LMFDB - Embedded newform 6028.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58174 q^{3} -1.88933 q^{5} +3.30001 q^{7} -0.498107 q^{9} +O(q^{10})$	$q-1.58174 q^{3} -1.88933 q^{5} +3.30001 q^{7} -0.498107 q^{9} -1.00000 q^{11} +6.53303 q^{13} +2.98842 q^{15} +3.23405 q^{17} +7.69079 q^{19} -5.21975 q^{21} +8.23717 q^{23} -1.43044 q^{25} +5.53309 q^{27} +9.00755 q^{29} -7.30339 q^{31} +1.58174 q^{33} -6.23480 q^{35} -8.63750 q^{37} -10.3335 q^{39} +9.80718 q^{41} +0.243468 q^{43} +0.941088 q^{45} -6.37815 q^{47} +3.89008 q^{49} -5.11542 q^{51} +8.36988 q^{53} +1.88933 q^{55} -12.1648 q^{57} -11.3401 q^{59} -7.24684 q^{61} -1.64376 q^{63} -12.3430 q^{65} -4.98815 q^{67} -13.0290 q^{69} +2.19115 q^{71} -2.42628 q^{73} +2.26258 q^{75} -3.30001 q^{77} +10.2652 q^{79} -7.25757 q^{81} +16.5003 q^{83} -6.11018 q^{85} -14.2476 q^{87} +6.56546 q^{89} +21.5591 q^{91} +11.5520 q^{93} -14.5304 q^{95} -9.38342 q^{97} +0.498107 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.58174	−0.913216	−0.456608	−	0.889668i	$-0.650936\pi$
$3$	−1.58174	−0.913216	−0.456608	+	0.889668i	$0.650936\pi$
$4$	0	0
$5$	−1.88933	−0.844933	−0.422466	−	0.906379i	$-0.638836\pi$
$5$	−1.88933	−0.844933	−0.422466	+	0.906379i	$0.638836\pi$
$6$	0	0
$7$	3.30001	1.24729	0.623644	−	0.781709i	$-0.285652\pi$
$7$	3.30001	1.24729	0.623644	+	0.781709i	$0.285652\pi$
$8$	0	0
$9$	−0.498107	−0.166036
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.53303	1.81194	0.905968	−	0.423346i	$-0.139144\pi$
$13$	6.53303	1.81194	0.905968	+	0.423346i	$0.139144\pi$
$14$	0	0
$15$	2.98842	0.771607
$16$	0	0
$17$	3.23405	0.784373	0.392186	−	0.919886i	$-0.371719\pi$
$17$	3.23405	0.784373	0.392186	+	0.919886i	$0.371719\pi$
$18$	0	0
$19$	7.69079	1.76439	0.882194	−	0.470886i	$-0.156066\pi$
$19$	7.69079	1.76439	0.882194	+	0.470886i	$0.156066\pi$
$20$	0	0
$21$	−5.21975	−1.13904
$22$	0	0
$23$	8.23717	1.71757	0.858784	−	0.512337i	$-0.171220\pi$
$23$	8.23717	1.71757	0.858784	+	0.512337i	$0.171220\pi$
$24$	0	0
$25$	−1.43044	−0.286088
$26$	0	0
$27$	5.53309	1.06484
$28$	0	0
$29$	9.00755	1.67266	0.836330	−	0.548227i	$-0.184697\pi$
$29$	9.00755	1.67266	0.836330	+	0.548227i	$0.184697\pi$
$30$	0	0
$31$	−7.30339	−1.31173	−0.655864	−	0.754879i	$-0.727695\pi$
$31$	−7.30339	−1.31173	−0.655864	+	0.754879i	$0.727695\pi$
$32$	0	0
$33$	1.58174	0.275345
$34$	0	0
$35$	−6.23480	−1.05387
$36$	0	0
$37$	−8.63750	−1.42000	−0.709998	−	0.704203i	$-0.751304\pi$
$37$	−8.63750	−1.42000	−0.709998	+	0.704203i	$0.751304\pi$
$38$	0	0
$39$	−10.3335	−1.65469
$40$	0	0
$41$	9.80718	1.53162	0.765812	−	0.643064i	$-0.222337\pi$
$41$	9.80718	1.53162	0.765812	+	0.643064i	$0.222337\pi$
$42$	0	0
$43$	0.243468	0.0371285	0.0185642	−	0.999828i	$-0.494090\pi$
$43$	0.243468	0.0371285	0.0185642	+	0.999828i	$0.494090\pi$
$44$	0	0
$45$	0.941088	0.140289
$46$	0	0
$47$	−6.37815	−0.930349	−0.465175	−	0.885219i	$-0.654008\pi$
$47$	−6.37815	−0.930349	−0.465175	+	0.885219i	$0.654008\pi$
$48$	0	0
$49$	3.89008	0.555725
$50$	0	0
$51$	−5.11542	−0.716302
$52$	0	0
$53$	8.36988	1.14969	0.574846	−	0.818262i	$-0.305062\pi$
$53$	8.36988	1.14969	0.574846	+	0.818262i	$0.305062\pi$
$54$	0	0
$55$	1.88933	0.254757
$56$	0	0
$57$	−12.1648	−1.61127
$58$	0	0
$59$	−11.3401	−1.47635	−0.738175	−	0.674609i	$-0.764312\pi$
$59$	−11.3401	−1.47635	−0.738175	+	0.674609i	$0.764312\pi$
$60$	0	0
$61$	−7.24684	−0.927863	−0.463931	−	0.885871i	$-0.653562\pi$
$61$	−7.24684	−0.927863	−0.463931	+	0.885871i	$0.653562\pi$
$62$	0	0
$63$	−1.64376	−0.207094
$64$	0	0
$65$	−12.3430	−1.53096
$66$	0	0
$67$	−4.98815	−0.609400	−0.304700	−	0.952448i	$-0.598556\pi$
$67$	−4.98815	−0.609400	−0.304700	+	0.952448i	$0.598556\pi$
$68$	0	0
$69$	−13.0290	−1.56851
$70$	0	0
$71$	2.19115	0.260041	0.130021	−	0.991511i	$-0.458496\pi$
$71$	2.19115	0.260041	0.130021	+	0.991511i	$0.458496\pi$
$72$	0	0
$73$	−2.42628	−0.283975	−0.141988	−	0.989868i	$-0.545349\pi$
$73$	−2.42628	−0.283975	−0.141988	+	0.989868i	$0.545349\pi$
$74$	0	0
$75$	2.26258	0.261261
$76$	0	0
$77$	−3.30001	−0.376071
$78$	0	0
$79$	10.2652	1.15493	0.577465	−	0.816416i	$-0.304042\pi$
$79$	10.2652	1.15493	0.577465	+	0.816416i	$0.304042\pi$
$80$	0	0
$81$	−7.25757	−0.806396
$82$	0	0
$83$	16.5003	1.81114	0.905570	−	0.424196i	$-0.139443\pi$
$83$	16.5003	1.81114	0.905570	+	0.424196i	$0.139443\pi$
$84$	0	0
$85$	−6.11018	−0.662742
$86$	0	0
$87$	−14.2476	−1.52750
$88$	0	0
