−24.7242	−1.50189	−0.750945	+	0.660365i	$0.770402\pi$
$272$	4.84611	0.293839
$273$	0	0
$274$	5.15372	0.311348
$275$	13.7137	0.826968
$276$	0	0
$277$	27.1986	1.63421	0.817103	−	0.576492i	$-0.195579\pi$
$277$	27.1986	1.63421	0.817103	+	0.576492i	$0.195579\pi$
$278$	1.42214	0.0852941
$279$	0	0
$280$	0.308626	0.0184439
$281$	−21.1941	−1.26433	−0.632167	−	0.774832i	$-0.717834\pi$
$281$	−21.1941	−1.26433	−0.632167	+	0.774832i	$0.717834\pi$
$282$	0	0
$283$	31.0745	1.84719	0.923593	−	0.383374i	$-0.125238\pi$
$283$	31.0745	1.84719	0.923593	+	0.383374i	$0.125238\pi$
$284$	−23.1194	−1.37188
$285$	0	0
$286$	−4.47993	−0.264904
$287$	1.46345	0.0863850
$288$	0	0
$289$	−15.2275	−0.895735
$290$	−0.184194	−0.0108163
$291$	0	0
$292$	26.8776	1.57289
$293$	18.9325	1.10605	0.553024	−	0.833165i	$-0.313474\pi$
$293$	18.9325	1.10605	0.553024	+	0.833165i	$0.313474\pi$
$294$	0	0
$295$	1.59914	0.0931052
$296$	1.03670	0.0602570
$297$	0	0
$298$	−1.39261	−0.0806718
$299$	−34.4079	−1.98986
$300$	0	0
$301$	8.04774	0.463864
$302$	0.616133	0.0354545
$303$	0	0
$304$	15.8823	0.910910
$305$	−0.792026	−0.0453513
$306$	0	0
$307$	−4.25979	−0.243119	−0.121559	−	0.992584i	$-0.538790\pi$
$307$	−4.25979	−0.243119	−0.121559	+	0.992584i	$0.538790\pi$
$308$	5.42918	0.309356
$309$	0	0
$310$	0.373268	0.0212002
$311$	−21.6483	−1.22757	−0.613783	−	0.789475i	$-0.710353\pi$
$311$	−21.6483	−1.22757	−0.613783	+	0.789475i	$0.710353\pi$
$312$	0	0
$313$	−0.443304	−0.0250570	−0.0125285	−	0.999922i	$-0.503988\pi$
$313$	−0.443304	−0.0250570	−0.0125285	+	0.999922i	$0.503988\pi$
$314$	3.53037	0.199230
$315$	0	0
$316$	2.89974	0.163123
$317$	−9.20138	−0.516801	−0.258401	−	0.966038i	$-0.583195\pi$
$317$	−9.20138	−0.516801	−0.258401	+	0.966038i	$0.583195\pi$
$318$	0	0
$319$	−6.58176	−0.368508
$320$	−2.09437	−0.117079
$321$	0	0
$322$	−1.30329	−0.0726293
$323$	5.80908	0.323226
$324$	0	0
$325$	31.8416	1.76625
$326$	4.13095	0.228792
$327$	0	0
$328$	1.41938	0.0783722
$329$	7.16642	0.395098
$330$	0	0
$331$	20.2676	1.11401	0.557005	−	0.830509i	$-0.311951\pi$
$331$	20.2676	1.11401	0.557005	+	0.830509i	$0.311951\pi$
$332$	−33.2184	−1.82310
$333$	0	0
$334$	−2.70772	−0.148160
$335$	0.174958	0.00955899
$336$	0	0
$337$	4.22202	0.229988	0.114994	−	0.993366i	$-0.463315\pi$
$337$	4.22202	0.229988	0.114994	+	0.993366i	$0.463315\pi$
$338$	−7.20124	−0.391696
$339$	0	0
$340$	−0.821619	−0.0445586
$341$	13.3379	0.722287
$342$	0	0
$343$	1.00000	0.0539949
$344$	7.80538	0.420838
$345$	0	0
$346$	3.67596	0.197621
$347$	−23.1846	−1.24462	−0.622308	−	0.782773i	$-0.713805\pi$
$347$	−23.1846	−1.24462	−0.622308	+	0.782773i	$0.713805\pi$
$348$	0	0
$349$	10.0741	0.539255	0.269628	−	0.962965i	$-0.413099\pi$
$349$	10.0741	0.539255	0.269628	+	0.962965i	$0.413099\pi$
$350$	1.20608	0.0644677
$351$	0	0
$352$	7.93902	0.423151
$353$	26.1314	1.39084	0.695418	−	0.718606i	$-0.255219\pi$
$353$	26.1314	1.39084	0.695418	+	0.718606i	$0.255219\pi$
$354$	0	0
$355$	3.79336	0.201331
$356$	25.0038	1.32520
$357$	0	0
$358$	2.60947	0.137915
$359$	−12.9795	−0.685033	−0.342517	−	0.939512i	$-0.611279\pi$
$359$	−12.9795	−0.685033	−0.342517	+	0.939512i	$0.611279\pi$
$360$	0	0
$361$	0.0382221	0.00201169
$362$	−3.60704	−0.189582
$363$	0	0
$364$	12.6059	0.660729
$365$	−4.41001	−0.230830
$366$	0	0
$367$	−10.0344	−0.523792	−0.261896	−	0.965096i	$-0.584348\pi$
