LMFDB - Embedded newform 6039.2.a.k.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$-2.08441$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08441 q^{2} +2.34475 q^{4} -2.98183 q^{5} +3.78550 q^{7} +0.718591 q^{8} +O(q^{10})$	\(q+2.08441 q^{2} +2.34475 q^{4} -2.98183 q^{5} +3.78550 q^{7} +0.718591 q^{8} -6.21535 q^{10} -1.00000 q^{11} +4.41489 q^{13} +7.89053 q^{14} -3.19166 q^{16} -5.30382 q^{17} +0.311984 q^{19} -6.99164 q^{20} -2.08441 q^{22} +5.62868 q^{23} +3.89132 q^{25} +9.20243 q^{26} +8.87605 q^{28} +8.30650 q^{29} -6.85645 q^{31} -8.08989 q^{32} -11.0553 q^{34} -11.2877 q^{35} +4.98139 q^{37} +0.650300 q^{38} -2.14272 q^{40} +11.9877 q^{41} -0.135602 q^{43} -2.34475 q^{44} +11.7324 q^{46} +10.6430 q^{47} +7.33004 q^{49} +8.11109 q^{50} +10.3518 q^{52} -9.36007 q^{53} +2.98183 q^{55} +2.72023 q^{56} +17.3141 q^{58} +7.08123 q^{59} -1.00000 q^{61} -14.2916 q^{62} -10.4793 q^{64} -13.1645 q^{65} +3.08414 q^{67} -12.4361 q^{68} -23.5282 q^{70} -4.02057 q^{71} -10.7454 q^{73} +10.3832 q^{74} +0.731522 q^{76} -3.78550 q^{77} +15.6493 q^{79} +9.51698 q^{80} +24.9873 q^{82} -10.8494 q^{83} +15.8151 q^{85} -0.282649 q^{86} -0.718591 q^{88} +13.6121 q^{89} +16.7126 q^{91} +13.1978 q^{92} +22.1844 q^{94} -0.930282 q^{95} +11.5514 q^{97} +15.2788 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * 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 $

$2$	2.08441	1.47390	0.736949	−	0.675949i	$-0.236266\pi$
$2$	2.08441	1.47390	0.736949	+	0.675949i	$0.236266\pi$
$3$	0	0
$3$	0	0
$4$	2.34475	1.17237
$5$	−2.98183	−1.33352	−0.666758	−	0.745274i	$-0.732318\pi$
$5$	−2.98183	−1.33352	−0.666758	+	0.745274i	$0.732318\pi$
$6$	0	0
$7$	3.78550	1.43079	0.715393	−	0.698722i	$-0.246248\pi$
$7$	3.78550	1.43079	0.715393	+	0.698722i	$0.246248\pi$
$8$	0.718591	0.254060
$9$	0	0
$10$	−6.21535	−1.96547
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.41489	1.22447	0.612236	−	0.790675i	$-0.290270\pi$
$13$	4.41489	1.22447	0.612236	+	0.790675i	$0.290270\pi$
$14$	7.89053	2.10883
$15$	0	0
$16$	−3.19166	−0.797914
$17$	−5.30382	−1.28636	−0.643182	−	0.765713i	$-0.722386\pi$
$17$	−5.30382	−1.28636	−0.643182	+	0.765713i	$0.722386\pi$
$18$	0	0
$19$	0.311984	0.0715739	0.0357870	−	0.999359i	$-0.488606\pi$
$19$	0.311984	0.0715739	0.0357870	+	0.999359i	$0.488606\pi$
$20$	−6.99164	−1.56338
$21$	0	0
$22$	−2.08441	−0.444397
$23$	5.62868	1.17366	0.586830	−	0.809710i	$-0.300376\pi$
$23$	5.62868	1.17366	0.586830	+	0.809710i	$0.300376\pi$
$24$	0	0
$25$	3.89132	0.778264
$26$	9.20243	1.80475
$27$	0	0
$28$	8.87605	1.67742
$29$	8.30650	1.54248	0.771239	−	0.636545i	$-0.219637\pi$
$29$	8.30650	1.54248	0.771239	+	0.636545i	$0.219637\pi$
$30$	0	0
$31$	−6.85645	−1.23145	−0.615727	−	0.787959i	$-0.711138\pi$
$31$	−6.85645	−1.23145	−0.615727	+	0.787959i	$0.711138\pi$
$32$	−8.08989	−1.43010
$33$	0	0
$34$	−11.0553	−1.89597
$35$	−11.2877	−1.90798
$36$	0	0
$37$	4.98139	0.818936	0.409468	−	0.912325i	$-0.365714\pi$
$37$	4.98139	0.818936	0.409468	+	0.912325i	$0.365714\pi$
$38$	0.650300	0.105493
$39$	0	0
$40$	−2.14272	−0.338794
$41$	11.9877	1.87217	0.936085	−	0.351775i	$-0.114422\pi$
$41$	11.9877	1.87217	0.936085	+	0.351775i	$0.114422\pi$
$42$	0	0
$43$	−0.135602	−0.0206791	−0.0103395	−	0.999947i	$-0.503291\pi$
$43$	−0.135602	−0.0206791	−0.0103395	+	0.999947i	$0.503291\pi$
$44$	−2.34475	−0.353484
$45$	0	0
$46$	11.7324	1.72986
$47$	10.6430	1.55245	0.776223	−	0.630459i	$-0.217133\pi$
$47$	10.6430	1.55245	0.776223	+	0.630459i	$0.217133\pi$
$48$	0	0
$49$	7.33004	1.04715
$50$	8.11109	1.14708
$51$	0	0
$52$	10.3518	1.43554
$53$	−9.36007	−1.28570	−0.642852	−	0.765990i	$-0.722249\pi$
$53$	−9.36007	−1.28570	−0.642852	+	0.765990i	$0.722249\pi$
$54$	0	0
$55$	2.98183	0.402070
$56$	2.72023	0.363506
$57$	0	0
$58$	17.3141	2.27345
$59$	7.08123	0.921897	0.460949	−	0.887427i	$-0.347509\pi$
$59$	7.08123	0.921897	0.460949	+	0.887427i	$0.347509\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−14.2916	−1.81504
$63$	0	0
$64$	−10.4793	−1.30991
$65$	−13.1645	−1.63285
$66$	0	0
$67$	3.08414	0.376788	0.188394	−	0.982094i	$-0.439672\pi$
$67$	3.08414	0.376788	0.188394	+	0.982094i	$0.439672\pi$
$68$	−12.4361	−1.50810
$69$	0	0
$70$	−23.5282	−2.81216
$71$	−4.02057	−0.477154	−0.238577	−	0.971124i	$-0.576681\pi$
$71$	−4.02057	−0.477154	−0.238577	+	0.971124i	$0.576681\pi$
$72$	0	0
$73$	−10.7454	−1.25765	−0.628827	−	0.777545i	$-0.716465\pi$
$73$	−10.7454	−1.25765	−0.628827	+	0.777545i	$0.716465\pi$
$74$	10.3832	1.20703
$75$	0	0
$76$	0.731522	0.0839114
$77$	−3.78550	−0.431398
$78$	0	0
$79$	15.6493	1.76068	0.880339	−	0.474345i	$-0.157315\pi$
$79$	15.6493	1.76068	0.880339	+	0.474345i	$0.157315\pi$
$80$	9.51698	1.06403
$81$	0	0
$82$	24.9873	2.75939
$83$	−10.8494	−1.19088	−0.595440	−	0.803400i	$-0.703022\pi$
