LMFDB - Embedded newform 6024.2.a.r.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 $ x^20 - 9x^19 - 31x^18 + 471x^17 - 82x^16 - 9476x^15 + 12881x^14 + 94079x^13 - 181272x^12 - 504914x^11 + 1108012x^10 + 1591114x^9 - 3399127x^8 - 3224089x^7 + 5089722x^6 + 4287384x^5 - 2840216x^4 - 2820864x^3 - 208576x^2 + 204992*x + 24832
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.553985$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.553985 q^{5} -2.25153 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.553985 q^{5} -2.25153 q^{7} +1.00000 q^{9} -5.96165 q^{11} -5.06127 q^{13} -0.553985 q^{15} +6.67868 q^{17} -6.35866 q^{19} -2.25153 q^{21} -6.49733 q^{23} -4.69310 q^{25} +1.00000 q^{27} +6.89257 q^{29} +7.49230 q^{31} -5.96165 q^{33} +1.24731 q^{35} +3.64759 q^{37} -5.06127 q^{39} +9.03363 q^{41} -10.3576 q^{43} -0.553985 q^{45} -0.815440 q^{47} -1.93062 q^{49} +6.67868 q^{51} +8.57917 q^{53} +3.30267 q^{55} -6.35866 q^{57} +10.8345 q^{59} +5.97894 q^{61} -2.25153 q^{63} +2.80387 q^{65} +11.5414 q^{67} -6.49733 q^{69} -4.16201 q^{71} +10.3109 q^{73} -4.69310 q^{75} +13.4228 q^{77} +5.05864 q^{79} +1.00000 q^{81} -14.3162 q^{83} -3.69989 q^{85} +6.89257 q^{87} -5.22634 q^{89} +11.3956 q^{91} +7.49230 q^{93} +3.52260 q^{95} +13.7276 q^{97} -5.96165 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.553985	−0.247750	−0.123875	−	0.992298i	$-0.539532\pi$
$5$	−0.553985	−0.247750	−0.123875	+	0.992298i	$0.539532\pi$
$6$	0	0
$7$	−2.25153	−0.850997	−0.425499	−	0.904959i	$-0.639901\pi$
$7$	−2.25153	−0.850997	−0.425499	+	0.904959i	$0.639901\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.96165	−1.79751	−0.898753	−	0.438456i	$-0.855525\pi$
$11$	−5.96165	−1.79751	−0.898753	+	0.438456i	$0.855525\pi$
$12$	0	0
$13$	−5.06127	−1.40374	−0.701872	−	0.712303i	$-0.747652\pi$
$13$	−5.06127	−1.40374	−0.701872	+	0.712303i	$0.747652\pi$
$14$	0	0
$15$	−0.553985	−0.143038
$16$	0	0
$17$	6.67868	1.61982	0.809909	−	0.586556i	$-0.199517\pi$
$17$	6.67868	1.61982	0.809909	+	0.586556i	$0.199517\pi$
$18$	0	0
$19$	−6.35866	−1.45878	−0.729388	−	0.684100i	$-0.760195\pi$
$19$	−6.35866	−1.45878	−0.729388	+	0.684100i	$0.760195\pi$
$20$	0	0
$21$	−2.25153	−0.491324
$22$	0	0
$23$	−6.49733	−1.35479	−0.677394	−	0.735621i	$-0.736891\pi$
$23$	−6.49733	−1.35479	−0.677394	+	0.735621i	$0.736891\pi$
$24$	0	0
$25$	−4.69310	−0.938620
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.89257	1.27992	0.639959	−	0.768409i	$-0.278951\pi$
$29$	6.89257	1.27992	0.639959	+	0.768409i	$0.278951\pi$
$30$	0	0
$31$	7.49230	1.34566	0.672828	−	0.739799i	$-0.265079\pi$
$31$	7.49230	1.34566	0.672828	+	0.739799i	$0.265079\pi$
$32$	0	0
$33$	−5.96165	−1.03779
$34$	0	0
$35$	1.24731	0.210834
$36$	0	0
$37$	3.64759	0.599661	0.299830	−	0.953993i	$-0.403070\pi$
$37$	3.64759	0.599661	0.299830	+	0.953993i	$0.403070\pi$
$38$	0	0
$39$	−5.06127	−0.810452
$40$	0	0
$41$	9.03363	1.41082	0.705408	−	0.708802i	$-0.250764\pi$
$41$	9.03363	1.41082	0.705408	+	0.708802i	$0.250764\pi$
$42$	0	0
$43$	−10.3576	−1.57952	−0.789759	−	0.613418i	$-0.789794\pi$
$43$	−10.3576	−1.57952	−0.789759	+	0.613418i	$0.789794\pi$
$44$	0	0
$45$	−0.553985	−0.0825833
$46$	0	0
$47$	−0.815440	−0.118944	−0.0594721	−	0.998230i	$-0.518942\pi$
$47$	−0.815440	−0.118944	−0.0594721	+	0.998230i	$0.518942\pi$
$48$	0	0
$49$	−1.93062	−0.275804
$50$	0	0
$51$	6.67868	0.935202
$52$	0	0
$53$	8.57917	1.17844	0.589220	−	0.807973i	$-0.299435\pi$
$53$	8.57917	1.17844	0.589220	+	0.807973i	$0.299435\pi$
$54$	0	0
$55$	3.30267	0.445331
$56$	0	0
$57$	−6.35866	−0.842225
$58$	0	0
$59$	10.8345	1.41053	0.705267	−	0.708942i	$-0.250827\pi$
$59$	10.8345	1.41053	0.705267	+	0.708942i	$0.250827\pi$
$60$	0	0
$61$	5.97894	0.765525	0.382763	−	0.923847i	$-0.374973\pi$
$61$	5.97894	0.765525	0.382763	+	0.923847i	$0.374973\pi$
$62$	0	0
$63$	−2.25153	−0.283666
$64$	0	0
$65$	2.80387	0.347777
$66$	0	0
$67$	11.5414	1.41001	0.705005	−	0.709203i	$-0.250945\pi$
$67$	11.5414	1.41001	0.705005	+	0.709203i	$0.250945\pi$
$68$	0	0
$69$	−6.49733	−0.782187
$70$	0	0
$71$	−4.16201	−0.493940	−0.246970	−	0.969023i	$-0.579435\pi$
$71$	−4.16201	−0.493940	−0.246970	+	0.969023i	$0.579435\pi$
$72$	0	0
$73$	10.3109	1.20680	0.603402	−	0.797437i	$-0.293812\pi$
$73$	10.3109	1.20680	0.603402	+	0.797437i	$0.293812\pi$
$74$	0	0
$75$	−4.69310	−0.541913
$76$	0	0
$77$	13.4228	1.52967
$78$	0	0
$79$	5.05864	0.569141	0.284571	−	0.958655i	$-0.408149\pi$