$367$	−10.0344	−0.523792	−0.261896	+	0.965096i	$0.584348\pi$
$368$	19.2685	1.00444
$369$	0	0
$370$	−0.0837409	−0.00435348
$371$	−9.12591	−0.473793
$372$	0	0
$373$	−35.1339	−1.81916	−0.909581	−	0.415527i	$-0.863597\pi$
$373$	−35.1339	−1.81916	−0.909581	+	0.415527i	$0.863597\pi$
$374$	0.917605	0.0474482
$375$	0	0
$376$	6.95060	0.358450
$377$	−15.2821	−0.787066
$378$	0	0
$379$	13.4078	0.688710	0.344355	−	0.938840i	$-0.388098\pi$
$379$	13.4078	0.688710	0.344355	+	0.938840i	$0.388098\pi$
$380$	−2.69271	−0.138133
$381$	0	0
$382$	−4.35132	−0.222633
$383$	30.0733	1.53667	0.768337	−	0.640046i	$-0.221085\pi$
$383$	30.0733	1.53667	0.768337	+	0.640046i	$0.221085\pi$
$384$	0	0
$385$	−0.890805	−0.0453996
$386$	3.27723	0.166807
$387$	0	0
$388$	13.9650	0.708967
$389$	16.7975	0.851667	0.425833	−	0.904802i	$-0.359981\pi$
$389$	16.7975	0.851667	0.425833	+	0.904802i	$0.359981\pi$
$390$	0	0
$391$	7.04762	0.356414
$392$	0.969884	0.0489865
$393$	0	0
$394$	0.765310	0.0385558
$395$	−0.475781	−0.0239391
$396$	0	0
$397$	−21.2921	−1.06862	−0.534309	−	0.845289i	$-0.679428\pi$
$397$	−21.2921	−1.06862	−0.534309	+	0.845289i	$0.679428\pi$
$398$	5.19292	0.260297
$399$	0	0
$400$	−17.8313	−0.891567
$401$	6.43709	0.321453	0.160727	−	0.986999i	$-0.448616\pi$
$401$	6.43709	0.321453	0.160727	+	0.986999i	$0.448616\pi$
$402$	0	0
$403$	30.9690	1.54267
$404$	−1.73393	−0.0862663
$405$	0	0
$406$	−0.578846	−0.0287277
$407$	−2.99229	−0.148322
$408$	0	0
$409$	8.28563	0.409698	0.204849	−	0.978794i	$-0.434330\pi$
$409$	8.28563	0.409698	0.204849	+	0.978794i	$0.434330\pi$
$410$	−0.114652	−0.00566228
$411$	0	0
$412$	−19.8069	−0.975816
$413$	5.02542	0.247285
$414$	0	0
$415$	5.45039	0.267549
$416$	18.4335	0.903774
$417$	0	0
$418$	3.00729	0.147091
$419$	−27.2585	−1.33167	−0.665833	−	0.746101i	$-0.731924\pi$
$419$	−27.2585	−1.33167	−0.665833	+	0.746101i	$0.731924\pi$
$420$	0	0
$421$	−21.3651	−1.04127	−0.520635	−	0.853780i	$-0.674305\pi$
$421$	−21.3651	−1.04127	−0.520635	+	0.853780i	$0.674305\pi$
$422$	−0.767290	−0.0373511
$423$	0	0
$424$	−8.85107	−0.429846
$425$	−6.52197	−0.316362
$426$	0	0
$427$	−2.48901	−0.120452
$428$	3.85402	0.186291
$429$	0	0
$430$	−0.630490	−0.0304049
$431$	11.7546	0.566197	0.283099	−	0.959091i	$-0.408638\pi$
$431$	11.7546	0.566197	0.283099	+	0.959091i	$0.408638\pi$
$432$	0	0
$433$	5.70487	0.274159	0.137079	−	0.990560i	$-0.456228\pi$
$433$	5.70487	0.274159	0.137079	+	0.990560i	$0.456228\pi$
$434$	1.17303	0.0563071
$435$	0	0
$436$	35.3458	1.69276
$437$	23.0973	1.10489
$438$	0	0
$439$	2.94845	0.140722	0.0703609	−	0.997522i	$-0.477585\pi$
$439$	2.94845	0.140722	0.0703609	+	0.997522i	$0.477585\pi$
$440$	−0.863977	−0.0411885
$441$	0	0
$442$	2.13057	0.101341
$443$	4.63594	0.220260	0.110130	−	0.993917i	$-0.464873\pi$
$443$	4.63594	0.220260	0.110130	+	0.993917i	$0.464873\pi$
$444$	0	0
$445$	−4.10256	−0.194480
$446$	3.50869	0.166141
$447$	0	0
$448$	−6.58175	−0.310959
$449$	−39.5232	−1.86522	−0.932608	−	0.360890i	$-0.882473\pi$
$449$	−39.5232	−1.86522	−0.932608	+	0.360890i	$0.882473\pi$
$450$	0	0
$451$	−4.09685	−0.192913
$452$	−12.0938	−0.568845
$453$	0	0
$454$	3.52322	0.165353
$455$	−2.06834	−0.0969654
$456$	0	0
$457$	−28.2011	−1.31919	−0.659597	−	0.751620i	$-0.729273\pi$
$457$	−28.2011	−1.31919	−0.659597	+	0.751620i	$0.729273\pi$
$458$	−4.72458	−0.220765
$459$	0	0
