LMFDB - Embedded newform 6014.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6014,2,Mod(1,6014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6014 = 2 \cdot 31 \cdot 97 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6014.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:	$48.0220317756$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.380224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 5x + 2 $ x^5 - 8x^3 - 2x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.56491$ of defining polynomial
Character	$\chi$	$=$	6014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.56491 q^{3} +1.00000 q^{4} -1.19216 q^{5} -2.56491 q^{6} +1.00000 q^{8} +3.57875 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.56491 q^{3} +1.00000 q^{4} -1.19216 q^{5} -2.56491 q^{6} +1.00000 q^{8} +3.57875 q^{9} -1.19216 q^{10} -2.08587 q^{11} -2.56491 q^{12} -6.11557 q^{13} +3.05779 q^{15} +1.00000 q^{16} +3.68504 q^{17} +3.57875 q^{18} -2.27263 q^{19} -1.19216 q^{20} -2.08587 q^{22} -7.64233 q^{23} -2.56491 q^{24} -3.57875 q^{25} -6.11557 q^{26} -1.48443 q^{27} +2.05616 q^{29} +3.05779 q^{30} -1.00000 q^{31} +1.00000 q^{32} +5.35006 q^{33} +3.68504 q^{34} +3.57875 q^{36} -2.27263 q^{38} +15.6859 q^{39} -1.19216 q^{40} -11.0844 q^{41} +2.08587 q^{43} -2.08587 q^{44} -4.26645 q^{45} -7.64233 q^{46} +12.3643 q^{47} -2.56491 q^{48} -7.00000 q^{49} -3.57875 q^{50} -9.45179 q^{51} -6.11557 q^{52} +13.6947 q^{53} -1.48443 q^{54} +2.48669 q^{55} +5.82910 q^{57} +2.05616 q^{58} +9.01812 q^{59} +3.05779 q^{60} +3.03510 q^{61} -1.00000 q^{62} +1.00000 q^{64} +7.29076 q^{65} +5.35006 q^{66} -9.34010 q^{67} +3.68504 q^{68} +19.6019 q^{69} -14.1156 q^{71} +3.57875 q^{72} -13.4171 q^{73} +9.17916 q^{75} -2.27263 q^{76} +15.6859 q^{78} +2.02848 q^{79} -1.19216 q^{80} -6.92881 q^{81} -11.0844 q^{82} +6.95808 q^{83} -4.39317 q^{85} +2.08587 q^{86} -5.27387 q^{87} -2.08587 q^{88} +5.81057 q^{89} -4.26645 q^{90} -7.64233 q^{92} +2.56491 q^{93} +12.3643 q^{94} +2.70935 q^{95} -2.56491 q^{96} -1.00000 q^{97} -7.00000 q^{98} -7.46480 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9	$ 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9} - 4 q^{10} + 4 q^{11} + 5 q^{16} + 14 q^{17} + q^{18} - 10 q^{19} - 4 q^{20} + 4 q^{22} - q^{25} + 6 q^{27} + 12 q^{29} - 5 q^{31} + 5 q^{32} + 14 q^{34} + q^{36} - 10 q^{38} + 20 q^{39} - 4 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{44} + 2 q^{45} + 40 q^{47} - 35 q^{49} - q^{50} + 6 q^{51} + 30 q^{53} + 6 q^{54} - 12 q^{55} + 4 q^{57} + 12 q^{58} + 22 q^{59} - 16 q^{61} - 5 q^{62} + 5 q^{64} + 12 q^{65} + 4 q^{67} + 14 q^{68} + 34 q^{69} - 40 q^{71} + q^{72} + 18 q^{73} - 6 q^{75} - 10 q^{76} + 20 q^{78} + 20 q^{79} - 4 q^{80} + 9 q^{81} + 2 q^{82} + 38 q^{83} - 38 q^{85} - 4 q^{86} + 20 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} + 40 q^{94} + 8 q^{95} - 5 q^{97} - 35 q^{98} - 34 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9 - 4 * q^10 + 4 * q^11 + 5 * q^16 + 14 * q^17 + q^18 - 10 * q^19 - 4 * q^20 + 4 * q^22 - q^25 + 6 * q^27 + 12 * q^29 - 5 * q^31 + 5 * q^32 + 14 * q^34 + q^36 - 10 * q^38 + 20 * q^39 - 4 * q^40 + 2 * q^41 - 4 * q^43 + 4 * q^44 + 2 * q^45 + 40 * q^47 - 35 * q^49 - q^50 + 6 * q^51 + 30 * q^53 + 6 * q^54 - 12 * q^55 + 4 * q^57 + 12 * q^58 + 22 * q^59 - 16 * q^61 - 5 * q^62 + 5 * q^64 + 12 * q^65 + 4 * q^67 + 14 * q^68 + 34 * q^69 - 40 * q^71 + q^72 + 18 * q^73 - 6 * q^75 - 10 * q^76 + 20 * q^78 + 20 * q^79 - 4 * q^80 + 9 * q^81 + 2 * q^82 + 38 * q^83 - 38 * q^85 - 4 * q^86 + 20 * q^87 + 4 * q^88 + 18 * q^89 + 2 * q^90 + 40 * q^94 + 8 * q^95 - 5 * q^97 - 35 * q^98 - 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.56491	−1.48085	−0.740425	−	0.672139i	$-0.765376\pi$
$3$	−2.56491	−1.48085	−0.740425	+	0.672139i	$0.765376\pi$
$4$	1.00000	0.500000
$5$	−1.19216	−0.533151	−0.266576	−	0.963814i	$-0.585892\pi$
$5$	−1.19216	−0.533151	−0.266576	+	0.963814i	$0.585892\pi$
$6$	−2.56491	−1.04712
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	3.57875	1.19292
$10$	−1.19216	−0.376995
$11$	−2.08587	−0.628913	−0.314457	−	0.949272i	$-0.601822\pi$
$11$	−2.08587	−0.628913	−0.314457	+	0.949272i	$0.601822\pi$
$12$	−2.56491	−0.740425
$13$	−6.11557	−1.69615	−0.848077	−	0.529872i	$-0.822240\pi$
$13$	−6.11557	−1.69615	−0.848077	+	0.529872i	$0.822240\pi$
$14$	0	0
$15$	3.05779	0.789517
$16$	1.00000	0.250000
$17$	3.68504	0.893754	0.446877	−	0.894595i	$-0.352536\pi$
$17$	3.68504	0.893754	0.446877	+	0.894595i	$0.352536\pi$
$18$	3.57875	0.843519
$19$	−2.27263	−0.521378	−0.260689	−	0.965423i	$-0.583950\pi$
$19$	−2.27263	−0.521378	−0.260689	+	0.965423i	$0.583950\pi$
$20$	−1.19216	−0.266576
$21$	0	0
$22$	−2.08587	−0.444709
$23$	−7.64233	−1.59354	−0.796768	−	0.604285i	$-0.793459\pi$
$23$	−7.64233	−1.59354	−0.796768	+	0.604285i	$0.793459\pi$
$24$	−2.56491	−0.523559
$25$	−3.57875	−0.715750
$26$	−6.11557	−1.19936
$27$	−1.48443	−0.285680
$28$	0	0
$29$	2.05616	0.381820	0.190910	−	0.981608i	$-0.438856\pi$
$29$	2.05616	0.381820	0.190910	+	0.981608i	$0.438856\pi$
$30$	3.05779	0.558273
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	5.35006	0.931326
$34$	3.68504	0.631980
$35$	0	0
$36$	3.57875	0.596458
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−2.27263	−0.368670
$39$	15.6859	2.51175
$40$	−1.19216	−0.188497
$41$	−11.0844	−1.73110	−0.865550	−	0.500823i	$-0.833031\pi$
$41$	−11.0844	−1.73110	−0.865550	+	0.500823i	$0.833031\pi$
$42$	0	0
$43$	2.08587	0.318092	0.159046	−	0.987271i	$-0.449158\pi$
$43$	2.08587	0.318092	0.159046	+	0.987271i	$0.449158\pi$
$44$	−2.08587	−0.314457
$45$	−4.26645	−0.636005