$79$	5.05864	0.569141	0.284571	+	0.958655i	$0.408149\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.3162	−1.57141	−0.785705	−	0.618601i	$-0.787700\pi$
$83$	−14.3162	−1.57141	−0.785705	+	0.618601i	$0.787700\pi$
$84$	0	0
$85$	−3.69989	−0.401309
$86$	0	0
$87$	6.89257	0.738961
$88$	0	0
$89$	−5.22634	−0.553991	−0.276995	−	0.960871i	$-0.589339\pi$
$89$	−5.22634	−0.553991	−0.276995	+	0.960871i	$0.589339\pi$
$90$	0	0
$91$	11.3956	1.19458
$92$	0	0
$93$	7.49230	0.776915
$94$	0	0
$95$	3.52260	0.361412
$96$	0	0
$97$	13.7276	1.39383	0.696913	−	0.717156i	$-0.254556\pi$
$97$	13.7276	1.39383	0.696913	+	0.717156i	$0.254556\pi$
$98$	0	0
$99$	−5.96165	−0.599168
$100$	0	0
$101$	1.45146	0.144426	0.0722128	−	0.997389i	$-0.476994\pi$
$101$	1.45146	0.144426	0.0722128	+	0.997389i	$0.476994\pi$
$102$	0	0
$103$	5.64013	0.555739	0.277869	−	0.960619i	$-0.410372\pi$
$103$	5.64013	0.555739	0.277869	+	0.960619i	$0.410372\pi$
$104$	0	0
$105$	1.24731	0.121725
$106$	0	0
$107$	5.62802	0.544081	0.272041	−	0.962286i	$-0.412301\pi$
$107$	5.62802	0.544081	0.272041	+	0.962286i	$0.412301\pi$
$108$	0	0
$109$	−12.7681	−1.22296	−0.611479	−	0.791261i	$-0.709425\pi$
$109$	−12.7681	−1.22296	−0.611479	+	0.791261i	$0.709425\pi$
$110$	0	0
$111$	3.64759	0.346214
$112$	0	0
$113$	3.32122	0.312434	0.156217	−	0.987723i	$-0.450070\pi$
$113$	3.32122	0.312434	0.156217	+	0.987723i	$0.450070\pi$
$114$	0	0
$115$	3.59943	0.335648
$116$	0	0
$117$	−5.06127	−0.467915
$118$	0	0
$119$	−15.0372	−1.37846
$120$	0	0
$121$	24.5413	2.23102
$122$	0	0
$123$	9.03363	0.814535
$124$	0	0
$125$	5.36984	0.480293
$126$	0	0
$127$	19.2387	1.70716	0.853578	−	0.520965i	$-0.174428\pi$
$127$	19.2387	1.70716	0.853578	+	0.520965i	$0.174428\pi$
$128$	0	0
$129$	−10.3576	−0.911935
$130$	0	0
$131$	6.68384	0.583970	0.291985	−	0.956423i	$-0.405684\pi$
$131$	6.68384	0.583970	0.291985	+	0.956423i	$0.405684\pi$
$132$	0	0
$133$	14.3167	1.24142
$134$	0	0
$135$	−0.553985	−0.0476795
$136$	0	0
$137$	−13.4962	−1.15306	−0.576528	−	0.817077i	$-0.695593\pi$
$137$	−13.4962	−1.15306	−0.576528	+	0.817077i	$0.695593\pi$
$138$	0	0
$139$	−20.2288	−1.71578	−0.857891	−	0.513832i	$-0.828225\pi$
$139$	−20.2288	−1.71578	−0.857891	+	0.513832i	$0.828225\pi$
$140$	0	0
$141$	−0.815440	−0.0686725
$142$	0	0
$143$	30.1735	2.52324
$144$	0	0
$145$	−3.81838	−0.317100
$146$	0	0
$147$	−1.93062	−0.159235
$148$	0	0
$149$	−10.8739	−0.890824	−0.445412	−	0.895326i	$-0.646943\pi$
$149$	−10.8739	−0.890824	−0.445412	+	0.895326i	$0.646943\pi$
$150$	0	0
$151$	9.96848	0.811223	0.405612	−	0.914046i	$-0.367059\pi$
$151$	9.96848	0.811223	0.405612	+	0.914046i	$0.367059\pi$
$152$	0	0
$153$	6.67868	0.539939
$154$	0	0
$155$	−4.15062	−0.333386
$156$	0	0
$157$	−22.0080	−1.75643	−0.878216	−	0.478264i	$-0.841266\pi$
$157$	−22.0080	−1.75643	−0.878216	+	0.478264i	$0.841266\pi$
$158$	0	0
$159$	8.57917	0.680373
$160$	0	0
$161$	14.6289	1.15292
$162$	0	0
$163$	9.15722	0.717249	0.358624	−	0.933482i	$-0.383246\pi$
$163$	9.15722	0.717249	0.358624	+	0.933482i	$0.383246\pi$
$164$	0	0
$165$	3.30267	0.257112
$166$	0	0
$167$	5.89867	0.456453	0.228226	−	0.973608i	$-0.426707\pi$
$167$	5.89867	0.456453	0.228226	+	0.973608i	$0.426707\pi$
$168$	0	0
$169$	12.6165	0.970499
$170$	0	0
$171$	−6.35866	−0.486259
$172$	0	0
$173$	−4.50974	−0.342869	−0.171434	−	0.985196i	$-0.554840\pi$
$173$	−4.50974	−0.342869	−0.171434	+	0.985196i	$0.554840\pi$
$174$	0	0
$175$	10.5666	0.798763
$176$	0	0
$177$	10.8345	0.814372
$178$	0	0
$179$	−3.97122	−0.296823	−0.148411	−	0.988926i	$-0.547416\pi$
$179$	−3.97122	−0.296823	−0.148411	+	0.988926i	$0.547416\pi$
$180$	0	0
$181$	14.1212	1.04962	0.524810	−	0.851219i	$-0.324136\pi$
$181$	14.1212	1.04962	0.524810	+	0.851219i	$0.324136\pi$
$182$	0	0
$183$	5.97894	0.441976
$184$	0	0
$185$	−2.02071	−0.148566
$186$	0	0
$187$	−39.8159	−2.91163
$188$	0	0
$189$	−2.25153	−0.163775
$190$	0	0
$191$	−3.90748	−0.282736	−0.141368	−	0.989957i	$-0.545150\pi$
$191$	−3.90748	−0.282736	−0.141368	+	0.989957i	$0.545150\pi$
$192$	0	0
$193$	−6.75852	−0.486489	−0.243244	−	0.969965i	$-0.578212\pi$
$193$	−6.75852	−0.486489	−0.243244	+	0.969965i	$0.578212\pi$
$194$	0	0
$195$	2.80387	0.200789
$196$	0	0
$197$	7.93757	0.565529	0.282764	−	0.959189i	$-0.408749\pi$
$197$	7.93757	0.565529	0.282764	+	0.959189i	$0.408749\pi$
$198$	0	0
$199$	−17.5085	−1.24115	−0.620573	−	0.784149i	$-0.713100\pi$
$199$	−17.5085	−1.24115	−0.620573	+	0.784149i	$0.713100\pi$