$83$	−10.8494	−1.19088	−0.595440	+	0.803400i	$0.703022\pi$
$84$	0	0
$85$	15.8151	1.71539
$86$	−0.282649	−0.0304788
$87$	0	0
$88$	−0.718591	−0.0766021
$89$	13.6121	1.44288	0.721440	−	0.692477i	$-0.243481\pi$
$89$	13.6121	1.44288	0.721440	+	0.692477i	$0.243481\pi$
$90$	0	0
$91$	16.7126	1.75196
$92$	13.1978	1.37597
$93$	0	0
$94$	22.1844	2.28814
$95$	−0.930282	−0.0954449
$96$	0	0
$97$	11.5514	1.17287	0.586434	−	0.809997i	$-0.300531\pi$
$97$	11.5514	1.17287	0.586434	+	0.809997i	$0.300531\pi$
$98$	15.2788	1.54339
$99$	0	0
$100$	9.12416	0.912416
$101$	0.740757	0.0737081	0.0368540	−	0.999321i	$-0.488266\pi$
$101$	0.740757	0.0737081	0.0368540	+	0.999321i	$0.488266\pi$
$102$	0	0
$103$	6.18245	0.609175	0.304588	−	0.952484i	$-0.401481\pi$
$103$	6.18245	0.609175	0.304588	+	0.952484i	$0.401481\pi$
$104$	3.17251	0.311090
$105$	0	0
$106$	−19.5102	−1.89500
$107$	−1.63971	−0.158517	−0.0792583	−	0.996854i	$-0.525255\pi$
$107$	−1.63971	−0.158517	−0.0792583	+	0.996854i	$0.525255\pi$
$108$	0	0
$109$	14.2123	1.36130	0.680648	−	0.732611i	$-0.261698\pi$
$109$	14.2123	1.36130	0.680648	+	0.732611i	$0.261698\pi$
$110$	6.21535	0.592610
$111$	0	0
$112$	−12.0820	−1.14164
$113$	6.91858	0.650845	0.325423	−	0.945569i	$-0.394493\pi$
$113$	6.91858	0.650845	0.325423	+	0.945569i	$0.394493\pi$
$114$	0	0
$115$	−16.7838	−1.56509
$116$	19.4766	1.80836
$117$	0	0
$118$	14.7601	1.35878
$119$	−20.0776	−1.84051
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.08441	−0.188713
$123$	0	0
$124$	−16.0766	−1.44372
$125$	3.30590	0.295688
$126$	0	0
$127$	−9.92811	−0.880977	−0.440488	−	0.897758i	$-0.645195\pi$
$127$	−9.92811	−0.880977	−0.440488	+	0.897758i	$0.645195\pi$
$128$	−5.66333	−0.500572
$129$	0	0
$130$	−27.4401	−2.40666
$131$	−5.11152	−0.446595	−0.223298	−	0.974750i	$-0.571682\pi$
$131$	−5.11152	−0.446595	−0.223298	+	0.974750i	$0.571682\pi$
$132$	0	0
$133$	1.18101	0.102407
$134$	6.42861	0.555347
$135$	0	0
$136$	−3.81128	−0.326814
$137$	−8.07318	−0.689739	−0.344869	−	0.938651i	$-0.612077\pi$
$137$	−8.07318	−0.689739	−0.344869	+	0.938651i	$0.612077\pi$
$138$	0	0
$139$	−16.6673	−1.41371	−0.706853	−	0.707360i	$-0.749886\pi$
$139$	−16.6673	−1.41371	−0.706853	+	0.707360i	$0.749886\pi$
$140$	−26.4669	−2.23686
$141$	0	0
$142$	−8.38051	−0.703277
$143$	−4.41489	−0.369192
$144$	0	0
$145$	−24.7686	−2.05692
$146$	−22.3978	−1.85365
$147$	0	0
$148$	11.6801	0.960098
$149$	17.1005	1.40093	0.700466	−	0.713686i	$-0.252976\pi$
$149$	17.1005	1.40093	0.700466	+	0.713686i	$0.252976\pi$
$150$	0	0
$151$	14.4630	1.17699	0.588493	−	0.808502i	$-0.299722\pi$
$151$	14.4630	1.17699	0.588493	+	0.808502i	$0.299722\pi$
$152$	0.224189	0.0181841
$153$	0	0
$154$	−7.89053	−0.635837
$155$	20.4448	1.64216
$156$	0	0
$157$	5.84286	0.466311	0.233156	−	0.972439i	$-0.425095\pi$
$157$	5.84286	0.466311	0.233156	+	0.972439i	$0.425095\pi$
$158$	32.6194	2.59506
$159$	0	0
$160$	24.1227	1.90707
$161$	21.3074	1.67926
$162$	0	0
$163$	−10.1026	−0.791294	−0.395647	−	0.918403i	$-0.629480\pi$
$163$	−10.1026	−0.791294	−0.395647	+	0.918403i	$0.629480\pi$
$164$	28.1082	2.19488
$165$	0	0
$166$	−22.6146	−1.75523
$167$	16.1240	1.24771	0.623856	−	0.781540i	$-0.285565\pi$
$167$	16.1240	1.24771	0.623856	+	0.781540i	$0.285565\pi$
$168$	0	0
$169$	6.49130	0.499330
$170$	32.9651	2.52831
$171$	0	0
$172$	−0.317952	−0.0242436
$173$	−16.5027	−1.25467	−0.627337	−	0.778748i	$-0.715855\pi$
$173$	−16.5027	−1.25467	−0.627337	+	0.778748i	$0.715855\pi$
$174$	0	0
$175$	14.7306	1.11353
$176$	3.19166	0.240580
$177$	0	0
$178$	28.3731	2.12666
$179$	18.6695	1.39542	0.697711	−	0.716380i	$-0.254202\pi$
$179$	18.6695	1.39542	0.697711	+	0.716380i	$0.254202\pi$
$180$	0	0
$181$	19.3738	1.44004	0.720020	−	0.693954i	$-0.244133\pi$
$181$	19.3738	1.44004	0.720020	+	0.693954i	$0.244133\pi$
$182$	34.8358	2.58220
$183$	0	0
$184$	4.04472	0.298181
$185$	−14.8537	−1.09206
$186$	0	0
$187$	5.30382	0.387854
$188$	24.9552	1.82005
$189$	0	0
$190$	−1.93909	−0.140676
$191$	−9.92418	−0.718088	−0.359044	−	0.933321i	$-0.616897\pi$
$191$	−9.92418	−0.718088	−0.359044	+	0.933321i	$0.616897\pi$
$192$	0	0
$193$	2.24190	0.161376	0.0806879	−	0.996739i	$-0.474288\pi$
$193$	2.24190	0.161376	0.0806879	+	0.996739i	$0.474288\pi$
$194$	24.0778	1.72869
$195$	0	0
$196$	17.1871	1.22765
$197$	1.80883	0.128874	0.0644368	−	0.997922i	$-0.479475\pi$
$197$	1.80883	0.128874	0.0644368	+	0.997922i	$0.479475\pi$
$198$	0	0
$199$	−23.1528	−1.64126	−0.820629	−	0.571461i	$-0.806377\pi$
$199$	−23.1528	−1.64126	−0.820629	+	0.571461i	$0.806377\pi$
$200$	2.79627	0.197726
$201$	0	0
$202$	1.54404	0.108638
$203$	31.4443	2.20696
$204$	0	0
$205$	−35.7454	−2.49657
$206$	12.8867	0.897862
$207$	0	0
$208$	−14.0908	−0.977023
$209$	−0.311984	−0.0215804
$210$	0	0
$211$	20.7281	1.42698	0.713490	−	0.700666i	$-0.247114\pi$
$211$	20.7281	1.42698	0.713490	+	0.700666i	$0.247114\pi$