$46$	−7.64233	−1.12680
$47$	12.3643	1.80352	0.901759	−	0.432239i	$-0.142276\pi$
$47$	12.3643	1.80352	0.901759	+	0.432239i	$0.142276\pi$
$48$	−2.56491	−0.370212
$49$	−7.00000	−1.00000
$50$	−3.57875	−0.506111
$51$	−9.45179	−1.32352
$52$	−6.11557	−0.848077
$53$	13.6947	1.88111	0.940557	−	0.339635i	$-0.110304\pi$
$53$	13.6947	1.88111	0.940557	+	0.339635i	$0.110304\pi$
$54$	−1.48443	−0.202006
$55$	2.48669	0.335306
$56$	0	0
$57$	5.82910	0.772083
$58$	2.05616	0.269988
$59$	9.01812	1.17406	0.587030	−	0.809565i	$-0.300297\pi$
$59$	9.01812	1.17406	0.587030	+	0.809565i	$0.300297\pi$
$60$	3.05779	0.394759
$61$	3.03510	0.388605	0.194302	−	0.980942i	$-0.437756\pi$
$61$	3.03510	0.388605	0.194302	+	0.980942i	$0.437756\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	7.29076	0.904307
$66$	5.35006	0.658547
$67$	−9.34010	−1.14107	−0.570537	−	0.821272i	$-0.693265\pi$
$67$	−9.34010	−1.14107	−0.570537	+	0.821272i	$0.693265\pi$
$68$	3.68504	0.446877
$69$	19.6019	2.35979
$70$	0	0
$71$	−14.1156	−1.67521	−0.837605	−	0.546276i	$-0.816045\pi$
$71$	−14.1156	−1.67521	−0.837605	+	0.546276i	$0.816045\pi$
$72$	3.57875	0.421759
$73$	−13.4171	−1.57036	−0.785178	−	0.619270i	$-0.787428\pi$
$73$	−13.4171	−1.57036	−0.785178	+	0.619270i	$0.787428\pi$
$74$	0	0
$75$	9.17916	1.05992
$76$	−2.27263	−0.260689
$77$	0	0
$78$	15.6859	1.77608
$79$	2.02848	0.228222	0.114111	−	0.993468i	$-0.463598\pi$
$79$	2.02848	0.228222	0.114111	+	0.993468i	$0.463598\pi$
$80$	−1.19216	−0.133288
$81$	−6.92881	−0.769867
$82$	−11.0844	−1.22407
$83$	6.95808	0.763748	0.381874	−	0.924214i	$-0.375279\pi$
$83$	6.95808	0.763748	0.381874	+	0.924214i	$0.375279\pi$
$84$	0	0
$85$	−4.39317	−0.476506
$86$	2.08587	0.224925
$87$	−5.27387	−0.565418
$88$	−2.08587	−0.222354
$89$	5.81057	0.615920	0.307960	−	0.951399i	$-0.400354\pi$
$89$	5.81057	0.615920	0.307960	+	0.951399i	$0.400354\pi$
$90$	−4.26645	−0.449723
$91$	0	0
$92$	−7.64233	−0.796768
$93$	2.56491	0.265968
$94$	12.3643	1.27528
$95$	2.70935	0.277974
$96$	−2.56491	−0.261780
$97$	−1.00000	−0.101535
$97$	−1.00000	−0.101535
$98$	−7.00000	−0.707107
$99$	−7.46480	−0.750240
$100$	−3.57875	−0.357875
$101$	−7.52493	−0.748759	−0.374379	−	0.927276i	$-0.622144\pi$
$101$	−7.52493	−0.748759	−0.374379	+	0.927276i	$0.622144\pi$
$102$	−9.45179	−0.935867
$103$	7.73438	0.762092	0.381046	−	0.924556i	$-0.375564\pi$
$103$	7.73438	0.762092	0.381046	+	0.924556i	$0.375564\pi$
$104$	−6.11557	−0.599681
$105$	0	0
$106$	13.6947	1.33015
$107$	−9.15067	−0.884629	−0.442315	−	0.896860i	$-0.645843\pi$
$107$	−9.15067	−0.884629	−0.442315	+	0.896860i	$0.645843\pi$
$108$	−1.48443	−0.142840
$109$	−10.3435	−0.990725	−0.495363	−	0.868686i	$-0.664965\pi$
$109$	−10.3435	−0.990725	−0.495363	+	0.868686i	$0.664965\pi$
$110$	2.48669	0.237097
$111$	0	0
$112$	0	0
$113$	1.41859	0.133450	0.0667250	−	0.997771i	$-0.478745\pi$
$113$	1.41859	0.133450	0.0667250	+	0.997771i	$0.478745\pi$
$114$	5.82910	0.545945
$115$	9.11090	0.849596
$116$	2.05616	0.190910
$117$	−21.8861	−2.02337
$118$	9.01812	0.830186
$119$	0	0
$120$	3.05779	0.279136
$121$	−6.64915	−0.604468
$122$	3.03510	0.274785
$123$	28.4306	2.56350
$124$	−1.00000	−0.0898027
$125$	10.2273	0.914754
$126$	0	0
$127$	5.07040	0.449926	0.224963	−	0.974367i	$-0.427774\pi$
$127$	5.07040	0.449926	0.224963	+	0.974367i	$0.427774\pi$
$128$	1.00000	0.0883883
$129$	−5.35006	−0.471046
$130$	7.29076	0.639442
$131$	12.9065	1.12765	0.563824	−	0.825895i	$-0.309330\pi$
$131$	12.9065	1.12765	0.563824	+	0.825895i	$0.309330\pi$
$132$	5.35006	0.465663
$133$	0	0
$134$	−9.34010	−0.806862
$135$	1.76969	0.152310
$136$	3.68504	0.315990
$137$	5.29462	0.452350	0.226175	−	0.974087i	$-0.427378\pi$
$137$	5.29462	0.452350	0.226175	+	0.974087i	$0.427378\pi$
$138$	19.6019	1.66862
$139$	10.2311	0.867794	0.433897	−	0.900962i	$-0.357138\pi$
$139$	10.2311	0.867794	0.433897	+	0.900962i	$0.357138\pi$
$140$	0	0
$141$	−31.7133	−2.67074
$142$	−14.1156	−1.18455
$143$	12.7563	1.06673
$144$	3.57875	0.298229
$145$	−2.45128	−0.203568
$146$	−13.4171	−1.11041
$147$	17.9543	1.48085
$148$	0	0
$149$	1.67111	0.136903	0.0684515	−	0.997654i	$-0.478194\pi$
$149$	1.67111	0.136903	0.0684515	+	0.997654i	$0.478194\pi$
$150$	9.17916	0.749475
$151$	−3.51414	−0.285977	−0.142988	−	0.989724i	$-0.545671\pi$
$151$	−3.51414	−0.285977	−0.142988	+	0.989724i	$0.545671\pi$
$152$	−2.27263	−0.184335
$153$	13.1878	1.06617
$154$	0	0
$155$	1.19216	0.0957568
$156$	15.6859	1.25588
$157$	20.4307	1.63055	0.815275	−	0.579074i	$-0.196586\pi$
$157$	20.4307	1.63055	0.815275	+	0.579074i	$0.196586\pi$
$158$	2.02848	0.161377
$159$	−35.1257	−2.78565
$160$	−1.19216	−0.0942487
$161$	0	0
$162$	−6.92881	−0.544378
$163$	−4.04704	−0.316989	−0.158494	−	0.987360i	$-0.550664\pi$
$163$	−4.04704	−0.316989	−0.158494	+	0.987360i	$0.550664\pi$
$164$	−11.0844	−0.865550
$165$	−6.37814	−0.496538
$166$	6.95808	0.540052
$167$	−21.8407	−1.69009	−0.845043	−	0.534698i	$-0.820425\pi$
$167$	−21.8407	−1.69009	−0.845043	+	0.534698i	$0.820425\pi$
$168$	0	0
$169$	24.4002	1.87694
$170$	−4.39317	−0.336941
$171$	−8.13319	−0.621960
$172$	2.08587	0.159046
$173$	−8.26027	−0.628016	−0.314008	−	0.949420i	$-0.601672\pi$
$173$	−8.26027	−0.628016	−0.314008	+	0.949420i	$0.601672\pi$
$174$	−5.27387	−0.399811
$175$	0	0
$176$	−2.08587	−0.157228
$177$	−23.1306	−1.73861
$178$	5.81057	0.435521
$179$	15.4456	1.15446	0.577229	−	0.816582i	$-0.304134\pi$
$179$	15.4456	1.15446	0.577229	+	0.816582i	$0.304134\pi$
$180$	−4.26645	−0.318002
$181$	−4.38885	−0.326220	−0.163110	−	0.986608i	$-0.552153\pi$
$181$	−4.38885	−0.326220	−0.163110	+	0.986608i	$0.552153\pi$
$182$	0	0
$183$	−7.78475	−0.575465
$184$	−7.64233	−0.563400