$200$	0	0
$201$	11.5414	0.814069
$202$	0	0
$203$	−15.5188	−1.08921
$204$	0	0
$205$	−5.00450	−0.349529
$206$	0	0
$207$	−6.49733	−0.451596
$208$	0	0
$209$	37.9081	2.62216
$210$	0	0
$211$	24.9564	1.71807	0.859034	−	0.511919i	$-0.171065\pi$
$211$	24.9564	1.71807	0.859034	+	0.511919i	$0.171065\pi$
$212$	0	0
$213$	−4.16201	−0.285176
$214$	0	0
$215$	5.73795	0.391325
$216$	0	0
$217$	−16.8691	−1.14515
$218$	0	0
$219$	10.3109	0.696748
$220$	0	0
$221$	−33.8026	−2.27381
$222$	0	0
$223$	21.2134	1.42056	0.710278	−	0.703921i	$-0.248569\pi$
$223$	21.2134	1.42056	0.710278	+	0.703921i	$0.248569\pi$
$224$	0	0
$225$	−4.69310	−0.312873
$226$	0	0
$227$	−21.8985	−1.45345	−0.726726	−	0.686927i	$-0.758959\pi$
$227$	−21.8985	−1.45345	−0.726726	+	0.686927i	$0.758959\pi$
$228$	0	0
$229$	23.8544	1.57634	0.788170	−	0.615457i	$-0.211029\pi$
$229$	23.8544	1.57634	0.788170	+	0.615457i	$0.211029\pi$
$230$	0	0
$231$	13.4228	0.883157
$232$	0	0
$233$	−24.5585	−1.60888	−0.804441	−	0.594033i	$-0.797535\pi$
$233$	−24.5585	−1.60888	−0.804441	+	0.594033i	$0.797535\pi$
$234$	0	0
$235$	0.451742	0.0294684
$236$	0	0
$237$	5.05864	0.328594
$238$	0	0
$239$	9.10729	0.589102	0.294551	−	0.955636i	$-0.404830\pi$
$239$	9.10729	0.589102	0.294551	+	0.955636i	$0.404830\pi$
$240$	0	0
$241$	−19.9788	−1.28695	−0.643475	−	0.765467i	$-0.722508\pi$
$241$	−19.9788	−1.28695	−0.643475	+	0.765467i	$0.722508\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.06954	0.0683303
$246$	0	0
$247$	32.1829	2.04775
$248$	0	0
$249$	−14.3162	−0.907254
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	38.7348	2.43524
$254$	0	0
$255$	−3.69989	−0.231696
$256$	0	0
$257$	−20.0940	−1.25343	−0.626714	−	0.779250i	$-0.715600\pi$
$257$	−20.0940	−1.25343	−0.626714	+	0.779250i	$0.715600\pi$
$258$	0	0
$259$	−8.21266	−0.510310
$260$	0	0
$261$	6.89257	0.426640
$262$	0	0
$263$	−2.75173	−0.169679	−0.0848396	−	0.996395i	$-0.527038\pi$
$263$	−2.75173	−0.169679	−0.0848396	+	0.996395i	$0.527038\pi$
$264$	0	0
$265$	−4.75274	−0.291958
$266$	0	0
$267$	−5.22634	−0.319847
$268$	0	0
$269$	−12.8447	−0.783154	−0.391577	−	0.920145i	$-0.628070\pi$
$269$	−12.8447	−0.783154	−0.391577	+	0.920145i	$0.628070\pi$
$270$	0	0
$271$	15.0559	0.914579	0.457289	−	0.889318i	$-0.348820\pi$
$271$	15.0559	0.914579	0.457289	+	0.889318i	$0.348820\pi$
$272$	0	0
$273$	11.3956	0.689693
$274$	0	0
$275$	27.9786	1.68717
$276$	0	0
$277$	−19.4418	−1.16814	−0.584072	−	0.811702i	$-0.698541\pi$
$277$	−19.4418	−1.16814	−0.584072	+	0.811702i	$0.698541\pi$
$278$	0	0
$279$	7.49230	0.448552
$280$	0	0
$281$	28.7267	1.71369	0.856846	−	0.515573i	$-0.172421\pi$
$281$	28.7267	1.71369	0.856846	+	0.515573i	$0.172421\pi$
$282$	0	0
$283$	−23.6879	−1.40810	−0.704048	−	0.710152i	$-0.748626\pi$
$283$	−23.6879	−1.40810	−0.704048	+	0.710152i	$0.748626\pi$
$284$	0	0
$285$	3.52260	0.208661
$286$	0	0
$287$	−20.3395	−1.20060
$288$	0	0
$289$	27.6047	1.62381
$290$	0	0
$291$	13.7276	0.804726
$292$	0	0
$293$	2.80684	0.163977	0.0819887	−	0.996633i	$-0.473873\pi$
$293$	2.80684	0.163977	0.0819887	+	0.996633i	$0.473873\pi$
$294$	0	0
$295$	−6.00216	−0.349459
$296$	0	0
$297$	−5.96165	−0.345930
$298$	0	0
$299$	32.8848	1.90178
$300$	0	0
$301$	23.3204	1.34417
$302$	0	0
$303$	1.45146	0.0833842
$304$	0	0
$305$	−3.31225	−0.189659
$306$	0	0
$307$	−17.4676	−0.996928	−0.498464	−	0.866910i	$-0.666102\pi$
$307$	−17.4676	−0.996928	−0.498464	+	0.866910i	$0.666102\pi$
$308$	0	0
$309$	5.64013	0.320856
$310$	0	0
$311$	18.2285	1.03364	0.516821	−	0.856094i	$-0.327115\pi$
$311$	18.2285	1.03364	0.516821	+	0.856094i	$0.327115\pi$
$312$	0	0
$313$	−26.6246	−1.50491	−0.752455	−	0.658643i	$-0.771131\pi$
$313$	−26.6246	−1.50491	−0.752455	+	0.658643i	$0.771131\pi$
$314$	0	0
$315$	1.24731	0.0702781
$316$	0	0
$317$	10.9098	0.612757	0.306379	−	0.951910i	$-0.400883\pi$
$317$	10.9098	0.612757	0.306379	+	0.951910i	$0.400883\pi$
$318$	0	0
$319$	−41.0911	−2.30066
$320$	0	0
$321$	5.62802	0.314126
$322$	0	0
$323$	−42.4674	−2.36295
$324$	0	0
$325$	23.7531	1.31758
$326$	0	0
$327$	−12.7681	−0.706075
$328$	0	0
$329$	1.83599	0.101221
$330$	0	0
$331$	15.2235	0.836760	0.418380	−	0.908272i	$-0.362598\pi$
$331$	15.2235	0.836760	0.418380	+	0.908272i	$0.362598\pi$
$332$	0	0
$333$	3.64759	0.199887
$334$	0	0
$335$	−6.39378	−0.349330
$336$	0	0
$337$	9.56802	0.521203	0.260602	−	0.965446i	$-0.416079\pi$
$337$	9.56802	0.521203	0.260602	+	0.965446i	$0.416079\pi$
$338$	0	0
