LMFDB - Embedded newform 8001.2.a.p.1.12

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			1.12
Root			$2.23233$ 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)

$f(q)$	$=$	$q+2.23233 q^{2} +2.98331 q^{4} -2.48067 q^{5} -1.00000 q^{7} +2.19508 q^{8} +O(q^{10})$	\(q+2.23233 q^{2} +2.98331 q^{4} -2.48067 q^{5} -1.00000 q^{7} +2.19508 q^{8} -5.53768 q^{10} +4.84606 q^{11} -0.271647 q^{13} -2.23233 q^{14} -1.06648 q^{16} -0.603684 q^{17} +2.73138 q^{19} -7.40061 q^{20} +10.8180 q^{22} -8.55250 q^{23} +1.15372 q^{25} -0.606407 q^{26} -2.98331 q^{28} +7.31162 q^{29} +6.33495 q^{31} -6.77089 q^{32} -1.34762 q^{34} +2.48067 q^{35} -8.93572 q^{37} +6.09735 q^{38} -5.44526 q^{40} +6.46178 q^{41} +6.83740 q^{43} +14.4573 q^{44} -19.0920 q^{46} +10.1489 q^{47} +1.00000 q^{49} +2.57548 q^{50} -0.810407 q^{52} +3.31845 q^{53} -12.0215 q^{55} -2.19508 q^{56} +16.3220 q^{58} +10.6805 q^{59} +13.3236 q^{61} +14.1417 q^{62} -12.9819 q^{64} +0.673866 q^{65} -9.40201 q^{67} -1.80098 q^{68} +5.53768 q^{70} -13.4706 q^{71} -8.79161 q^{73} -19.9475 q^{74} +8.14856 q^{76} -4.84606 q^{77} +14.4807 q^{79} +2.64557 q^{80} +14.4248 q^{82} +0.343843 q^{83} +1.49754 q^{85} +15.2634 q^{86} +10.6375 q^{88} +1.84059 q^{89} +0.271647 q^{91} -25.5148 q^{92} +22.6557 q^{94} -6.77566 q^{95} -4.97484 q^{97} +2.23233 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.23233	1.57850	0.789249	−	0.614073i	$-0.210470\pi$
$2$	2.23233	1.57850	0.789249	+	0.614073i	$0.210470\pi$
$3$	0	0
$3$	0	0
$4$	2.98331	1.49166
$5$	−2.48067	−1.10939	−0.554694	−	0.832054i	$-0.687165\pi$
$5$	−2.48067	−1.10939	−0.554694	+	0.832054i	$0.687165\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.19508	0.776077
$9$	0	0
$10$	−5.53768	−1.75117
$11$	4.84606	1.46114	0.730571	−	0.682837i	$-0.239254\pi$
$11$	4.84606	1.46114	0.730571	+	0.682837i	$0.239254\pi$
$12$	0	0
$13$	−0.271647	−0.0753413	−0.0376707	−	0.999290i	$-0.511994\pi$
$13$	−0.271647	−0.0753413	−0.0376707	+	0.999290i	$0.511994\pi$
$14$	−2.23233	−0.596616
$15$	0	0
$16$	−1.06648	−0.266619
$17$	−0.603684	−0.146415	−0.0732074	−	0.997317i	$-0.523324\pi$
$17$	−0.603684	−0.146415	−0.0732074	+	0.997317i	$0.523324\pi$
$18$	0	0
$19$	2.73138	0.626622	0.313311	−	0.949651i	$-0.398562\pi$
$19$	2.73138	0.626622	0.313311	+	0.949651i	$0.398562\pi$
$20$	−7.40061	−1.65483
$21$	0	0
$22$	10.8180	2.30641
$23$	−8.55250	−1.78332	−0.891660	−	0.452707i	$-0.850458\pi$
$23$	−8.55250	−1.78332	−0.891660	+	0.452707i	$0.850458\pi$
$24$	0	0
$25$	1.15372	0.230744
$26$	−0.606407	−0.118926
$27$	0	0
$28$	−2.98331	−0.563793
$29$	7.31162	1.35773	0.678867	−	0.734262i	$-0.262471\pi$
$29$	7.31162	1.35773	0.678867	+	0.734262i	$0.262471\pi$
$30$	0	0
$31$	6.33495	1.13779	0.568895	−	0.822410i	$-0.307371\pi$
$31$	6.33495	1.13779	0.568895	+	0.822410i	$0.307371\pi$
$32$	−6.77089	−1.19694
$33$	0	0
$34$	−1.34762	−0.231115
$35$	2.48067	0.419310
$36$	0	0
$37$	−8.93572	−1.46902	−0.734512	−	0.678596i	$-0.762589\pi$
$37$	−8.93572	−1.46902	−0.734512	+	0.678596i	$0.762589\pi$
$38$	6.09735	0.989122
$39$	0	0
$40$	−5.44526	−0.860972
$41$	6.46178	1.00916	0.504580	−	0.863365i	$-0.331647\pi$
$41$	6.46178	1.00916	0.504580	+	0.863365i	$0.331647\pi$
$42$	0	0
$43$	6.83740	1.04269	0.521347	−	0.853345i	$-0.325430\pi$
$43$	6.83740	1.04269	0.521347	+	0.853345i	$0.325430\pi$
$44$	14.4573	2.17952
$45$	0	0
$46$	−19.0920	−2.81497
$47$	10.1489	1.48037	0.740185	−	0.672403i	$-0.234738\pi$
$47$	10.1489	1.48037	0.740185	+	0.672403i	$0.234738\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.57548	0.364229
$51$	0	0
$52$	−0.810407	−0.112383
$53$	3.31845	0.455825	0.227912	−	0.973682i	$-0.426810\pi$
$53$	3.31845	0.455825	0.227912	+	0.973682i	$0.426810\pi$
$54$	0	0
$55$	−12.0215	−1.62098
$56$	−2.19508	−0.293330
$57$	0	0
$58$	16.3220	2.14318
$59$	10.6805	1.39048	0.695241	−	0.718777i	$-0.255298\pi$
$59$	10.6805	1.39048	0.695241	+	0.718777i	$0.255298\pi$
$60$	0	0
$61$	13.3236	1.70591	0.852955	−	0.521985i	$-0.174808\pi$
$61$	13.3236	1.70591	0.852955	+	0.521985i	$0.174808\pi$
$62$	14.1417	1.79600
$63$	0	0
$64$	−12.9819	−1.62274
$65$	0.673866	0.0835828
$66$	0	0
$67$	−9.40201	−1.14864	−0.574319	−	0.818631i	$-0.694733\pi$
$67$	−9.40201	−1.14864	−0.574319	+	0.818631i	$0.694733\pi$
$68$	−1.80098	−0.218400
$69$	0	0
$70$	5.53768	0.661879
$71$	−13.4706	−1.59867	−0.799335	−	0.600886i	$-0.794815\pi$
$71$	−13.4706	−1.59867	−0.799335	+	0.600886i	$0.794815\pi$
$72$	0	0
$73$	−8.79161	−1.02898	−0.514490	−	0.857496i	$-0.672019\pi$
$73$	−8.79161	−1.02898	−0.514490	+	0.857496i	$0.672019\pi$
$74$	−19.9475	−2.31885
$75$	0	0
$76$	8.14856	0.934704
$77$	−4.84606	−0.552260
$78$	0	0
$79$	14.4807	1.62921	0.814603	−	0.580018i	$-0.196955\pi$
$79$	14.4807	1.62921	0.814603	+	0.580018i	$0.196955\pi$
$80$	2.64557	0.295784
$81$	0	0
$82$	14.4248	1.59296
