LMFDB - Embedded newform 6024.2.a.n.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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 7 x^{13} - 22 x^{12} + 214 x^{11} + 91 x^{10} - 2481 x^{9} + 1285 x^{8} + 13253 x^{7} + \cdots - 5832 $ x^14 - 7x^13 - 22x^12 + 214x^11 + 91x^10 - 2481x^9 + 1285x^8 + 13253x^7 - 14287x^6 - 29907x^5 + 49572x^4 + 10044x^3 - 53784x^2 + 32076*x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.804573$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.195427 q^{5} -0.953396 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.195427 q^{5} -0.953396 q^{7} +1.00000 q^{9} +2.78811 q^{11} +0.661622 q^{13} +0.195427 q^{15} +2.87531 q^{17} -0.297725 q^{19} +0.953396 q^{21} -2.03768 q^{23} -4.96181 q^{25} -1.00000 q^{27} -7.41700 q^{29} +0.177549 q^{31} -2.78811 q^{33} +0.186319 q^{35} -4.21680 q^{37} -0.661622 q^{39} +1.77549 q^{41} -0.688815 q^{43} -0.195427 q^{45} -0.681923 q^{47} -6.09104 q^{49} -2.87531 q^{51} -2.80771 q^{53} -0.544872 q^{55} +0.297725 q^{57} +14.2430 q^{59} +6.19607 q^{61} -0.953396 q^{63} -0.129299 q^{65} -6.86617 q^{67} +2.03768 q^{69} -6.57408 q^{71} +6.65772 q^{73} +4.96181 q^{75} -2.65818 q^{77} +9.77040 q^{79} +1.00000 q^{81} +2.12152 q^{83} -0.561913 q^{85} +7.41700 q^{87} -7.16560 q^{89} -0.630788 q^{91} -0.177549 q^{93} +0.0581834 q^{95} -3.02576 q^{97} +2.78811 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * 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
$5$	−0.195427	−0.0873974	−0.0436987	−	0.999045i	$-0.513914\pi$
$5$	−0.195427	−0.0873974	−0.0436987	+	0.999045i	$0.513914\pi$
$6$	0	0
$7$	−0.953396	−0.360350	−0.180175	−	0.983635i	$-0.557666\pi$
$7$	−0.953396	−0.360350	−0.180175	+	0.983635i	$0.557666\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.78811	0.840648	0.420324	−	0.907374i	$-0.361916\pi$
$11$	2.78811	0.840648	0.420324	+	0.907374i	$0.361916\pi$
$12$	0	0
$13$	0.661622	0.183501	0.0917505	−	0.995782i	$-0.470754\pi$
$13$	0.661622	0.183501	0.0917505	+	0.995782i	$0.470754\pi$
$14$	0	0
$15$	0.195427	0.0504589
$16$	0	0
$17$	2.87531	0.697366	0.348683	−	0.937241i	$-0.386629\pi$
$17$	2.87531	0.697366	0.348683	+	0.937241i	$0.386629\pi$
$18$	0	0
$19$	−0.297725	−0.0683028	−0.0341514	−	0.999417i	$-0.510873\pi$
$19$	−0.297725	−0.0683028	−0.0341514	+	0.999417i	$0.510873\pi$
$20$	0	0
$21$	0.953396	0.208048
$22$	0	0
$23$	−2.03768	−0.424886	−0.212443	−	0.977174i	$-0.568142\pi$
$23$	−2.03768	−0.424886	−0.212443	+	0.977174i	$0.568142\pi$
$24$	0	0
$25$	−4.96181	−0.992362
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.41700	−1.37730	−0.688651	−	0.725093i	$-0.741797\pi$