$89$	6.56546	0.695937	0.347969	−	0.937506i	$-0.386872\pi$
$89$	6.56546	0.695937	0.347969	+	0.937506i	$0.386872\pi$
$90$	0	0
$91$	21.5591	2.26000
$92$	0	0
$93$	11.5520	1.19789
$94$	0	0
$95$	−14.5304	−1.49079
$96$	0	0
$97$	−9.38342	−0.952742	−0.476371	−	0.879244i	$-0.658048\pi$
$97$	−9.38342	−0.952742	−0.476371	+	0.879244i	$0.658048\pi$
$98$	0	0
$99$	0.498107	0.0500617
$100$	0	0
$101$	−7.18494	−0.714928	−0.357464	−	0.933927i	$-0.616359\pi$
$101$	−7.18494	−0.714928	−0.357464	+	0.933927i	$0.616359\pi$
$102$	0	0
$103$	4.52338	0.445702	0.222851	−	0.974853i	$-0.428464\pi$
$103$	4.52338	0.445702	0.222851	+	0.974853i	$0.428464\pi$
$104$	0	0
$105$	9.86182	0.962415
$106$	0	0
$107$	−10.8066	−1.04471	−0.522355	−	0.852728i	$-0.674946\pi$
$107$	−10.8066	−1.04471	−0.522355	+	0.852728i	$0.674946\pi$
$108$	0	0
$109$	2.14834	0.205773	0.102887	−	0.994693i	$-0.467192\pi$
$109$	2.14834	0.205773	0.102887	+	0.994693i	$0.467192\pi$
$110$	0	0
$111$	13.6623	1.29676
$112$	0	0
$113$	9.40325	0.884584	0.442292	−	0.896871i	$-0.354166\pi$
$113$	9.40325	0.884584	0.442292	+	0.896871i	$0.354166\pi$
$114$	0	0
$115$	−15.5627	−1.45123
$116$	0	0
$117$	−3.25415	−0.300846
$118$	0	0
$119$	10.6724	0.978338
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−15.5124	−1.39870
$124$	0	0
$125$	12.1492	1.08666
$126$	0	0
$127$	−8.78599	−0.779631	−0.389815	−	0.920893i	$-0.627461\pi$
$127$	−8.78599	−0.779631	−0.389815	+	0.920893i	$0.627461\pi$
$128$	0	0
$129$	−0.385102	−0.0339063
$130$	0	0
$131$	−16.3486	−1.42838	−0.714192	−	0.699950i	$-0.753206\pi$
$131$	−16.3486	−1.42838	−0.714192	+	0.699950i	$0.753206\pi$
$132$	0	0
$133$	25.3797	2.20070
$134$	0	0
$135$	−10.4538	−0.899721
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	12.7891	1.08476	0.542381	−	0.840133i	$-0.317523\pi$
$139$	12.7891	1.08476	0.542381	+	0.840133i	$0.317523\pi$
$140$	0	0
$141$	10.0886	0.849610
$142$	0	0
$143$	−6.53303	−0.546319
$144$	0	0
$145$	−17.0182	−1.41328
$146$	0	0
$147$	−6.15308	−0.507497
$148$	0	0
$149$	13.3019	1.08974	0.544868	−	0.838522i	$-0.316580\pi$
$149$	13.3019	1.08974	0.544868	+	0.838522i	$0.316580\pi$
$150$	0	0
$151$	−0.366794	−0.0298493	−0.0149246	−	0.999889i	$-0.504751\pi$
$151$	−0.366794	−0.0298493	−0.0149246	+	0.999889i	$0.504751\pi$
$152$	0	0
$153$	−1.61090	−0.130234
$154$	0	0
$155$	13.7985	1.10832
$156$	0	0
$157$	−14.8923	−1.18854	−0.594268	−	0.804267i	$-0.702558\pi$
$157$	−14.8923	−1.18854	−0.594268	+	0.804267i	$0.702558\pi$
$158$	0	0
$159$	−13.2390	−1.04992
$160$	0	0
$161$	27.1828	2.14230
$162$	0	0
$163$	3.53314	0.276737	0.138368	−	0.990381i	$-0.455814\pi$
$163$	3.53314	0.276737	0.138368	+	0.990381i	$0.455814\pi$
$164$	0	0
$165$	−2.98842	−0.232648
$166$	0	0
$167$	24.2257	1.87464	0.937320	−	0.348471i	$-0.113299\pi$
$167$	24.2257	1.87464	0.937320	+	0.348471i	$0.113299\pi$
$168$	0	0
$169$	29.6805	2.28311
$170$	0	0
$171$	−3.83084	−0.292952
$172$	0	0
$173$	−9.38009	−0.713155	−0.356578	−	0.934266i	$-0.616056\pi$
$173$	−9.38009	−0.713155	−0.356578	+	0.934266i	$0.616056\pi$
$174$	0	0
$175$	−4.72048	−0.356834
$176$	0	0
$177$	17.9370	1.34823
$178$	0	0
$179$	−22.8198	−1.70563	−0.852816	−	0.522211i	$-0.825107\pi$
$179$	−22.8198	−1.70563	−0.852816	+	0.522211i	$0.825107\pi$
$180$	0	0
$181$	−6.21564	−0.462005	−0.231003	−	0.972953i	$-0.574201\pi$
$181$	−6.21564	−0.462005	−0.231003	+	0.972953i	$0.574201\pi$
$182$	0	0
$183$	11.4626	0.847339
$184$	0	0
$185$	16.3191	1.19980
$186$	0	0
$187$	−3.23405	−0.236497
$188$	0	0
$189$	18.2593	1.32817
$190$	0	0
$191$	0.882563	0.0638600	0.0319300	−	0.999490i	$-0.489835\pi$
$191$	0.882563	0.0638600	0.0319300	+	0.999490i	$0.489835\pi$
$192$	0	0
$193$	−5.23180	−0.376593	−0.188297	−	0.982112i	$-0.560297\pi$
$193$	−5.23180	−0.376593	−0.188297	+	0.982112i	$0.560297\pi$
$194$	0	0
$195$	19.5234	1.39810
$196$	0	0
$197$	0.184988	0.0131798	0.00658992	−	0.999978i	$-0.497902\pi$
$197$	0.184988	0.0131798	0.00658992	+	0.999978i	$0.497902\pi$
$198$	0	0
$199$	15.9781	1.13266	0.566330	−	0.824179i	$-0.308363\pi$
$199$	15.9781	1.13266	0.566330	+	0.824179i	$0.308363\pi$
$200$	0	0
$201$	7.88994	0.556514
$202$	0	0
$203$	29.7250	2.08629
$204$	0	0
$205$	−18.5290	−1.29412
$206$	0	0
$207$	−4.10300	−0.285178
$208$	0	0
$209$	−7.69079	−0.531983
$210$	0	0
$211$	12.0701	0.830942	0.415471	−	0.909606i	$-0.363617\pi$
$211$	12.0701	0.830942	0.415471	+	0.909606i	$0.363617\pi$
$212$	0	0
$213$	−3.46582	−0.237474
$214$	0	0
$215$	−0.459990	−0.0313711