$83$	0.343843	0.0377417	0.0188708	−	0.999822i	$-0.493993\pi$
$83$	0.343843	0.0377417	0.0188708	+	0.999822i	$0.493993\pi$
$84$	0	0
$85$	1.49754	0.162431
$86$	15.2634	1.64589
$87$	0	0
$88$	10.6375	1.13396
$89$	1.84059	0.195102	0.0975509	−	0.995231i	$-0.468899\pi$
$89$	1.84059	0.195102	0.0975509	+	0.995231i	$0.468899\pi$
$90$	0	0
$91$	0.271647	0.0284763
$92$	−25.5148	−2.66010
$93$	0	0
$94$	22.6557	2.33676
$95$	−6.77566	−0.695168
$96$	0	0
$97$	−4.97484	−0.505118	−0.252559	−	0.967581i	$-0.581272\pi$
$97$	−4.97484	−0.505118	−0.252559	+	0.967581i	$0.581272\pi$
$98$	2.23233	0.225500
$99$	0	0
$100$	3.44190	0.344190
$101$	9.26356	0.921758	0.460879	−	0.887463i	$-0.347534\pi$
$101$	9.26356	0.921758	0.460879	+	0.887463i	$0.347534\pi$
$102$	0	0
$103$	9.77276	0.962939	0.481470	−	0.876463i	$-0.340103\pi$
$103$	9.77276	0.962939	0.481470	+	0.876463i	$0.340103\pi$
$104$	−0.596286	−0.0584707
$105$	0	0
$106$	7.40790	0.719518
$107$	14.6134	1.41273	0.706365	−	0.707848i	$-0.250334\pi$
$107$	14.6134	1.41273	0.706365	+	0.707848i	$0.250334\pi$
$108$	0	0
$109$	8.28924	0.793965	0.396983	−	0.917826i	$-0.370057\pi$
$109$	8.28924	0.793965	0.396983	+	0.917826i	$0.370057\pi$
$110$	−26.8359	−2.55871
$111$	0	0
$112$	1.06648	0.100773
$113$	5.84002	0.549383	0.274692	−	0.961532i	$-0.411424\pi$
$113$	5.84002	0.549383	0.274692	+	0.961532i	$0.411424\pi$
$114$	0	0
$115$	21.2159	1.97839
$116$	21.8128	2.02527
$117$	0	0
$118$	23.8424	2.19487
$119$	0.603684	0.0553396
$120$	0	0
$121$	12.4843	1.13494
$122$	29.7427	2.69277
$123$	0	0
$124$	18.8991	1.69719
$125$	9.54135	0.853404
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−15.4382	−1.36456
$129$	0	0
$130$	1.50429	0.131935
$131$	−0.343786	−0.0300367	−0.0150184	−	0.999887i	$-0.504781\pi$
$131$	−0.343786	−0.0300367	−0.0150184	+	0.999887i	$0.504781\pi$
$132$	0	0
$133$	−2.73138	−0.236841
$134$	−20.9884	−1.81312
$135$	0	0
$136$	−1.32513	−0.113629
$137$	−17.5617	−1.50039	−0.750197	−	0.661215i	$-0.770041\pi$
$137$	−17.5617	−1.50039	−0.750197	+	0.661215i	$0.770041\pi$
$138$	0	0
$139$	−15.1648	−1.28626	−0.643129	−	0.765758i	$-0.722364\pi$
$139$	−15.1648	−1.28626	−0.643129	+	0.765758i	$0.722364\pi$
$140$	7.40061	0.625466
$141$	0	0
$142$	−30.0709	−2.52350
$143$	−1.31642	−0.110084
$144$	0	0
$145$	−18.1377	−1.50625
$146$	−19.6258	−1.62424
$147$	0	0
$148$	−26.6580	−2.19128
$149$	18.3434	1.50275	0.751376	−	0.659874i	$-0.229390\pi$
$149$	18.3434	1.50275	0.751376	+	0.659874i	$0.229390\pi$
$150$	0	0
$151$	5.86394	0.477201	0.238600	−	0.971118i	$-0.423311\pi$
$151$	5.86394	0.477201	0.238600	+	0.971118i	$0.423311\pi$
$152$	5.99560	0.486307
$153$	0	0
$154$	−10.8180	−0.871741
$155$	−15.7149	−1.26225
$156$	0	0
$157$	23.5647	1.88066	0.940332	−	0.340257i	$-0.110514\pi$
$157$	23.5647	1.88066	0.940332	+	0.340257i	$0.110514\pi$
$158$	32.3258	2.57170
$159$	0	0
$160$	16.7963	1.32787
$161$	8.55250	0.674031
$162$	0	0
$163$	−9.45585	−0.740639	−0.370320	−	0.928904i	$-0.620752\pi$
$163$	−9.45585	−0.740639	−0.370320	+	0.928904i	$0.620752\pi$
$164$	19.2775	1.50532
$165$	0	0
$166$	0.767572	0.0595751
$167$	16.1790	1.25197	0.625984	−	0.779836i	$-0.284697\pi$
$167$	16.1790	1.25197	0.625984	+	0.779836i	$0.284697\pi$
$168$	0	0
$169$	−12.9262	−0.994324
$170$	3.34301	0.256397
$171$	0	0
$172$	20.3981	1.55534
$173$	4.95191	0.376487	0.188243	−	0.982122i	$-0.439721\pi$
$173$	4.95191	0.376487	0.188243	+	0.982122i	$0.439721\pi$
$174$	0	0
$175$	−1.15372	−0.0872129
$176$	−5.16821	−0.389568
$177$	0	0
$178$	4.10880	0.307968
$179$	−7.43803	−0.555944	−0.277972	−	0.960589i	$-0.589662\pi$
$179$	−7.43803	−0.555944	−0.277972	+	0.960589i	$0.589662\pi$
$180$	0	0
$181$	−2.79650	−0.207862	−0.103931	−	0.994584i	$-0.533142\pi$
$181$	−2.79650	−0.207862	−0.103931	+	0.994584i	$0.533142\pi$
$182$	0.606407	0.0449498
$183$	0	0
$184$	−18.7734	−1.38399
$185$	22.1666	1.62972
$186$	0	0
$187$	−2.92549	−0.213933
$188$	30.2773	2.20820
$189$	0	0
$190$	−15.1255	−1.09732
$191$	−10.0695	−0.728600	−0.364300	−	0.931282i	$-0.618692\pi$
$191$	−10.0695	−0.728600	−0.364300	+	0.931282i	$0.618692\pi$
$192$	0	0
$193$	−15.5080	−1.11629	−0.558145	−	0.829743i	$-0.688487\pi$
$193$	−15.5080	−1.11629	−0.558145	+	0.829743i	$0.688487\pi$
$194$	−11.1055	−0.797328
$195$	0	0
$196$	2.98331	0.213094
$197$	8.94674	0.637429	0.318714	−	0.947851i	$-0.396749\pi$
$197$	8.94674	0.637429	0.318714	+	0.947851i	$0.396749\pi$
$198$	0	0
$199$	−7.89357	−0.559560	−0.279780	−	0.960064i	$-0.590262\pi$
$199$	−7.89357	−0.559560	−0.279780	+	0.960064i	$0.590262\pi$
$200$	2.53250	0.179075
$201$	0	0
$202$	20.6793	1.45499
$203$	−7.31162	−0.513175
$204$	0	0
$205$	−16.0295	−1.11955
$206$	21.8161	1.52000
$207$	0	0
$208$	0.289705	0.0200874
$209$	13.2364	0.915584
$210$	0	0
$211$	9.61958	0.662239	0.331120	−	0.943589i	$-0.392574\pi$