$460$	−3.26682	−0.152316
$461$	−25.8036	−1.20179	−0.600897	−	0.799327i	$-0.705190\pi$
$461$	−25.8036	−1.20179	−0.600897	+	0.799327i	$0.705190\pi$
$462$	0	0
$463$	−11.7873	−0.547801	−0.273900	−	0.961758i	$-0.588314\pi$
$463$	−11.7873	−0.547801	−0.273900	+	0.961758i	$0.588314\pi$
$464$	8.55798	0.397294
$465$	0	0
$466$	−2.86108	−0.132537
$467$	21.0242	0.972884	0.486442	−	0.873713i	$-0.338294\pi$
$467$	21.0242	0.972884	0.486442	+	0.873713i	$0.338294\pi$
$468$	0	0
$469$	0.549822	0.0253884
$470$	−0.561444	−0.0258975
$471$	0	0
$472$	4.87408	0.224348
$473$	−22.5291	−1.03589
$474$	0	0
$475$	−21.3746	−0.980734
$476$	−2.58201	−0.118346
$477$	0	0
$478$	0.0734936	0.00336152
$479$	−29.9975	−1.37062	−0.685311	−	0.728251i	$-0.740334\pi$
$479$	−29.9975	−1.37062	−0.685311	+	0.728251i	$0.740334\pi$
$480$	0	0
$481$	−6.94774	−0.316790
$482$	−4.53860	−0.206727
$483$	0	0
$484$	6.13460	0.278845
$485$	−2.29134	−0.104045
$486$	0	0
$487$	12.1673	0.551354	0.275677	−	0.961250i	$-0.411098\pi$
$487$	12.1673	0.551354	0.275677	+	0.961250i	$0.411098\pi$
$488$	−2.41405	−0.109279
$489$	0	0
$490$	−0.0783436	−0.00353920
$491$	−16.0438	−0.724048	−0.362024	−	0.932169i	$-0.617914\pi$
$491$	−16.0438	−0.724048	−0.362024	+	0.932169i	$0.617914\pi$
$492$	0	0
$493$	3.13016	0.140975
$494$	6.98256	0.314160
$495$	0	0
$496$	−17.3427	−0.778709
$497$	11.9210	0.534729
$498$	0	0
$499$	−26.3857	−1.18119	−0.590593	−	0.806970i	$-0.701106\pi$
$499$	−26.3857	−1.18119	−0.590593	+	0.806970i	$0.701106\pi$
$500$	6.10881	0.273194
$501$	0	0
$502$	6.80500	0.303722
$503$	−18.3203	−0.816862	−0.408431	−	0.912789i	$-0.633924\pi$
$503$	−18.3203	−0.816862	−0.408431	+	0.912789i	$0.633924\pi$
$504$	0	0
$505$	0.284499	0.0126600
$506$	3.64846	0.162194
$507$	0	0
$508$	1.93938	0.0860463
$509$	−36.9703	−1.63868	−0.819339	−	0.573309i	$-0.805660\pi$
$509$	−36.9703	−1.63868	−0.819339	+	0.573309i	$0.805660\pi$
$510$	0	0
$511$	−13.8588	−0.613079
$512$	−17.3835	−0.768249
$513$	0	0
$514$	0.589940	0.0260211
$515$	3.24986	0.143206
$516$	0	0
$517$	−20.0619	−0.882323
$518$	−0.263163	−0.0115627
$519$	0	0
$520$	−2.00605	−0.0879712
$521$	2.13160	0.0933871	0.0466936	−	0.998909i	$-0.485132\pi$
$521$	2.13160	0.0933871	0.0466936	+	0.998909i	$0.485132\pi$
$522$	0	0
$523$	−24.4693	−1.06997	−0.534983	−	0.844863i	$-0.679682\pi$
$523$	−24.4693	−1.06997	−0.534983	+	0.844863i	$0.679682\pi$
$524$	−36.4390	−1.59185
$525$	0	0
$526$	4.73368	0.206398
$527$	−6.34324	−0.276316
$528$	0	0
$529$	5.02182	0.218340
$530$	0.714957	0.0310557
$531$	0	0
$532$	−8.46208	−0.366878
$533$	−9.51238	−0.412027
$534$	0	0
$535$	−0.632358	−0.0273392
$536$	0.533263	0.0230335
$537$	0	0
$538$	3.95892	0.170681
$539$	−2.79943	−0.120580
$540$	0	0
$541$	−3.07605	−0.132250	−0.0661250	−	0.997811i	$-0.521064\pi$
$541$	−3.07605	−0.132250	−0.0661250	+	0.997811i	$0.521064\pi$
$542$	6.08715	0.261466
$543$	0	0
$544$	−3.77564	−0.161879
$545$	−5.79944	−0.248421
$546$	0	0
$547$	38.7810	1.65816	0.829078	−	0.559133i	$-0.188866\pi$
$547$	38.7810	1.65816	0.829078	+	0.559133i	$0.188866\pi$
$548$	40.5969	1.73422
$549$	0	0
$550$	−3.37634	−0.143968
$551$	10.2585	0.437028
$552$	0	0
$553$	−1.49518	−0.0635817
$554$	−6.69635	−0.284500
$555$	0	0
$556$	11.2025	0.475091
$557$	−15.3795	−0.651651	−0.325825	−	0.945430i	$-0.605642\pi$
$557$	−15.3795	−0.651651	−0.325825	+	0.945430i	$0.605642\pi$
$558$	0	0