$216$	0	0
$217$	−24.1013	−1.63610
$218$	0	0
$219$	3.83774	0.259331
$220$	0	0
$221$	21.1281	1.42123
$222$	0	0
$223$	−11.8347	−0.792512	−0.396256	−	0.918140i	$-0.629691\pi$
$223$	−11.8347	−0.792512	−0.396256	+	0.918140i	$0.629691\pi$
$224$	0	0
$225$	0.712514	0.0475009
$226$	0	0
$227$	2.72610	0.180938	0.0904688	−	0.995899i	$-0.471163\pi$
$227$	2.72610	0.180938	0.0904688	+	0.995899i	$0.471163\pi$
$228$	0	0
$229$	−9.53103	−0.629828	−0.314914	−	0.949120i	$-0.601976\pi$
$229$	−9.53103	−0.629828	−0.314914	+	0.949120i	$0.601976\pi$
$230$	0	0
$231$	5.21975	0.343434
$232$	0	0
$233$	0.235262	0.0154125	0.00770627	−	0.999970i	$-0.497547\pi$
$233$	0.235262	0.0154125	0.00770627	+	0.999970i	$0.497547\pi$
$234$	0	0
$235$	12.0504	0.786083
$236$	0	0
$237$	−16.2369	−1.05470
$238$	0	0
$239$	13.4193	0.868020	0.434010	−	0.900908i	$-0.357098\pi$
$239$	13.4193	0.868020	0.434010	+	0.900908i	$0.357098\pi$
$240$	0	0
$241$	18.0948	1.16559	0.582796	−	0.812619i	$-0.301959\pi$
$241$	18.0948	1.16559	0.582796	+	0.812619i	$0.301959\pi$
$242$	0	0
$243$	−5.11970	−0.328429
$244$	0	0
$245$	−7.34963	−0.469551
$246$	0	0
$247$	50.2441	3.19696
$248$	0	0
$249$	−26.0991	−1.65396
$250$	0	0
$251$	−3.87201	−0.244399	−0.122199	−	0.992506i	$-0.538995\pi$
$251$	−3.87201	−0.244399	−0.122199	+	0.992506i	$0.538995\pi$
$252$	0	0
$253$	−8.23717	−0.517866
$254$	0	0
$255$	9.66470	0.605227
$256$	0	0
$257$	23.2433	1.44988	0.724940	−	0.688812i	$-0.241868\pi$
$257$	23.2433	1.44988	0.724940	+	0.688812i	$0.241868\pi$
$258$	0	0
$259$	−28.5039	−1.77114
$260$	0	0
$261$	−4.48673	−0.277721
$262$	0	0
$263$	5.00816	0.308816	0.154408	−	0.988007i	$-0.450653\pi$
$263$	5.00816	0.308816	0.154408	+	0.988007i	$0.450653\pi$
$264$	0	0
$265$	−15.8135	−0.971413
$266$	0	0
$267$	−10.3848	−0.635541
$268$	0	0
$269$	14.4914	0.883557	0.441779	−	0.897124i	$-0.354348\pi$
$269$	14.4914	0.883557	0.441779	+	0.897124i	$0.354348\pi$
$270$	0	0
$271$	−11.2408	−0.682831	−0.341416	−	0.939912i	$-0.610906\pi$
$271$	−11.2408	−0.682831	−0.341416	+	0.939912i	$0.610906\pi$
$272$	0	0
$273$	−34.1008	−2.06387
$274$	0	0
$275$	1.43044	0.0862589
$276$	0	0
$277$	−24.8897	−1.49548	−0.747739	−	0.663993i	$-0.768860\pi$
$277$	−24.8897	−1.49548	−0.747739	+	0.663993i	$0.768860\pi$
$278$	0	0
$279$	3.63787	0.217794
$280$	0	0
$281$	−12.9529	−0.772703	−0.386352	−	0.922352i	$-0.626265\pi$
$281$	−12.9529	−0.772703	−0.386352	+	0.922352i	$0.626265\pi$
$282$	0	0
$283$	−22.2792	−1.32436	−0.662180	−	0.749345i	$-0.730369\pi$
$283$	−22.2792	−1.32436	−0.662180	+	0.749345i	$0.730369\pi$
$284$	0	0
$285$	22.9833	1.36141
$286$	0	0
$287$	32.3638	1.91038
$288$	0	0
$289$	−6.54092	−0.384760
$290$	0	0
$291$	14.8421	0.870060
$292$	0	0
$293$	−15.7788	−0.921806	−0.460903	−	0.887450i	$-0.652474\pi$
$293$	−15.7788	−0.921806	−0.460903	+	0.887450i	$0.652474\pi$
$294$	0	0
$295$	21.4251	1.24742
$296$	0	0
$297$	−5.53309	−0.321062
$298$	0	0
$299$	53.8137	3.11213
$300$	0	0
$301$	0.803447	0.0463099
$302$	0	0
$303$	11.3647	0.652884
$304$	0	0
$305$	13.6917	0.783982
$306$	0	0
$307$	5.66491	0.323313	0.161657	−	0.986847i	$-0.448316\pi$
$307$	5.66491	0.323313	0.161657	+	0.986847i	$0.448316\pi$
$308$	0	0
$309$	−7.15479	−0.407022
$310$	0	0
$311$	−5.67145	−0.321599	−0.160799	−	0.986987i	$-0.551407\pi$
$311$	−5.67145	−0.321599	−0.160799	+	0.986987i	$0.551407\pi$
$312$	0	0
$313$	−29.8468	−1.68704	−0.843521	−	0.537096i	$-0.819521\pi$
$313$	−29.8468	−1.68704	−0.843521	+	0.537096i	$0.819521\pi$
$314$	0	0
$315$	3.10560	0.174981
$316$	0	0
$317$	−19.8827	−1.11673	−0.558363	−	0.829597i	$-0.688570\pi$
$317$	−19.8827	−1.11673	−0.558363	+	0.829597i	$0.688570\pi$
$318$	0	0
$319$	−9.00755	−0.504326
$320$	0	0
$321$	17.0931	0.954046
$322$	0	0
$323$	24.8724	1.38394
$324$	0	0
$325$	−9.34512	−0.518374
$326$	0	0
$327$	−3.39810	−0.187916
$328$	0	0
$329$	−21.0480	−1.16041
$330$	0	0
$331$	12.1333	0.666908	0.333454	−	0.942766i	$-0.391786\pi$
$331$	12.1333	0.666908	0.333454	+	0.942766i	$0.391786\pi$
$332$	0	0
$333$	4.30240	0.235770
$334$	0	0
$335$	9.42425	0.514902
$336$	0	0
$337$	−3.57810	−0.194911	−0.0974557	−	0.995240i	$-0.531070\pi$
$337$	−3.57810	−0.194911	−0.0974557	+	0.995240i	$0.531070\pi$
$338$	0	0
$339$	−14.8735	−0.807816
$340$	0	0
$341$	7.30339	0.395501
$342$	0	0
$343$	−10.2628	−0.554138
$344$	0	0
$345$	24.6161	1.32529
$346$	0	0
$347$	30.7812	1.65242	0.826211	−	0.563361i	$-0.190492\pi$
$347$	30.7812	1.65242	0.826211	+	0.563361i	$0.190492\pi$