$185$	0	0
$186$	2.56491	0.188068
$187$	−7.68651	−0.562094
$188$	12.3643	0.901759
$189$	0	0
$190$	2.70935	0.196557
$191$	3.85595	0.279006	0.139503	−	0.990222i	$-0.455449\pi$
$191$	3.85595	0.279006	0.139503	+	0.990222i	$0.455449\pi$
$192$	−2.56491	−0.185106
$193$	9.77052	0.703298	0.351649	−	0.936132i	$-0.385621\pi$
$193$	9.77052	0.703298	0.351649	+	0.936132i	$0.385621\pi$
$194$	−1.00000	−0.0717958
$195$	−18.7001	−1.33914
$196$	−7.00000	−0.500000
$197$	16.4138	1.16943	0.584716	−	0.811238i	$-0.301206\pi$
$197$	16.4138	1.16943	0.584716	+	0.811238i	$0.301206\pi$
$198$	−7.46480	−0.530500
$199$	27.2570	1.93220	0.966099	−	0.258171i	$-0.0831198\pi$
$199$	27.2570	1.93220	0.966099	+	0.258171i	$0.0831198\pi$
$200$	−3.57875	−0.253056
$201$	23.9565	1.68976
$202$	−7.52493	−0.529452
$203$	0	0
$204$	−9.45179	−0.661758
$205$	13.2145	0.922938
$206$	7.73438	0.538880
$207$	−27.3500	−1.90095
$208$	−6.11557	−0.424039
$209$	4.74042	0.327902
$210$	0	0
$211$	22.5045	1.54927	0.774637	−	0.632406i	$-0.217933\pi$
$211$	22.5045	1.54927	0.774637	+	0.632406i	$0.217933\pi$
$212$	13.6947	0.940557
$213$	36.2051	2.48073
$214$	−9.15067	−0.625527
$215$	−2.48669	−0.169591
$216$	−1.48443	−0.101003
$217$	0	0
$218$	−10.3435	−0.700548
$219$	34.4137	2.32546
$220$	2.48669	0.167653
$221$	−22.5361	−1.51595
$222$	0	0
$223$	12.7329	0.852656	0.426328	−	0.904569i	$-0.359807\pi$
$223$	12.7329	0.852656	0.426328	+	0.904569i	$0.359807\pi$
$224$	0	0
$225$	−12.8074	−0.853829
$226$	1.41859	0.0943634
$227$	11.0167	0.731203	0.365602	−	0.930771i	$-0.380863\pi$
$227$	11.0167	0.731203	0.365602	+	0.930771i	$0.380863\pi$
$228$	5.82910	0.386041
$229$	6.60088	0.436199	0.218099	−	0.975927i	$-0.430014\pi$
$229$	6.60088	0.436199	0.218099	+	0.975927i	$0.430014\pi$
$230$	9.11090	0.600755
$231$	0	0
$232$	2.05616	0.134994
$233$	2.51982	0.165079	0.0825393	−	0.996588i	$-0.473697\pi$
$233$	2.51982	0.165079	0.0825393	+	0.996588i	$0.473697\pi$
$234$	−21.8861	−1.43074
$235$	−14.7403	−0.961548
$236$	9.01812	0.587030
$237$	−5.20287	−0.337963
$238$	0	0
$239$	−7.44291	−0.481442	−0.240721	−	0.970594i	$-0.577384\pi$
$239$	−7.44291	−0.481442	−0.240721	+	0.970594i	$0.577384\pi$
$240$	3.05779	0.197379
$241$	17.4348	1.12308	0.561538	−	0.827451i	$-0.310210\pi$
$241$	17.4348	1.12308	0.561538	+	0.827451i	$0.310210\pi$
$242$	−6.64915	−0.427424
$243$	22.2250	1.42574
$244$	3.03510	0.194302
$245$	8.34514	0.533151
$246$	28.4306	1.81267
$247$	13.8985	0.884338
$248$	−1.00000	−0.0635001
$249$	−17.8468	−1.13100
$250$	10.2273	0.646829
$251$	20.8519	1.31616	0.658081	−	0.752947i	$-0.271368\pi$
$251$	20.8519	1.31616	0.658081	+	0.752947i	$0.271368\pi$
$252$	0	0
$253$	15.9409	1.00220
$254$	5.07040	0.318145
$255$	11.2681	0.705634
$256$	1.00000	0.0625000
$257$	13.4424	0.838511	0.419256	−	0.907868i	$-0.362291\pi$
$257$	13.4424	0.838511	0.419256	+	0.907868i	$0.362291\pi$
$258$	−5.35006	−0.333080
$259$	0	0
$260$	7.29076	0.452154
$261$	7.35849	0.455479
$262$	12.9065	0.797368
$263$	−23.0932	−1.42399	−0.711995	−	0.702185i	$-0.752208\pi$
$263$	−23.0932	−1.42399	−0.711995	+	0.702185i	$0.752208\pi$
$264$	5.35006	0.329273
$265$	−16.3263	−1.00292
$266$	0	0
$267$	−14.9036	−0.912084
$268$	−9.34010	−0.570537
$269$	8.38325	0.511136	0.255568	−	0.966791i	$-0.417738\pi$
$269$	8.38325	0.511136	0.255568	+	0.966791i	$0.417738\pi$
$270$	1.76969	0.107700
$271$	27.1207	1.64746	0.823732	−	0.566979i	$-0.191888\pi$
$271$	27.1207	1.64746	0.823732	+	0.566979i	$0.191888\pi$
$272$	3.68504	0.223439
$273$	0	0
$274$	5.29462	0.319860
$275$	7.46480	0.450144
$276$	19.6019	1.17989
$277$	−16.4306	−0.987217	−0.493608	−	0.869684i	$-0.664322\pi$
$277$	−16.4306	−0.987217	−0.493608	+	0.869684i	$0.664322\pi$
$278$	10.2311	0.613623
$279$	−3.57875	−0.214254
$280$	0	0
$281$	17.0159	1.01508	0.507542	−	0.861627i	$-0.330554\pi$
$281$	17.0159	1.01508	0.507542	+	0.861627i	$0.330554\pi$
$282$	−31.7133	−1.88850
$283$	−6.37196	−0.378773	−0.189387	−	0.981903i	$-0.560650\pi$
$283$	−6.37196	−0.378773	−0.189387	+	0.981903i	$0.560650\pi$
$284$	−14.1156	−0.837605
$285$	−6.94923	−0.411637
$286$	12.7563	0.754295
$287$	0	0
$288$	3.57875	0.210880
$289$	−3.42046	−0.201204
$290$	−2.45128	−0.143944
$291$	2.56491	0.150358
$292$	−13.4171	−0.785178
$293$	−27.9531	−1.63303	−0.816517	−	0.577321i	$-0.804098\pi$
$293$	−27.9531	−1.63303	−0.816517	+	0.577321i	$0.804098\pi$
$294$	17.9543	1.04712
$295$	−10.7511	−0.625952
$296$	0	0
$297$	3.09634	0.179668
$298$	1.67111	0.0968050
$299$	46.7372	2.70288
$300$	9.17916	0.529959
$301$	0	0
$302$	−3.51414	−0.202216
$303$	19.3008	1.10880
$304$	−2.27263	−0.130345
$305$	−3.61833	−0.207185
$306$	13.1878	0.753899
$307$	−0.407488	−0.0232566	−0.0116283	−	0.999932i	$-0.503701\pi$
$307$	−0.407488	−0.0232566	−0.0116283	+	0.999932i	$0.503701\pi$
$308$	0	0
$309$	−19.8380	−1.12854
$310$	1.19216	0.0677103
$311$	−9.69040	−0.549492	−0.274746	−	0.961517i	$-0.588594\pi$
$311$	−9.69040	−0.549492	−0.274746	+	0.961517i	$0.588594\pi$
$312$	15.6859	0.888038
$313$	4.28486	0.242195	0.121097	−	0.992641i	$-0.461359\pi$
$313$	4.28486	0.242195	0.121097	+	0.992641i	$0.461359\pi$
$314$	20.4307	1.15297
$315$	0	0
$316$	2.02848	0.114111
$317$	−11.1685	−0.627283	−0.313641	−	0.949542i	$-0.601549\pi$
$317$	−11.1685	−0.627283	−0.313641	+	0.949542i	$0.601549\pi$
$318$	−35.1257	−1.96975
$319$	−4.28889	−0.240132
$320$	−1.19216	−0.0666439
$321$	23.4706	1.31000
$322$	0	0
$323$	−8.37476	−0.465984
$324$	−6.92881	−0.384934
$325$	21.8861	1.21402
$326$	−4.04704	−0.224145
$327$	26.5300	1.46712
$328$	−11.0844	−0.612036
$329$	0	0
$330$	−6.37814	−0.351105
$331$	26.4422	1.45339	0.726696	−	0.686959i	$-0.241055\pi$
$331$	26.4422	1.45339	0.726696	+	0.686959i	$0.241055\pi$