$212$	−21.9470	−1.50732
$213$	0	0
$214$	−3.41782	−0.233637
$215$	0.404342	0.0275759
$216$	0	0
$217$	−25.9551	−1.76195
$218$	29.6243	2.00641
$219$	0	0
$220$	6.99164	0.471376
$221$	−23.4158	−1.57512
$222$	0	0
$223$	−7.99293	−0.535246	−0.267623	−	0.963524i	$-0.586238\pi$
$223$	−7.99293	−0.535246	−0.267623	+	0.963524i	$0.586238\pi$
$224$	−30.6243	−2.04617
$225$	0	0
$226$	14.4211	0.959279
$227$	−15.2803	−1.01419	−0.507094	−	0.861891i	$-0.669280\pi$
$227$	−15.2803	−1.01419	−0.507094	+	0.861891i	$0.669280\pi$
$228$	0	0
$229$	6.31189	0.417102	0.208551	−	0.978012i	$-0.433125\pi$
$229$	6.31189	0.417102	0.208551	+	0.978012i	$0.433125\pi$
$230$	−34.9842	−2.30679
$231$	0	0
$232$	5.96898	0.391883
$233$	−1.04962	−0.0687630	−0.0343815	−	0.999409i	$-0.510946\pi$
$233$	−1.04962	−0.0687630	−0.0343815	+	0.999409i	$0.510946\pi$
$234$	0	0
$235$	−31.7357	−2.07021
$236$	16.6037	1.08081
$237$	0	0
$238$	−41.8499	−2.71273
$239$	19.8108	1.28146	0.640728	−	0.767768i	$-0.278633\pi$
$239$	19.8108	1.28146	0.640728	+	0.767768i	$0.278633\pi$
$240$	0	0
$241$	6.82853	0.439864	0.219932	−	0.975515i	$-0.429416\pi$
$241$	6.82853	0.439864	0.219932	+	0.975515i	$0.429416\pi$
$242$	2.08441	0.133991
$243$	0	0
$244$	−2.34475	−0.150107
$245$	−21.8569	−1.39639
$246$	0	0
$247$	1.37737	0.0876402
$248$	−4.92699	−0.312864
$249$	0	0
$250$	6.89083	0.435814
$251$	−6.08527	−0.384099	−0.192049	−	0.981385i	$-0.561513\pi$
$251$	−6.08527	−0.384099	−0.192049	+	0.981385i	$0.561513\pi$
$252$	0	0
$253$	−5.62868	−0.353872
$254$	−20.6942	−1.29847
$255$	0	0
$256$	9.15393	0.572120
$257$	−4.93147	−0.307617	−0.153808	−	0.988101i	$-0.549154\pi$
$257$	−4.93147	−0.307617	−0.153808	+	0.988101i	$0.549154\pi$
$258$	0	0
$259$	18.8571	1.17172
$260$	−30.8674	−1.91431
$261$	0	0
$262$	−10.6545	−0.658236
$263$	7.52597	0.464071	0.232035	−	0.972707i	$-0.425461\pi$
$263$	7.52597	0.464071	0.232035	+	0.972707i	$0.425461\pi$
$264$	0	0
$265$	27.9101	1.71451
$266$	2.46171	0.150937
$267$	0	0
$268$	7.23154	0.441736
$269$	−10.0727	−0.614141	−0.307070	−	0.951687i	$-0.599349\pi$
$269$	−10.0727	−0.614141	−0.307070	+	0.951687i	$0.599349\pi$
$270$	0	0
$271$	16.2243	0.985556	0.492778	−	0.870155i	$-0.335982\pi$
$271$	16.2243	0.985556	0.492778	+	0.870155i	$0.335982\pi$
$272$	16.9280	1.02641
$273$	0	0
$274$	−16.8278	−1.01660
$275$	−3.89132	−0.234655
$276$	0	0
$277$	22.1944	1.33353	0.666767	−	0.745266i	$-0.267678\pi$
$277$	22.1944	1.33353	0.666767	+	0.745266i	$0.267678\pi$
$278$	−34.7415	−2.08366
$279$	0	0
$280$	−8.11127	−0.484741
$281$	−27.7353	−1.65455	−0.827276	−	0.561796i	$-0.810111\pi$
$281$	−27.7353	−1.65455	−0.827276	+	0.561796i	$0.810111\pi$
$282$	0	0
$283$	−26.0469	−1.54833	−0.774165	−	0.632984i	$-0.781830\pi$
$283$	−26.0469	−1.54833	−0.774165	+	0.632984i	$0.781830\pi$
$284$	−9.42723	−0.559403
$285$	0	0
$286$	−9.20243	−0.544151
$287$	45.3796	2.67867
$288$	0	0
$289$	11.1305	0.654734
$290$	−51.6278	−3.03169
$291$	0	0
$292$	−25.1952	−1.47444
$293$	−2.89726	−0.169260	−0.0846300	−	0.996412i	$-0.526971\pi$
$293$	−2.89726	−0.169260	−0.0846300	+	0.996412i	$0.526971\pi$
$294$	0	0
$295$	−21.1150	−1.22936
$296$	3.57959	0.208059
$297$	0	0
$298$	35.6445	2.06483
$299$	24.8500	1.43711
$300$	0	0
$301$	−0.513321	−0.0295873
$302$	30.1468	1.73476
$303$	0	0
$304$	−0.995744	−0.0571099
$305$	2.98183	0.170739
$306$	0	0
$307$	−22.9909	−1.31216	−0.656081	−	0.754691i	$-0.727787\pi$
$307$	−22.9909	−1.31216	−0.656081	+	0.754691i	$0.727787\pi$
$308$	−8.87605	−0.505760
$309$	0	0
$310$	42.6152	2.42038
$311$	3.59123	0.203640	0.101820	−	0.994803i	$-0.467533\pi$
$311$	3.59123	0.203640	0.101820	+	0.994803i	$0.467533\pi$
$312$	0	0
$313$	−14.8121	−0.837228	−0.418614	−	0.908164i	$-0.637484\pi$
$313$	−14.8121	−0.837228	−0.418614	+	0.908164i	$0.637484\pi$
$314$	12.1789	0.687295
$315$	0	0
$316$	36.6935	2.06417
$317$	−3.69043	−0.207275	−0.103638	−	0.994615i	$-0.533048\pi$
$317$	−3.69043	−0.207275	−0.103638	+	0.994615i	$0.533048\pi$
$318$	0	0
$319$	−8.30650	−0.465075
$320$	31.2475	1.74679
$321$	0	0
$322$	44.4132	2.47505
$323$	−1.65470	−0.0920702
$324$	0	0
$325$	17.1798	0.952962
$326$	−21.0578	−1.16629
$327$	0	0
$328$	8.61428	0.475644
$329$	40.2892	2.22122
$330$	0	0
$331$	−0.830413	−0.0456437	−0.0228218	−	0.999740i	$-0.507265\pi$
$331$	−0.830413	−0.0456437	−0.0228218	+	0.999740i	$0.507265\pi$
$332$	−25.4392	−1.39616
$333$	0	0
$334$	33.6089	1.83900
$335$	−9.19640	−0.502453
$336$	0	0
$337$	−13.2432	−0.721404	−0.360702	−	0.932681i	$-0.617463\pi$
$337$	−13.2432	−0.721404	−0.360702	+	0.932681i	$0.617463\pi$
$338$	13.5305	0.735962
$339$	0	0
$340$	37.0824	2.01107
$341$	6.85645	0.371298
$342$	0	0
$343$	1.24937	0.0674595
$344$	−0.0974422	−0.00525373
$345$	0	0
$346$	−34.3982	−1.84926
$347$	−18.6321	−1.00023	−0.500113	−	0.865960i	$-0.666708\pi$