$559$	−52.3099	−2.21247
$560$	1.15827	0.0489461
$561$	0	0
$562$	5.21803	0.220109
$563$	−17.9471	−0.756380	−0.378190	−	0.925728i	$-0.623453\pi$
$563$	−17.9471	−0.756380	−0.378190	+	0.925728i	$0.623453\pi$
$564$	0	0
$565$	1.98432	0.0834809
$566$	−7.65060	−0.321579
$567$	0	0
$568$	11.5620	0.485129
$569$	−10.7314	−0.449884	−0.224942	−	0.974372i	$-0.572219\pi$
$569$	−10.7314	−0.449884	−0.224942	+	0.974372i	$0.572219\pi$
$570$	0	0
$571$	−18.4705	−0.772966	−0.386483	−	0.922296i	$-0.626310\pi$
$571$	−18.4705	−0.772966	−0.386483	+	0.922296i	$0.626310\pi$
$572$	−35.2894	−1.47552
$573$	0	0
$574$	−0.360305	−0.0150389
$575$	−25.9318	−1.08143
$576$	0	0
$577$	−0.892310	−0.0371473	−0.0185737	−	0.999827i	$-0.505913\pi$
$577$	−0.892310	−0.0371473	−0.0185737	+	0.999827i	$0.505913\pi$
$578$	3.74904	0.155939
$579$	0	0
$580$	−1.45094	−0.0602469
$581$	17.1283	0.710603
$582$	0	0
$583$	25.5474	1.05806
$584$	−13.4415	−0.556212
$585$	0	0
$586$	−4.66121	−0.192553
$587$	30.5797	1.26216	0.631080	−	0.775718i	$-0.282612\pi$
$587$	30.5797	1.26216	0.631080	+	0.775718i	$0.282612\pi$
$588$	0	0
$589$	−20.7888	−0.856589
$590$	−0.393710	−0.0162088
$591$	0	0
$592$	3.89075	0.159909
$593$	−33.7801	−1.38718	−0.693590	−	0.720370i	$-0.743972\pi$
$593$	−33.7801	−1.38718	−0.693590	+	0.720370i	$0.743972\pi$
$594$	0	0
$595$	0.423650	0.0173679
$596$	−10.9699	−0.449344
$597$	0	0
$598$	8.47129	0.346417
$599$	8.67399	0.354410	0.177205	−	0.984174i	$-0.443294\pi$
$599$	8.67399	0.354410	0.177205	+	0.984174i	$0.443294\pi$
$600$	0	0
$601$	−19.4904	−0.795031	−0.397516	−	0.917595i	$-0.630128\pi$
$601$	−19.4904	−0.795031	−0.397516	+	0.917595i	$0.630128\pi$
$602$	−1.98137	−0.0807546
$603$	0	0
$604$	4.85341	0.197483
$605$	−1.00655	−0.0409220
$606$	0	0
$607$	−32.4809	−1.31836	−0.659179	−	0.751986i	$-0.729096\pi$
$607$	−32.4809	−1.31836	−0.659179	+	0.751986i	$0.729096\pi$
$608$	−12.3740	−0.501832
$609$	0	0
$610$	0.194998	0.00789524
$611$	−46.5814	−1.88448
$612$	0	0
$613$	−7.74253	−0.312718	−0.156359	−	0.987700i	$-0.549976\pi$
$613$	−7.74253	−0.312718	−0.156359	+	0.987700i	$0.549976\pi$
$614$	1.04877	0.0423248
$615$	0	0
$616$	−2.71513	−0.109396
$617$	−36.9258	−1.48658	−0.743288	−	0.668971i	$-0.766735\pi$
$617$	−36.9258	−1.48658	−0.743288	+	0.668971i	$0.766735\pi$
$618$	0	0
$619$	5.50391	0.221221	0.110610	−	0.993864i	$-0.464719\pi$
$619$	5.50391	0.221221	0.110610	+	0.993864i	$0.464719\pi$
$620$	2.94031	0.118086
$621$	0	0
$622$	5.32986	0.213708
$623$	−12.8927	−0.516533
$624$	0	0
$625$	23.4914	0.939656
$626$	0.109142	0.00436221
$627$	0	0
$628$	27.8095	1.10972
$629$	1.42308	0.0567417
$630$	0	0
$631$	30.7518	1.22421	0.612105	−	0.790777i	$-0.290323\pi$
$631$	30.7518	1.22421	0.612105	+	0.790777i	$0.290323\pi$
$632$	−1.45015	−0.0576840
$633$	0	0
$634$	2.26540	0.0899704
$635$	−0.318209	−0.0126277
$636$	0	0
$637$	−6.49995	−0.257537
$638$	1.62044	0.0641539
$639$	0	0
$640$	2.32048	0.0917250
$641$	−11.7811	−0.465324	−0.232662	−	0.972558i	$-0.574744\pi$
$641$	−11.7811	−0.465324	−0.232662	+	0.972558i	$0.574744\pi$
$642$	0	0
$643$	−34.3871	−1.35609	−0.678047	−	0.735019i	$-0.737173\pi$
$643$	−34.3871	−1.35609	−0.678047	+	0.735019i	$0.737173\pi$
$644$	−10.2663	−0.404547
$645$	0	0
$646$	−1.43021	−0.0562707
$647$	6.03048	0.237083	0.118541	−	0.992949i	$-0.462178\pi$
$647$	6.03048	0.237083	0.118541	+	0.992949i	$0.462178\pi$