$211$	9.61958	0.662239	0.331120	+	0.943589i	$0.392574\pi$
$212$	9.89998	0.679934
$213$	0	0
$214$	32.6220	2.22999
$215$	−16.9613	−1.15675
$216$	0	0
$217$	−6.33495	−0.430044
$218$	18.5043	1.25327
$219$	0	0
$220$	−35.8638	−2.41794
$221$	0.163989	0.0110311
$222$	0	0
$223$	−21.2094	−1.42029	−0.710145	−	0.704056i	$-0.751370\pi$
$223$	−21.2094	−1.42029	−0.710145	+	0.704056i	$0.751370\pi$
$224$	6.77089	0.452399
$225$	0	0
$226$	13.0369	0.867200
$227$	19.0683	1.26561	0.632805	−	0.774311i	$-0.281903\pi$
$227$	19.0683	1.26561	0.632805	+	0.774311i	$0.281903\pi$
$228$	0	0
$229$	−7.11891	−0.470431	−0.235216	−	0.971943i	$-0.575580\pi$
$229$	−7.11891	−0.470431	−0.235216	+	0.971943i	$0.575580\pi$
$230$	47.3610	3.12289
$231$	0	0
$232$	16.0496	1.05371
$233$	6.92852	0.453903	0.226951	−	0.973906i	$-0.427124\pi$
$233$	6.92852	0.453903	0.226951	+	0.973906i	$0.427124\pi$
$234$	0	0
$235$	−25.1761	−1.64231
$236$	31.8632	2.07412
$237$	0	0
$238$	1.34762	0.0873534
$239$	−4.81696	−0.311583	−0.155791	−	0.987790i	$-0.549793\pi$
$239$	−4.81696	−0.311583	−0.155791	+	0.987790i	$0.549793\pi$
$240$	0	0
$241$	22.4216	1.44430	0.722150	−	0.691736i	$-0.243154\pi$
$241$	22.4216	1.44430	0.722150	+	0.691736i	$0.243154\pi$
$242$	27.8691	1.79150
$243$	0	0
$244$	39.7484	2.54463
$245$	−2.48067	−0.158484
$246$	0	0
$247$	−0.741972	−0.0472105
$248$	13.9057	0.883013
$249$	0	0
$250$	21.2995	1.34710
$251$	10.7991	0.681631	0.340815	−	0.940130i	$-0.389297\pi$
$251$	10.7991	0.681631	0.340815	+	0.940130i	$0.389297\pi$
$252$	0	0
$253$	−41.4459	−2.60568
$254$	−2.23233	−0.140069
$255$	0	0
$256$	−8.49937	−0.531210
$257$	−11.8527	−0.739351	−0.369675	−	0.929161i	$-0.620531\pi$
$257$	−11.8527	−0.739351	−0.369675	+	0.929161i	$0.620531\pi$
$258$	0	0
$259$	8.93572	0.555239
$260$	2.01035	0.124677
$261$	0	0
$262$	−0.767445	−0.0474129
$263$	−3.00747	−0.185449	−0.0927244	−	0.995692i	$-0.529558\pi$
$263$	−3.00747	−0.185449	−0.0927244	+	0.995692i	$0.529558\pi$
$264$	0	0
$265$	−8.23199	−0.505687
$266$	−6.09735	−0.373853
$267$	0	0
$268$	−28.0491	−1.71337
$269$	16.6439	1.01480	0.507398	−	0.861712i	$-0.330607\pi$
$269$	16.6439	1.01480	0.507398	+	0.861712i	$0.330607\pi$
$270$	0	0
$271$	26.0154	1.58033	0.790163	−	0.612897i	$-0.209996\pi$
$271$	26.0154	1.58033	0.790163	+	0.612897i	$0.209996\pi$
$272$	0.643814	0.0390370
$273$	0	0
$274$	−39.2035	−2.36837
$275$	5.59099	0.337149
$276$	0	0
$277$	16.1504	0.970386	0.485193	−	0.874407i	$-0.338749\pi$
$277$	16.1504	0.970386	0.485193	+	0.874407i	$0.338749\pi$
$278$	−33.8528	−2.03036
$279$	0	0
$280$	5.44526	0.325417
$281$	24.7010	1.47354	0.736768	−	0.676146i	$-0.236351\pi$
$281$	24.7010	1.47354	0.736768	+	0.676146i	$0.236351\pi$
$282$	0	0
$283$	4.11619	0.244682	0.122341	−	0.992488i	$-0.460960\pi$
$283$	4.11619	0.244682	0.122341	+	0.992488i	$0.460960\pi$
$284$	−40.1871	−2.38466
$285$	0	0
$286$	−2.93868	−0.173768
$287$	−6.46178	−0.381427
$288$	0	0
$289$	−16.6356	−0.978563
$290$	−40.4894	−2.37762
$291$	0	0
$292$	−26.2281	−1.53488
$293$	7.56164	0.441756	0.220878	−	0.975301i	$-0.429108\pi$
$293$	7.56164	0.441756	0.220878	+	0.975301i	$0.429108\pi$
$294$	0	0
$295$	−26.4948	−1.54258
$296$	−19.6146	−1.14008
$297$	0	0
$298$	40.9487	2.37209
$299$	2.32326	0.134358
$300$	0	0
$301$	−6.83740	−0.394101
$302$	13.0903	0.753260
$303$	0	0
$304$	−2.91295	−0.167069
$305$	−33.0514	−1.89252
$306$	0	0
$307$	−28.4570	−1.62413	−0.812063	−	0.583570i	$-0.801656\pi$
$307$	−28.4570	−1.62413	−0.812063	+	0.583570i	$0.801656\pi$
$308$	−14.4573	−0.823782
$309$	0	0
$310$	−35.0809	−1.99246
$311$	−27.8485	−1.57914	−0.789571	−	0.613659i	$-0.789697\pi$
$311$	−27.8485	−1.57914	−0.789571	+	0.613659i	$0.789697\pi$
$312$	0	0
$313$	23.8022	1.34538	0.672690	−	0.739924i	$-0.265139\pi$
$313$	23.8022	1.34538	0.672690	+	0.739924i	$0.265139\pi$
$314$	52.6042	2.96863
$315$	0	0
$316$	43.2005	2.43022
$317$	21.7210	1.21997	0.609986	−	0.792412i	$-0.291175\pi$
$317$	21.7210	1.21997	0.609986	+	0.792412i	$0.291175\pi$
$318$	0	0
$319$	35.4325	1.98384
$320$	32.2039	1.80025
$321$	0	0
$322$	19.0920	1.06396
$323$	−1.64889	−0.0917468
$324$	0	0
$325$	−0.313404	−0.0173845
$326$	−21.1086	−1.16910
$327$	0	0
$328$	14.1841	0.783186
$329$	−10.1489	−0.559527
$330$	0	0
$331$	22.9326	1.26049	0.630245	−	0.776396i	$-0.282954\pi$
$331$	22.9326	1.26049	0.630245	+	0.776396i	$0.282954\pi$
$332$	1.02579	0.0562976
$333$	0	0
$334$	36.1169	1.97623
$335$	23.3233	1.27429
$336$	0	0
$337$	−31.7497	−1.72952	−0.864758	−	0.502189i	$-0.832528\pi$
$337$	−31.7497	−1.72952	−0.864758	+	0.502189i	$0.832528\pi$
$338$	−28.8556	−1.56954
$339$	0	0
$340$	4.46763	0.242291
$341$	30.6995	1.66247
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	15.0086	0.809211
$345$	0	0
$346$	11.0543	0.594283
$347$	27.8404	1.49455	0.747275	−	0.664515i	$-0.231362\pi$
$347$	27.8404	1.49455	0.747275	+	0.664515i	$0.231362\pi$