$347$	−18.6321	−1.00023	−0.500113	+	0.865960i	$0.666708\pi$
$348$	0	0
$349$	4.87651	0.261034	0.130517	−	0.991446i	$-0.458336\pi$
$349$	4.87651	0.261034	0.130517	+	0.991446i	$0.458336\pi$
$350$	30.7046	1.64123
$351$	0	0
$352$	8.08989	0.431193
$353$	−22.1701	−1.17999	−0.589997	−	0.807406i	$-0.700871\pi$
$353$	−22.1701	−1.17999	−0.589997	+	0.807406i	$0.700871\pi$
$354$	0	0
$355$	11.9887	0.636293
$356$	31.9169	1.69159
$357$	0	0
$358$	38.9147	2.05671
$359$	−3.41958	−0.180478	−0.0902392	−	0.995920i	$-0.528763\pi$
$359$	−3.41958	−0.180478	−0.0902392	+	0.995920i	$0.528763\pi$
$360$	0	0
$361$	−18.9027	−0.994877
$362$	40.3828	2.12247
$363$	0	0
$364$	39.1868	2.05395
$365$	32.0410	1.67710
$366$	0	0
$367$	17.0313	0.889025	0.444512	−	0.895773i	$-0.353377\pi$
$367$	17.0313	0.889025	0.444512	+	0.895773i	$0.353377\pi$
$368$	−17.9648	−0.936481
$369$	0	0
$370$	−30.9611	−1.60959
$371$	−35.4326	−1.83957
$372$	0	0
$373$	19.9234	1.03160	0.515798	−	0.856710i	$-0.327496\pi$
$373$	19.9234	1.03160	0.515798	+	0.856710i	$0.327496\pi$
$374$	11.0553	0.571656
$375$	0	0
$376$	7.64799	0.394415
$377$	36.6723	1.88872
$378$	0	0
$379$	8.27906	0.425267	0.212633	−	0.977132i	$-0.431796\pi$
$379$	8.27906	0.425267	0.212633	+	0.977132i	$0.431796\pi$
$380$	−2.18128	−0.111897
$381$	0	0
$382$	−20.6860	−1.05839
$383$	22.9227	1.17129	0.585647	−	0.810566i	$-0.300840\pi$
$383$	22.9227	1.17129	0.585647	+	0.810566i	$0.300840\pi$
$384$	0	0
$385$	11.2877	0.575276
$386$	4.67304	0.237851
$387$	0	0
$388$	27.0851	1.37504
$389$	16.2780	0.825327	0.412663	−	0.910884i	$-0.364599\pi$
$389$	16.2780	0.825327	0.412663	+	0.910884i	$0.364599\pi$
$390$	0	0
$391$	−29.8535	−1.50976
$392$	5.26730	0.266039
$393$	0	0
$394$	3.77033	0.189946
$395$	−46.6634	−2.34789
$396$	0	0
$397$	11.0033	0.552239	0.276119	−	0.961123i	$-0.410952\pi$
$397$	11.0033	0.552239	0.276119	+	0.961123i	$0.410952\pi$
$398$	−48.2598	−2.41905
$399$	0	0
$400$	−12.4198	−0.620988
$401$	−29.1596	−1.45616	−0.728080	−	0.685492i	$-0.759587\pi$
$401$	−29.1596	−1.45616	−0.728080	+	0.685492i	$0.759587\pi$
$402$	0	0
$403$	−30.2705	−1.50788
$404$	1.73689	0.0864134
$405$	0	0
$406$	65.5426	3.25283
$407$	−4.98139	−0.246918
$408$	0	0
$409$	−19.0595	−0.942433	−0.471216	−	0.882018i	$-0.656185\pi$
$409$	−19.0595	−0.942433	−0.471216	+	0.882018i	$0.656185\pi$
$410$	−74.5079	−3.67968
$411$	0	0
$412$	14.4963	0.714181
$413$	26.8060	1.31904
$414$	0	0
$415$	32.3512	1.58806
$416$	−35.7160	−1.75112
$417$	0	0
$418$	−0.650300	−0.0318072
$419$	19.6633	0.960616	0.480308	−	0.877100i	$-0.340525\pi$
$419$	19.6633	0.960616	0.480308	+	0.877100i	$0.340525\pi$
$420$	0	0
$421$	−18.0228	−0.878380	−0.439190	−	0.898394i	$-0.644734\pi$
$421$	−18.0228	−0.878380	−0.439190	+	0.898394i	$0.644734\pi$
$422$	43.2057	2.10322
$423$	0	0
$424$	−6.72606	−0.326646
$425$	−20.6389	−1.00113
$426$	0	0
$427$	−3.78550	−0.183193
$428$	−3.84470	−0.185841
$429$	0	0
$430$	0.842812	0.0406440
$431$	−22.5762	−1.08746	−0.543728	−	0.839261i	$-0.682988\pi$
$431$	−22.5762	−1.08746	−0.543728	+	0.839261i	$0.682988\pi$
$432$	0	0
$433$	8.30030	0.398887	0.199444	−	0.979909i	$-0.436087\pi$
$433$	8.30030	0.398887	0.199444	+	0.979909i	$0.436087\pi$
$434$	−54.1010	−2.59693
$435$	0	0
$436$	33.3244	1.59595
$437$	1.75605	0.0840035
$438$	0	0
$439$	6.53799	0.312041	0.156021	−	0.987754i	$-0.450133\pi$
$439$	6.53799	0.312041	0.156021	+	0.987754i	$0.450133\pi$
$440$	2.14272	0.102150
$441$	0	0
$442$	−48.8080	−2.32156
$443$	−6.53605	−0.310537	−0.155269	−	0.987872i	$-0.549624\pi$
$443$	−6.53605	−0.310537	−0.155269	+	0.987872i	$0.549624\pi$
$444$	0	0
$445$	−40.5890	−1.92410
$446$	−16.6605	−0.788898
$447$	0	0
$448$	−39.6694	−1.87420
$449$	32.0478	1.51243	0.756215	−	0.654323i	$-0.227046\pi$
$449$	32.0478	1.51243	0.756215	+	0.654323i	$0.227046\pi$
$450$	0	0
$451$	−11.9877	−0.564480
$452$	16.2223	0.763034
$453$	0	0
$454$	−31.8503	−1.49481
$455$	−49.8342	−2.33626
$456$	0	0
$457$	−12.5441	−0.586788	−0.293394	−	0.955992i	$-0.594785\pi$
$457$	−12.5441	−0.586788	−0.293394	+	0.955992i	$0.594785\pi$
$458$	13.1565	0.614765
$459$	0	0
$460$	−39.3537	−1.83488
$461$	−32.6919	−1.52261	−0.761307	−	0.648392i	$-0.775442\pi$
$461$	−32.6919	−1.52261	−0.761307	+	0.648392i	$0.775442\pi$
$462$	0	0
$463$	−10.4702	−0.486593	−0.243297	−	0.969952i	$-0.578229\pi$
$463$	−10.4702	−0.486593	−0.243297	+	0.969952i	$0.578229\pi$
$464$	−26.5115	−1.23077
$465$	0	0
$466$	−2.18784	−0.101350
$467$	6.16377	0.285225	0.142613	−	0.989779i	$-0.454450\pi$
$467$	6.16377	0.285225	0.142613	+	0.989779i	$0.454450\pi$
$468$	0	0
$469$	11.6750	0.539103
$470$	−66.1501	−3.05128
$471$	0	0
$472$	5.08851	0.234218
$473$	0.135602	0.00623497
$474$	0	0
$475$	1.21403	0.0557034
$476$	−47.0769	−2.15777
$477$	0	0
$478$	41.2938	1.88873
$479$	14.8725	0.679541	0.339770	−	0.940508i	$-0.389651\pi$