$332$	6.95808	0.381874
$333$	0	0
$334$	−21.8407	−1.19507
$335$	11.1349	0.608366
$336$	0	0
$337$	−25.3166	−1.37908	−0.689542	−	0.724246i	$-0.742188\pi$
$337$	−25.3166	−1.37908	−0.689542	+	0.724246i	$0.742188\pi$
$338$	24.4002	1.32720
$339$	−3.63856	−0.197619
$340$	−4.39317	−0.238253
$341$	2.08587	0.112956
$342$	−8.13319	−0.439792
$343$	0	0
$344$	2.08587	0.112462
$345$	−23.3686	−1.25812
$346$	−8.26027	−0.444075
$347$	−5.16584	−0.277317	−0.138658	−	0.990340i	$-0.544279\pi$
$347$	−5.16584	−0.277317	−0.138658	+	0.990340i	$0.544279\pi$
$348$	−5.27387	−0.282709
$349$	−27.9387	−1.49552	−0.747762	−	0.663967i	$-0.768871\pi$
$349$	−27.9387	−1.49552	−0.747762	+	0.663967i	$0.768871\pi$
$350$	0	0
$351$	9.07817	0.484557
$352$	−2.08587	−0.111177
$353$	1.59124	0.0846931	0.0423465	−	0.999103i	$-0.486517\pi$
$353$	1.59124	0.0846931	0.0423465	+	0.999103i	$0.486517\pi$
$354$	−23.1306	−1.22938
$355$	16.8281	0.893141
$356$	5.81057	0.307960
$357$	0	0
$358$	15.4456	0.816326
$359$	−19.4572	−1.02691	−0.513455	−	0.858116i	$-0.671635\pi$
$359$	−19.4572	−1.02691	−0.513455	+	0.858116i	$0.671635\pi$
$360$	−4.26645	−0.224862
$361$	−13.8351	−0.728165
$362$	−4.38885	−0.230673
$363$	17.0545	0.895127
$364$	0	0
$365$	15.9954	0.837237
$366$	−7.78475	−0.406915
$367$	0.227067	0.0118528	0.00592639	−	0.999982i	$-0.498114\pi$
$367$	0.227067	0.0118528	0.00592639	+	0.999982i	$0.498114\pi$
$368$	−7.64233	−0.398384
$369$	−39.6684	−2.06506
$370$	0	0
$371$	0	0
$372$	2.56491	0.132984
$373$	−4.04998	−0.209700	−0.104850	−	0.994488i	$-0.533436\pi$
$373$	−4.04998	−0.209700	−0.104850	+	0.994488i	$0.533436\pi$
$374$	−7.68651	−0.397460
$375$	−26.2320	−1.35461
$376$	12.3643	0.637640
$377$	−12.5746	−0.647626
$378$	0	0
$379$	23.2947	1.19657	0.598283	−	0.801285i	$-0.295850\pi$
$379$	23.2947	1.19657	0.598283	+	0.801285i	$0.295850\pi$
$380$	2.70935	0.138987
$381$	−13.0051	−0.666272
$382$	3.85595	0.197287
$383$	−14.5107	−0.741462	−0.370731	−	0.928740i	$-0.620893\pi$
$383$	−14.5107	−0.741462	−0.370731	+	0.928740i	$0.620893\pi$
$384$	−2.56491	−0.130890
$385$	0	0
$386$	9.77052	0.497306
$387$	7.46480	0.379457
$388$	−1.00000	−0.0507673
$389$	25.8846	1.31240	0.656200	−	0.754587i	$-0.272163\pi$
$389$	25.8846	1.31240	0.656200	+	0.754587i	$0.272163\pi$
$390$	−18.7001	−0.946917
$391$	−28.1623	−1.42423
$392$	−7.00000	−0.353553
$393$	−33.1040	−1.66988
$394$	16.4138	0.826913
$395$	−2.41828	−0.121677
$396$	−7.46480	−0.375120
$397$	7.08040	0.355355	0.177677	−	0.984089i	$-0.443142\pi$
$397$	7.08040	0.355355	0.177677	+	0.984089i	$0.443142\pi$
$398$	27.2570	1.36627
$399$	0	0
$400$	−3.57875	−0.178937
$401$	−7.32147	−0.365617	−0.182808	−	0.983149i	$-0.558519\pi$
$401$	−7.32147	−0.365617	−0.182808	+	0.983149i	$0.558519\pi$
$402$	23.9565	1.19484
$403$	6.11557	0.304638
$404$	−7.52493	−0.374379
$405$	8.26027	0.410456
$406$	0	0
$407$	0	0
$408$	−9.45179	−0.467933
$409$	0.184355	0.00911577	0.00455789	−	0.999990i	$-0.498549\pi$
$409$	0.184355	0.00911577	0.00455789	+	0.999990i	$0.498549\pi$
$410$	13.2145	0.652616
$411$	−13.5802	−0.669862
$412$	7.73438	0.381046
$413$	0	0
$414$	−27.3500	−1.34418
$415$	−8.29516	−0.407193
$416$	−6.11557	−0.299841
$417$	−26.2419	−1.28507
$418$	4.74042	0.231861
$419$	4.06696	0.198684	0.0993419	−	0.995053i	$-0.468326\pi$
$419$	4.06696	0.198684	0.0993419	+	0.995053i	$0.468326\pi$
$420$	0	0
$421$	−18.9535	−0.923736	−0.461868	−	0.886949i	$-0.652821\pi$
$421$	−18.9535	−0.923736	−0.461868	+	0.886949i	$0.652821\pi$
$422$	22.5045	1.09550
$423$	44.2487	2.15145
$424$	13.6947	0.665074
$425$	−13.1878	−0.639704
$426$	36.2051	1.75414
$427$	0	0
$428$	−9.15067	−0.442315
$429$	−32.7187	−1.57967
$430$	−2.48669	−0.119919
$431$	18.0337	0.868653	0.434326	−	0.900756i	$-0.356986\pi$
$431$	18.0337	0.868653	0.434326	+	0.900756i	$0.356986\pi$
$432$	−1.48443	−0.0714199
$433$	21.6920	1.04245	0.521226	−	0.853419i	$-0.325475\pi$
$433$	21.6920	1.04245	0.521226	+	0.853419i	$0.325475\pi$
$434$	0	0
$435$	6.28731	0.301453
$436$	−10.3435	−0.495363
$437$	17.3682	0.830835
$438$	34.4137	1.64435
$439$	32.1935	1.53651	0.768257	−	0.640142i	$-0.221124\pi$
$439$	32.1935	1.53651	0.768257	+	0.640142i	$0.221124\pi$
$440$	2.48669	0.118549
$441$	−25.0512	−1.19292
$442$	−22.5361	−1.07194
$443$	3.38872	0.161003	0.0805014	−	0.996754i	$-0.474348\pi$
$443$	3.38872	0.161003	0.0805014	+	0.996754i	$0.474348\pi$
$444$	0	0
$445$	−6.92715	−0.328378
$446$	12.7329	0.602919
$447$	−4.28625	−0.202733
$448$	0	0
$449$	19.0605	0.899520	0.449760	−	0.893150i	$-0.351510\pi$
$449$	19.0605	0.899520	0.449760	+	0.893150i	$0.351510\pi$
$450$	−12.8074	−0.603748
$451$	23.1207	1.08871
$452$	1.41859	0.0667250
$453$	9.01344	0.423488
$454$	11.0167	0.517039
$455$	0	0
$456$	5.82910	0.272972
$457$	−15.2182	−0.711876	−0.355938	−	0.934509i	$-0.615839\pi$
$457$	−15.2182	−0.711876	−0.355938	+	0.934509i	$0.615839\pi$
$458$	6.60088	0.308439
$459$	−5.47021	−0.255327
$460$	9.11090	0.424798
$461$	8.09186	0.376876	0.188438	−	0.982085i	$-0.439658\pi$
$461$	8.09186	0.376876	0.188438	+	0.982085i	$0.439658\pi$
$462$	0	0
$463$	10.0145	0.465414	0.232707	−	0.972547i	$-0.425242\pi$
$463$	10.0145	0.465414	0.232707	+	0.972547i	$0.425242\pi$
$464$	2.05616	0.0954550
$465$	−3.05779	−0.141801
$466$	2.51982	0.116728
$467$	−1.62322	−0.0751137	−0.0375569	−	0.999294i	$-0.511958\pi$
$467$	−1.62322	−0.0751137	−0.0375569	+	0.999294i	$0.511958\pi$
$468$	−21.8861	−1.01169
$469$	0	0
$470$	−14.7403	−0.679917
$471$	−52.4029	−2.41460
$472$	9.01812	0.415093
$473$	−4.35085	−0.200052
$474$	−5.20287	−0.238976
$475$	8.13319	0.373176
$476$	0	0
$477$	49.0100	2.24401
$478$	−7.44291	−0.340431
$479$	3.41250	0.155921	0.0779604	−	0.996956i	$-0.475159\pi$