$348$	0	0
$349$	8.62834	0.461864	0.230932	−	0.972970i	$-0.425822\pi$
$349$	8.62834	0.461864	0.230932	+	0.972970i	$0.425822\pi$
$350$	0	0
$351$	36.1478	1.92943
$352$	0	0
$353$	1.86790	0.0994184	0.0497092	−	0.998764i	$-0.484171\pi$
$353$	1.86790	0.0994184	0.0497092	+	0.998764i	$0.484171\pi$
$354$	0	0
$355$	−4.13979	−0.219717
$356$	0	0
$357$	−16.8809	−0.893434
$358$	0	0
$359$	34.2209	1.80611	0.903056	−	0.429523i	$-0.141318\pi$
$359$	34.2209	1.80611	0.903056	+	0.429523i	$0.141318\pi$
$360$	0	0
$361$	40.1482	2.11306
$362$	0	0
$363$	−1.58174	−0.0830197
$364$	0	0
$365$	4.58405	0.239940
$366$	0	0
$367$	25.3918	1.32544	0.662719	−	0.748868i	$-0.269402\pi$
$367$	25.3918	1.32544	0.662719	+	0.748868i	$0.269402\pi$
$368$	0	0
$369$	−4.88503	−0.254305
$370$	0	0
$371$	27.6207	1.43400
$372$	0	0
$373$	16.3373	0.845912	0.422956	−	0.906150i	$-0.360992\pi$
$373$	16.3373	0.845912	0.422956	+	0.906150i	$0.360992\pi$
$374$	0	0
$375$	−19.2169	−0.992354
$376$	0	0
$377$	58.8466	3.03075
$378$	0	0
$379$	−29.5040	−1.51552	−0.757759	−	0.652535i	$-0.773706\pi$
$379$	−29.5040	−1.51552	−0.757759	+	0.652535i	$0.773706\pi$
$380$	0	0
$381$	13.8971	0.711972
$382$	0	0
$383$	1.08213	0.0552943	0.0276471	−	0.999618i	$-0.491199\pi$
$383$	1.08213	0.0552943	0.0276471	+	0.999618i	$0.491199\pi$
$384$	0	0
$385$	6.23480	0.317755
$386$	0	0
$387$	−0.121273	−0.00616466
$388$	0	0
$389$	10.8697	0.551115	0.275557	−	0.961285i	$-0.411138\pi$
$389$	10.8697	0.551115	0.275557	+	0.961285i	$0.411138\pi$
$390$	0	0
$391$	26.6394	1.34721
$392$	0	0
$393$	25.8592	1.30442
$394$	0	0
$395$	−19.3944	−0.975838
$396$	0	0
$397$	−33.5658	−1.68462	−0.842310	−	0.538994i	$-0.818805\pi$
$397$	−33.5658	−1.68462	−0.842310	+	0.538994i	$0.818805\pi$
$398$	0	0
$399$	−40.1440	−2.00971
$400$	0	0
$401$	−11.2069	−0.559644	−0.279822	−	0.960052i	$-0.590275\pi$
$401$	−11.2069	−0.559644	−0.279822	+	0.960052i	$0.590275\pi$
$402$	0	0
$403$	−47.7133	−2.37677
$404$	0	0
$405$	13.7119	0.681351
$406$	0	0
$407$	8.63750	0.428145
$408$	0	0
$409$	1.70160	0.0841385	0.0420692	−	0.999115i	$-0.486605\pi$
$409$	1.70160	0.0841385	0.0420692	+	0.999115i	$0.486605\pi$
$410$	0	0
$411$	1.58174	0.0780213
$412$	0	0
$413$	−37.4223	−1.84143
$414$	0	0
$415$	−31.1744	−1.53029
$416$	0	0
$417$	−20.2291	−0.990622
$418$	0	0
$419$	8.68679	0.424377	0.212189	−	0.977229i	$-0.431941\pi$
$419$	8.68679	0.424377	0.212189	+	0.977229i	$0.431941\pi$
$420$	0	0
$421$	36.8668	1.79678	0.898389	−	0.439201i	$-0.144739\pi$
$421$	36.8668	1.79678	0.898389	+	0.439201i	$0.144739\pi$
$422$	0	0
$423$	3.17701	0.154471
$424$	0	0
$425$	−4.62612	−0.224400
$426$	0	0
$427$	−23.9147	−1.15731
$428$	0	0
$429$	10.3335	0.498908
$430$	0	0
$431$	23.9577	1.15400	0.577001	−	0.816743i	$-0.304223\pi$
$431$	23.9577	1.15400	0.577001	+	0.816743i	$0.304223\pi$
$432$	0	0
$433$	3.76845	0.181100	0.0905501	−	0.995892i	$-0.471137\pi$
$433$	3.76845	0.181100	0.0905501	+	0.995892i	$0.471137\pi$
$434$	0	0
$435$	26.9183	1.29063
$436$	0	0
$437$	63.3503	3.03046
$438$	0	0
$439$	32.3785	1.54534	0.772670	−	0.634808i	$-0.218921\pi$
$439$	32.3785	1.54534	0.772670	+	0.634808i	$0.218921\pi$
$440$	0	0
$441$	−1.93768	−0.0922703
$442$	0	0
$443$	36.4213	1.73043	0.865214	−	0.501403i	$-0.167183\pi$
$443$	36.4213	1.73043	0.865214	+	0.501403i	$0.167183\pi$
$444$	0	0
$445$	−12.4043	−0.588020
$446$	0	0
$447$	−21.0402	−0.995165
$448$	0	0
$449$	−15.6638	−0.739220	−0.369610	−	0.929187i	$-0.620509\pi$
$449$	−15.6638	−0.739220	−0.369610	+	0.929187i	$0.620509\pi$
$450$	0	0
$451$	−9.80718	−0.461802
$452$	0	0
$453$	0.580172	0.0272589
$454$	0	0
$455$	−40.7321	−1.90955
$456$	0	0
$457$	−36.6517	−1.71449	−0.857247	−	0.514905i	$-0.827827\pi$
$457$	−36.6517	−1.71449	−0.857247	+	0.514905i	$0.827827\pi$
$458$	0	0
$459$	17.8943	0.835234
$460$	0	0
$461$	14.0687	0.655247	0.327623	−	0.944808i	$-0.393752\pi$
$461$	14.0687	0.655247	0.327623	+	0.944808i	$0.393752\pi$
$462$	0	0
$463$	10.0120	0.465299	0.232649	−	0.972561i	$-0.425261\pi$
$463$	10.0120	0.465299	0.232649	+	0.972561i	$0.425261\pi$
$464$	0	0
$465$	−21.8256	−1.01214
$466$	0	0
$467$	−6.25323	−0.289365	−0.144683	−	0.989478i	$-0.546216\pi$
$467$	−6.25323	−0.289365	−0.144683	+	0.989478i	$0.546216\pi$
$468$	0	0
$469$	−16.4610	−0.760096
$470$	0	0
$471$	23.5557	1.08539
$472$	0	0
$473$	−0.243468	−0.0111947
$474$	0	0
$475$	−11.0012	−0.504771
$476$	0	0
$477$	−4.16910	−0.190890
$478$	0	0
$479$	5.69884	0.260386	0.130193	−	0.991489i	$-0.458440\pi$