$479$	14.8725	0.679541	0.339770	+	0.940508i	$0.389651\pi$
$480$	0	0
$481$	21.9923	1.00276
$482$	14.2334	0.648315
$483$	0	0
$484$	2.34475	0.106579
$485$	−34.4444	−1.56404
$486$	0	0
$487$	11.9464	0.541342	0.270671	−	0.962672i	$-0.412755\pi$
$487$	11.9464	0.541342	0.270671	+	0.962672i	$0.412755\pi$
$488$	−0.718591	−0.0325291
$489$	0	0
$490$	−45.5587	−2.05813
$491$	10.7791	0.486453	0.243227	−	0.969969i	$-0.421794\pi$
$491$	10.7791	0.486453	0.243227	+	0.969969i	$0.421794\pi$
$492$	0	0
$493$	−44.0562	−1.98419
$494$	2.87101	0.129173
$495$	0	0
$496$	21.8834	0.982595
$497$	−15.2199	−0.682706
$498$	0	0
$499$	−3.60513	−0.161388	−0.0806940	−	0.996739i	$-0.525714\pi$
$499$	−3.60513	−0.161388	−0.0806940	+	0.996739i	$0.525714\pi$
$500$	7.75149	0.346657
$501$	0	0
$502$	−12.6842	−0.566122
$503$	36.2506	1.61634	0.808168	−	0.588952i	$-0.200459\pi$
$503$	36.2506	1.61634	0.808168	+	0.588952i	$0.200459\pi$
$504$	0	0
$505$	−2.20881	−0.0982909
$506$	−11.7324	−0.521571
$507$	0	0
$508$	−23.2789	−1.03283
$509$	−12.8068	−0.567652	−0.283826	−	0.958876i	$-0.591604\pi$
$509$	−12.8068	−0.567652	−0.283826	+	0.958876i	$0.591604\pi$
$510$	0	0
$511$	−40.6767	−1.79943
$512$	30.4071	1.34382
$513$	0	0
$514$	−10.2792	−0.453395
$515$	−18.4350	−0.812345
$516$	0	0
$517$	−10.6430	−0.468080
$518$	39.3058	1.72700
$519$	0	0
$520$	−9.45988	−0.414843
$521$	−7.62484	−0.334050	−0.167025	−	0.985953i	$-0.553416\pi$
$521$	−7.62484	−0.334050	−0.167025	+	0.985953i	$0.553416\pi$
$522$	0	0
$523$	7.28355	0.318487	0.159244	−	0.987239i	$-0.449094\pi$
$523$	7.28355	0.318487	0.159244	+	0.987239i	$0.449094\pi$
$524$	−11.9852	−0.523576
$525$	0	0
$526$	15.6872	0.683993
$527$	36.3654	1.58410
$528$	0	0
$529$	8.68202	0.377479
$530$	58.1760	2.52701
$531$	0	0
$532$	2.76918	0.120059
$533$	52.9246	2.29242
$534$	0	0
$535$	4.88934	0.211384
$536$	2.21624	0.0957270
$537$	0	0
$538$	−20.9955	−0.905180
$539$	−7.33004	−0.315727
$540$	0	0
$541$	−13.0431	−0.560765	−0.280383	−	0.959888i	$-0.590461\pi$
$541$	−13.0431	−0.560765	−0.280383	+	0.959888i	$0.590461\pi$
$542$	33.8180	1.45261
$543$	0	0
$544$	42.9073	1.83964
$545$	−42.3788	−1.81531
$546$	0	0
$547$	5.68800	0.243201	0.121601	−	0.992579i	$-0.461197\pi$
$547$	5.68800	0.243201	0.121601	+	0.992579i	$0.461197\pi$
$548$	−18.9296	−0.808631
$549$	0	0
$550$	−8.11109	−0.345858
$551$	2.59149	0.110401
$552$	0	0
$553$	59.2403	2.51915
$554$	46.2622	1.96549
$555$	0	0
$556$	−39.0807	−1.65739
$557$	−34.6291	−1.46728	−0.733641	−	0.679538i	$-0.762180\pi$
$557$	−34.6291	−1.46728	−0.733641	+	0.679538i	$0.762180\pi$
$558$	0	0
$559$	−0.598667	−0.0253209
$560$	36.0266	1.52240
$561$	0	0
$562$	−57.8117	−2.43864
$563$	−26.7927	−1.12918	−0.564588	−	0.825373i	$-0.690965\pi$
$563$	−26.7927	−1.12918	−0.564588	+	0.825373i	$0.690965\pi$
$564$	0	0
$565$	−20.6300	−0.867913
$566$	−54.2924	−2.28208
$567$	0	0
$568$	−2.88915	−0.121226
$569$	−26.8823	−1.12696	−0.563482	−	0.826129i	$-0.690538\pi$
$569$	−26.8823	−1.12696	−0.563482	+	0.826129i	$0.690538\pi$
$570$	0	0
$571$	11.6787	0.488740	0.244370	−	0.969682i	$-0.421419\pi$
$571$	11.6787	0.488740	0.244370	+	0.969682i	$0.421419\pi$
$572$	−10.3518	−0.432831
$573$	0	0
$574$	94.5895	3.94809
$575$	21.9030	0.913418
$576$	0	0
$577$	−14.5191	−0.604438	−0.302219	−	0.953238i	$-0.597727\pi$
$577$	−14.5191	−0.604438	−0.302219	+	0.953238i	$0.597727\pi$
$578$	23.2004	0.965011
$579$	0	0
$580$	−58.0761	−2.41148
$581$	−41.0706	−1.70389
$582$	0	0
$583$	9.36007	0.387654
$584$	−7.72155	−0.319520
$585$	0	0
$586$	−6.03907	−0.249472
$587$	−42.4867	−1.75361	−0.876806	−	0.480844i	$-0.840330\pi$
$587$	−42.4867	−1.75361	−0.876806	+	0.480844i	$0.840330\pi$
$588$	0	0
$589$	−2.13910	−0.0881400
$590$	−44.0123	−1.81196
$591$	0	0
$592$	−15.8989	−0.653441
$593$	−26.9749	−1.10772	−0.553862	−	0.832608i	$-0.686847\pi$
$593$	−26.9749	−1.10772	−0.553862	+	0.832608i	$0.686847\pi$
$594$	0	0
$595$	59.8681	2.45435
$596$	40.0964	1.64241
$597$	0	0
$598$	51.7975	2.11816
$599$	−0.252191	−0.0103042	−0.00515212	−	0.999987i	$-0.501640\pi$
$599$	−0.252191	−0.0103042	−0.00515212	+	0.999987i	$0.501640\pi$
$600$	0	0
$601$	10.0959	0.411821	0.205910	−	0.978571i	$-0.433984\pi$
$601$	10.0959	0.411821	0.205910	+	0.978571i	$0.433984\pi$
$602$	−1.06997	−0.0436087
$603$	0	0
$604$	33.9122	1.37987
$605$	−2.98183	−0.121229
$606$	0	0
$607$	−28.8159	−1.16960	−0.584801	−	0.811177i	$-0.698827\pi$
$607$	−28.8159	−1.16960	−0.584801	+	0.811177i	$0.698827\pi$
$608$	−2.52391	−0.102358
$609$	0	0
$610$	6.21535	0.251652
$611$	46.9879	1.90092
$612$	0	0
$613$	−19.4842	−0.786961	−0.393481	−	0.919333i	$-0.628729\pi$
$613$	−19.4842	−0.786961	−0.393481	+	0.919333i	$0.628729\pi$
$614$	−47.9224	−1.93399
$615$	0	0
$616$	−2.72023	−0.109601
$617$	−13.6018	−0.547589	−0.273794	−	0.961788i	$-0.588279\pi$
$617$	−13.6018	−0.547589	−0.273794	+	0.961788i	$0.588279\pi$
$618$	0	0