$339$	3.32122	0.180384
$340$	0	0
$341$	−44.6664	−2.41882
$342$	0	0
$343$	20.1075	1.08571
$344$	0	0
$345$	3.59943	0.193787
$346$	0	0
$347$	−15.5199	−0.833154	−0.416577	−	0.909100i	$-0.636770\pi$
$347$	−15.5199	−0.833154	−0.416577	+	0.909100i	$0.636770\pi$
$348$	0	0
$349$	7.58828	0.406191	0.203096	−	0.979159i	$-0.434900\pi$
$349$	7.58828	0.406191	0.203096	+	0.979159i	$0.434900\pi$
$350$	0	0
$351$	−5.06127	−0.270151
$352$	0	0
$353$	−14.6231	−0.778309	−0.389154	−	0.921173i	$-0.627233\pi$
$353$	−14.6231	−0.778309	−0.389154	+	0.921173i	$0.627233\pi$
$354$	0	0
$355$	2.30569	0.122374
$356$	0	0
$357$	−15.0372	−0.795854
$358$	0	0
$359$	19.3395	1.02070	0.510350	−	0.859967i	$-0.329516\pi$
$359$	19.3395	1.02070	0.510350	+	0.859967i	$0.329516\pi$
$360$	0	0
$361$	21.4326	1.12803
$362$	0	0
$363$	24.5413	1.28808
$364$	0	0
$365$	−5.71210	−0.298985
$366$	0	0
$367$	15.6584	0.817364	0.408682	−	0.912677i	$-0.365988\pi$
$367$	15.6584	0.817364	0.408682	+	0.912677i	$0.365988\pi$
$368$	0	0
$369$	9.03363	0.470272
$370$	0	0
$371$	−19.3162	−1.00285
$372$	0	0
$373$	22.8143	1.18128	0.590639	−	0.806936i	$-0.298876\pi$
$373$	22.8143	1.18128	0.590639	+	0.806936i	$0.298876\pi$
$374$	0	0
$375$	5.36984	0.277297
$376$	0	0
$377$	−34.8852	−1.79668
$378$	0	0
$379$	−11.6256	−0.597165	−0.298582	−	0.954384i	$-0.596514\pi$
$379$	−11.6256	−0.597165	−0.298582	+	0.954384i	$0.596514\pi$
$380$	0	0
$381$	19.2387	0.985627
$382$	0	0
$383$	16.6811	0.852362	0.426181	−	0.904638i	$-0.359859\pi$
$383$	16.6811	0.852362	0.426181	+	0.904638i	$0.359859\pi$
$384$	0	0
$385$	−7.43604	−0.378976
$386$	0	0
$387$	−10.3576	−0.526506
$388$	0	0
$389$	−25.0930	−1.27227	−0.636133	−	0.771580i	$-0.719467\pi$
$389$	−25.0930	−1.27227	−0.636133	+	0.771580i	$0.719467\pi$
$390$	0	0
$391$	−43.3936	−2.19451
$392$	0	0
$393$	6.68384	0.337155
$394$	0	0
$395$	−2.80241	−0.141005
$396$	0	0
$397$	10.4175	0.522837	0.261419	−	0.965226i	$-0.415810\pi$
$397$	10.4175	0.522837	0.261419	+	0.965226i	$0.415810\pi$
$398$	0	0
$399$	14.3167	0.716731
$400$	0	0
$401$	17.7344	0.885612	0.442806	−	0.896617i	$-0.353983\pi$
$401$	17.7344	0.885612	0.442806	+	0.896617i	$0.353983\pi$
$402$	0	0
$403$	−37.9206	−1.88896
$404$	0	0
$405$	−0.553985	−0.0275278
$406$	0	0
$407$	−21.7457	−1.07789
$408$	0	0
$409$	6.19302	0.306225	0.153113	−	0.988209i	$-0.451070\pi$
$409$	6.19302	0.306225	0.153113	+	0.988209i	$0.451070\pi$
$410$	0	0
$411$	−13.4962	−0.665717
$412$	0	0
$413$	−24.3942	−1.20036
$414$	0	0
$415$	7.93098	0.389317
$416$	0	0
$417$	−20.2288	−0.990607
$418$	0	0
$419$	−16.4358	−0.802941	−0.401471	−	0.915872i	$-0.631501\pi$
$419$	−16.4358	−0.802941	−0.401471	+	0.915872i	$0.631501\pi$
$420$	0	0
$421$	1.17576	0.0573028	0.0286514	−	0.999589i	$-0.490879\pi$
$421$	1.17576	0.0573028	0.0286514	+	0.999589i	$0.490879\pi$
$422$	0	0
$423$	−0.815440	−0.0396481
$424$	0	0
$425$	−31.3437	−1.52039
$426$	0	0
$427$	−13.4618	−0.651460
$428$	0	0
$429$	30.1735	1.45679
$430$	0	0
$431$	24.9358	1.20111	0.600556	−	0.799582i	$-0.294946\pi$
$431$	24.9358	1.20111	0.600556	+	0.799582i	$0.294946\pi$
$432$	0	0
$433$	7.35853	0.353629	0.176814	−	0.984244i	$-0.443421\pi$
$433$	7.35853	0.353629	0.176814	+	0.984244i	$0.443421\pi$
$434$	0	0
$435$	−3.81838	−0.183078
$436$	0	0
$437$	41.3143	1.97633
$438$	0	0
$439$	−29.2388	−1.39549	−0.697746	−	0.716346i	$-0.745813\pi$
$439$	−29.2388	−1.39549	−0.697746	+	0.716346i	$0.745813\pi$
$440$	0	0
$441$	−1.93062	−0.0919345
$442$	0	0
$443$	7.98448	0.379354	0.189677	−	0.981847i	$-0.439256\pi$
$443$	7.98448	0.379354	0.189677	+	0.981847i	$0.439256\pi$
$444$	0	0
$445$	2.89532	0.137251
$446$	0	0
$447$	−10.8739	−0.514317
$448$	0	0
$449$	−20.3039	−0.958201	−0.479100	−	0.877760i	$-0.659037\pi$
$449$	−20.3039	−0.958201	−0.479100	+	0.877760i	$0.659037\pi$
$450$	0	0
$451$	−53.8553	−2.53595
$452$	0	0
$453$	9.96848	0.468360
$454$	0	0
$455$	−6.31299	−0.295958
$456$	0	0
$457$	28.5177	1.33400	0.667000	−	0.745058i	$-0.267578\pi$
$457$	28.5177	1.33400	0.667000	+	0.745058i	$0.267578\pi$
$458$	0	0
$459$	6.67868	0.311734
$460$	0	0
$461$	22.9249	1.06772	0.533859	−	0.845573i	$-0.320741\pi$
$461$	22.9249	1.06772	0.533859	+	0.845573i	$0.320741\pi$
$462$	0	0
$463$	22.8154	1.06032	0.530160	−	0.847898i	$-0.322132\pi$
$463$	22.8154	1.06032	0.530160	+	0.847898i	$0.322132\pi$
$464$	0	0
$465$	−4.15062	−0.192480
$466$	0	0
$467$	24.2038	1.12002	0.560009	−	0.828487i	$-0.310798\pi$
$467$	24.2038	1.12002	0.560009	+	0.828487i	$0.310798\pi$