$479$	5.69884	0.260386	0.130193	+	0.991489i	$0.458440\pi$
$480$	0	0
$481$	−56.4290	−2.57294
$482$	0	0
$483$	−42.9960	−1.95638
$484$	0	0
$485$	17.7284	0.805003
$486$	0	0
$487$	20.7094	0.938431	0.469216	−	0.883084i	$-0.344537\pi$
$487$	20.7094	0.938431	0.469216	+	0.883084i	$0.344537\pi$
$488$	0	0
$489$	−5.58850	−0.252721
$490$	0	0
$491$	−25.8039	−1.16451	−0.582256	−	0.813005i	$-0.697830\pi$
$491$	−25.8039	−1.16451	−0.582256	+	0.813005i	$0.697830\pi$
$492$	0	0
$493$	29.1309	1.31199
$494$	0	0
$495$	−0.941088	−0.0422988
$496$	0	0
$497$	7.23081	0.324346
$498$	0	0
$499$	22.9963	1.02946	0.514728	−	0.857354i	$-0.327893\pi$
$499$	22.9963	1.02946	0.514728	+	0.857354i	$0.327893\pi$
$500$	0	0
$501$	−38.3187	−1.71195
$502$	0	0
$503$	−40.4793	−1.80488	−0.902442	−	0.430811i	$-0.858227\pi$
$503$	−40.4793	−1.80488	−0.902442	+	0.430811i	$0.858227\pi$
$504$	0	0
$505$	13.5747	0.604066
$506$	0	0
$507$	−46.9467	−2.08498
$508$	0	0
$509$	−38.5287	−1.70776	−0.853878	−	0.520474i	$-0.825755\pi$
$509$	−38.5287	−1.70776	−0.853878	+	0.520474i	$0.825755\pi$
$510$	0	0
$511$	−8.00677	−0.354199
$512$	0	0
$513$	42.5538	1.87880
$514$	0	0
$515$	−8.54614	−0.376588
$516$	0	0
$517$	6.37815	0.280511
$518$	0	0
$519$	14.8368	0.651265
$520$	0	0
$521$	−3.00901	−0.131827	−0.0659135	−	0.997825i	$-0.520996\pi$
$521$	−3.00901	−0.131827	−0.0659135	+	0.997825i	$0.520996\pi$
$522$	0	0
$523$	7.69363	0.336419	0.168210	−	0.985751i	$-0.446201\pi$
$523$	7.69363	0.336419	0.168210	+	0.985751i	$0.446201\pi$
$524$	0	0
$525$	7.46655	0.325867
$526$	0	0
$527$	−23.6195	−1.02888
$528$	0	0
$529$	44.8510	1.95004
$530$	0	0
$531$	5.64857	0.245127
$532$	0	0
$533$	64.0706	2.77521
$534$	0	0
$535$	20.4171	0.882710
$536$	0	0
$537$	36.0949	1.55761
$538$	0	0
$539$	−3.89008	−0.167557
$540$	0	0
$541$	−2.27947	−0.0980019	−0.0490010	−	0.998799i	$-0.515604\pi$
$541$	−2.27947	−0.0980019	−0.0490010	+	0.998799i	$0.515604\pi$
$542$	0	0
$543$	9.83151	0.421911
$544$	0	0
$545$	−4.05891	−0.173865
$546$	0	0
$547$	−39.3365	−1.68191	−0.840955	−	0.541106i	$-0.818006\pi$
$547$	−39.3365	−1.68191	−0.840955	+	0.541106i	$0.818006\pi$
$548$	0	0
$549$	3.60970	0.154058
$550$	0	0
$551$	69.2751	2.95122
$552$	0	0
$553$	33.8754	1.44053
$554$	0	0
$555$	−25.8125	−1.09568
$556$	0	0
$557$	28.8961	1.22437	0.612184	−	0.790715i	$-0.290291\pi$
$557$	28.8961	1.22437	0.612184	+	0.790715i	$0.290291\pi$
$558$	0	0
$559$	1.59058	0.0672745
$560$	0	0
$561$	5.11542	0.215973
$562$	0	0
$563$	4.73633	0.199613	0.0998063	−	0.995007i	$-0.468178\pi$
$563$	4.73633	0.199613	0.0998063	+	0.995007i	$0.468178\pi$
$564$	0	0
$565$	−17.7658	−0.747414
$566$	0	0
$567$	−23.9501	−1.00581
$568$	0	0
$569$	−0.534479	−0.0224065	−0.0112033	−	0.999937i	$-0.503566\pi$
$569$	−0.534479	−0.0224065	−0.0112033	+	0.999937i	$0.503566\pi$
$570$	0	0
$571$	7.87567	0.329587	0.164793	−	0.986328i	$-0.447304\pi$
$571$	7.87567	0.329587	0.164793	+	0.986328i	$0.447304\pi$
$572$	0	0
$573$	−1.39598	−0.0583180
$574$	0	0
$575$	−11.7828	−0.491377
$576$	0	0
$577$	12.5520	0.522548	0.261274	−	0.965265i	$-0.415857\pi$
$577$	12.5520	0.522548	0.261274	+	0.965265i	$0.415857\pi$
$578$	0	0
$579$	8.27533	0.343911
$580$	0	0
$581$	54.4511	2.25901
$582$	0	0
$583$	−8.36988	−0.346645
$584$	0	0
$585$	6.14816	0.254195
$586$	0	0
$587$	33.8520	1.39722	0.698610	−	0.715503i	$-0.253802\pi$
$587$	33.8520	1.39722	0.698610	+	0.715503i	$0.253802\pi$
$588$	0	0
$589$	−56.1688	−2.31440
$590$	0	0
$591$	−0.292602	−0.0120360
$592$	0	0
$593$	15.4600	0.634865	0.317433	−	0.948281i	$-0.397179\pi$
$593$	15.4600	0.634865	0.317433	+	0.948281i	$0.397179\pi$
$594$	0	0
$595$	−20.1637	−0.826630
$596$	0	0
$597$	−25.2732	−1.03436
$598$	0	0
$599$	−22.9966	−0.939614	−0.469807	−	0.882769i	$-0.655676\pi$
$599$	−22.9966	−0.939614	−0.469807	+	0.882769i	$0.655676\pi$
$600$	0	0
$601$	−18.9106	−0.771379	−0.385690	−	0.922629i	$-0.626036\pi$
$601$	−18.9106	−0.771379	−0.385690	+	0.922629i	$0.626036\pi$
$602$	0	0
$603$	2.48463	0.101182
$604$	0	0
$605$	−1.88933	−0.0768121
$606$	0	0
$607$	46.6574	1.89376	0.946882	−	0.321581i	$-0.104214\pi$
$607$	46.6574	1.89376	0.946882	+	0.321581i	$0.104214\pi$
$608$	0	0
$609$	−47.0172	−1.90523
$610$	0	0
$611$	−41.6687	−1.68573
$612$	0	0
$613$	−15.8397	−0.639758	−0.319879	−	0.947458i	$-0.603642\pi$
$613$	−15.8397	−0.639758	−0.319879	+	0.947458i	$0.603642\pi$
$614$	0	0
$615$	29.3080	1.18181
$616$	0	0
$617$	18.8098	0.757254	0.378627	−	0.925549i	$-0.376396\pi$