$479$	3.41250	0.155921	0.0779604	+	0.996956i	$0.475159\pi$
$480$	3.05779	0.139568
$481$	0	0
$482$	17.4348	0.794134
$483$	0	0
$484$	−6.64915	−0.302234
$485$	1.19216	0.0541333
$486$	22.2250	1.00815
$487$	2.35204	0.106581	0.0532905	−	0.998579i	$-0.483029\pi$
$487$	2.35204	0.106581	0.0532905	+	0.998579i	$0.483029\pi$
$488$	3.03510	0.137393
$489$	10.3803	0.469412
$490$	8.34514	0.376995
$491$	28.5822	1.28990	0.644949	−	0.764225i	$-0.276878\pi$
$491$	28.5822	1.28990	0.644949	+	0.764225i	$0.276878\pi$
$492$	28.4306	1.28175
$493$	7.57705	0.341253
$494$	13.8985	0.625322
$495$	8.89925	0.399992
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	−17.8468	−0.799735
$499$	7.26228	0.325104	0.162552	−	0.986700i	$-0.448027\pi$
$499$	7.26228	0.325104	0.162552	+	0.986700i	$0.448027\pi$
$500$	10.2273	0.457377
$501$	56.0194	2.50276
$502$	20.8519	0.930668
$503$	0.490857	0.0218862	0.0109431	−	0.999940i	$-0.496517\pi$
$503$	0.490857	0.0218862	0.0109431	+	0.999940i	$0.496517\pi$
$504$	0	0
$505$	8.97094	0.399202
$506$	15.9409	0.708659
$507$	−62.5843	−2.77947
$508$	5.07040	0.224963
$509$	24.0558	1.06626	0.533128	−	0.846035i	$-0.321017\pi$
$509$	24.0558	1.06626	0.533128	+	0.846035i	$0.321017\pi$
$510$	11.2681	0.498959
$511$	0	0
$512$	1.00000	0.0441942
$513$	3.37358	0.148947
$514$	13.4424	0.592917
$515$	−9.22064	−0.406310
$516$	−5.35006	−0.235523
$517$	−25.7903	−1.13426
$518$	0	0
$519$	21.1868	0.929998
$520$	7.29076	0.319721
$521$	13.6556	0.598265	0.299132	−	0.954212i	$-0.403303\pi$
$521$	13.6556	0.598265	0.299132	+	0.954212i	$0.403303\pi$
$522$	7.35849	0.322072
$523$	−2.49341	−0.109029	−0.0545146	−	0.998513i	$-0.517361\pi$
$523$	−2.49341	−0.109029	−0.0545146	+	0.998513i	$0.517361\pi$
$524$	12.9065	0.563824
$525$	0	0
$526$	−23.0932	−1.00691
$527$	−3.68504	−0.160523
$528$	5.35006	0.232831
$529$	35.4052	1.53936
$530$	−16.3263	−0.709171
$531$	32.2736	1.40055
$532$	0	0
$533$	67.7877	2.93621
$534$	−14.9036	−0.644941
$535$	10.9091	0.471641
$536$	−9.34010	−0.403431
$537$	−39.6165	−1.70958
$538$	8.38325	0.361427
$539$	14.6011	0.628913
$540$	1.76969	0.0761552
$541$	−11.9200	−0.512479	−0.256240	−	0.966613i	$-0.582484\pi$
$541$	−11.9200	−0.512479	−0.256240	+	0.966613i	$0.582484\pi$
$542$	27.1207	1.16493
$543$	11.2570	0.483083
$544$	3.68504	0.157995
$545$	12.3311	0.528206
$546$	0	0
$547$	3.71721	0.158936	0.0794682	−	0.996837i	$-0.474678\pi$
$547$	3.71721	0.158936	0.0794682	+	0.996837i	$0.474678\pi$
$548$	5.29462	0.226175
$549$	10.8619	0.463573
$550$	7.46480	0.318300
$551$	−4.67291	−0.199073
$552$	19.6019	0.834311
$553$	0	0
$554$	−16.4306	−0.698068
$555$	0	0
$556$	10.2311	0.433897
$557$	9.54412	0.404398	0.202199	−	0.979345i	$-0.435191\pi$
$557$	9.54412	0.404398	0.202199	+	0.979345i	$0.435191\pi$
$558$	−3.57875	−0.151500
$559$	−12.7563	−0.539533
$560$	0	0
$561$	19.7152	0.832376
$562$	17.0159	0.717773
$563$	−43.5226	−1.83426	−0.917130	−	0.398588i	$-0.869500\pi$
$563$	−43.5226	−1.83426	−0.917130	+	0.398588i	$0.869500\pi$
$564$	−31.7133	−1.33537
$565$	−1.69119	−0.0711490
$566$	−6.37196	−0.267833
$567$	0	0
$568$	−14.1156	−0.592276
$569$	−20.2543	−0.849105	−0.424552	−	0.905403i	$-0.639569\pi$
$569$	−20.2543	−0.849105	−0.424552	+	0.905403i	$0.639569\pi$
$570$	−6.94923	−0.291071
$571$	−5.32277	−0.222751	−0.111375	−	0.993778i	$-0.535526\pi$
$571$	−5.32277	−0.222751	−0.111375	+	0.993778i	$0.535526\pi$
$572$	12.7563	0.533367
$573$	−9.89014	−0.413167
$574$	0	0
$575$	27.3500	1.14057
$576$	3.57875	0.149115
$577$	−27.5604	−1.14735	−0.573677	−	0.819082i	$-0.694483\pi$
$577$	−27.5604	−1.14735	−0.573677	+	0.819082i	$0.694483\pi$
$578$	−3.42046	−0.142273
$579$	−25.0605	−1.04148
$580$	−2.45128	−0.101784
$581$	0	0
$582$	2.56491	0.106319
$583$	−28.5654	−1.18306
$584$	−13.4171	−0.555205
$585$	26.0918	1.07876
$586$	−27.9531	−1.15473
$587$	26.9219	1.11119	0.555593	−	0.831454i	$-0.312491\pi$
$587$	26.9219	1.11119	0.555593	+	0.831454i	$0.312491\pi$
$588$	17.9543	0.740425
$589$	2.27263	0.0936423
$590$	−10.7511	−0.442615
$591$	−42.0997	−1.73175
$592$	0	0
$593$	−6.80410	−0.279411	−0.139705	−	0.990193i	$-0.544616\pi$
$593$	−6.80410	−0.279411	−0.139705	+	0.990193i	$0.544616\pi$
$594$	3.09634	0.127044
$595$	0	0
$596$	1.67111	0.0684515
$597$	−69.9117	−2.86130
$598$	46.7372	1.91123
$599$	−43.0298	−1.75815	−0.879074	−	0.476685i	$-0.841838\pi$
$599$	−43.0298	−1.75815	−0.879074	+	0.476685i	$0.841838\pi$
$600$	9.17916	0.374737
$601$	−6.77237	−0.276251	−0.138125	−	0.990415i	$-0.544108\pi$
$601$	−6.77237	−0.276251	−0.138125	+	0.990415i	$0.544108\pi$
$602$	0	0
$603$	−33.4259	−1.36121
$604$	−3.51414	−0.142988
$605$	7.92687	0.322273
$606$	19.3008	0.784039
$607$	−20.5497	−0.834087	−0.417043	−	0.908887i	$-0.636934\pi$
$607$	−20.5497	−0.834087	−0.417043	+	0.908887i	$0.636934\pi$
$608$	−2.27263	−0.0921675
$609$	0	0
$610$	−3.61833	−0.146502
$611$	−75.6148	−3.05905
$612$	13.1878	0.533087
$613$	23.1101	0.933408	0.466704	−	0.884414i	$-0.345441\pi$
$613$	23.1101	0.933408	0.466704	+	0.884414i	$0.345441\pi$
$614$	−0.407488	−0.0164449
$615$	−33.8939	−1.36673
$616$	0	0
$617$	8.84104	0.355927	0.177963	−	0.984037i	$-0.443049\pi$
$617$	8.84104	0.355927	0.177963	+	0.984037i	$0.443049\pi$
$618$	−19.8380	−0.798000
$619$	41.3065	1.66025	0.830123	−	0.557580i	$-0.188270\pi$
$619$	41.3065	1.66025	0.830123	+	0.557580i	$0.188270\pi$
$620$	1.19216	0.0478784
$621$	11.3445	0.455241
$622$	−9.69040	−0.388549
$623$	0	0
$624$	15.6859	0.627938
$625$	5.70118	0.228047
$626$	4.28486	0.171258
$627$	−12.1587	−0.485573
$628$	20.4307	0.815275
$629$	0	0
$630$	0	0
$631$	45.4336	1.80868	0.904342	−	0.426808i	$-0.140362\pi$
$631$	45.4336	1.80868	0.904342	+	0.426808i	$0.140362\pi$