$468$	0	0
$469$	−25.9858	−1.19991
$470$	0	0
$471$	−22.0080	−1.01408
$472$	0	0
$473$	61.7483	2.83919
$474$	0	0
$475$	29.8418	1.36924
$476$	0	0
$477$	8.57917	0.392813
$478$	0	0
$479$	−23.9005	−1.09204	−0.546021	−	0.837772i	$-0.683858\pi$
$479$	−23.9005	−1.09204	−0.546021	+	0.837772i	$0.683858\pi$
$480$	0	0
$481$	−18.4615	−0.841770
$482$	0	0
$483$	14.6289	0.665639
$484$	0	0
$485$	−7.60488	−0.345320
$486$	0	0
$487$	−27.4824	−1.24535	−0.622674	−	0.782481i	$-0.713954\pi$
$487$	−27.4824	−1.24535	−0.622674	+	0.782481i	$0.713954\pi$
$488$	0	0
$489$	9.15722	0.414104
$490$	0	0
$491$	−12.7378	−0.574850	−0.287425	−	0.957803i	$-0.592799\pi$
$491$	−12.7378	−0.574850	−0.287425	+	0.957803i	$0.592799\pi$
$492$	0	0
$493$	46.0333	2.07323
$494$	0	0
$495$	3.30267	0.148444
$496$	0	0
$497$	9.37088	0.420342
$498$	0	0
$499$	25.0947	1.12339	0.561696	−	0.827343i	$-0.310149\pi$
$499$	25.0947	1.12339	0.561696	+	0.827343i	$0.310149\pi$
$500$	0	0
$501$	5.89867	0.263533
$502$	0	0
$503$	15.3970	0.686517	0.343259	−	0.939241i	$-0.388469\pi$
$503$	15.3970	0.686517	0.343259	+	0.939241i	$0.388469\pi$
$504$	0	0
$505$	−0.804087	−0.0357814
$506$	0	0
$507$	12.6165	0.560318
$508$	0	0
$509$	36.8147	1.63178	0.815890	−	0.578207i	$-0.196247\pi$
$509$	36.8147	1.63178	0.815890	+	0.578207i	$0.196247\pi$
$510$	0	0
$511$	−23.2153	−1.02699
$512$	0	0
$513$	−6.35866	−0.280742
$514$	0	0
$515$	−3.12455	−0.137684
$516$	0	0
$517$	4.86137	0.213803
$518$	0	0
$519$	−4.50974	−0.197955
$520$	0	0
$521$	−19.7612	−0.865754	−0.432877	−	0.901453i	$-0.642502\pi$
$521$	−19.7612	−0.865754	−0.432877	+	0.901453i	$0.642502\pi$
$522$	0	0
$523$	−12.6749	−0.554233	−0.277116	−	0.960836i	$-0.589379\pi$
$523$	−12.6749	−0.554233	−0.277116	+	0.960836i	$0.589379\pi$
$524$	0	0
$525$	10.5666	0.461166
$526$	0	0
$527$	50.0386	2.17972
$528$	0	0
$529$	19.2153	0.835448
$530$	0	0
$531$	10.8345	0.470178
$532$	0	0
$533$	−45.7217	−1.98042
$534$	0	0
$535$	−3.11784	−0.134796
$536$	0	0
$537$	−3.97122	−0.171371
$538$	0	0
$539$	11.5097	0.495758
$540$	0	0
$541$	4.73245	0.203464	0.101732	−	0.994812i	$-0.467562\pi$
$541$	4.73245	0.203464	0.101732	+	0.994812i	$0.467562\pi$
$542$	0	0
$543$	14.1212	0.605998
$544$	0	0
$545$	7.07331	0.302987
$546$	0	0
$547$	30.2273	1.29242	0.646212	−	0.763158i	$-0.276352\pi$
$547$	30.2273	1.29242	0.646212	+	0.763158i	$0.276352\pi$
$548$	0	0
$549$	5.97894	0.255175
$550$	0	0
$551$	−43.8275	−1.86712
$552$	0	0
$553$	−11.3897	−0.484338
$554$	0	0
$555$	−2.02071	−0.0857745
$556$	0	0
$557$	32.0594	1.35840	0.679199	−	0.733954i	$-0.262327\pi$
$557$	32.0594	1.35840	0.679199	+	0.733954i	$0.262327\pi$
$558$	0	0
$559$	52.4226	2.21724
$560$	0	0
$561$	−39.8159	−1.68103
$562$	0	0
$563$	−7.14260	−0.301025	−0.150512	−	0.988608i	$-0.548092\pi$
$563$	−7.14260	−0.301025	−0.150512	+	0.988608i	$0.548092\pi$
$564$	0	0
$565$	−1.83991	−0.0774055
$566$	0	0
$567$	−2.25153	−0.0945553
$568$	0	0
$569$	−29.0626	−1.21837	−0.609183	−	0.793029i	$-0.708503\pi$
$569$	−29.0626	−1.21837	−0.609183	+	0.793029i	$0.708503\pi$
$570$	0	0
$571$	13.1225	0.549161	0.274580	−	0.961564i	$-0.411461\pi$
$571$	13.1225	0.549161	0.274580	+	0.961564i	$0.411461\pi$
$572$	0	0
$573$	−3.90748	−0.163237
$574$	0	0
$575$	30.4926	1.27163
$576$	0	0
$577$	20.8634	0.868557	0.434278	−	0.900779i	$-0.357003\pi$
$577$	20.8634	0.868557	0.434278	+	0.900779i	$0.357003\pi$
$578$	0	0
$579$	−6.75852	−0.280874
$580$	0	0
$581$	32.2334	1.33727
$582$	0	0
$583$	−51.1460	−2.11825
$584$	0	0
$585$	2.80387	0.115926
$586$	0	0
$587$	10.3203	0.425966	0.212983	−	0.977056i	$-0.431682\pi$
$587$	10.3203	0.425966	0.212983	+	0.977056i	$0.431682\pi$
$588$	0	0
$589$	−47.6410	−1.96301
$590$	0	0
$591$	7.93757	0.326508
$592$	0	0
$593$	−30.9403	−1.27056	−0.635282	−	0.772280i	$-0.719116\pi$
$593$	−30.9403	−1.27056	−0.635282	+	0.772280i	$0.719116\pi$
$594$	0	0
$595$	8.33040	0.341513
$596$	0	0
$597$	−17.5085	−0.716576
$598$	0	0
$599$	12.9338	0.528462	0.264231	−	0.964459i	$-0.414882\pi$
$599$	12.9338	0.528462	0.264231	+	0.964459i	$0.414882\pi$
$600$	0	0
$601$	27.7354	1.13135	0.565676	−	0.824628i	$-0.308615\pi$
$601$	27.7354	1.13135	0.565676	+	0.824628i	$0.308615\pi$
$602$	0	0
$603$	11.5414	0.470003
$604$	0	0
$605$	−13.5955	−0.552736
$606$	0	0
$607$	−25.4913	−1.03466	−0.517330	−	0.855786i	$-0.673074\pi$
$607$	−25.4913	−1.03466	−0.517330	+	0.855786i	$0.673074\pi$
$608$	0	0
$609$	−15.5188	−0.628854
$610$	0	0
$611$	4.12717	0.166967