$348$	0	0
$349$	−34.4480	−1.84396	−0.921979	−	0.387240i	$-0.873429\pi$
$349$	−34.4480	−1.84396	−0.921979	+	0.387240i	$0.873429\pi$
$350$	−2.57548	−0.137665
$351$	0	0
$352$	−32.8121	−1.74889
$353$	23.5834	1.25522	0.627608	−	0.778530i	$-0.284034\pi$
$353$	23.5834	1.25522	0.627608	+	0.778530i	$0.284034\pi$
$354$	0	0
$355$	33.4162	1.77355
$356$	5.49104	0.291025
$357$	0	0
$358$	−16.6042	−0.877557
$359$	32.1624	1.69746	0.848732	−	0.528823i	$-0.177366\pi$
$359$	32.1624	1.69746	0.848732	+	0.528823i	$0.177366\pi$
$360$	0	0
$361$	−11.5396	−0.607345
$362$	−6.24272	−0.328110
$363$	0	0
$364$	0.810407	0.0424769
$365$	21.8091	1.14154
$366$	0	0
$367$	−25.0720	−1.30875	−0.654375	−	0.756171i	$-0.727068\pi$
$367$	−25.0720	−1.30875	−0.654375	+	0.756171i	$0.727068\pi$
$368$	9.12103	0.475467
$369$	0	0
$370$	49.4832	2.57251
$371$	−3.31845	−0.172286
$372$	0	0
$373$	−28.7664	−1.48947	−0.744734	−	0.667361i	$-0.767424\pi$
$373$	−28.7664	−1.48947	−0.744734	+	0.667361i	$0.767424\pi$
$374$	−6.53066	−0.337693
$375$	0	0
$376$	22.2776	1.14888
$377$	−1.98618	−0.102293
$378$	0	0
$379$	18.1442	0.932007	0.466004	−	0.884783i	$-0.345693\pi$
$379$	18.1442	0.932007	0.466004	+	0.884783i	$0.345693\pi$
$380$	−20.2139	−1.03695
$381$	0	0
$382$	−22.4784	−1.15009
$383$	8.96217	0.457946	0.228973	−	0.973433i	$-0.426463\pi$
$383$	8.96217	0.457946	0.228973	+	0.973433i	$0.426463\pi$
$384$	0	0
$385$	12.0215	0.612671
$386$	−34.6190	−1.76206
$387$	0	0
$388$	−14.8415	−0.753463
$389$	11.7846	0.597501	0.298751	−	0.954331i	$-0.403430\pi$
$389$	11.7846	0.597501	0.298751	+	0.954331i	$0.403430\pi$
$390$	0	0
$391$	5.16300	0.261104
$392$	2.19508	0.110868
$393$	0	0
$394$	19.9721	1.00618
$395$	−35.9218	−1.80742
$396$	0	0
$397$	0.149178	0.00748703	0.00374352	−	0.999993i	$-0.498808\pi$
$397$	0.149178	0.00748703	0.00374352	+	0.999993i	$0.498808\pi$
$398$	−17.6211	−0.883264
$399$	0	0
$400$	−1.23041	−0.0615207
$401$	−6.95960	−0.347546	−0.173773	−	0.984786i	$-0.555596\pi$
$401$	−6.95960	−0.347546	−0.173773	+	0.984786i	$0.555596\pi$
$402$	0	0
$403$	−1.72087	−0.0857226
$404$	27.6361	1.37495
$405$	0	0
$406$	−16.3220	−0.810046
$407$	−43.3030	−2.14645
$408$	0	0
$409$	−24.1483	−1.19405	−0.597027	−	0.802221i	$-0.703652\pi$
$409$	−24.1483	−1.19405	−0.597027	+	0.802221i	$0.703652\pi$
$410$	−35.7833	−1.76721
$411$	0	0
$412$	29.1552	1.43637
$413$	−10.6805	−0.525553
$414$	0	0
$415$	−0.852961	−0.0418702
$416$	1.83929	0.0901787
$417$	0	0
$418$	29.5482	1.44525
$419$	−19.8940	−0.971887	−0.485943	−	0.873990i	$-0.661524\pi$
$419$	−19.8940	−0.971887	−0.485943	+	0.873990i	$0.661524\pi$
$420$	0	0
$421$	4.35419	0.212210	0.106105	−	0.994355i	$-0.466162\pi$
$421$	4.35419	0.212210	0.106105	+	0.994355i	$0.466162\pi$
$422$	21.4741	1.04534
$423$	0	0
$424$	7.28427	0.353755
$425$	−0.696481	−0.0337843
$426$	0	0
$427$	−13.3236	−0.644773
$428$	43.5963	2.10731
$429$	0	0
$430$	−37.8633	−1.82593
$431$	−33.3171	−1.60483	−0.802414	−	0.596768i	$-0.796451\pi$
$431$	−33.3171	−1.60483	−0.802414	+	0.596768i	$0.796451\pi$
$432$	0	0
$433$	−15.7980	−0.759205	−0.379603	−	0.925150i	$-0.623939\pi$
$433$	−15.7980	−0.759205	−0.379603	+	0.925150i	$0.623939\pi$
$434$	−14.1417	−0.678824
$435$	0	0
$436$	24.7294	1.18432
$437$	−23.3601	−1.11747
$438$	0	0
$439$	−1.01491	−0.0484392	−0.0242196	−	0.999707i	$-0.507710\pi$
$439$	−1.01491	−0.0484392	−0.0242196	+	0.999707i	$0.507710\pi$
$440$	−26.3881	−1.25800
$441$	0	0
$442$	0.366078	0.0174125
$443$	5.24695	0.249290	0.124645	−	0.992201i	$-0.460221\pi$
$443$	5.24695	0.249290	0.124645	+	0.992201i	$0.460221\pi$
$444$	0	0
$445$	−4.56589	−0.216444
$446$	−47.3465	−2.24192
$447$	0	0
$448$	12.9819	0.613338
$449$	27.0115	1.27475	0.637377	−	0.770552i	$-0.280020\pi$
$449$	27.0115	1.27475	0.637377	+	0.770552i	$0.280020\pi$
$450$	0	0
$451$	31.3142	1.47453
$452$	17.4226	0.819491
$453$	0	0
$454$	42.5669	1.99776
$455$	−0.673866	−0.0315913
$456$	0	0
$457$	−22.0104	−1.02960	−0.514801	−	0.857310i	$-0.672134\pi$
$457$	−22.0104	−1.02960	−0.514801	+	0.857310i	$0.672134\pi$
$458$	−15.8918	−0.742574
$459$	0	0
$460$	63.2937	2.95108
$461$	−5.65948	−0.263588	−0.131794	−	0.991277i	$-0.542074\pi$
$461$	−5.65948	−0.263588	−0.131794	+	0.991277i	$0.542074\pi$
$462$	0	0
$463$	30.7836	1.43064	0.715318	−	0.698799i	$-0.246282\pi$
$463$	30.7836	1.43064	0.715318	+	0.698799i	$0.246282\pi$
$464$	−7.79767	−0.361998
$465$	0	0
$466$	15.4668	0.716485
$467$	14.1888	0.656579	0.328290	−	0.944577i	$-0.393528\pi$
$467$	14.1888	0.656579	0.328290	+	0.944577i	$0.393528\pi$
$468$	0	0
$469$	9.40201	0.434145
$470$	−56.2014	−2.59238
$471$	0	0
$472$	23.4445	1.07912
$473$	33.1345	1.52352
$474$	0	0
$475$	3.15125	0.144589
$476$	1.80098	0.0825476
$477$	0	0
$478$	−10.7531	−0.491833
$479$	−11.0792	−0.506223	−0.253112	−	0.967437i	$-0.581454\pi$