$632$	2.02848	0.0806887
$633$	−57.7220	−2.29424
$634$	−11.1685	−0.443556
$635$	−6.04475	−0.239878
$636$	−35.1257	−1.39282
$637$	42.8090	1.69615
$638$	−4.28889	−0.169799
$639$	−50.5161	−1.99839
$640$	−1.19216	−0.0471244
$641$	−6.89279	−0.272249	−0.136124	−	0.990692i	$-0.543465\pi$
$641$	−6.89279	−0.272249	−0.136124	+	0.990692i	$0.543465\pi$
$642$	23.4706	0.926312
$643$	−41.4292	−1.63381	−0.816903	−	0.576775i	$-0.804311\pi$
$643$	−41.4292	−1.63381	−0.816903	+	0.576775i	$0.804311\pi$
$644$	0	0
$645$	6.37814	0.251139
$646$	−8.37476	−0.329500
$647$	−20.6340	−0.811207	−0.405604	−	0.914049i	$-0.632939\pi$
$647$	−20.6340	−0.811207	−0.405604	+	0.914049i	$0.632939\pi$
$648$	−6.92881	−0.272189
$649$	−18.8106	−0.738381
$650$	21.8861	0.858443
$651$	0	0
$652$	−4.04704	−0.158494
$653$	−18.4520	−0.722082	−0.361041	−	0.932550i	$-0.617579\pi$
$653$	−18.4520	−0.722082	−0.361041	+	0.932550i	$0.617579\pi$
$654$	26.5300	1.03741
$655$	−15.3867	−0.601207
$656$	−11.0844	−0.432775
$657$	−48.0165	−1.87330
$658$	0	0
$659$	−17.7305	−0.690684	−0.345342	−	0.938477i	$-0.612237\pi$
$659$	−17.7305	−0.690684	−0.345342	+	0.938477i	$0.612237\pi$
$660$	−6.37814	−0.248269
$661$	48.0942	1.87065	0.935324	−	0.353791i	$-0.115108\pi$
$661$	48.0942	1.87065	0.935324	+	0.353791i	$0.115108\pi$
$662$	26.4422	1.02770
$663$	57.8031	2.24489
$664$	6.95808	0.270026
$665$	0	0
$666$	0	0
$667$	−15.7139	−0.608444
$668$	−21.8407	−0.845043
$669$	−32.6586	−1.26266
$670$	11.1349	0.430179
$671$	−6.33082	−0.244399
$672$	0	0
$673$	−26.8810	−1.03619	−0.518094	−	0.855324i	$-0.673358\pi$
$673$	−26.8810	−1.03619	−0.518094	+	0.855324i	$0.673358\pi$
$674$	−25.3166	−0.975160
$675$	5.31242	0.204475
$676$	24.4002	0.938471
$677$	37.8051	1.45297	0.726484	−	0.687183i	$-0.241153\pi$
$677$	37.8051	1.45297	0.726484	+	0.687183i	$0.241153\pi$
$678$	−3.63856	−0.139738
$679$	0	0
$680$	−4.39317	−0.168470
$681$	−28.2568	−1.08280
$682$	2.08587	0.0798720
$683$	26.3991	1.01013	0.505066	−	0.863081i	$-0.331468\pi$
$683$	26.3991	1.01013	0.505066	+	0.863081i	$0.331468\pi$
$684$	−8.13319	−0.310980
$685$	−6.31205	−0.241171
$686$	0	0
$687$	−16.9307	−0.645945
$688$	2.08587	0.0795230
$689$	−83.7511	−3.19066
$690$	−23.3686	−0.889628
$691$	−5.50175	−0.209296	−0.104648	−	0.994509i	$-0.533372\pi$
$691$	−5.50175	−0.209296	−0.104648	+	0.994509i	$0.533372\pi$
$692$	−8.26027	−0.314008
$693$	0	0
$694$	−5.16584	−0.196093
$695$	−12.1972	−0.462666
$696$	−5.27387	−0.199905
$697$	−40.8466	−1.54718
$698$	−27.9387	−1.05749
$699$	−6.46309	−0.244457
$700$	0	0
$701$	−36.3343	−1.37233	−0.686165	−	0.727446i	$-0.740707\pi$
$701$	−36.3343	−1.37233	−0.686165	+	0.727446i	$0.740707\pi$
$702$	9.07817	0.342633
$703$	0	0
$704$	−2.08587	−0.0786141
$705$	37.8074	1.42391
$706$	1.59124	0.0598870
$707$	0	0
$708$	−23.1306	−0.869303
$709$	22.8444	0.857939	0.428970	−	0.903319i	$-0.358877\pi$
$709$	22.8444	0.857939	0.428970	+	0.903319i	$0.358877\pi$
$710$	16.8281	0.631546
$711$	7.25943	0.272250
$712$	5.81057	0.217760
$713$	7.64233	0.286208
$714$	0	0
$715$	−15.2076	−0.568731
$716$	15.4456	0.577229
$717$	19.0904	0.712943
$718$	−19.4572	−0.726135
$719$	−7.68219	−0.286497	−0.143249	−	0.989687i	$-0.545755\pi$
$719$	−7.68219	−0.286497	−0.143249	+	0.989687i	$0.545755\pi$
$720$	−4.26645	−0.159001
$721$	0	0
$722$	−13.8351	−0.514890
$723$	−44.7187	−1.66311
$724$	−4.38885	−0.163110
$725$	−7.35849	−0.273288
$726$	17.0545	0.632950
$727$	28.5024	1.05710	0.528549	−	0.848903i	$-0.322736\pi$
$727$	28.5024	1.05710	0.528549	+	0.848903i	$0.322736\pi$
$728$	0	0
$729$	−36.2188	−1.34144
$730$	15.9954	0.592016
$731$	7.68651	0.284296
$732$	−7.78475	−0.287733
$733$	22.6338	0.835999	0.417999	−	0.908447i	$-0.362731\pi$
$733$	22.6338	0.835999	0.417999	+	0.908447i	$0.362731\pi$
$734$	0.227067	0.00838118
$735$	−21.4045	−0.789517
$736$	−7.64233	−0.281700
$737$	19.4822	0.717637
$738$	−39.6684	−1.46022
$739$	12.0210	0.442200	0.221100	−	0.975251i	$-0.429035\pi$
$739$	12.0210	0.442200	0.221100	+	0.975251i	$0.429035\pi$
$740$	0	0
$741$	−35.6483	−1.30957
$742$	0	0
$743$	24.6006	0.902510	0.451255	−	0.892395i	$-0.350977\pi$
$743$	24.6006	0.902510	0.451255	+	0.892395i	$0.350977\pi$
$744$	2.56491	0.0940341
$745$	−1.99224	−0.0729900
$746$	−4.04998	−0.148280
$747$	24.9012	0.911087
$748$	−7.68651	−0.281047
$749$	0	0
$750$	−26.2320	−0.957856
$751$	23.3525	0.852146	0.426073	−	0.904689i	$-0.359897\pi$
$751$	23.3525	0.852146	0.426073	+	0.904689i	$0.359897\pi$
$752$	12.3643	0.450880
$753$	−53.4833	−1.94904
$754$	−12.5746	−0.457941
$755$	4.18943	0.152469
$756$	0	0
$757$	−18.6625	−0.678301	−0.339150	−	0.940732i	$-0.610140\pi$
$757$	−18.6625	−0.678301	−0.339150	+	0.940732i	$0.610140\pi$
$758$	23.2947	0.846100
$759$	−40.8869	−1.48410
$760$	2.70935	0.0982785
$761$	−6.96272	−0.252398	−0.126199	−	0.992005i	$-0.540278\pi$
$761$	−6.96272	−0.252398	−0.126199	+	0.992005i	$0.540278\pi$
$762$	−13.0051	−0.471126
$763$	0	0
$764$	3.85595	0.139503
$765$	−15.7220	−0.568432
$766$	−14.5107	−0.524293
$767$	−55.1510	−1.99139
$768$	−2.56491	−0.0925531
$769$	−4.55829	−0.164376	−0.0821881	−	0.996617i	$-0.526191\pi$
$769$	−4.55829	−0.164376	−0.0821881	+	0.996617i	$0.526191\pi$
$770$	0	0
$771$	−34.4784	−1.24171
$772$	9.77052	0.351649
$773$	−28.7553	−1.03426	−0.517128	−	0.855908i	$-0.672999\pi$
$773$	−28.7553	−1.03426	−0.517128	+	0.855908i	$0.672999\pi$
$774$	7.46480	0.268317
$775$	3.57875	0.128552
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	25.8846	0.928008
$779$	25.1909	0.902557
$780$	−18.7001	−0.669572
$781$	29.4432	1.05356
$782$	−28.1623	−1.00708
$783$	−3.05224	−0.109078
$784$	−7.00000	−0.250000
$785$	−24.3568	−0.869330