$648$	0	0
$649$	−14.0683	−0.552231
$650$	−7.83946	−0.307489
$651$	0	0
$652$	32.5404	1.27438
$653$	−6.35698	−0.248768	−0.124384	−	0.992234i	$-0.539695\pi$
$653$	−6.35698	−0.248768	−0.124384	+	0.992234i	$0.539695\pi$
$654$	0	0
$655$	5.97882	0.233612
$656$	5.32695	0.207982
$657$	0	0
$658$	−1.76439	−0.0687830
$659$	−22.5576	−0.878720	−0.439360	−	0.898311i	$-0.644795\pi$
$659$	−22.5576	−0.878720	−0.439360	+	0.898311i	$0.644795\pi$
$660$	0	0
$661$	−44.8294	−1.74366	−0.871830	−	0.489809i	$-0.837067\pi$
$661$	−44.8294	−1.74366	−0.871830	+	0.489809i	$0.837067\pi$
$662$	−4.98993	−0.193939
$663$	0	0
$664$	16.6125	0.644690
$665$	1.38844	0.0538412
$666$	0	0
$667$	12.4457	0.481900
$668$	−21.3293	−0.825255
$669$	0	0
$670$	−0.0430750	−0.00166413
$671$	6.96782	0.268990
$672$	0	0
$673$	25.0877	0.967059	0.483529	−	0.875328i	$-0.339355\pi$
$673$	25.0877	0.967059	0.483529	+	0.875328i	$0.339355\pi$
$674$	−1.03947	−0.0400388
$675$	0	0
$676$	−56.7257	−2.18176
$677$	34.4793	1.32515	0.662573	−	0.748997i	$-0.269464\pi$
$677$	34.4793	1.32515	0.662573	+	0.748997i	$0.269464\pi$
$678$	0	0
$679$	−7.20075	−0.276339
$680$	0.410891	0.0157569
$681$	0	0
$682$	−3.28381	−0.125744
$683$	−12.6147	−0.482690	−0.241345	−	0.970439i	$-0.577589\pi$
$683$	−12.6147	−0.482690	−0.241345	+	0.970439i	$0.577589\pi$
$684$	0	0
$685$	−6.66104	−0.254505
$686$	−0.246202	−0.00940003
$687$	0	0
$688$	29.2936	1.11681
$689$	59.3179	2.25983
$690$	0	0
$691$	−41.2619	−1.56968	−0.784839	−	0.619700i	$-0.787254\pi$
$691$	−41.2619	−1.56968	−0.784839	+	0.619700i	$0.787254\pi$
$692$	28.9563	1.10075
$693$	0	0
$694$	5.70810	0.216676
$695$	−1.83807	−0.0697220
$696$	0	0
$697$	1.94838	0.0738002
$698$	−2.48027	−0.0938795
$699$	0	0
$700$	9.50055	0.359087
$701$	34.9290	1.31925	0.659624	−	0.751595i	$-0.270715\pi$
$701$	34.9290	1.31925	0.659624	+	0.751595i	$0.270715\pi$
$702$	0	0
$703$	4.66387	0.175901
$704$	18.4252	0.694425
$705$	0	0
$706$	−6.43361	−0.242132
$707$	0.894063	0.0336247
$708$	0	0
$709$	26.8327	1.00772	0.503861	−	0.863785i	$-0.331912\pi$
$709$	26.8327	1.00772	0.503861	+	0.863785i	$0.331912\pi$
$710$	−0.933933	−0.0350499
$711$	0	0
$712$	−12.5044	−0.468621
$713$	−25.2212	−0.944540
$714$	0	0
$715$	5.79019	0.216541
$716$	20.5554	0.768191
$717$	0	0
$718$	3.19558	0.119258
$719$	−20.7889	−0.775296	−0.387648	−	0.921808i	$-0.626712\pi$
$719$	−20.7889	−0.775296	−0.387648	+	0.921808i	$0.626712\pi$
$720$	0	0
$721$	10.2130	0.380351
$722$	−0.00941036	−0.000350217 0
$723$	0	0
$724$	−28.4135	−1.05598
$725$	−11.5175	−0.427748
$726$	0	0
$727$	22.0538	0.817928	0.408964	−	0.912550i	$-0.365890\pi$
$727$	22.0538	0.817928	0.408964	+	0.912550i	$0.365890\pi$
$728$	−6.30420	−0.233649
$729$	0	0
$730$	1.08575	0.0401855
$731$	10.7144	0.396287
$732$	0	0
$733$	−31.0700	−1.14760	−0.573798	−	0.818997i	$-0.694531\pi$
$733$	−31.0700	−1.14760	−0.573798	+	0.818997i	$0.694531\pi$
$734$	2.47049	0.0911875
$735$	0	0
$736$	−15.0122	−0.553358
$737$	−1.53919	−0.0566968
$738$	0	0
$739$	47.8405	1.75984	0.879921	−	0.475120i	$-0.157595\pi$
$739$	47.8405	1.75984	0.879921	+	0.475120i	$0.157595\pi$
$740$	−0.659645	−0.0242490
$741$	0	0
$742$	2.24682	0.0824832
$743$	−17.0228	−0.624507	−0.312254	−	0.949999i	$-0.601084\pi$
$743$	−17.0228	−0.624507	−0.312254	+	0.949999i	$0.601084\pi$
$744$	0	0
$745$	1.79991	0.0659436
$746$	8.65002	0.316700