$612$	0	0
$613$	−43.0602	−1.73919	−0.869593	−	0.493770i	$-0.835618\pi$
$613$	−43.0602	−1.73919	−0.869593	+	0.493770i	$0.835618\pi$
$614$	0	0
$615$	−5.00450	−0.201801
$616$	0	0
$617$	27.3858	1.10251	0.551256	−	0.834336i	$-0.314149\pi$
$617$	27.3858	1.10251	0.551256	+	0.834336i	$0.314149\pi$
$618$	0	0
$619$	18.8685	0.758391	0.379195	−	0.925317i	$-0.376201\pi$
$619$	18.8685	0.758391	0.379195	+	0.925317i	$0.376201\pi$
$620$	0	0
$621$	−6.49733	−0.260729
$622$	0	0
$623$	11.7672	0.471445
$624$	0	0
$625$	20.4907	0.819628
$626$	0	0
$627$	37.9081	1.51390
$628$	0	0
$629$	24.3611	0.971341
$630$	0	0
$631$	21.3836	0.851267	0.425634	−	0.904896i	$-0.360051\pi$
$631$	21.3836	0.851267	0.425634	+	0.904896i	$0.360051\pi$
$632$	0	0
$633$	24.9564	0.991927
$634$	0	0
$635$	−10.6579	−0.422947
$636$	0	0
$637$	9.77142	0.387158
$638$	0	0
$639$	−4.16201	−0.164647
$640$	0	0
$641$	11.9332	0.471335	0.235667	−	0.971834i	$-0.424272\pi$
$641$	11.9332	0.471335	0.235667	+	0.971834i	$0.424272\pi$
$642$	0	0
$643$	40.8210	1.60982	0.804911	−	0.593395i	$-0.202213\pi$
$643$	40.8210	1.60982	0.804911	+	0.593395i	$0.202213\pi$
$644$	0	0
$645$	5.73795	0.225932
$646$	0	0
$647$	14.1098	0.554712	0.277356	−	0.960767i	$-0.410542\pi$
$647$	14.1098	0.554712	0.277356	+	0.960767i	$0.410542\pi$
$648$	0	0
$649$	−64.5916	−2.53544
$650$	0	0
$651$	−16.8691	−0.661153
$652$	0	0
$653$	7.80199	0.305315	0.152658	−	0.988279i	$-0.451217\pi$
$653$	7.80199	0.305315	0.152658	+	0.988279i	$0.451217\pi$
$654$	0	0
$655$	−3.70275	−0.144678
$656$	0	0
$657$	10.3109	0.402268
$658$	0	0
$659$	10.3824	0.404440	0.202220	−	0.979340i	$-0.435184\pi$
$659$	10.3824	0.404440	0.202220	+	0.979340i	$0.435184\pi$
$660$	0	0
$661$	23.4316	0.911384	0.455692	−	0.890137i	$-0.349392\pi$
$661$	23.4316	0.911384	0.455692	+	0.890137i	$0.349392\pi$
$662$	0	0
$663$	−33.8026	−1.31278
$664$	0	0
$665$	−7.93124	−0.307560
$666$	0	0
$667$	−44.7833	−1.73402
$668$	0	0
$669$	21.2134	0.820159
$670$	0	0
$671$	−35.6444	−1.37604
$672$	0	0
$673$	36.3460	1.40104	0.700518	−	0.713635i	$-0.252952\pi$
$673$	36.3460	1.40104	0.700518	+	0.713635i	$0.252952\pi$
$674$	0	0
$675$	−4.69310	−0.180638
$676$	0	0
$677$	17.2559	0.663197	0.331599	−	0.943421i	$-0.392412\pi$
$677$	17.2559	0.663197	0.331599	+	0.943421i	$0.392412\pi$
$678$	0	0
$679$	−30.9080	−1.18614
$680$	0	0
$681$	−21.8985	−0.839151
$682$	0	0
$683$	8.22620	0.314767	0.157383	−	0.987538i	$-0.449694\pi$
$683$	8.22620	0.314767	0.157383	+	0.987538i	$0.449694\pi$
$684$	0	0
$685$	7.47668	0.285669
$686$	0	0
$687$	23.8544	0.910101
$688$	0	0
$689$	−43.4215	−1.65423
$690$	0	0
$691$	−20.2004	−0.768459	−0.384230	−	0.923238i	$-0.625533\pi$
$691$	−20.2004	−0.768459	−0.384230	+	0.923238i	$0.625533\pi$
$692$	0	0
$693$	13.4228	0.509891
$694$	0	0
$695$	11.2064	0.425084
$696$	0	0
$697$	60.3327	2.28526
$698$	0	0
$699$	−24.5585	−0.928888
$700$	0	0
$701$	−23.1259	−0.873454	−0.436727	−	0.899594i	$-0.643862\pi$
$701$	−23.1259	−0.873454	−0.436727	+	0.899594i	$0.643862\pi$
$702$	0	0
$703$	−23.1938	−0.874771
$704$	0	0
$705$	0.451742	0.0170136
$706$	0	0
$707$	−3.26800	−0.122906
$708$	0	0
$709$	−41.2938	−1.55082	−0.775411	−	0.631457i	$-0.782457\pi$
$709$	−41.2938	−1.55082	−0.775411	+	0.631457i	$0.782457\pi$
$710$	0	0
$711$	5.05864	0.189714
$712$	0	0
$713$	−48.6799	−1.82308
$714$	0	0
$715$	−16.7157	−0.625132
$716$	0	0
$717$	9.10729	0.340118
$718$	0	0
$719$	−30.3060	−1.13022	−0.565112	−	0.825014i	$-0.691167\pi$
$719$	−30.3060	−1.13022	−0.565112	+	0.825014i	$0.691167\pi$
$720$	0	0
$721$	−12.6989	−0.472932
$722$	0	0
$723$	−19.9788	−0.743021
$724$	0	0
$725$	−32.3475	−1.20136
$726$	0	0
$727$	−45.9360	−1.70367	−0.851835	−	0.523810i	$-0.824510\pi$
$727$	−45.9360	−1.70367	−0.851835	+	0.523810i	$0.824510\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−69.1750	−2.55853
$732$	0	0
$733$	29.4109	1.08631	0.543157	−	0.839631i	$-0.317229\pi$
$733$	29.4109	1.08631	0.543157	+	0.839631i	$0.317229\pi$
$734$	0	0
$735$	1.06954	0.0394505
$736$	0	0
$737$	−68.8059	−2.53450
$738$	0	0
$739$	31.6913	1.16578	0.582891	−	0.812550i	$-0.301921\pi$
$739$	31.6913	1.16578	0.582891	+	0.812550i	$0.301921\pi$
$740$	0	0
$741$	32.1829	1.18227
$742$	0	0
$743$	3.37886	0.123958	0.0619791	−	0.998077i	$-0.480259\pi$
$743$	3.37886	0.123958	0.0619791	+	0.998077i	$0.480259\pi$
$744$	0	0
$745$	6.02398	0.220701
$746$	0	0
$747$	−14.3162	−0.523804
$748$	0	0
$749$	−12.6716	−0.463012
$750$	0	0