$29$	−7.41700	−1.37730	−0.688651	+	0.725093i	$0.741797\pi$
$30$	0	0
$31$	0.177549	0.0318887	0.0159444	−	0.999873i	$-0.494925\pi$
$31$	0.177549	0.0318887	0.0159444	+	0.999873i	$0.494925\pi$
$32$	0	0
$33$	−2.78811	−0.485348
$34$	0	0
$35$	0.186319	0.0314936
$36$	0	0
$37$	−4.21680	−0.693237	−0.346618	−	0.938006i	$-0.612670\pi$
$37$	−4.21680	−0.693237	−0.346618	+	0.938006i	$0.612670\pi$
$38$	0	0
$39$	−0.661622	−0.105944
$40$	0	0
$41$	1.77549	0.277284	0.138642	−	0.990343i	$-0.455726\pi$
$41$	1.77549	0.277284	0.138642	+	0.990343i	$0.455726\pi$
$42$	0	0
$43$	−0.688815	−0.105043	−0.0525217	−	0.998620i	$-0.516726\pi$
$43$	−0.688815	−0.105043	−0.0525217	+	0.998620i	$0.516726\pi$
$44$	0	0
$45$	−0.195427	−0.0291325
$46$	0	0
$47$	−0.681923	−0.0994687	−0.0497343	−	0.998762i	$-0.515837\pi$
$47$	−0.681923	−0.0994687	−0.0497343	+	0.998762i	$0.515837\pi$
$48$	0	0
$49$	−6.09104	−0.870148
$50$	0	0
$51$	−2.87531	−0.402625
$52$	0	0
$53$	−2.80771	−0.385669	−0.192834	−	0.981231i	$-0.561768\pi$
$53$	−2.80771	−0.385669	−0.192834	+	0.981231i	$0.561768\pi$
$54$	0	0
$55$	−0.544872	−0.0734705
$56$	0	0
$57$	0.297725	0.0394347
$58$	0	0
$59$	14.2430	1.85428	0.927140	−	0.374715i	$-0.122259\pi$
$59$	14.2430	1.85428	0.927140	+	0.374715i	$0.122259\pi$
$60$	0	0
$61$	6.19607	0.793326	0.396663	−	0.917964i	$-0.370168\pi$
$61$	6.19607	0.793326	0.396663	+	0.917964i	$0.370168\pi$
$62$	0	0
$63$	−0.953396	−0.120117
$64$	0	0
$65$	−0.129299	−0.0160375
$66$	0	0
$67$	−6.86617	−0.838836	−0.419418	−	0.907793i	$-0.637766\pi$
$67$	−6.86617	−0.838836	−0.419418	+	0.907793i	$0.637766\pi$
$68$	0	0
$69$	2.03768	0.245308
$70$	0	0
$71$	−6.57408	−0.780200	−0.390100	−	0.920772i	$-0.627560\pi$
$71$	−6.57408	−0.780200	−0.390100	+	0.920772i	$0.627560\pi$
$72$	0	0
$73$	6.65772	0.779227	0.389613	−	0.920978i	$-0.372609\pi$
$73$	6.65772	0.779227	0.389613	+	0.920978i	$0.372609\pi$
$74$	0	0
$75$	4.96181	0.572940
$76$	0	0
$77$	−2.65818	−0.302927
$78$	0	0
$79$	9.77040	1.09926	0.549628	−	0.835410i	$-0.314769\pi$
$79$	9.77040	1.09926	0.549628	+	0.835410i	$0.314769\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.12152	0.232867	0.116434	−	0.993198i	$-0.462854\pi$
$83$	2.12152	0.232867	0.116434	+	0.993198i	$0.462854\pi$
$84$	0	0
$85$	−0.561913	−0.0609480
$86$	0	0
$87$	7.41700	0.795186
$88$	0	0
$89$	−7.16560	−0.759552	−0.379776	−	0.925078i	$-0.623999\pi$
$89$	−7.16560	−0.759552	−0.379776	+	0.925078i	$0.623999\pi$
$90$	0	0
$91$	−0.630788	−0.0661245
$92$	0	0
$93$	−0.177549	−0.0184110
$94$	0	0
$95$	0.0581834	0.00596949
$96$	0	0
$97$	−3.02576	−0.307220	−0.153610	−	0.988132i	$-0.549090\pi$