$479$	−11.0792	−0.506223	−0.253112	+	0.967437i	$0.581454\pi$
$480$	0	0
$481$	2.42736	0.110678
$482$	50.0524	2.27983
$483$	0	0
$484$	37.2446	1.69294
$485$	12.3409	0.560373
$486$	0	0
$487$	−4.78994	−0.217053	−0.108526	−	0.994094i	$-0.534613\pi$
$487$	−4.78994	−0.217053	−0.108526	+	0.994094i	$0.534613\pi$
$488$	29.2463	1.32392
$489$	0	0
$490$	−5.53768	−0.250167
$491$	−20.8353	−0.940283	−0.470142	−	0.882591i	$-0.655797\pi$
$491$	−20.8353	−0.940283	−0.470142	+	0.882591i	$0.655797\pi$
$492$	0	0
$493$	−4.41390	−0.198792
$494$	−1.65633	−0.0745217
$495$	0	0
$496$	−6.75607	−0.303357
$497$	13.4706	0.604240
$498$	0	0
$499$	28.5667	1.27882	0.639410	−	0.768866i	$-0.279179\pi$
$499$	28.5667	1.27882	0.639410	+	0.768866i	$0.279179\pi$
$500$	28.4648	1.27299
$501$	0	0
$502$	24.1071	1.07595
$503$	−2.78014	−0.123960	−0.0619802	−	0.998077i	$-0.519742\pi$
$503$	−2.78014	−0.123960	−0.0619802	+	0.998077i	$0.519742\pi$
$504$	0	0
$505$	−22.9798	−1.02259
$506$	−92.5211	−4.11307
$507$	0	0
$508$	−2.98331	−0.132363
$509$	−10.1807	−0.451253	−0.225627	−	0.974214i	$-0.572443\pi$
$509$	−10.1807	−0.451253	−0.225627	+	0.974214i	$0.572443\pi$
$510$	0	0
$511$	8.79161	0.388918
$512$	11.9030	0.526043
$513$	0	0
$514$	−26.4592	−1.16706
$515$	−24.2430	−1.06827
$516$	0	0
$517$	49.1822	2.16303
$518$	19.9475	0.876443
$519$	0	0
$520$	1.47919	0.0648667
$521$	−20.5075	−0.898451	−0.449225	−	0.893418i	$-0.648300\pi$
$521$	−20.5075	−0.898451	−0.449225	+	0.893418i	$0.648300\pi$
$522$	0	0
$523$	−42.8438	−1.87343	−0.936713	−	0.350097i	$-0.886149\pi$
$523$	−42.8438	−1.87343	−0.936713	+	0.350097i	$0.886149\pi$
$524$	−1.02562	−0.0448045
$525$	0	0
$526$	−6.71368	−0.292730
$527$	−3.82431	−0.166589
$528$	0	0
$529$	50.1452	2.18023
$530$	−18.3765	−0.798226
$531$	0	0
$532$	−8.14856	−0.353285
$533$	−1.75532	−0.0760315
$534$	0	0
$535$	−36.2510	−1.56727
$536$	−20.6382	−0.891432
$537$	0	0
$538$	37.1547	1.60185
$539$	4.84606	0.208735
$540$	0	0
$541$	15.0797	0.648328	0.324164	−	0.946001i	$-0.394917\pi$
$541$	15.0797	0.648328	0.324164	+	0.946001i	$0.394917\pi$
$542$	58.0751	2.49454
$543$	0	0
$544$	4.08747	0.175249
$545$	−20.5629	−0.880816
$546$	0	0
$547$	−19.1805	−0.820099	−0.410049	−	0.912063i	$-0.634489\pi$
$547$	−19.1805	−0.820099	−0.410049	+	0.912063i	$0.634489\pi$
$548$	−52.3919	−2.23807
$549$	0	0
$550$	12.4810	0.532190
$551$	19.9708	0.850786
$552$	0	0
$553$	−14.4807	−0.615782
$554$	36.0532	1.53175
$555$	0	0
$556$	−45.2412	−1.91866
$557$	23.9231	1.01366	0.506828	−	0.862047i	$-0.330818\pi$
$557$	23.9231	1.01366	0.506828	+	0.862047i	$0.330818\pi$
$558$	0	0
$559$	−1.85736	−0.0785579
$560$	−2.64557	−0.111796
$561$	0	0
$562$	55.1408	2.32597
$563$	−30.3803	−1.28038	−0.640189	−	0.768217i	$-0.721144\pi$
$563$	−30.3803	−1.28038	−0.640189	+	0.768217i	$0.721144\pi$
$564$	0	0
$565$	−14.4872	−0.609480
$566$	9.18870	0.386230
$567$	0	0
$568$	−29.5691	−1.24069
$569$	−36.9558	−1.54927	−0.774633	−	0.632411i	$-0.782066\pi$
$569$	−36.9558	−1.54927	−0.774633	+	0.632411i	$0.782066\pi$
$570$	0	0
$571$	−44.0003	−1.84135	−0.920677	−	0.390324i	$-0.872363\pi$
$571$	−44.0003	−1.84135	−0.920677	+	0.390324i	$0.872363\pi$
$572$	−3.92728	−0.164208
$573$	0	0
$574$	−14.4248	−0.602081
$575$	−9.86718	−0.411490
$576$	0	0
$577$	22.6870	0.944471	0.472235	−	0.881472i	$-0.343447\pi$
$577$	22.6870	0.944471	0.472235	+	0.881472i	$0.343447\pi$
$578$	−37.1361	−1.54466
$579$	0	0
$580$	−54.1104	−2.24681
$581$	−0.343843	−0.0142650
$582$	0	0
$583$	16.0814	0.666025
$584$	−19.2983	−0.798568
$585$	0	0
$586$	16.8801	0.697310
$587$	−43.6145	−1.80016	−0.900081	−	0.435723i	$-0.856493\pi$
$587$	−43.6145	−1.80016	−0.900081	+	0.435723i	$0.856493\pi$
$588$	0	0
$589$	17.3032	0.712964
$590$	−59.1451	−2.43497
$591$	0	0
$592$	9.52973	0.391670
$593$	−1.97648	−0.0811643	−0.0405821	−	0.999176i	$-0.512921\pi$
$593$	−1.97648	−0.0811643	−0.0405821	+	0.999176i	$0.512921\pi$
$594$	0	0
$595$	−1.49754	−0.0613931
$596$	54.7242	2.24159
$597$	0	0
$598$	5.18629	0.212083
$599$	15.7618	0.644010	0.322005	−	0.946738i	$-0.395643\pi$
$599$	15.7618	0.644010	0.322005	+	0.946738i	$0.395643\pi$
$600$	0	0
$601$	−45.8667	−1.87094	−0.935471	−	0.353405i	$-0.885024\pi$
$601$	−45.8667	−1.87094	−0.935471	+	0.353405i	$0.885024\pi$
$602$	−15.2634	−0.622088
$603$	0	0
$604$	17.4940	0.711819
$605$	−30.9694	−1.25909
$606$	0	0
$607$	−12.7712	−0.518367	−0.259184	−	0.965828i	$-0.583453\pi$
$607$	−12.7712	−0.518367	−0.259184	+	0.965828i	$0.583453\pi$
$608$	−18.4939	−0.750026
$609$	0	0
$610$	−73.7817	−2.98733
$611$	−2.75692	−0.111533
$612$	0	0
$613$	−31.3590	−1.26658	−0.633290	−	0.773915i	$-0.718296\pi$
$613$	−31.3590	−1.26658	−0.633290	+	0.773915i	$0.718296\pi$
$614$	−63.5255	−2.56368
$615$	0	0
$616$	−10.6375	−0.428596
$617$	9.19226	0.370067	0.185033	−	0.982732i	$-0.440761\pi$
$617$	9.19226	0.370067	0.185033	+	0.982732i	$0.440761\pi$