$751$	40.6771	1.48433	0.742165	−	0.670218i	$-0.233799\pi$
$751$	40.6771	1.48433	0.742165	+	0.670218i	$0.233799\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−5.52239	−0.200980
$756$	0	0
$757$	−18.9731	−0.689590	−0.344795	−	0.938678i	$-0.612052\pi$
$757$	−18.9731	−0.689590	−0.344795	+	0.938678i	$0.612052\pi$
$758$	0	0
$759$	38.7348	1.40598
$760$	0	0
$761$	4.25145	0.154115	0.0770574	−	0.997027i	$-0.475448\pi$
$761$	4.25145	0.154115	0.0770574	+	0.997027i	$0.475448\pi$
$762$	0	0
$763$	28.7476	1.04073
$764$	0	0
$765$	−3.69989	−0.133770
$766$	0	0
$767$	−54.8365	−1.98003
$768$	0	0
$769$	1.15458	0.0416353	0.0208176	−	0.999783i	$-0.493373\pi$
$769$	1.15458	0.0416353	0.0208176	+	0.999783i	$0.493373\pi$
$770$	0	0
$771$	−20.0940	−0.723667
$772$	0	0
$773$	32.1342	1.15579	0.577893	−	0.816113i	$-0.303875\pi$
$773$	32.1342	1.15579	0.577893	+	0.816113i	$0.303875\pi$
$774$	0	0
$775$	−35.1621	−1.26306
$776$	0	0
$777$	−8.21266	−0.294627
$778$	0	0
$779$	−57.4418	−2.05806
$780$	0	0
$781$	24.8125	0.887860
$782$	0	0
$783$	6.89257	0.246320
$784$	0	0
$785$	12.1921	0.435156
$786$	0	0
$787$	−23.8002	−0.848385	−0.424193	−	0.905572i	$-0.639442\pi$
$787$	−23.8002	−0.848385	−0.424193	+	0.905572i	$0.639442\pi$
$788$	0	0
$789$	−2.75173	−0.0979643
$790$	0	0
$791$	−7.47782	−0.265881
$792$	0	0
$793$	−30.2611	−1.07460
$794$	0	0
$795$	−4.75274	−0.168562
$796$	0	0
$797$	15.3498	0.543719	0.271860	−	0.962337i	$-0.412361\pi$
$797$	15.3498	0.543719	0.271860	+	0.962337i	$0.412361\pi$
$798$	0	0
$799$	−5.44606	−0.192668
$800$	0	0
$801$	−5.22634	−0.184664
$802$	0	0
$803$	−61.4702	−2.16924
$804$	0	0
$805$	−8.10421	−0.285636
$806$	0	0
$807$	−12.8447	−0.452154
$808$	0	0
$809$	12.4995	0.439457	0.219729	−	0.975561i	$-0.429483\pi$
$809$	12.4995	0.439457	0.219729	+	0.975561i	$0.429483\pi$
$810$	0	0
$811$	15.1249	0.531107	0.265554	−	0.964096i	$-0.414445\pi$
$811$	15.1249	0.531107	0.265554	+	0.964096i	$0.414445\pi$
$812$	0	0
$813$	15.0559	0.528032
$814$	0	0
$815$	−5.07296	−0.177698
$816$	0	0
$817$	65.8604	2.30416
$818$	0	0
$819$	11.3956	0.398194
$820$	0	0
$821$	18.8290	0.657136	0.328568	−	0.944480i	$-0.393434\pi$
$821$	18.8290	0.657136	0.328568	+	0.944480i	$0.393434\pi$
$822$	0	0
$823$	22.0736	0.769437	0.384719	−	0.923034i	$-0.374298\pi$
$823$	22.0736	0.769437	0.384719	+	0.923034i	$0.374298\pi$
$824$	0	0
$825$	27.9786	0.974091
$826$	0	0
$827$	−5.04548	−0.175448	−0.0877242	−	0.996145i	$-0.527959\pi$
$827$	−5.04548	−0.175448	−0.0877242	+	0.996145i	$0.527959\pi$
$828$	0	0
$829$	26.5310	0.921459	0.460729	−	0.887541i	$-0.347588\pi$
$829$	26.5310	0.921459	0.460729	+	0.887541i	$0.347588\pi$
$830$	0	0
$831$	−19.4418	−0.674428
$832$	0	0
$833$	−12.8940	−0.446751
$834$	0	0
$835$	−3.26778	−0.113086
$836$	0	0
$837$	7.49230	0.258972
$838$	0	0
$839$	−6.00548	−0.207332	−0.103666	−	0.994612i	$-0.533057\pi$
$839$	−6.00548	−0.207332	−0.103666	+	0.994612i	$0.533057\pi$
$840$	0	0
$841$	18.5076	0.638192
$842$	0	0
$843$	28.7267	0.989400
$844$	0	0
$845$	−6.98935	−0.240441
$846$	0	0
$847$	−55.2553	−1.89860
$848$	0	0
$849$	−23.6879	−0.812965
$850$	0	0
$851$	−23.6996	−0.812413
$852$	0	0
$853$	0.785449	0.0268933	0.0134466	−	0.999910i	$-0.495720\pi$
$853$	0.785449	0.0268933	0.0134466	+	0.999910i	$0.495720\pi$
$854$	0	0
$855$	3.52260	0.120471
$856$	0	0
$857$	51.9443	1.77438	0.887191	−	0.461402i	$-0.152653\pi$
$857$	51.9443	1.77438	0.887191	+	0.461402i	$0.152653\pi$
$858$	0	0
$859$	−24.0417	−0.820292	−0.410146	−	0.912020i	$-0.634522\pi$
$859$	−24.0417	−0.820292	−0.410146	+	0.912020i	$0.634522\pi$
$860$	0	0
$861$	−20.3395	−0.693167
$862$	0	0
$863$	7.21297	0.245532	0.122766	−	0.992436i	$-0.460823\pi$
$863$	7.21297	0.245532	0.122766	+	0.992436i	$0.460823\pi$
$864$	0	0
$865$	2.49833	0.0849457
$866$	0	0
$867$	27.6047	0.937506
$868$	0	0
$869$	−30.1578	−1.02303
$870$	0	0
$871$	−58.4143	−1.97929
$872$	0	0
$873$	13.7276	0.464609
$874$	0	0
$875$	−12.0903	−0.408728
$876$	0	0
$877$	1.02139	0.0344900	0.0172450	−	0.999851i	$-0.494510\pi$
$877$	1.02139	0.0344900	0.0172450	+	0.999851i	$0.494510\pi$
$878$	0	0
$879$	2.80684	0.0946724
$880$	0	0
$881$	−11.1814	−0.376712	−0.188356	−	0.982101i	$-0.560316\pi$
$881$	−11.1814	−0.376712	−0.188356	+	0.982101i	$0.560316\pi$
$882$	0	0
$883$	−25.0497	−0.842989	−0.421494	−	0.906831i	$-0.638494\pi$
$883$	−25.0497	−0.842989	−0.421494	+	0.906831i	$0.638494\pi$
$884$	0	0
$885$	−6.00216	−0.201761
$886$	0	0
$887$	−51.2115	−1.71951	−0.859757	−	0.510703i	$-0.829385\pi$