$786$	−33.1040	−1.18078
$787$	−38.4061	−1.36903	−0.684514	−	0.729000i	$-0.739986\pi$
$787$	−38.4061	−1.36903	−0.684514	+	0.729000i	$0.739986\pi$
$788$	16.4138	0.584716
$789$	59.2320	2.10871
$790$	−2.41828	−0.0860386
$791$	0	0
$792$	−7.46480	−0.265250
$793$	−18.5614	−0.659134
$794$	7.08040	0.251274
$795$	41.8755	1.48517
$796$	27.2570	0.966099
$797$	−20.7502	−0.735009	−0.367505	−	0.930022i	$-0.619788\pi$
$797$	−20.7502	−0.735009	−0.367505	+	0.930022i	$0.619788\pi$
$798$	0	0
$799$	45.5630	1.61190
$800$	−3.57875	−0.126528
$801$	20.7946	0.734740
$802$	−7.32147	−0.258530
$803$	27.9864	0.987617
$804$	23.9565	0.844880
$805$	0	0
$806$	6.11557	0.215412
$807$	−21.5022	−0.756915
$808$	−7.52493	−0.264726
$809$	−47.7120	−1.67746	−0.838732	−	0.544544i	$-0.816703\pi$
$809$	−47.7120	−1.67746	−0.838732	+	0.544544i	$0.816703\pi$
$810$	8.26027	0.290236
$811$	15.7718	0.553823	0.276911	−	0.960895i	$-0.410689\pi$
$811$	15.7718	0.553823	0.276911	+	0.960895i	$0.410689\pi$
$812$	0	0
$813$	−69.5620	−2.43965
$814$	0	0
$815$	4.82473	0.169003
$816$	−9.45179	−0.330879
$817$	−4.74042	−0.165846
$818$	0.184355	0.00644582
$819$	0	0
$820$	13.2145	0.461469
$821$	1.93852	0.0676547	0.0338274	−	0.999428i	$-0.489230\pi$
$821$	1.93852	0.0676547	0.0338274	+	0.999428i	$0.489230\pi$
$822$	−13.5802	−0.473664
$823$	−2.47025	−0.0861073	−0.0430537	−	0.999073i	$-0.513709\pi$
$823$	−2.47025	−0.0861073	−0.0430537	+	0.999073i	$0.513709\pi$
$824$	7.73438	0.269440
$825$	−19.1465	−0.666596
$826$	0	0
$827$	−39.8386	−1.38532	−0.692661	−	0.721263i	$-0.743562\pi$
$827$	−39.8386	−1.38532	−0.692661	+	0.721263i	$0.743562\pi$
$828$	−27.3500	−0.950477
$829$	37.3375	1.29678	0.648392	−	0.761307i	$-0.275442\pi$
$829$	37.3375	1.29678	0.648392	+	0.761307i	$0.275442\pi$
$830$	−8.29516	−0.287929
$831$	42.1429	1.46192
$832$	−6.11557	−0.212019
$833$	−25.7953	−0.893754
$834$	−26.2419	−0.908684
$835$	26.0377	0.901072
$836$	4.74042	0.163951
$837$	1.48443	0.0513096
$838$	4.06696	0.140491
$839$	35.0853	1.21128	0.605640	−	0.795739i	$-0.292917\pi$
$839$	35.0853	1.21128	0.605640	+	0.795739i	$0.292917\pi$
$840$	0	0
$841$	−24.7722	−0.854213
$842$	−18.9535	−0.653180
$843$	−43.6442	−1.50319
$844$	22.5045	0.774637
$845$	−29.0891	−1.00069
$846$	44.2487	1.52130
$847$	0	0
$848$	13.6947	0.470279
$849$	16.3435	0.560907
$850$	−13.1878	−0.452339
$851$	0	0
$852$	36.2051	1.24037
$853$	−44.4943	−1.52345	−0.761727	−	0.647898i	$-0.775648\pi$
$853$	−44.4943	−1.52345	−0.761727	+	0.647898i	$0.775648\pi$
$854$	0	0
$855$	9.69608	0.331599
$856$	−9.15067	−0.312764
$857$	2.91643	0.0996232	0.0498116	−	0.998759i	$-0.484138\pi$
$857$	2.91643	0.0996232	0.0498116	+	0.998759i	$0.484138\pi$
$858$	−32.7187	−1.11700
$859$	−4.23028	−0.144335	−0.0721676	−	0.997393i	$-0.522992\pi$
$859$	−4.23028	−0.144335	−0.0721676	+	0.997393i	$0.522992\pi$
$860$	−2.48669	−0.0847956
$861$	0	0
$862$	18.0337	0.614230
$863$	−55.8204	−1.90015	−0.950074	−	0.312024i	$-0.898993\pi$
$863$	−55.8204	−1.90015	−0.950074	+	0.312024i	$0.898993\pi$
$864$	−1.48443	−0.0505015
$865$	9.84758	0.334828
$866$	21.6920	0.737124
$867$	8.77317	0.297952
$868$	0	0
$869$	−4.23115	−0.143532
$870$	6.28731	0.213160
$871$	57.1201	1.93544
$872$	−10.3435	−0.350274
$873$	−3.57875	−0.121122
$874$	17.3682	0.587489
$875$	0	0
$876$	34.4137	1.16273
$877$	−10.8092	−0.365001	−0.182500	−	0.983206i	$-0.558419\pi$
$877$	−10.8092	−0.365001	−0.182500	+	0.983206i	$0.558419\pi$
$878$	32.1935	1.08648
$879$	71.6970	2.41828
$880$	2.48669	0.0838265
$881$	−12.6077	−0.424764	−0.212382	−	0.977187i	$-0.568122\pi$
$881$	−12.6077	−0.424764	−0.212382	+	0.977187i	$0.568122\pi$
$882$	−25.0512	−0.843519
$883$	−30.7096	−1.03346	−0.516731	−	0.856148i	$-0.672851\pi$
$883$	−30.7096	−1.03346	−0.516731	+	0.856148i	$0.672851\pi$
$884$	−22.5361	−0.757973
$885$	27.5755	0.926940
$886$	3.38872	0.113846
$887$	−9.30765	−0.312520	−0.156260	−	0.987716i	$-0.549944\pi$
$887$	−9.30765	−0.312520	−0.156260	+	0.987716i	$0.549944\pi$
$888$	0	0
$889$	0	0
$890$	−6.92715	−0.232199
$891$	14.4526	0.484180
$892$	12.7329	0.426328
$893$	−28.0995	−0.940315
$894$	−4.28625	−0.143354
$895$	−18.4137	−0.615501
$896$	0	0
$897$	−119.877	−4.00257
$898$	19.0605	0.636056
$899$	−2.05616	−0.0685769
$900$	−12.8074	−0.426915
$901$	50.4656	1.68125
$902$	23.1207	0.769835
$903$	0	0
$904$	1.41859	0.0471817
$905$	5.23222	0.173925
$906$	9.01344	0.299452
$907$	−7.88530	−0.261827	−0.130914	−	0.991394i	$-0.541791\pi$
$907$	−7.88530	−0.261827	−0.130914	+	0.991394i	$0.541791\pi$
$908$	11.0167	0.365602
$909$	−26.9298	−0.893206
$910$	0	0
$911$	−39.5162	−1.30923	−0.654615	−	0.755962i	$-0.727169\pi$
$911$	−39.5162	−1.30923	−0.654615	+	0.755962i	$0.727169\pi$
$912$	5.82910	0.193021
$913$	−14.5136	−0.480331
$914$	−15.2182	−0.503373
$915$	9.28069	0.306810
$916$	6.60088	0.218099
$917$	0	0
$918$	−5.47021	−0.180544
$919$	30.1318	0.993958	0.496979	−	0.867763i	$-0.334443\pi$
$919$	30.1318	0.993958	0.496979	+	0.867763i	$0.334443\pi$
$920$	9.11090	0.300378
$921$	1.04517	0.0344395
$922$	8.09186	0.266491
$923$	86.3248	2.84142
$924$	0	0
$925$	0	0
$926$	10.0145	0.329097
$927$	27.6794	0.909111
$928$	2.05616	0.0674969
$929$	−10.4067	−0.341433	−0.170717	−	0.985320i	$-0.554608\pi$
$929$	−10.4067	−0.341433	−0.170717	+	0.985320i	$0.554608\pi$
$930$	−3.05779	−0.100269
$931$	15.9084	0.521378
$932$	2.51982	0.0825393
$933$	24.8550	0.813715
$934$	−1.62322	−0.0531134
$935$	9.16357	0.299681
$936$	−21.8861	−0.715369
$937$	−25.5898	−0.835981	−0.417991	−	0.908451i	$-0.637266\pi$
$937$	−25.5898	−0.835981	−0.417991	+	0.908451i	$0.637266\pi$
$938$	0	0
$939$	−10.9903	−0.358654
$940$	−14.7403	−0.480774
$941$	−21.6260	−0.704987	−0.352494	−	0.935814i	$-0.614666\pi$