$619$	−28.6157	−1.15016	−0.575081	−	0.818097i	$-0.695029\pi$
$619$	−28.6157	−1.15016	−0.575081	+	0.818097i	$0.695029\pi$
$620$	47.9378	1.92523
$621$	0	0
$622$	7.48558	0.300145
$623$	51.5287	2.06445
$624$	0	0
$625$	−29.3142	−1.17257
$626$	−30.8744	−1.23399
$627$	0	0
$628$	13.7000	0.546691
$629$	−26.4204	−1.05345
$630$	0	0
$631$	0.321585	0.0128021	0.00640104	−	0.999980i	$-0.497962\pi$
$631$	0.321585	0.0128021	0.00640104	+	0.999980i	$0.497962\pi$
$632$	11.2454	0.447319
$633$	0	0
$634$	−7.69235	−0.305502
$635$	29.6039	1.17480
$636$	0	0
$637$	32.3614	1.28220
$638$	−17.3141	−0.685472
$639$	0	0
$640$	16.8871	0.667521
$641$	41.8152	1.65160	0.825800	−	0.563963i	$-0.190724\pi$
$641$	41.8152	1.65160	0.825800	+	0.563963i	$0.190724\pi$
$642$	0	0
$643$	−20.5122	−0.808923	−0.404461	−	0.914555i	$-0.632541\pi$
$643$	−20.5122	−0.808923	−0.404461	+	0.914555i	$0.632541\pi$
$644$	49.9604	1.96872
$645$	0	0
$646$	−3.44907	−0.135702
$647$	29.8808	1.17474	0.587368	−	0.809320i	$-0.300164\pi$
$647$	29.8808	1.17474	0.587368	+	0.809320i	$0.300164\pi$
$648$	0	0
$649$	−7.08123	−0.277962
$650$	35.8096	1.40457
$651$	0	0
$652$	−23.6880	−0.927692
$653$	−34.4146	−1.34675	−0.673373	−	0.739303i	$-0.735155\pi$
$653$	−34.4146	−1.34675	−0.673373	+	0.739303i	$0.735155\pi$
$654$	0	0
$655$	15.2417	0.595542
$656$	−38.2607	−1.49383
$657$	0	0
$658$	83.9791	3.27385
$659$	−1.74540	−0.0679911	−0.0339955	−	0.999422i	$-0.510823\pi$
$659$	−1.74540	−0.0679911	−0.0339955	+	0.999422i	$0.510823\pi$
$660$	0	0
$661$	−21.6073	−0.840425	−0.420212	−	0.907426i	$-0.638044\pi$
$661$	−21.6073	−0.840425	−0.420212	+	0.907426i	$0.638044\pi$
$662$	−1.73092	−0.0672741
$663$	0	0
$664$	−7.79631	−0.302555
$665$	−3.52159	−0.136561
$666$	0	0
$667$	46.7546	1.81035
$668$	37.8066	1.46278
$669$	0	0
$670$	−19.1690	−0.740564
$671$	1.00000	0.0386046
$672$	0	0
$673$	35.8905	1.38348	0.691738	−	0.722149i	$-0.256845\pi$
$673$	35.8905	1.38348	0.691738	+	0.722149i	$0.256845\pi$
$674$	−27.6043	−1.06328
$675$	0	0
$676$	15.2204	0.585402
$677$	16.5765	0.637088	0.318544	−	0.947908i	$-0.396806\pi$
$677$	16.5765	0.637088	0.318544	+	0.947908i	$0.396806\pi$
$678$	0	0
$679$	43.7279	1.67812
$680$	11.3646	0.435812
$681$	0	0
$682$	14.2916	0.547254
$683$	−29.9616	−1.14645	−0.573224	−	0.819399i	$-0.694307\pi$
$683$	−29.9616	−1.14645	−0.573224	+	0.819399i	$0.694307\pi$
$684$	0	0
$685$	24.0729	0.919777
$686$	2.60419	0.0994283
$687$	0	0
$688$	0.432794	0.0165001
$689$	−41.3237	−1.57431
$690$	0	0
$691$	−9.35132	−0.355741	−0.177871	−	0.984054i	$-0.556921\pi$
$691$	−9.35132	−0.355741	−0.177871	+	0.984054i	$0.556921\pi$
$692$	−38.6945	−1.47095
$693$	0	0
$694$	−38.8370	−1.47423
$695$	49.6992	1.88520
$696$	0	0
$697$	−63.5808	−2.40829
$698$	10.1646	0.384737
$699$	0	0
$700$	34.5395	1.30547
$701$	11.0613	0.417779	0.208890	−	0.977939i	$-0.433015\pi$
$701$	11.0613	0.417779	0.208890	+	0.977939i	$0.433015\pi$
$702$	0	0
$703$	1.55411	0.0586144
$704$	10.4793	0.394953
$705$	0	0
$706$	−46.2114	−1.73919
$707$	2.80414	0.105460
$708$	0	0
$709$	4.66820	0.175318	0.0876589	−	0.996151i	$-0.472061\pi$
$709$	4.66820	0.175318	0.0876589	+	0.996151i	$0.472061\pi$
$710$	24.9893	0.937830
$711$	0	0
$712$	9.78154	0.366579
$713$	−38.5927	−1.44531
$714$	0	0
$715$	13.1645	0.492323
$716$	43.7752	1.63595
$717$	0	0
$718$	−7.12779	−0.266007
$719$	−30.4581	−1.13589	−0.567947	−	0.823065i	$-0.692262\pi$
$719$	−30.4581	−1.13589	−0.567947	+	0.823065i	$0.692262\pi$
$720$	0	0
$721$	23.4037	0.871600
$722$	−39.4008	−1.46635
$723$	0	0
$724$	45.4265	1.68826
$725$	32.3232	1.20046
$726$	0	0
$727$	40.3806	1.49763	0.748816	−	0.662778i	$-0.230623\pi$
$727$	40.3806	1.49763	0.748816	+	0.662778i	$0.230623\pi$
$728$	12.0095	0.445103
$729$	0	0
$730$	66.7864	2.47187
$731$	0.719207	0.0266008
$732$	0	0
$733$	30.9302	1.14243	0.571217	−	0.820799i	$-0.306472\pi$
$733$	30.9302	1.14243	0.571217	+	0.820799i	$0.306472\pi$
$734$	35.5001	1.31033
$735$	0	0
$736$	−45.5354	−1.67846
$737$	−3.08414	−0.113606
$738$	0	0
$739$	29.5134	1.08567	0.542834	−	0.839840i	$-0.317351\pi$
$739$	29.5134	1.08567	0.542834	+	0.839840i	$0.317351\pi$
$740$	−34.8281	−1.28031
$741$	0	0
$742$	−73.8558	−2.71133
$743$	12.2720	0.450216	0.225108	−	0.974334i	$-0.427727\pi$
$743$	12.2720	0.450216	0.225108	+	0.974334i	$0.427727\pi$
$744$	0	0
$745$	−50.9909	−1.86816
$746$	41.5285	1.52047
$747$	0	0
$748$	12.4361	0.454709
$749$	−6.20713	−0.226803
$750$	0	0
$751$	8.28527	0.302334	0.151167	−	0.988508i	$-0.451697\pi$
$751$	8.28527	0.302334	0.151167	+	0.988508i	$0.451697\pi$
$752$	−33.9689	−1.23872
$753$	0	0
$754$	76.4400	2.78378
$755$	−43.1264	−1.56953
$756$	0	0
$757$	−3.05443	−0.111015	−0.0555076	−	0.998458i	$-0.517678\pi$
$757$	−3.05443	−0.111015	−0.0555076	+	0.998458i	$0.517678\pi$
$758$	17.2569	0.626799
$759$	0	0
$760$	−0.668493	−0.0242488