$747$	0	0
$748$	7.22817	0.264288
$749$	−1.98724	−0.0726122
$750$	0	0
$751$	−24.1639	−0.881754	−0.440877	−	0.897568i	$-0.645332\pi$
$751$	−24.1639	−0.881754	−0.440877	+	0.897568i	$0.645332\pi$
$752$	26.0857	0.951246
$753$	0	0
$754$	3.76247	0.137021
$755$	−0.796335	−0.0289816
$756$	0	0
$757$	−35.9690	−1.30732	−0.653658	−	0.756790i	$-0.726767\pi$
$757$	−35.9690	−1.30732	−0.653658	+	0.756790i	$0.726767\pi$
$758$	−3.30101	−0.119898
$759$	0	0
$760$	1.34662	0.0488471
$761$	28.0258	1.01593	0.507967	−	0.861376i	$-0.330397\pi$
$761$	28.0258	1.01593	0.507967	+	0.861376i	$0.330397\pi$
$762$	0	0
$763$	−18.2252	−0.659798
$764$	−34.2763	−1.24007
$765$	0	0
$766$	−7.40410	−0.267521
$767$	−32.6650	−1.17946
$768$	0	0
$769$	−15.0578	−0.542997	−0.271498	−	0.962439i	$-0.587519\pi$
$769$	−15.0578	−0.542997	−0.271498	+	0.962439i	$0.587519\pi$
$770$	0.219318	0.00790367
$771$	0	0
$772$	25.8155	0.929119
$773$	−29.8191	−1.07252	−0.536260	−	0.844053i	$-0.680163\pi$
$773$	−29.8191	−1.07252	−0.536260	+	0.844053i	$0.680163\pi$
$774$	0	0
$775$	23.3400	0.838399
$776$	−6.98389	−0.250707
$777$	0	0
$778$	−4.13557	−0.148268
$779$	6.38546	0.228783
$780$	0	0
$781$	−33.3720	−1.19414
$782$	−1.73514	−0.0620484
$783$	0	0
$784$	3.63998	0.129999
$785$	−4.56290	−0.162857
$786$	0	0
$787$	22.2169	0.791947	0.395974	−	0.918262i	$-0.370407\pi$
$787$	22.2169	0.791947	0.395974	+	0.918262i	$0.370407\pi$
$788$	6.02851	0.214757
$789$	0	0
$790$	0.117138	0.00416759
$791$	6.23590	0.221723
$792$	0	0
$793$	16.1784	0.574513
$794$	5.24215	0.186037
$795$	0	0
$796$	40.9057	1.44987
$797$	−10.9848	−0.389102	−0.194551	−	0.980892i	$-0.562325\pi$
$797$	−10.9848	−0.389102	−0.194551	+	0.980892i	$0.562325\pi$
$798$	0	0
$799$	9.54106	0.337539
$800$	13.8925	0.491175
$801$	0	0
$802$	−1.58482	−0.0559621
$803$	38.7969	1.36911
$804$	0	0
$805$	1.68446	0.0593694
$806$	−7.62462	−0.268566
$807$	0	0
$808$	0.867137	0.0305058
$809$	−15.6198	−0.549162	−0.274581	−	0.961564i	$-0.588539\pi$
$809$	−15.6198	−0.549162	−0.274581	+	0.961564i	$0.588539\pi$
$810$	0	0
$811$	−33.9143	−1.19089	−0.595447	−	0.803395i	$-0.703025\pi$
$811$	−33.9143	−1.19089	−0.595447	+	0.803395i	$0.703025\pi$
$812$	−4.55970	−0.160014
$813$	0	0
$814$	0.736708	0.0258216
$815$	−5.33914	−0.187022
$816$	0	0
$817$	35.1146	1.22850
$818$	−2.03994	−0.0713247
$819$	0	0
$820$	−0.903141	−0.0315391
$821$	−15.7908	−0.551102	−0.275551	−	0.961286i	$-0.588860\pi$
$821$	−15.7908	−0.551102	−0.275551	+	0.961286i	$0.588860\pi$
$822$	0	0
$823$	21.7601	0.758509	0.379254	−	0.925292i	$-0.376181\pi$
$823$	21.7601	0.758509	0.379254	+	0.925292i	$0.376181\pi$
$824$	9.90540	0.345071
$825$	0	0
$826$	−1.23727	−0.0430501
$827$	35.0663	1.21937	0.609687	−	0.792642i	$-0.291295\pi$
$827$	35.0663	1.21937	0.609687	+	0.792642i	$0.291295\pi$
$828$	0	0
$829$	9.47354	0.329030	0.164515	−	0.986375i	$-0.447394\pi$
$829$	9.47354	0.329030	0.164515	+	0.986375i	$0.447394\pi$
$830$	−1.34190	−0.0465779
$831$	0	0
$832$	42.7811	1.48317
$833$	1.33136	0.0461288
$834$	0	0
$835$	3.49965	0.121110
$836$	23.6890	0.819302
$837$	0	0
$838$	6.71109	0.231831
$839$	−0.187162	−0.00646157	−0.00323078	−	0.999995i	$-0.501028\pi$
$839$	−0.187162	−0.00646157	−0.00323078	+	0.999995i	$0.501028\pi$
$840$	0	0
$841$	−23.4723	−0.809390
$842$	5.26012	0.181276
$843$	0	0
$844$	−6.04411	−0.208047
$845$	9.30740	0.320184
$846$	0	0