$618$	0	0
$619$	−34.4399	−1.38426	−0.692129	−	0.721774i	$-0.743327\pi$
$619$	−34.4399	−1.38426	−0.692129	+	0.721774i	$0.743327\pi$
$620$	−46.8825	−1.88285
$621$	0	0
$622$	−62.1671	−2.49267
$623$	−1.84059	−0.0737415
$624$	0	0
$625$	−29.4375	−1.17750
$626$	53.1345	2.12368
$627$	0	0
$628$	70.3007	2.80530
$629$	5.39435	0.215087
$630$	0	0
$631$	−32.1497	−1.27986	−0.639929	−	0.768434i	$-0.721036\pi$
$631$	−32.1497	−1.27986	−0.639929	+	0.768434i	$0.721036\pi$
$632$	31.7863	1.26439
$633$	0	0
$634$	48.4885	1.92572
$635$	2.48067	0.0984423
$636$	0	0
$637$	−0.271647	−0.0107630
$638$	79.0972	3.13149
$639$	0	0
$640$	38.2971	1.51382
$641$	−11.3723	−0.449181	−0.224590	−	0.974453i	$-0.572104\pi$
$641$	−11.3723	−0.449181	−0.224590	+	0.974453i	$0.572104\pi$
$642$	0	0
$643$	13.3776	0.527560	0.263780	−	0.964583i	$-0.415031\pi$
$643$	13.3776	0.527560	0.263780	+	0.964583i	$0.415031\pi$
$644$	25.5148	1.00542
$645$	0	0
$646$	−3.68087	−0.144822
$647$	−27.5660	−1.08373	−0.541866	−	0.840465i	$-0.682282\pi$
$647$	−27.5660	−1.08373	−0.541866	+	0.840465i	$0.682282\pi$
$648$	0	0
$649$	51.7583	2.03169
$650$	−0.699623	−0.0274415
$651$	0	0
$652$	−28.2097	−1.10478
$653$	−17.4422	−0.682568	−0.341284	−	0.939960i	$-0.610862\pi$
$653$	−17.4422	−0.682568	−0.341284	+	0.939960i	$0.610862\pi$
$654$	0	0
$655$	0.852820	0.0333224
$656$	−6.89133	−0.269061
$657$	0	0
$658$	−22.6557	−0.883212
$659$	33.9551	1.32270	0.661351	−	0.750076i	$-0.269983\pi$
$659$	33.9551	1.32270	0.661351	+	0.750076i	$0.269983\pi$
$660$	0	0
$661$	20.8865	0.812391	0.406195	−	0.913786i	$-0.366855\pi$
$661$	20.8865	0.812391	0.406195	+	0.913786i	$0.366855\pi$
$662$	51.1932	1.98968
$663$	0	0
$664$	0.754762	0.0292905
$665$	6.77566	0.262749
$666$	0	0
$667$	−62.5326	−2.42127
$668$	48.2670	1.86751
$669$	0	0
$670$	52.0653	2.01146
$671$	64.5669	2.49258
$672$	0	0
$673$	−16.2952	−0.628136	−0.314068	−	0.949401i	$-0.601692\pi$
$673$	−16.2952	−0.628136	−0.314068	+	0.949401i	$0.601692\pi$
$674$	−70.8759	−2.73004
$675$	0	0
$676$	−38.5629	−1.48319
$677$	25.6066	0.984140	0.492070	−	0.870556i	$-0.336240\pi$
$677$	25.6066	0.984140	0.492070	+	0.870556i	$0.336240\pi$
$678$	0	0
$679$	4.97484	0.190917
$680$	3.28722	0.126059
$681$	0	0
$682$	68.5316	2.62421
$683$	7.30405	0.279482	0.139741	−	0.990188i	$-0.455373\pi$
$683$	7.30405	0.279482	0.139741	+	0.990188i	$0.455373\pi$
$684$	0	0
$685$	43.5646	1.66452
$686$	−2.23233	−0.0852309
$687$	0	0
$688$	−7.29192	−0.278002
$689$	−0.901448	−0.0343424
$690$	0	0
$691$	1.34746	0.0512596	0.0256298	−	0.999672i	$-0.491841\pi$
$691$	1.34746	0.0512596	0.0256298	+	0.999672i	$0.491841\pi$
$692$	14.7731	0.561588
$693$	0	0
$694$	62.1490	2.35914
$695$	37.6188	1.42696
$696$	0	0
$697$	−3.90087	−0.147756
$698$	−76.8994	−2.91068
$699$	0	0
$700$	−3.44190	−0.130092
$701$	−11.4220	−0.431404	−0.215702	−	0.976459i	$-0.569204\pi$
$701$	−11.4220	−0.431404	−0.215702	+	0.976459i	$0.569204\pi$
$702$	0	0
$703$	−24.4069	−0.920522
$704$	−62.9112	−2.37105
$705$	0	0
$706$	52.6459	1.98136
$707$	−9.26356	−0.348392
$708$	0	0
$709$	34.0463	1.27864	0.639318	−	0.768942i	$-0.279217\pi$
$709$	34.0463	1.27864	0.639318	+	0.768942i	$0.279217\pi$
$710$	74.5960	2.79954
$711$	0	0
$712$	4.04023	0.151414
$713$	−54.1796	−2.02904
$714$	0	0
$715$	3.26560	0.122126
$716$	−22.1899	−0.829277
$717$	0	0
$718$	71.7971	2.67944
$719$	7.18031	0.267781	0.133890	−	0.990996i	$-0.457253\pi$
$719$	7.18031	0.267781	0.133890	+	0.990996i	$0.457253\pi$
$720$	0	0
$721$	−9.77276	−0.363957
$722$	−25.7601	−0.958693
$723$	0	0
$724$	−8.34284	−0.310059
$725$	8.43555	0.313288
$726$	0	0
$727$	−17.6583	−0.654911	−0.327456	−	0.944867i	$-0.606191\pi$
$727$	−17.6583	−0.654911	−0.327456	+	0.944867i	$0.606191\pi$
$728$	0.596286	0.0220998
$729$	0	0
$730$	48.6851	1.80192
$731$	−4.12763	−0.152666
$732$	0	0
$733$	23.8353	0.880377	0.440188	−	0.897905i	$-0.354912\pi$
$733$	23.8353	0.880377	0.440188	+	0.897905i	$0.354912\pi$
$734$	−55.9691	−2.06586
$735$	0	0
$736$	57.9080	2.13452
$737$	−45.5627	−1.67832
$738$	0	0
$739$	−28.3026	−1.04113	−0.520564	−	0.853823i	$-0.674278\pi$
$739$	−28.3026	−1.04113	−0.520564	+	0.853823i	$0.674278\pi$
$740$	66.1298	2.43098
$741$	0	0
$742$	−7.40790	−0.271952
$743$	−24.5932	−0.902237	−0.451118	−	0.892464i	$-0.648975\pi$
$743$	−24.5932	−0.902237	−0.451118	+	0.892464i	$0.648975\pi$
$744$	0	0
$745$	−45.5040	−1.66714
$746$	−64.2162	−2.35112
$747$	0	0
$748$	−8.72764	−0.319114
$749$	−14.6134	−0.533962
$750$	0	0
$751$	20.9109	0.763049	0.381524	−	0.924359i	$-0.375399\pi$
$751$	20.9109	0.763049	0.381524	+	0.924359i	$0.375399\pi$
$752$	−10.8236	−0.394695
$753$	0	0
$754$	−4.43381	−0.161470
$755$	−14.5465	−0.529401
$756$	0	0
$757$	−26.3896	−0.959147	−0.479574	−	0.877502i	$-0.659209\pi$
$757$	−26.3896	−0.959147	−0.479574	+	0.877502i	$0.659209\pi$