$617$	18.8098	0.757254	0.378627	+	0.925549i	$0.376396\pi$
$618$	0	0
$619$	−3.44139	−0.138321	−0.0691606	−	0.997606i	$-0.522032\pi$
$619$	−3.44139	−0.138321	−0.0691606	+	0.997606i	$0.522032\pi$
$620$	0	0
$621$	45.5770	1.82894
$622$	0	0
$623$	21.6661	0.868034
$624$	0	0
$625$	−15.8016	−0.632065
$626$	0	0
$627$	12.1648	0.485816
$628$	0	0
$629$	−27.9341	−1.11381
$630$	0	0
$631$	36.0671	1.43581	0.717905	−	0.696141i	$-0.245101\pi$
$631$	36.0671	1.43581	0.717905	+	0.696141i	$0.245101\pi$
$632$	0	0
$633$	−19.0918	−0.758830
$634$	0	0
$635$	16.5996	0.658736
$636$	0	0
$637$	25.4140	1.00694
$638$	0	0
$639$	−1.09143	−0.0431762
$640$	0	0
$641$	−17.9436	−0.708729	−0.354365	−	0.935107i	$-0.615303\pi$
$641$	−17.9436	−0.708729	−0.354365	+	0.935107i	$0.615303\pi$
$642$	0	0
$643$	−1.14950	−0.0453319	−0.0226660	−	0.999743i	$-0.507215\pi$
$643$	−1.14950	−0.0453319	−0.0226660	+	0.999743i	$0.507215\pi$
$644$	0	0
$645$	0.727584	0.0286486
$646$	0	0
$647$	31.5381	1.23989	0.619945	−	0.784645i	$-0.287155\pi$
$647$	31.5381	1.23989	0.619945	+	0.784645i	$0.287155\pi$
$648$	0	0
$649$	11.3401	0.445136
$650$	0	0
$651$	38.1219	1.49411
$652$	0	0
$653$	−29.7143	−1.16281	−0.581405	−	0.813614i	$-0.697497\pi$
$653$	−29.7143	−1.16281	−0.581405	+	0.813614i	$0.697497\pi$
$654$	0	0
$655$	30.8879	1.20689
$656$	0	0
$657$	1.20855	0.0471500
$658$	0	0
$659$	1.53995	0.0599878	0.0299939	−	0.999550i	$-0.490451\pi$
$659$	1.53995	0.0599878	0.0299939	+	0.999550i	$0.490451\pi$
$660$	0	0
$661$	11.2119	0.436092	0.218046	−	0.975938i	$-0.430032\pi$
$661$	11.2119	0.436092	0.218046	+	0.975938i	$0.430032\pi$
$662$	0	0
$663$	−33.4192	−1.29789
$664$	0	0
$665$	−47.9505	−1.85944
$666$	0	0
$667$	74.1967	2.87291
$668$	0	0
$669$	18.7194	0.723735
$670$	0	0
$671$	7.24684	0.279761
$672$	0	0
$673$	−34.2609	−1.32066	−0.660330	−	0.750975i	$-0.729584\pi$
$673$	−34.2609	−1.32066	−0.660330	+	0.750975i	$0.729584\pi$
$674$	0	0
$675$	−7.91476	−0.304639
$676$	0	0
$677$	−27.2540	−1.04745	−0.523727	−	0.851886i	$-0.675459\pi$
$677$	−27.2540	−1.04745	−0.523727	+	0.851886i	$0.675459\pi$
$678$	0	0
$679$	−30.9654	−1.18834
$680$	0	0
$681$	−4.31197	−0.165235
$682$	0	0
$683$	8.52064	0.326033	0.163017	−	0.986623i	$-0.447878\pi$
$683$	8.52064	0.326033	0.163017	+	0.986623i	$0.447878\pi$
$684$	0	0
$685$	1.88933	0.0721875
$686$	0	0
$687$	15.0756	0.575170
$688$	0	0
$689$	54.6807	2.08317
$690$	0	0
$691$	−20.9272	−0.796109	−0.398055	−	0.917362i	$-0.630315\pi$
$691$	−20.9272	−0.796109	−0.398055	+	0.917362i	$0.630315\pi$
$692$	0	0
$693$	1.64376	0.0624413
$694$	0	0
$695$	−24.1629	−0.916551
$696$	0	0
$697$	31.7169	1.20136
$698$	0	0
$699$	−0.372123	−0.0140750
$700$	0	0
$701$	0.112863	0.00426278	0.00213139	−	0.999998i	$-0.499322\pi$
$701$	0.112863	0.00426278	0.00213139	+	0.999998i	$0.499322\pi$
$702$	0	0
$703$	−66.4292	−2.50542
$704$	0	0
$705$	−19.0606	−0.717864
$706$	0	0
$707$	−23.7104	−0.891721
$708$	0	0
$709$	15.9935	0.600649	0.300324	−	0.953837i	$-0.402905\pi$
$709$	15.9935	0.600649	0.300324	+	0.953837i	$0.402905\pi$
$710$	0	0
$711$	−5.11319	−0.191760
$712$	0	0
$713$	−60.1593	−2.25298
$714$	0	0
$715$	12.3430	0.461603
$716$	0	0
$717$	−21.2258	−0.792690
$718$	0	0
$719$	12.1230	0.452110	0.226055	−	0.974114i	$-0.427417\pi$
$719$	12.1230	0.452110	0.226055	+	0.974114i	$0.427417\pi$
$720$	0	0
$721$	14.9272	0.555918
$722$	0	0
$723$	−28.6213	−1.06444
$724$	0	0
$725$	−12.8848	−0.478529
$726$	0	0
$727$	21.7666	0.807280	0.403640	−	0.914918i	$-0.367745\pi$
$727$	21.7666	0.807280	0.403640	+	0.914918i	$0.367745\pi$
$728$	0	0
$729$	29.8707	1.10632
$730$	0	0
$731$	0.787387	0.0291226
$732$	0	0
$733$	44.9308	1.65956	0.829778	−	0.558094i	$-0.188467\pi$
$733$	44.9308	1.65956	0.829778	+	0.558094i	$0.188467\pi$
$734$	0	0
$735$	11.6252	0.428801
$736$	0	0
$737$	4.98815	0.183741
$738$	0	0
$739$	−21.6202	−0.795312	−0.397656	−	0.917535i	$-0.630176\pi$
$739$	−21.6202	−0.795312	−0.397656	+	0.917535i	$0.630176\pi$
$740$	0	0
$741$	−79.4730	−2.91951
$742$	0	0
$743$	29.4068	1.07883	0.539415	−	0.842040i	$-0.318645\pi$
$743$	29.4068	1.07883	0.539415	+	0.842040i	$0.318645\pi$
$744$	0	0
$745$	−25.1317	−0.920754
$746$	0	0
$747$	−8.21891	−0.300714
$748$	0	0
$749$	−35.6618	−1.30305
$750$	0	0
$751$	−12.1309	−0.442661	−0.221331	−	0.975199i	$-0.571040\pi$
$751$	−12.1309	−0.442661	−0.221331	+	0.975199i	$0.571040\pi$
$752$	0	0
$753$	6.12450	0.223189
$754$	0	0
$755$	0.692994	0.0252206