$941$	−21.6260	−0.704987	−0.352494	+	0.935814i	$0.614666\pi$
$942$	−52.4029	−1.70738
$943$	84.7110	2.75857
$944$	9.01812	0.293515
$945$	0	0
$946$	−4.35085	−0.141458
$947$	20.3742	0.662072	0.331036	−	0.943618i	$-0.392602\pi$
$947$	20.3742	0.662072	0.331036	+	0.943618i	$0.392602\pi$
$948$	−5.20287	−0.168981
$949$	82.0534	2.66357
$950$	8.13319	0.263875
$951$	28.6460	0.928911
$952$	0	0
$953$	−0.356643	−0.0115528	−0.00577641	−	0.999983i	$-0.501839\pi$
$953$	−0.356643	−0.0115528	−0.00577641	+	0.999983i	$0.501839\pi$
$954$	49.0100	1.58676
$955$	−4.59691	−0.148753
$956$	−7.44291	−0.240721
$957$	11.0006	0.355599
$958$	3.41250	0.110253
$959$	0	0
$960$	3.05779	0.0986896
$961$	1.00000	0.0322581
$962$	0	0
$963$	−32.7480	−1.05529
$964$	17.4348	0.561538
$965$	−11.6481	−0.374964
$966$	0	0
$967$	−38.3797	−1.23421	−0.617104	−	0.786881i	$-0.711694\pi$
$967$	−38.3797	−1.23421	−0.617104	+	0.786881i	$0.711694\pi$
$968$	−6.64915	−0.213712
$969$	21.4805	0.690052
$970$	1.19216	0.0382780
$971$	−34.2357	−1.09868	−0.549339	−	0.835600i	$-0.685120\pi$
$971$	−34.2357	−1.09868	−0.549339	+	0.835600i	$0.685120\pi$
$972$	22.2250	0.712869
$973$	0	0
$974$	2.35204	0.0753642
$975$	−56.1358	−1.79778
$976$	3.03510	0.0971512
$977$	−38.7541	−1.23985	−0.619926	−	0.784660i	$-0.712837\pi$
$977$	−38.7541	−1.23985	−0.619926	+	0.784660i	$0.712837\pi$
$978$	10.3803	0.331925
$979$	−12.1201	−0.387360
$980$	8.34514	0.266576
$981$	−37.0167	−1.18185
$982$	28.5822	0.912096
$983$	−5.95988	−0.190091	−0.0950453	−	0.995473i	$-0.530300\pi$
$983$	−5.95988	−0.190091	−0.0950453	+	0.995473i	$0.530300\pi$
$984$	28.4306	0.906333
$985$	−19.5679	−0.623484
$986$	7.57705	0.241302
$987$	0	0
$988$	13.8985	0.442169
$989$	−15.9409	−0.506891
$990$	8.89925	0.282837
$991$	−52.1100	−1.65533	−0.827665	−	0.561223i	$-0.810331\pi$
$991$	−52.1100	−1.65533	−0.827665	+	0.561223i	$0.810331\pi$
$992$	−1.00000	−0.0317500
$993$	−67.8217	−2.15226
$994$	0	0
$995$	−32.4948	−1.03015
$996$	−17.8468	−0.565498
$997$	−25.7028	−0.814015	−0.407008	−	0.913425i	$-0.633428\pi$
$997$	−25.7028	−0.814015	−0.407008	+	0.913425i	$0.633428\pi$
$998$	7.26228	0.229883
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6014.2.a.d.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6014.2.a.d.1.1	✓	5	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:	\( N \)	\(=\)	\( 6014 = 2 \cdot 31 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.56491 q^{3} +1.00000 q^{4} -1.19216 q^{5} -2.56491 q^{6} +1.00000 q^{8} +3.57875 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.56491 q^{3} +1.00000 q^{4} -1.19216 q^{5} -2.56491 q^{6} +1.00000 q^{8} +3.57875 q^{9} -1.19216 q^{10} -2.08587 q^{11} -2.56491 q^{12} -6.11557 q^{13} +3.05779 q^{15} +1.00000 q^{16} +3.68504 q^{17} +3.57875 q^{18} -2.27263 q^{19} -1.19216 q^{20} -2.08587 q^{22} -7.64233 q^{23} -2.56491 q^{24} -3.57875 q^{25} -6.11557 q^{26} -1.48443 q^{27} +2.05616 q^{29} +3.05779 q^{30} -1.00000 q^{31} +1.00000 q^{32} +5.35006 q^{33} +3.68504 q^{34} +3.57875 q^{36} -2.27263 q^{38} +15.6859 q^{39} -1.19216 q^{40} -11.0844 q^{41} +2.08587 q^{43} -2.08587 q^{44} -4.26645 q^{45} -7.64233 q^{46} +12.3643 q^{47} -2.56491 q^{48} -7.00000 q^{49} -3.57875 q^{50} -9.45179 q^{51} -6.11557 q^{52} +13.6947 q^{53} -1.48443 q^{54} +2.48669 q^{55} +5.82910 q^{57} +2.05616 q^{58} +9.01812 q^{59} +3.05779 q^{60} +3.03510 q^{61} -1.00000 q^{62} +1.00000 q^{64} +7.29076 q^{65} +5.35006 q^{66} -9.34010 q^{67} +3.68504 q^{68} +19.6019 q^{69} -14.1156 q^{71} +3.57875 q^{72} -13.4171 q^{73} +9.17916 q^{75} -2.27263 q^{76} +15.6859 q^{78} +2.02848 q^{79} -1.19216 q^{80} -6.92881 q^{81} -11.0844 q^{82} +6.95808 q^{83} -4.39317 q^{85} +2.08587 q^{86} -5.27387 q^{87} -2.08587 q^{88} +5.81057 q^{89} -4.26645 q^{90} -7.64233 q^{92} +2.56491 q^{93} +12.3643 q^{94} +2.70935 q^{95} -2.56491 q^{96} -1.00000 q^{97} -7.00000 q^{98} -7.46480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9	\( 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9} - 4 q^{10} + 4 q^{11} + 5 q^{16} + 14 q^{17} + q^{18} - 10 q^{19} - 4 q^{20} + 4 q^{22} - q^{25} + 6 q^{27} + 12 q^{29} - 5 q^{31} + 5 q^{32} + 14 q^{34} + q^{36} - 10 q^{38} + 20 q^{39} - 4 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{44} + 2 q^{45} + 40 q^{47} - 35 q^{49} - q^{50} + 6 q^{51} + 30 q^{53} + 6 q^{54} - 12 q^{55} + 4 q^{57} + 12 q^{58} + 22 q^{59} - 16 q^{61} - 5 q^{62} + 5 q^{64} + 12 q^{65} + 4 q^{67} + 14 q^{68} + 34 q^{69} - 40 q^{71} + q^{72} + 18 q^{73} - 6 q^{75} - 10 q^{76} + 20 q^{78} + 20 q^{79} - 4 q^{80} + 9 q^{81} + 2 q^{82} + 38 q^{83} - 38 q^{85} - 4 q^{86} + 20 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} + 40 q^{94} + 8 q^{95} - 5 q^{97} - 35 q^{98} - 34 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9 - 4 * q^10 + 4 * q^11 + 5 * q^16 + 14 * q^17 + q^18 - 10 * q^19 - 4 * q^20 + 4 * q^22 - q^25 + 6 * q^27 + 12 * q^29 - 5 * q^31 + 5 * q^32 + 14 * q^34 + q^36 - 10 * q^38 + 20 * q^39 - 4 * q^40 + 2 * q^41 - 4 * q^43 + 4 * q^44 + 2 * q^45 + 40 * q^47 - 35 * q^49 - q^50 + 6 * q^51 + 30 * q^53 + 6 * q^54 - 12 * q^55 + 4 * q^57 + 12 * q^58 + 22 * q^59 - 16 * q^61 - 5 * q^62 + 5 * q^64 + 12 * q^65 + 4 * q^67 + 14 * q^68 + 34 * q^69 - 40 * q^71 + q^72 + 18 * q^73 - 6 * q^75 - 10 * q^76 + 20 * q^78 + 20 * q^79 - 4 * q^80 + 9 * q^81 + 2 * q^82 + 38 * q^83 - 38 * q^85 - 4 * q^86 + 20 * q^87 + 4 * q^88 + 18 * q^89 + 2 * q^90 + 40 * q^94 + 8 * q^95 - 5 * q^97 - 35 * q^98 - 34 * q^99

Introduction

L-functions

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Database

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