$761$	−43.7061	−1.58434	−0.792172	−	0.610298i	$-0.791050\pi$
$761$	−43.7061	−1.58434	−0.792172	+	0.610298i	$0.791050\pi$
$762$	0	0
$763$	53.8009	1.94772
$764$	−23.2697	−0.841867
$765$	0	0
$766$	47.7802	1.72637
$767$	31.2629	1.12884
$768$	0	0
$769$	−13.1128	−0.472861	−0.236431	−	0.971648i	$-0.575978\pi$
$769$	−13.1128	−0.472861	−0.236431	+	0.971648i	$0.575978\pi$
$770$	23.5282	0.847898
$771$	0	0
$772$	5.25669	0.189193
$773$	−46.5477	−1.67420	−0.837102	−	0.547048i	$-0.815752\pi$
$773$	−46.5477	−1.67420	−0.837102	+	0.547048i	$0.815752\pi$
$774$	0	0
$775$	−26.6806	−0.958397
$776$	8.30075	0.297979
$777$	0	0
$778$	33.9299	1.21645
$779$	3.73997	0.133999
$780$	0	0
$781$	4.02057	0.143867
$782$	−62.2268	−2.22522
$783$	0	0
$784$	−23.3950	−0.835535
$785$	−17.4224	−0.621834
$786$	0	0
$787$	30.5629	1.08945	0.544725	−	0.838615i	$-0.316634\pi$
$787$	30.5629	1.08945	0.544725	+	0.838615i	$0.316634\pi$
$788$	4.24124	0.151088
$789$	0	0
$790$	−97.2655	−3.46055
$791$	26.1903	0.931220
$792$	0	0
$793$	−4.41489	−0.156778
$794$	22.9353	0.813943
$795$	0	0
$796$	−54.2875	−1.92417
$797$	−3.15070	−0.111603	−0.0558017	−	0.998442i	$-0.517771\pi$
$797$	−3.15070	−0.111603	−0.0558017	+	0.998442i	$0.517771\pi$
$798$	0	0
$799$	−56.4487	−1.99701
$800$	−31.4804	−1.11300
$801$	0	0
$802$	−60.7804	−2.14623
$803$	10.7454	0.379197
$804$	0	0
$805$	−63.5350	−2.23932
$806$	−63.0960	−2.22246
$807$	0	0
$808$	0.532302	0.0187263
$809$	−7.42365	−0.261002	−0.130501	−	0.991448i	$-0.541659\pi$
$809$	−7.42365	−0.261002	−0.130501	+	0.991448i	$0.541659\pi$
$810$	0	0
$811$	−47.6535	−1.67334	−0.836670	−	0.547708i	$-0.815501\pi$
$811$	−47.6535	−1.67334	−0.836670	+	0.547708i	$0.815501\pi$
$812$	73.7289	2.58738
$813$	0	0
$814$	−10.3832	−0.363932
$815$	30.1242	1.05520
$816$	0	0
$817$	−0.0423055	−0.00148008
$818$	−39.7278	−1.38905
$819$	0	0
$820$	−83.8139	−2.92691
$821$	43.3269	1.51212	0.756059	−	0.654503i	$-0.227122\pi$
$821$	43.3269	1.51212	0.756059	+	0.654503i	$0.227122\pi$
$822$	0	0
$823$	−29.7284	−1.03627	−0.518133	−	0.855300i	$-0.673373\pi$
$823$	−29.7284	−1.03627	−0.518133	+	0.855300i	$0.673373\pi$
$824$	4.44266	0.154767
$825$	0	0
$826$	55.8746	1.94413
$827$	−20.1947	−0.702240	−0.351120	−	0.936330i	$-0.614199\pi$
$827$	−20.1947	−0.702240	−0.351120	+	0.936330i	$0.614199\pi$
$828$	0	0
$829$	−29.4893	−1.02421	−0.512103	−	0.858924i	$-0.671134\pi$
$829$	−29.4893	−1.02421	−0.512103	+	0.858924i	$0.671134\pi$
$830$	67.4330	2.34063
$831$	0	0
$832$	−46.2650	−1.60395
$833$	−38.8772	−1.34701
$834$	0	0
$835$	−48.0790	−1.66384
$836$	−0.731522	−0.0253002
$837$	0	0
$838$	40.9863	1.41585
$839$	34.6484	1.19619	0.598097	−	0.801424i	$-0.295924\pi$
$839$	34.6484	1.19619	0.598097	+	0.801424i	$0.295924\pi$
$840$	0	0
$841$	39.9979	1.37924
$842$	−37.5669	−1.29464
$843$	0	0
$844$	48.6021	1.67295
$845$	−19.3559	−0.665865
$846$	0	0
$847$	3.78550	0.130071
$848$	29.8741	1.02588
$849$	0	0
$850$	−43.0197	−1.47556
$851$	28.0387	0.961153
$852$	0	0
$853$	0.382697	0.0131033	0.00655165	−	0.999979i	$-0.497915\pi$
$853$	0.382697	0.0131033	0.00655165	+	0.999979i	$0.497915\pi$
$854$	−7.89053	−0.270008
$855$	0	0
$856$	−1.17828	−0.0402728
$857$	−27.0449	−0.923835	−0.461917	−	0.886923i	$-0.652838\pi$
$857$	−27.0449	−0.923835	−0.461917	+	0.886923i	$0.652838\pi$
$858$	0	0
$859$	−35.5906	−1.21434	−0.607168	−	0.794573i	$-0.707695\pi$
$859$	−35.5906	−1.21434	−0.607168	+	0.794573i	$0.707695\pi$
$860$	0.948078	0.0323292
$861$	0	0
$862$	−47.0579	−1.60280
$863$	19.7203	0.671285	0.335643	−	0.941989i	$-0.391047\pi$
$863$	19.7203	0.671285	0.335643	+	0.941989i	$0.391047\pi$
$864$	0	0
$865$	49.2081	1.67313
$866$	17.3012	0.587919
$867$	0	0
$868$	−60.8582	−2.06566
$869$	−15.6493	−0.530865
$870$	0	0
$871$	13.6162	0.461366
$872$	10.2129	0.345851
$873$	0	0
$874$	3.66033	0.123813
$875$	12.5145	0.423067
$876$	0	0
$877$	−21.3828	−0.722045	−0.361022	−	0.932557i	$-0.617572\pi$
$877$	−21.3828	−0.722045	−0.361022	+	0.932557i	$0.617572\pi$
$878$	13.6278	0.459917
$879$	0	0
$880$	−9.51698	−0.320817
$881$	34.7349	1.17025	0.585124	−	0.810944i	$-0.301046\pi$
$881$	34.7349	1.17025	0.585124	+	0.810944i	$0.301046\pi$
$882$	0	0
$883$	−23.9852	−0.807167	−0.403583	−	0.914943i	$-0.632235\pi$
$883$	−23.9852	−0.807167	−0.403583	+	0.914943i	$0.632235\pi$
$884$	−54.9041	−1.84662
$885$	0	0
$886$	−13.6238	−0.457700
$887$	37.2407	1.25042	0.625211	−	0.780456i	$-0.285013\pi$
$887$	37.2407	1.25042	0.625211	+	0.780456i	$0.285013\pi$
$888$	0	0
$889$	−37.5829	−1.26049
$890$	−84.6039	−2.83593
$891$	0	0
$892$	−18.7414	−0.627508
$893$	3.32045	0.111115
$894$	0	0
$895$	−55.6692	−1.86082
$896$	−21.4385	−0.716212
$897$	0	0
$898$	66.8006	2.22917
$899$	−56.9531	−1.89949
$900$	0	0
$901$	49.6441	1.65388
$902$	−24.9873	−0.831986
$903$	0	0
$904$	4.97163	0.165354
$905$	−57.7693	−1.92032
$906$	0	0