$847$	−3.16317	−0.108688
$848$	−33.2181	−1.14072
$849$	0	0
$850$	1.60572	0.0550758
$851$	5.65825	0.193962
$852$	0	0
$853$	−17.7583	−0.608034	−0.304017	−	0.952667i	$-0.598328\pi$
$853$	−17.7583	−0.608034	−0.304017	+	0.952667i	$0.598328\pi$
$854$	0.612799	0.0209696
$855$	0	0
$856$	−1.92739	−0.0658769
$857$	24.4576	0.835455	0.417727	−	0.908572i	$-0.362827\pi$
$857$	24.4576	0.835455	0.417727	+	0.908572i	$0.362827\pi$
$858$	0	0
$859$	6.32700	0.215874	0.107937	−	0.994158i	$-0.465575\pi$
$859$	6.32700	0.215874	0.107937	+	0.994158i	$0.465575\pi$
$860$	−4.96650	−0.169356
$861$	0	0
$862$	−2.89399	−0.0985698
$863$	0.857036	0.0291738	0.0145869	−	0.999894i	$-0.495357\pi$
$863$	0.857036	0.0291738	0.0145869	+	0.999894i	$0.495357\pi$
$864$	0	0
$865$	−4.75108	−0.161541
$866$	−1.40455	−0.0477286
$867$	0	0
$868$	9.24019	0.313632
$869$	4.18567	0.141989
$870$	0	0
$871$	−3.57381	−0.121094
$872$	−17.6764	−0.598598
$873$	0	0
$874$	−5.68660	−0.192352
$875$	−3.14987	−0.106485
$876$	0	0
$877$	6.88110	0.232358	0.116179	−	0.993228i	$-0.462935\pi$
$877$	6.88110	0.232358	0.116179	+	0.993228i	$0.462935\pi$
$878$	−0.725913	−0.0244984
$879$	0	0
$880$	−3.24251	−0.109305
$881$	51.7694	1.74416	0.872078	−	0.489367i	$-0.162772\pi$
$881$	51.7694	1.74416	0.872078	+	0.489367i	$0.162772\pi$
$882$	0	0
$883$	−18.4022	−0.619283	−0.309641	−	0.950853i	$-0.600209\pi$
$883$	−18.4022	−0.619283	−0.309641	+	0.950853i	$0.600209\pi$
$884$	16.7829	0.564472
$885$	0	0
$886$	−1.14138	−0.0383453
$887$	54.6988	1.83661	0.918303	−	0.395878i	$-0.129560\pi$
$887$	54.6988	1.83661	0.918303	+	0.395878i	$0.129560\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	1.01006	0.0338572
$891$	0	0
$892$	27.6387	0.925413
$893$	31.2691	1.04638
$894$	0	0
$895$	−3.37267	−0.112736
$896$	7.29232	0.243619
$897$	0	0
$898$	9.73069	0.324717
$899$	−11.2018	−0.373602
$900$	0	0
$901$	−12.1498	−0.404770
$902$	1.00865	0.0335844
$903$	0	0
$904$	6.04810	0.201157
$905$	4.66200	0.154970
$906$	0	0
$907$	39.2425	1.30303	0.651514	−	0.758637i	$-0.274134\pi$
$907$	39.2425	1.30303	0.651514	+	0.758637i	$0.274134\pi$
$908$	27.7532	0.921021
$909$	0	0
$910$	0.509230	0.0168808
$911$	−7.34578	−0.243377	−0.121688	−	0.992568i	$-0.538831\pi$
$911$	−7.34578	−0.243377	−0.121688	+	0.992568i	$0.538831\pi$
$912$	0	0
$913$	−47.9496	−1.58690
$914$	6.94317	0.229660
$915$	0	0
$916$	−37.2165	−1.22967
$917$	18.7890	0.620467
$918$	0	0
$919$	58.3954	1.92629	0.963144	−	0.268986i	$-0.0866887\pi$
$919$	58.3954	1.92629	0.963144	+	0.268986i	$0.0866887\pi$
$920$	1.63373	0.0538625
$921$	0	0
$922$	6.35289	0.209221
$923$	−77.4858	−2.55048
$924$	0	0
$925$	−5.23623	−0.172166
$926$	2.90205	0.0953672
$927$	0	0
$928$	−6.66759	−0.218874
$929$	43.5784	1.42976	0.714880	−	0.699247i	$-0.246481\pi$
$929$	43.5784	1.42976	0.714880	+	0.699247i	$0.246481\pi$
$930$	0	0
$931$	4.36328	0.143001
$932$	−22.5374	−0.738236
$933$	0	0
$934$	−5.17620	−0.169370
$935$	−1.18598	−0.0387857
$936$	0	0
$937$	19.8795	0.649435	0.324718	−	0.945811i	$-0.394731\pi$
$937$	19.8795	0.649435	0.324718	+	0.945811i	$0.394731\pi$
$938$	−0.135367	−0.00441989
$939$	0	0
$940$	−4.42261	−0.144250
$941$	−8.08530	−0.263573	−0.131787	−	0.991278i	$-0.542071\pi$
$941$	−8.08530	−0.263573	−0.131787	+	0.991278i	$0.542071\pi$
$942$	0	0
$943$	7.74689	0.252274
$944$	18.2925	0.595369
$945$	0	0
$946$	5.54671	0.180339