$887$	−51.2115	−1.71951	−0.859757	+	0.510703i	$0.829385\pi$
$888$	0	0
$889$	−43.3164	−1.45278
$890$	0	0
$891$	−5.96165	−0.199723
$892$	0	0
$893$	5.18511	0.173513
$894$	0	0
$895$	2.19999	0.0735377
$896$	0	0
$897$	32.8848	1.09799
$898$	0	0
$899$	51.6412	1.72233
$900$	0	0
$901$	57.2975	1.90886
$902$	0	0
$903$	23.3204	0.776054
$904$	0	0
$905$	−7.82293	−0.260043
$906$	0	0
$907$	24.7931	0.823243	0.411621	−	0.911355i	$-0.364963\pi$
$907$	24.7931	0.823243	0.411621	+	0.911355i	$0.364963\pi$
$908$	0	0
$909$	1.45146	0.0481419
$910$	0	0
$911$	−0.964044	−0.0319402	−0.0159701	−	0.999872i	$-0.505084\pi$
$911$	−0.964044	−0.0319402	−0.0159701	+	0.999872i	$0.505084\pi$
$912$	0	0
$913$	85.3484	2.82462
$914$	0	0
$915$	−3.31225	−0.109500
$916$	0	0
$917$	−15.0489	−0.496957
$918$	0	0
$919$	38.4728	1.26910	0.634550	−	0.772882i	$-0.281185\pi$
$919$	38.4728	1.26910	0.634550	+	0.772882i	$0.281185\pi$
$920$	0	0
$921$	−17.4676	−0.575577
$922$	0	0
$923$	21.0651	0.693366
$924$	0	0
$925$	−17.1185	−0.562854
$926$	0	0
$927$	5.64013	0.185246
$928$	0	0
$929$	44.7746	1.46901	0.734503	−	0.678605i	$-0.237415\pi$
$929$	44.7746	1.46901	0.734503	+	0.678605i	$0.237415\pi$
$930$	0	0
$931$	12.2762	0.402336
$932$	0	0
$933$	18.2285	0.596773
$934$	0	0
$935$	22.0574	0.721356
$936$	0	0
$937$	36.3117	1.18625	0.593126	−	0.805109i	$-0.297893\pi$
$937$	36.3117	1.18625	0.593126	+	0.805109i	$0.297893\pi$
$938$	0	0
$939$	−26.6246	−0.868860
$940$	0	0
$941$	−51.5207	−1.67953	−0.839763	−	0.542953i	$-0.817306\pi$
$941$	−51.5207	−1.67953	−0.839763	+	0.542953i	$0.817306\pi$
$942$	0	0
$943$	−58.6945	−1.91135
$944$	0	0
$945$	1.24731	0.0405751
$946$	0	0
$947$	−57.1868	−1.85832	−0.929160	−	0.369677i	$-0.879468\pi$
$947$	−57.1868	−1.85832	−0.929160	+	0.369677i	$0.879468\pi$
$948$	0	0
$949$	−52.1864	−1.69404
$950$	0	0
$951$	10.9098	0.353775
$952$	0	0
$953$	−19.0854	−0.618235	−0.309118	−	0.951024i	$-0.600034\pi$
$953$	−19.0854	−0.618235	−0.309118	+	0.951024i	$0.600034\pi$
$954$	0	0
$955$	2.16469	0.0700477
$956$	0	0
$957$	−41.0911	−1.32829
$958$	0	0
$959$	30.3870	0.981248
$960$	0	0
$961$	25.1345	0.810790
$962$	0	0
$963$	5.62802	0.181360
$964$	0	0
$965$	3.74412	0.120527
$966$	0	0
$967$	−18.5918	−0.597872	−0.298936	−	0.954273i	$-0.596632\pi$
$967$	−18.5918	−0.597872	−0.298936	+	0.954273i	$0.596632\pi$
$968$	0	0
$969$	−42.4674	−1.36425
$970$	0	0
$971$	0.103303	0.00331516	0.00165758	−	0.999999i	$-0.499472\pi$
$971$	0.103303	0.00331516	0.00165758	+	0.999999i	$0.499472\pi$
$972$	0	0
$973$	45.5456	1.46013
$974$	0	0
$975$	23.7531	0.760707
$976$	0	0
$977$	35.9665	1.15067	0.575335	−	0.817918i	$-0.304872\pi$
$977$	35.9665	1.15067	0.575335	+	0.817918i	$0.304872\pi$
$978$	0	0
$979$	31.1576	0.995801
$980$	0	0
$981$	−12.7681	−0.407653
$982$	0	0
$983$	14.0289	0.447452	0.223726	−	0.974652i	$-0.428178\pi$
$983$	14.0289	0.447452	0.223726	+	0.974652i	$0.428178\pi$
$984$	0	0
$985$	−4.39730	−0.140110
$986$	0	0
$987$	1.83599	0.0584401
$988$	0	0
$989$	67.2967	2.13991
$990$	0	0
$991$	−23.4421	−0.744661	−0.372331	−	0.928100i	$-0.621441\pi$
$991$	−23.4421	−0.744661	−0.372331	+	0.928100i	$0.621441\pi$
$992$	0	0
$993$	15.2235	0.483104
$994$	0	0
$995$	9.69946	0.307494
$996$	0	0
$997$	48.3178	1.53024	0.765120	−	0.643888i	$-0.222680\pi$
$997$	48.3178	1.53024	0.765120	+	0.643888i	$0.222680\pi$
$998$	0	0
$999$	3.64759	0.115405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.8	✓	20	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.553985 q^{5} -2.25153 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.553985 q^{5} -2.25153 q^{7} +1.00000 q^{9} -5.96165 q^{11} -5.06127 q^{13} -0.553985 q^{15} +6.67868 q^{17} -6.35866 q^{19} -2.25153 q^{21} -6.49733 q^{23} -4.69310 q^{25} +1.00000 q^{27} +6.89257 q^{29} +7.49230 q^{31} -5.96165 q^{33} +1.24731 q^{35} +3.64759 q^{37} -5.06127 q^{39} +9.03363 q^{41} -10.3576 q^{43} -0.553985 q^{45} -0.815440 q^{47} -1.93062 q^{49} +6.67868 q^{51} +8.57917 q^{53} +3.30267 q^{55} -6.35866 q^{57} +10.8345 q^{59} +5.97894 q^{61} -2.25153 q^{63} +2.80387 q^{65} +11.5414 q^{67} -6.49733 q^{69} -4.16201 q^{71} +10.3109 q^{73} -4.69310 q^{75} +13.4228 q^{77} +5.05864 q^{79} +1.00000 q^{81} -14.3162 q^{83} -3.69989 q^{85} +6.89257 q^{87} -5.22634 q^{89} +11.3956 q^{91} +7.49230 q^{93} +3.52260 q^{95} +13.7276 q^{97} -5.96165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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