$756$	0	0
$757$	−35.2821	−1.28235	−0.641176	−	0.767394i	$-0.721553\pi$
$757$	−35.2821	−1.28235	−0.641176	+	0.767394i	$0.721553\pi$
$758$	0	0
$759$	13.0290	0.472924
$760$	0	0
$761$	−36.0970	−1.30851	−0.654257	−	0.756272i	$-0.727019\pi$
$761$	−36.0970	−1.30851	−0.654257	+	0.756272i	$0.727019\pi$
$762$	0	0
$763$	7.08953	0.256658
$764$	0	0
$765$	3.04353	0.110039
$766$	0	0
$767$	−74.0849	−2.67505
$768$	0	0
$769$	−49.6741	−1.79129	−0.895647	−	0.444765i	$-0.853287\pi$
$769$	−49.6741	−1.79129	−0.895647	+	0.444765i	$0.853287\pi$
$770$	0	0
$771$	−36.7648	−1.32405
$772$	0	0
$773$	−35.7732	−1.28667	−0.643335	−	0.765584i	$-0.722450\pi$
$773$	−35.7732	−1.28667	−0.643335	+	0.765584i	$0.722450\pi$
$774$	0	0
$775$	10.4471	0.375270
$776$	0	0
$777$	45.0856	1.61744
$778$	0	0
$779$	75.4250	2.70238
$780$	0	0
$781$	−2.19115	−0.0784054
$782$	0	0
$783$	49.8395	1.78112
$784$	0	0
$785$	28.1365	1.00423
$786$	0	0
$787$	−19.4828	−0.694485	−0.347243	−	0.937775i	$-0.612882\pi$
$787$	−19.4828	−0.694485	−0.347243	+	0.937775i	$0.612882\pi$
$788$	0	0
$789$	−7.92159	−0.282016
$790$	0	0
$791$	31.0309	1.10333
$792$	0	0
$793$	−47.3438	−1.68123
$794$	0	0
$795$	25.0127	0.887110
$796$	0	0
$797$	−6.28033	−0.222461	−0.111230	−	0.993795i	$-0.535479\pi$
$797$	−6.28033	−0.222461	−0.111230	+	0.993795i	$0.535479\pi$
$798$	0	0
$799$	−20.6273	−0.729741
$800$	0	0
$801$	−3.27030	−0.115551
$802$	0	0
$803$	2.42628	0.0856217
$804$	0	0
$805$	−51.3571	−1.81010
$806$	0	0
$807$	−22.9216	−0.806879
$808$	0	0
$809$	25.8362	0.908352	0.454176	−	0.890912i	$-0.349934\pi$
$809$	25.8362	0.908352	0.454176	+	0.890912i	$0.349934\pi$
$810$	0	0
$811$	3.24613	0.113987	0.0569935	−	0.998375i	$-0.481849\pi$
$811$	3.24613	0.113987	0.0569935	+	0.998375i	$0.481849\pi$
$812$	0	0
$813$	17.7800	0.623573
$814$	0	0
$815$	−6.67525	−0.233824
$816$	0	0
$817$	1.87246	0.0655091
$818$	0	0
$819$	−10.7387	−0.375242
$820$	0	0
$821$	−33.1756	−1.15784	−0.578919	−	0.815385i	$-0.696525\pi$
$821$	−33.1756	−1.15784	−0.578919	+	0.815385i	$0.696525\pi$
$822$	0	0
$823$	7.17288	0.250031	0.125015	−	0.992155i	$-0.460102\pi$
$823$	7.17288	0.250031	0.125015	+	0.992155i	$0.460102\pi$
$824$	0	0
$825$	−2.26258	−0.0787731
$826$	0	0
$827$	−8.76632	−0.304835	−0.152417	−	0.988316i	$-0.548706\pi$
$827$	−8.76632	−0.304835	−0.152417	+	0.988316i	$0.548706\pi$
$828$	0	0
$829$	−4.91410	−0.170674	−0.0853369	−	0.996352i	$-0.527197\pi$
$829$	−4.91410	−0.170674	−0.0853369	+	0.996352i	$0.527197\pi$
$830$	0	0
$831$	39.3690	1.36569
$832$	0	0
$833$	12.5807	0.435896
$834$	0	0
$835$	−45.7702	−1.58394
$836$	0	0
$837$	−40.4103	−1.39678
$838$	0	0
$839$	4.04256	0.139565	0.0697823	−	0.997562i	$-0.477770\pi$
$839$	4.04256	0.139565	0.0697823	+	0.997562i	$0.477770\pi$
$840$	0	0
$841$	52.1359	1.79779
$842$	0	0
$843$	20.4880	0.705645
$844$	0	0
$845$	−56.0761	−1.92908
$846$	0	0
$847$	3.30001	0.113390
$848$	0	0
$849$	35.2398	1.20943
$850$	0	0
$851$	−71.1486	−2.43894
$852$	0	0
$853$	3.39867	0.116368	0.0581842	−	0.998306i	$-0.481469\pi$
$853$	3.39867	0.116368	0.0581842	+	0.998306i	$0.481469\pi$
$854$	0	0
$855$	7.23771	0.247524
$856$	0	0
$857$	−31.3041	−1.06933	−0.534665	−	0.845064i	$-0.679562\pi$
$857$	−31.3041	−1.06933	−0.534665	+	0.845064i	$0.679562\pi$
$858$	0	0
$859$	−46.1203	−1.57361	−0.786803	−	0.617204i	$-0.788265\pi$
$859$	−46.1203	−1.57361	−0.786803	+	0.617204i	$0.788265\pi$
$860$	0	0
$861$	−51.1911	−1.74459
$862$	0	0
$863$	−42.1697	−1.43547	−0.717736	−	0.696315i	$-0.754822\pi$
$863$	−42.1697	−1.43547	−0.717736	+	0.696315i	$0.754822\pi$
$864$	0	0
$865$	17.7221	0.602568
$866$	0	0
$867$	10.3460	0.351369
$868$	0	0
$869$	−10.2652	−0.348224
$870$	0	0
$871$	−32.5877	−1.10419
$872$	0	0
$873$	4.67395	0.158189
$874$	0	0
$875$	40.0925	1.35538
$876$	0	0
$877$	−11.8517	−0.400203	−0.200102	−	0.979775i	$-0.564127\pi$
$877$	−11.8517	−0.400203	−0.200102	+	0.979775i	$0.564127\pi$
$878$	0	0
$879$	24.9579	0.841809
$880$	0	0
$881$	1.37716	0.0463978	0.0231989	−	0.999731i	$-0.492615\pi$
$881$	1.37716	0.0463978	0.0231989	+	0.999731i	$0.492615\pi$
$882$	0	0
$883$	−52.5277	−1.76770	−0.883848	−	0.467774i	$-0.845056\pi$
$883$	−52.5277	−1.76770	−0.883848	+	0.467774i	$0.845056\pi$
$884$	0	0
$885$	−33.8889	−1.13916
$886$	0	0
$887$	37.5583	1.26108	0.630542	−	0.776155i	$-0.282833\pi$
$887$	37.5583	1.26108	0.630542	+	0.776155i	$0.282833\pi$
$888$	0	0
$889$	−28.9939	−0.972424
$890$	0	0
$891$	7.25757	0.243138
$892$	0	0