$758$	40.5040	1.47117
$759$	0	0
$760$	−14.8731	−0.539504
$761$	−20.9653	−0.759991	−0.379995	−	0.924988i	$-0.624074\pi$
$761$	−20.9653	−0.759991	−0.379995	+	0.924988i	$0.624074\pi$
$762$	0	0
$763$	−8.28924	−0.300091
$764$	−30.0403	−1.08682
$765$	0	0
$766$	20.0066	0.722866
$767$	−2.90132	−0.104761
$768$	0	0
$769$	−23.9395	−0.863280	−0.431640	−	0.902046i	$-0.642065\pi$
$769$	−23.9395	−0.863280	−0.431640	+	0.902046i	$0.642065\pi$
$770$	26.8359	0.967100
$771$	0	0
$772$	−46.2652	−1.66512
$773$	−8.31326	−0.299007	−0.149504	−	0.988761i	$-0.547768\pi$
$773$	−8.31326	−0.299007	−0.149504	+	0.988761i	$0.547768\pi$
$774$	0	0
$775$	7.30875	0.262538
$776$	−10.9202	−0.392011
$777$	0	0
$778$	26.3071	0.943155
$779$	17.6496	0.632362
$780$	0	0
$781$	−65.2795	−2.33588
$782$	11.5255	0.412153
$783$	0	0
$784$	−1.06648	−0.0380884
$785$	−58.4561	−2.08639
$786$	0	0
$787$	−13.6594	−0.486904	−0.243452	−	0.969913i	$-0.578280\pi$
$787$	−13.6594	−0.486904	−0.243452	+	0.969913i	$0.578280\pi$
$788$	26.6909	0.950824
$789$	0	0
$790$	−80.1895	−2.85301
$791$	−5.84002	−0.207647
$792$	0	0
$793$	−3.61931	−0.128525
$794$	0.333015	0.0118183
$795$	0	0
$796$	−23.5490	−0.834671
$797$	−14.4902	−0.513270	−0.256635	−	0.966508i	$-0.582614\pi$
$797$	−14.4902	−0.513270	−0.256635	+	0.966508i	$0.582614\pi$
$798$	0	0
$799$	−6.12673	−0.216748
$800$	−7.81170	−0.276185
$801$	0	0
$802$	−15.5362	−0.548601
$803$	−42.6047	−1.50349
$804$	0	0
$805$	−21.2159	−0.747763
$806$	−3.84155	−0.135313
$807$	0	0
$808$	20.3342	0.715356
$809$	31.4471	1.10562	0.552810	−	0.833307i	$-0.313556\pi$
$809$	31.4471	1.10562	0.552810	+	0.833307i	$0.313556\pi$
$810$	0	0
$811$	−9.36239	−0.328758	−0.164379	−	0.986397i	$-0.552562\pi$
$811$	−9.36239	−0.328758	−0.164379	+	0.986397i	$0.552562\pi$
$812$	−21.8128	−0.765480
$813$	0	0
$814$	−96.6668	−3.38817
$815$	23.4568	0.821657
$816$	0	0
$817$	18.6755	0.653375
$818$	−53.9070	−1.88481
$819$	0	0
$820$	−47.8211	−1.66998
$821$	−23.8608	−0.832747	−0.416373	−	0.909194i	$-0.636699\pi$
$821$	−23.8608	−0.832747	−0.416373	+	0.909194i	$0.636699\pi$
$822$	0	0
$823$	11.4438	0.398904	0.199452	−	0.979908i	$-0.436084\pi$
$823$	11.4438	0.398904	0.199452	+	0.979908i	$0.436084\pi$
$824$	21.4520	0.747315
$825$	0	0
$826$	−23.8424	−0.829584
$827$	−9.70579	−0.337503	−0.168752	−	0.985659i	$-0.553974\pi$
$827$	−9.70579	−0.337503	−0.168752	+	0.985659i	$0.553974\pi$
$828$	0	0
$829$	55.1410	1.91513	0.957563	−	0.288223i	$-0.0930644\pi$
$829$	55.1410	1.91513	0.957563	+	0.288223i	$0.0930644\pi$
$830$	−1.90409	−0.0660920
$831$	0	0
$832$	3.52650	0.122259
$833$	−0.603684	−0.0209164
$834$	0	0
$835$	−40.1348	−1.38892
$836$	39.4884	1.36574
$837$	0	0
$838$	−44.4101	−1.53412
$839$	−33.0511	−1.14105	−0.570525	−	0.821280i	$-0.693260\pi$
$839$	−33.0511	−1.14105	−0.570525	+	0.821280i	$0.693260\pi$
$840$	0	0
$841$	24.4598	0.843440
$842$	9.72000	0.334973
$843$	0	0
$844$	28.6982	0.987833
$845$	32.0656	1.10309
$846$	0	0
$847$	−12.4843	−0.428966
$848$	−3.53905	−0.121532
$849$	0	0
$850$	−1.55478	−0.0533285
$851$	76.4227	2.61974
$852$	0	0
$853$	48.3364	1.65501	0.827503	−	0.561461i	$-0.189761\pi$
$853$	48.3364	1.65501	0.827503	+	0.561461i	$0.189761\pi$
$854$	−29.7427	−1.01777
$855$	0	0
$856$	32.0775	1.09639
$857$	19.1924	0.655600	0.327800	−	0.944747i	$-0.393693\pi$
$857$	19.1924	0.655600	0.327800	+	0.944747i	$0.393693\pi$
$858$	0	0
$859$	21.5900	0.736643	0.368321	−	0.929699i	$-0.379933\pi$
$859$	21.5900	0.736643	0.368321	+	0.929699i	$0.379933\pi$
$860$	−50.6009	−1.72548
$861$	0	0
$862$	−74.3748	−2.53322
$863$	−23.7114	−0.807144	−0.403572	−	0.914948i	$-0.632231\pi$
$863$	−23.7114	−0.807144	−0.403572	+	0.914948i	$0.632231\pi$
$864$	0	0
$865$	−12.2840	−0.417670
$866$	−35.2665	−1.19840
$867$	0	0
$868$	−18.8991	−0.641478
$869$	70.1744	2.38050
$870$	0	0
$871$	2.55403	0.0865399
$872$	18.1955	0.616178
$873$	0	0
$874$	−52.1476	−1.76392
$875$	−9.54135	−0.322557
$876$	0	0
$877$	−13.6769	−0.461836	−0.230918	−	0.972973i	$-0.574173\pi$
$877$	−13.6769	−0.461836	−0.230918	+	0.972973i	$0.574173\pi$
$878$	−2.26563	−0.0764612
$879$	0	0
$880$	12.8206	0.432183
$881$	33.2161	1.11908	0.559540	−	0.828803i	$-0.310978\pi$
$881$	33.2161	1.11908	0.559540	+	0.828803i	$0.310978\pi$
$882$	0	0
$883$	−7.32919	−0.246647	−0.123323	−	0.992367i	$-0.539355\pi$
$883$	−7.32919	−0.246647	−0.123323	+	0.992367i	$0.539355\pi$
$884$	0.489230	0.0164546
$885$	0	0
$886$	11.7129	0.393504
$887$	−12.2837	−0.412445	−0.206223	−	0.978505i	$-0.566117\pi$
$887$	−12.2837	−0.412445	−0.206223	+	0.978505i	$0.566117\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−10.1926	−0.341656
$891$	0	0
$892$	−63.2744	−2.11858
$893$	27.7205	0.927632
$894$	0	0
$895$	18.4513	0.616758
$896$	15.4382	0.515754
$897$	0	0
$898$	60.2988	2.01220
$899$	46.3187	1.54482
$900$	0	0
$901$	−2.00330	−0.0667395