$947$	−40.1421	−1.30444	−0.652221	−	0.758028i	$-0.726163\pi$
$947$	−40.1421	−1.30444	−0.652221	+	0.758028i	$0.726163\pi$
$948$	0	0
$949$	90.0818	2.92418
$950$	5.26246	0.170737
$951$	0	0
$952$	1.29126	0.0418500
$953$	−29.1867	−0.945449	−0.472725	−	0.881210i	$-0.656729\pi$
$953$	−29.1867	−0.945449	−0.472725	+	0.881210i	$0.656729\pi$
$954$	0	0
$955$	5.62396	0.181987
$956$	0.578925	0.0187238
$957$	0	0
$958$	7.38544	0.238613
$959$	−20.9329	−0.675959
$960$	0	0
$961$	−8.29959	−0.267729
$962$	1.71055	0.0551502
$963$	0	0
$964$	−35.7515	−1.15148
$965$	−4.23573	−0.136353
$966$	0	0
$967$	−15.3154	−0.492511	−0.246256	−	0.969205i	$-0.579200\pi$
$967$	−15.3154	−0.492511	−0.246256	+	0.969205i	$0.579200\pi$
$968$	−3.06791	−0.0986062
$969$	0	0
$970$	0.564133	0.0181132
$971$	50.0150	1.60506	0.802529	−	0.596613i	$-0.203487\pi$
$971$	50.0150	1.60506	0.802529	+	0.596613i	$0.203487\pi$
$972$	0	0
$973$	−5.77630	−0.185180
$974$	−2.99562	−0.0959858
$975$	0	0
$976$	−9.05995	−0.290002
$977$	−11.7653	−0.376406	−0.188203	−	0.982130i	$-0.560266\pi$
$977$	−11.7653	−0.376406	−0.188203	+	0.982130i	$0.560266\pi$
$978$	0	0
$979$	36.0921	1.15351
$980$	−0.617130	−0.0197135
$981$	0	0
$982$	3.95002	0.126050
$983$	10.5544	0.336633	0.168316	−	0.985733i	$-0.446167\pi$
$983$	10.5544	0.336633	0.168316	+	0.985733i	$0.446167\pi$
$984$	0	0
$985$	−0.989142	−0.0315167
$986$	−0.770651	−0.0245425
$987$	0	0
$988$	55.0031	1.74988
$989$	42.6013	1.35464
$990$	0	0
$991$	2.89587	0.0919905	0.0459952	−	0.998942i	$-0.485354\pi$
$991$	2.89587	0.0919905	0.0459952	+	0.998942i	$0.485354\pi$
$992$	13.5118	0.429000
$993$	0	0
$994$	−2.93497	−0.0930915
$995$	−6.71170	−0.212775
$996$	0	0
$997$	−13.5622	−0.429518	−0.214759	−	0.976667i	$-0.568897\pi$
$997$	−13.5622	−0.429518	−0.214759	+	0.976667i	$0.568897\pi$
$998$	6.49621	0.205634
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.l.1.4		7
3.2	odd	2		2667.2.a.j.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.j.1.4	✓	7	3.2	odd	2
8001.2.a.l.1.4		7	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)

\(f(q)\)	\(=\)	\(q-0.246202 q^{2} -1.93938 q^{4} +0.318209 q^{5} +1.00000 q^{7} +0.969884 q^{8} +O(q^{10})\)	\(q-0.246202 q^{2} -1.93938 q^{4} +0.318209 q^{5} +1.00000 q^{7} +0.969884 q^{8} -0.0783436 q^{10} -2.79943 q^{11} -6.49995 q^{13} -0.246202 q^{14} +3.63998 q^{16} +1.33136 q^{17} +4.36328 q^{19} -0.617130 q^{20} +0.689226 q^{22} +5.29356 q^{23} -4.89874 q^{25} +1.60030 q^{26} -1.93938 q^{28} +2.35110 q^{29} -4.76449 q^{31} -2.83594 q^{32} -0.327782 q^{34} +0.318209 q^{35} +1.06889 q^{37} -1.07425 q^{38} +0.308626 q^{40} +1.46345 q^{41} +8.04774 q^{43} +5.42918 q^{44} -1.30329 q^{46} +7.16642 q^{47} +1.00000 q^{49} +1.20608 q^{50} +12.6059 q^{52} -9.12591 q^{53} -0.890805 q^{55} +0.969884 q^{56} -0.578846 q^{58} +5.02542 q^{59} -2.48901 q^{61} +1.17303 q^{62} -6.58175 q^{64} -2.06834 q^{65} +0.549822 q^{67} -2.58201 q^{68} -0.0783436 q^{70} +11.9210 q^{71} -13.8588 q^{73} -0.263163 q^{74} -8.46208 q^{76} -2.79943 q^{77} -1.49518 q^{79} +1.15827 q^{80} -0.360305 q^{82} +17.1283 q^{83} +0.423650 q^{85} -1.98137 q^{86} -2.71513 q^{88} -12.8927 q^{89} -6.49995 q^{91} -10.2663 q^{92} -1.76439 q^{94} +1.38844 q^{95} -7.20075 q^{97} -0.246202 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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