$907$	8.37749	0.278170	0.139085	−	0.990280i	$-0.455584\pi$
$907$	8.37749	0.278170	0.139085	+	0.990280i	$0.455584\pi$
$908$	−35.8284	−1.18901
$909$	0	0
$910$	−103.875	−3.44341
$911$	3.99772	0.132450	0.0662252	−	0.997805i	$-0.478904\pi$
$911$	3.99772	0.132450	0.0662252	+	0.997805i	$0.478904\pi$
$912$	0	0
$913$	10.8494	0.359064
$914$	−26.1470	−0.864865
$915$	0	0
$916$	14.7998	0.488999
$917$	−19.3497	−0.638982
$918$	0	0
$919$	9.13809	0.301438	0.150719	−	0.988577i	$-0.451841\pi$
$919$	9.13809	0.301438	0.150719	+	0.988577i	$0.451841\pi$
$920$	−12.0607	−0.397629
$921$	0	0
$922$	−68.1432	−2.24418
$923$	−17.7504	−0.584262
$924$	0	0
$925$	19.3842	0.637348
$926$	−21.8242	−0.717189
$927$	0	0
$928$	−67.1987	−2.20590
$929$	−51.7590	−1.69816	−0.849079	−	0.528267i	$-0.822842\pi$
$929$	−51.7590	−1.69816	−0.849079	+	0.528267i	$0.822842\pi$
$930$	0	0
$931$	2.28685	0.0749485
$932$	−2.46110	−0.0806159
$933$	0	0
$934$	12.8478	0.420393
$935$	−15.8151	−0.517209
$936$	0	0
$937$	−27.9536	−0.913203	−0.456601	−	0.889671i	$-0.650933\pi$
$937$	−27.9536	−0.913203	−0.456601	+	0.889671i	$0.650933\pi$
$938$	24.3355	0.794583
$939$	0	0
$940$	−74.4122	−2.42706
$941$	−47.4243	−1.54599	−0.772994	−	0.634414i	$-0.781242\pi$
$941$	−47.4243	−1.54599	−0.772994	+	0.634414i	$0.781242\pi$
$942$	0	0
$943$	67.4751	2.19729
$944$	−22.6008	−0.735595
$945$	0	0
$946$	0.282649	0.00918971
$947$	−19.9896	−0.649574	−0.324787	−	0.945787i	$-0.605293\pi$
$947$	−19.9896	−0.649574	−0.324787	+	0.945787i	$0.605293\pi$
$948$	0	0
$949$	−47.4398	−1.53996
$950$	2.53053	0.0821011
$951$	0	0
$952$	−14.4276	−0.467601
$953$	14.9169	0.483206	0.241603	−	0.970375i	$-0.422327\pi$
$953$	14.9169	0.483206	0.241603	+	0.970375i	$0.422327\pi$
$954$	0	0
$955$	29.5922	0.957582
$956$	46.4513	1.50234
$957$	0	0
$958$	31.0003	1.00157
$959$	−30.5611	−0.986868
$960$	0	0
$961$	16.0109	0.516481
$962$	45.8409	1.47797
$963$	0	0
$964$	16.0112	0.515685
$965$	−6.68498	−0.215197
$966$	0	0
$967$	28.9409	0.930675	0.465338	−	0.885133i	$-0.345933\pi$
$967$	28.9409	0.930675	0.465338	+	0.885133i	$0.345933\pi$
$968$	0.718591	0.0230964
$969$	0	0
$970$	−71.7960	−2.30523
$971$	−7.32733	−0.235145	−0.117573	−	0.993064i	$-0.537511\pi$
$971$	−7.32733	−0.235145	−0.117573	+	0.993064i	$0.537511\pi$
$972$	0	0
$973$	−63.0943	−2.02271
$974$	24.9011	0.797882
$975$	0	0
$976$	3.19166	0.102162
$977$	−5.07070	−0.162226	−0.0811131	−	0.996705i	$-0.525847\pi$
$977$	−5.07070	−0.162226	−0.0811131	+	0.996705i	$0.525847\pi$
$978$	0	0
$979$	−13.6121	−0.435045
$980$	−51.2490	−1.63709
$981$	0	0
$982$	22.4680	0.716982
$983$	−1.61384	−0.0514735	−0.0257368	−	0.999669i	$-0.508193\pi$
$983$	−1.61384	−0.0514735	−0.0257368	+	0.999669i	$0.508193\pi$
$984$	0	0
$985$	−5.39362	−0.171855
$986$	−91.8309	−2.92449
$987$	0	0
$988$	3.22959	0.102747
$989$	−0.763258	−0.0242702
$990$	0	0
$991$	−23.6526	−0.751350	−0.375675	−	0.926752i	$-0.622589\pi$
$991$	−23.6526	−0.751350	−0.375675	+	0.926752i	$0.622589\pi$
$992$	55.4679	1.76111
$993$	0	0
$994$	−31.7244	−1.00624
$995$	69.0378	2.18864
$996$	0	0
$997$	3.40590	0.107866	0.0539330	−	0.998545i	$-0.482824\pi$
$997$	3.40590	0.107866	0.0539330	+	0.998545i	$0.482824\pi$
$998$	−7.51456	−0.237869
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.17		19
3.2	odd	2		671.2.a.c.1.3	✓	19
33.32	even	2		7381.2.a.i.1.17		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.3	✓	19	3.2	odd	2
6039.2.a.k.1.17		19	1.1	even	1	trivial
7381.2.a.i.1.17		19	33.32	even	2

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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08441 q^{2} +2.34475 q^{4} -2.98183 q^{5} +3.78550 q^{7} +0.718591 q^{8} +O(q^{10})\)	\(q+2.08441 q^{2} +2.34475 q^{4} -2.98183 q^{5} +3.78550 q^{7} +0.718591 q^{8} -6.21535 q^{10} -1.00000 q^{11} +4.41489 q^{13} +7.89053 q^{14} -3.19166 q^{16} -5.30382 q^{17} +0.311984 q^{19} -6.99164 q^{20} -2.08441 q^{22} +5.62868 q^{23} +3.89132 q^{25} +9.20243 q^{26} +8.87605 q^{28} +8.30650 q^{29} -6.85645 q^{31} -8.08989 q^{32} -11.0553 q^{34} -11.2877 q^{35} +4.98139 q^{37} +0.650300 q^{38} -2.14272 q^{40} +11.9877 q^{41} -0.135602 q^{43} -2.34475 q^{44} +11.7324 q^{46} +10.6430 q^{47} +7.33004 q^{49} +8.11109 q^{50} +10.3518 q^{52} -9.36007 q^{53} +2.98183 q^{55} +2.72023 q^{56} +17.3141 q^{58} +7.08123 q^{59} -1.00000 q^{61} -14.2916 q^{62} -10.4793 q^{64} -13.1645 q^{65} +3.08414 q^{67} -12.4361 q^{68} -23.5282 q^{70} -4.02057 q^{71} -10.7454 q^{73} +10.3832 q^{74} +0.731522 q^{76} -3.78550 q^{77} +15.6493 q^{79} +9.51698 q^{80} +24.9873 q^{82} -10.8494 q^{83} +15.8151 q^{85} -0.282649 q^{86} -0.718591 q^{88} +13.6121 q^{89} +16.7126 q^{91} +13.1978 q^{92} +22.1844 q^{94} -0.930282 q^{95} +11.5514 q^{97} +15.2788 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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