$902$	69.9037	2.32754
$903$	0	0
$904$	12.8193	0.426364
$905$	6.93720	0.230600
$906$	0	0
$907$	−29.1730	−0.968673	−0.484336	−	0.874882i	$-0.660939\pi$
$907$	−29.1730	−0.968673	−0.484336	+	0.874882i	$0.660939\pi$
$908$	56.8868	1.88785
$909$	0	0
$910$	−1.50429	−0.0498669
$911$	18.3870	0.609188	0.304594	−	0.952482i	$-0.401479\pi$
$911$	18.3870	0.609188	0.304594	+	0.952482i	$0.401479\pi$
$912$	0	0
$913$	1.66628	0.0551459
$914$	−49.1345	−1.62522
$915$	0	0
$916$	−21.2379	−0.701721
$917$	0.343786	0.0113528
$918$	0	0
$919$	34.0371	1.12278	0.561389	−	0.827552i	$-0.310267\pi$
$919$	34.0371	1.12278	0.561389	+	0.827552i	$0.310267\pi$
$920$	46.5706	1.53539
$921$	0	0
$922$	−12.6338	−0.416073
$923$	3.65926	0.120446
$924$	0	0
$925$	−10.3093	−0.338968
$926$	68.7193	2.25826
$927$	0	0
$928$	−49.5061	−1.62512
$929$	0.443035	0.0145355	0.00726776	−	0.999974i	$-0.497687\pi$
$929$	0.443035	0.0145355	0.00726776	+	0.999974i	$0.497687\pi$
$930$	0	0
$931$	2.73138	0.0895174
$932$	20.6699	0.677067
$933$	0	0
$934$	31.6741	1.03641
$935$	7.25717	0.237335
$936$	0	0
$937$	16.1142	0.526429	0.263215	−	0.964737i	$-0.415217\pi$
$937$	16.1142	0.526429	0.263215	+	0.964737i	$0.415217\pi$
$938$	20.9884	0.685296
$939$	0	0
$940$	−75.1081	−2.44975
$941$	12.7977	0.417193	0.208596	−	0.978002i	$-0.433110\pi$
$941$	12.7977	0.417193	0.208596	+	0.978002i	$0.433110\pi$
$942$	0	0
$943$	−55.2643	−1.79965
$944$	−11.3905	−0.370729
$945$	0	0
$946$	73.9671	2.40488
$947$	−10.8195	−0.351588	−0.175794	−	0.984427i	$-0.556249\pi$
$947$	−10.8195	−0.351588	−0.175794	+	0.984427i	$0.556249\pi$
$948$	0	0
$949$	2.38821	0.0775247
$950$	7.03463	0.228234
$951$	0	0
$952$	1.32513	0.0429478
$953$	−41.4215	−1.34178	−0.670888	−	0.741559i	$-0.734087\pi$
$953$	−41.4215	−1.34178	−0.670888	+	0.741559i	$0.734087\pi$
$954$	0	0
$955$	24.9790	0.808301
$956$	−14.3705	−0.464775
$957$	0	0
$958$	−24.7326	−0.799073
$959$	17.5617	0.567095
$960$	0	0
$961$	9.13157	0.294567
$962$	5.41868	0.174705
$963$	0	0
$964$	66.8905	2.15440
$965$	38.4702	1.23840
$966$	0	0
$967$	50.3115	1.61791	0.808954	−	0.587872i	$-0.200034\pi$
$967$	50.3115	1.61791	0.808954	+	0.587872i	$0.200034\pi$
$968$	27.4040	0.880799
$969$	0	0
$970$	27.5491	0.884547
$971$	12.2383	0.392746	0.196373	−	0.980529i	$-0.437084\pi$
$971$	12.2383	0.392746	0.196373	+	0.980529i	$0.437084\pi$
$972$	0	0
$973$	15.1648	0.486160
$974$	−10.6927	−0.342618
$975$	0	0
$976$	−14.2093	−0.454828
$977$	−21.8393	−0.698700	−0.349350	−	0.936992i	$-0.613598\pi$
$977$	−21.8393	−0.698700	−0.349350	+	0.936992i	$0.613598\pi$
$978$	0	0
$979$	8.91959	0.285071
$980$	−7.40061	−0.236404
$981$	0	0
$982$	−46.5113	−1.48424
$983$	−19.5952	−0.624990	−0.312495	−	0.949919i	$-0.601165\pi$
$983$	−19.5952	−0.624990	−0.312495	+	0.949919i	$0.601165\pi$
$984$	0	0
$985$	−22.1939	−0.707156
$986$	−9.85331	−0.313793
$987$	0	0
$988$	−2.21353	−0.0704219
$989$	−58.4768	−1.85946
$990$	0	0
$991$	−25.3643	−0.805724	−0.402862	−	0.915261i	$-0.631985\pi$
$991$	−25.3643	−0.805724	−0.402862	+	0.915261i	$0.631985\pi$
$992$	−42.8932	−1.36186
$993$	0	0
$994$	30.0709	0.953792
$995$	19.5813	0.620770
$996$	0	0
$997$	50.0141	1.58396	0.791981	−	0.610545i	$-0.209050\pi$
$997$	50.0141	1.58396	0.791981	+	0.610545i	$0.209050\pi$
$998$	63.7704	2.01862
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.12		14
3.2	odd	2		2667.2.a.m.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.3	✓	14	3.2	odd	2
8001.2.a.p.1.12		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)

\(f(q)\)	\(=\)	\(q+2.23233 q^{2} +2.98331 q^{4} -2.48067 q^{5} -1.00000 q^{7} +2.19508 q^{8} +O(q^{10})\)	\(q+2.23233 q^{2} +2.98331 q^{4} -2.48067 q^{5} -1.00000 q^{7} +2.19508 q^{8} -5.53768 q^{10} +4.84606 q^{11} -0.271647 q^{13} -2.23233 q^{14} -1.06648 q^{16} -0.603684 q^{17} +2.73138 q^{19} -7.40061 q^{20} +10.8180 q^{22} -8.55250 q^{23} +1.15372 q^{25} -0.606407 q^{26} -2.98331 q^{28} +7.31162 q^{29} +6.33495 q^{31} -6.77089 q^{32} -1.34762 q^{34} +2.48067 q^{35} -8.93572 q^{37} +6.09735 q^{38} -5.44526 q^{40} +6.46178 q^{41} +6.83740 q^{43} +14.4573 q^{44} -19.0920 q^{46} +10.1489 q^{47} +1.00000 q^{49} +2.57548 q^{50} -0.810407 q^{52} +3.31845 q^{53} -12.0215 q^{55} -2.19508 q^{56} +16.3220 q^{58} +10.6805 q^{59} +13.3236 q^{61} +14.1417 q^{62} -12.9819 q^{64} +0.673866 q^{65} -9.40201 q^{67} -1.80098 q^{68} +5.53768 q^{70} -13.4706 q^{71} -8.79161 q^{73} -19.9475 q^{74} +8.14856 q^{76} -4.84606 q^{77} +14.4807 q^{79} +2.64557 q^{80} +14.4248 q^{82} +0.343843 q^{83} +1.49754 q^{85} +15.2634 q^{86} +10.6375 q^{88} +1.84059 q^{89} +0.271647 q^{91} -25.5148 q^{92} +22.6557 q^{94} -6.77566 q^{95} -4.97484 q^{97} +2.23233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 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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