$893$	−49.0530	−1.64150
$894$	0	0
$895$	43.1141	1.44114
$896$	0	0
$897$	−85.1191	−2.84204
$898$	0	0
$899$	−65.7856	−2.19407
$900$	0	0
$901$	27.0686	0.901787
$902$	0	0
$903$	−1.27084	−0.0422909
$904$	0	0
$905$	11.7434	0.390363
$906$	0	0
$907$	32.7075	1.08603	0.543017	−	0.839722i	$-0.317282\pi$
$907$	32.7075	1.08603	0.543017	+	0.839722i	$0.317282\pi$
$908$	0	0
$909$	3.57887	0.118704
$910$	0	0
$911$	−39.5968	−1.31190	−0.655951	−	0.754804i	$-0.727732\pi$
$911$	−39.5968	−1.31190	−0.655951	+	0.754804i	$0.727732\pi$
$912$	0	0
$913$	−16.5003	−0.546080
$914$	0	0
$915$	−21.6566	−0.715945
$916$	0	0
$917$	−53.9506	−1.78161
$918$	0	0
$919$	11.9037	0.392666	0.196333	−	0.980537i	$-0.437097\pi$
$919$	11.9037	0.392666	0.196333	+	0.980537i	$0.437097\pi$
$920$	0	0
$921$	−8.96040	−0.295255
$922$	0	0
$923$	14.3148	0.471178
$924$	0	0
$925$	12.3554	0.406245
$926$	0	0
$927$	−2.25313	−0.0740024
$928$	0	0
$929$	−22.9830	−0.754047	−0.377023	−	0.926204i	$-0.623052\pi$
$929$	−22.9830	−0.754047	−0.377023	+	0.926204i	$0.623052\pi$
$930$	0	0
$931$	29.9178	0.980515
$932$	0	0
$933$	8.97075	0.293689
$934$	0	0
$935$	6.11018	0.199824
$936$	0	0
$937$	20.9464	0.684288	0.342144	−	0.939648i	$-0.388847\pi$
$937$	20.9464	0.684288	0.342144	+	0.939648i	$0.388847\pi$
$938$	0	0
$939$	47.2098	1.54063
$940$	0	0
$941$	−32.2244	−1.05049	−0.525243	−	0.850952i	$-0.676026\pi$
$941$	−32.2244	−1.05049	−0.525243	+	0.850952i	$0.676026\pi$
$942$	0	0
$943$	80.7834	2.63067
$944$	0	0
$945$	−34.4977	−1.12221
$946$	0	0
$947$	14.8009	0.480966	0.240483	−	0.970653i	$-0.422694\pi$
$947$	14.8009	0.480966	0.240483	+	0.970653i	$0.422694\pi$
$948$	0	0
$949$	−15.8510	−0.514545
$950$	0	0
$951$	31.4492	1.01981
$952$	0	0
$953$	−8.75524	−0.283610	−0.141805	−	0.989895i	$-0.545291\pi$
$953$	−8.75524	−0.283610	−0.141805	+	0.989895i	$0.545291\pi$
$954$	0	0
$955$	−1.66745	−0.0539574
$956$	0	0
$957$	14.2476	0.460559
$958$	0	0
$959$	−3.30001	−0.106563
$960$	0	0
$961$	22.3395	0.720629
$962$	0	0
$963$	5.38283	0.173459
$964$	0	0
$965$	9.88459	0.318196
$966$	0	0
$967$	−19.7108	−0.633857	−0.316929	−	0.948449i	$-0.602652\pi$
$967$	−19.7108	−0.633857	−0.316929	+	0.948449i	$0.602652\pi$
$968$	0	0
$969$	−39.3416	−1.26383
$970$	0	0
$971$	17.1264	0.549612	0.274806	−	0.961500i	$-0.411386\pi$
$971$	17.1264	0.549612	0.274806	+	0.961500i	$0.411386\pi$
$972$	0	0
$973$	42.2043	1.35301
$974$	0	0
$975$	14.7815	0.473388
$976$	0	0
$977$	−61.1703	−1.95701	−0.978506	−	0.206220i	$-0.933884\pi$
$977$	−61.1703	−1.95701	−0.978506	+	0.206220i	$0.933884\pi$
$978$	0	0
$979$	−6.56546	−0.209833
$980$	0	0
$981$	−1.07010	−0.0341657
$982$	0	0
$983$	19.1940	0.612193	0.306097	−	0.952000i	$-0.400977\pi$
$983$	19.1940	0.612193	0.306097	+	0.952000i	$0.400977\pi$
$984$	0	0
$985$	−0.349503	−0.0111361
$986$	0	0
$987$	33.2924	1.05971
$988$	0	0
$989$	2.00549	0.0637707
$990$	0	0
$991$	−5.92562	−0.188233	−0.0941167	−	0.995561i	$-0.530003\pi$
$991$	−5.92562	−0.188233	−0.0941167	+	0.995561i	$0.530003\pi$
$992$	0	0
$993$	−19.1917	−0.609031
$994$	0	0
$995$	−30.1879	−0.957021
$996$	0	0
$997$	−20.2319	−0.640752	−0.320376	−	0.947291i	$-0.603809\pi$
$997$	−20.2319	−0.640752	−0.320376	+	0.947291i	$0.603809\pi$
$998$	0	0
$999$	−47.7920	−1.51207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.7	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.7	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-1.58174 q^{3} -1.88933 q^{5} +3.30001 q^{7} -0.498107 q^{9} +O(q^{10})\)	\(q-1.58174 q^{3} -1.88933 q^{5} +3.30001 q^{7} -0.498107 q^{9} -1.00000 q^{11} +6.53303 q^{13} +2.98842 q^{15} +3.23405 q^{17} +7.69079 q^{19} -5.21975 q^{21} +8.23717 q^{23} -1.43044 q^{25} +5.53309 q^{27} +9.00755 q^{29} -7.30339 q^{31} +1.58174 q^{33} -6.23480 q^{35} -8.63750 q^{37} -10.3335 q^{39} +9.80718 q^{41} +0.243468 q^{43} +0.941088 q^{45} -6.37815 q^{47} +3.89008 q^{49} -5.11542 q^{51} +8.36988 q^{53} +1.88933 q^{55} -12.1648 q^{57} -11.3401 q^{59} -7.24684 q^{61} -1.64376 q^{63} -12.3430 q^{65} -4.98815 q^{67} -13.0290 q^{69} +2.19115 q^{71} -2.42628 q^{73} +2.26258 q^{75} -3.30001 q^{77} +10.2652 q^{79} -7.25757 q^{81} +16.5003 q^{83} -6.11018 q^{85} -14.2476 q^{87} +6.56546 q^{89} +21.5591 q^{91} +11.5520 q^{93} -14.5304 q^{95} -9.38342 q^{97} +0.498107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

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Database

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