$97$	−3.02576	−0.307220	−0.153610	+	0.988132i	$0.549090\pi$
$98$	0	0
$99$	2.78811	0.280216
$100$	0	0
$101$	−5.40469	−0.537787	−0.268893	−	0.963170i	$-0.586658\pi$
$101$	−5.40469	−0.537787	−0.268893	+	0.963170i	$0.586658\pi$
$102$	0	0
$103$	12.0964	1.19189	0.595947	−	0.803024i	$-0.296777\pi$
$103$	12.0964	1.19189	0.595947	+	0.803024i	$0.296777\pi$
$104$	0	0
$105$	−0.186319	−0.0181829
$106$	0	0
$107$	15.3618	1.48508	0.742539	−	0.669803i	$-0.233622\pi$
$107$	15.3618	1.48508	0.742539	+	0.669803i	$0.233622\pi$
$108$	0	0
$109$	−7.73808	−0.741173	−0.370587	−	0.928798i	$-0.620843\pi$
$109$	−7.73808	−0.741173	−0.370587	+	0.928798i	$0.620843\pi$
$110$	0	0
$111$	4.21680	0.400241
$112$	0	0
$113$	5.61883	0.528575	0.264287	−	0.964444i	$-0.414863\pi$
$113$	5.61883	0.528575	0.264287	+	0.964444i	$0.414863\pi$
$114$	0	0
$115$	0.398217	0.0371339
$116$	0	0
$117$	0.661622	0.0611670
$118$	0	0
$119$	−2.74131	−0.251296
$120$	0	0
$121$	−3.22642	−0.293311
$122$	0	0
$123$	−1.77549	−0.160090
$124$	0	0
$125$	1.94680	0.174127
$126$	0	0
$127$	−0.186938	−0.0165881	−0.00829403	−	0.999966i	$-0.502640\pi$
$127$	−0.186938	−0.0165881	−0.00829403	+	0.999966i	$0.502640\pi$
$128$	0	0
$129$	0.688815	0.0606468
$130$	0	0
$131$	−18.2430	−1.59390	−0.796951	−	0.604044i	$-0.793555\pi$
$131$	−18.2430	−1.59390	−0.796951	+	0.604044i	$0.793555\pi$
$132$	0	0
$133$	0.283850	0.0246129
$134$	0	0
$135$	0.195427	0.0168196
$136$	0	0
$137$	6.84740	0.585013	0.292506	−	0.956264i	$-0.405511\pi$
$137$	6.84740	0.585013	0.292506	+	0.956264i	$0.405511\pi$
$138$	0	0
$139$	4.35161	0.369099	0.184549	−	0.982823i	$-0.440917\pi$
$139$	4.35161	0.369099	0.184549	+	0.982823i	$0.440917\pi$
$140$	0	0
$141$	0.681923	0.0574283
$142$	0	0
$143$	1.84468	0.154260
$144$	0	0
$145$	1.44948	0.120373
$146$	0	0
$147$	6.09104	0.502380
$148$	0	0
$149$	−20.6293	−1.69002	−0.845009	−	0.534752i	$-0.820405\pi$
$149$	−20.6293	−1.69002	−0.845009	+	0.534752i	$0.820405\pi$
$150$	0	0
$151$	−9.22216	−0.750488	−0.375244	−	0.926926i	$-0.622441\pi$
$151$	−9.22216	−0.750488	−0.375244	+	0.926926i	$0.622441\pi$
$152$	0	0
$153$	2.87531	0.232455
$154$	0	0
$155$	−0.0346977	−0.00278699
$156$	0	0
$157$	−4.91211	−0.392029	−0.196014	−	0.980601i	$-0.562800\pi$
$157$	−4.91211	−0.392029	−0.196014	+	0.980601i	$0.562800\pi$
$158$	0	0
$159$	2.80771	0.222666
$160$	0	0
$161$	1.94272	0.153107
$162$	0	0
$163$	−10.0923	−0.790494	−0.395247	−	0.918575i	$-0.629341\pi$
$163$	−10.0923	−0.790494	−0.395247	+	0.918575i	$0.629341\pi$
$164$	0	0
$165$	0.544872	0.0424182
$166$	0	0
$167$	−20.2214	−1.56478	−0.782391	−	0.622788i	$-0.786000\pi$
$167$	−20.2214	−1.56478	−0.782391	+	0.622788i	$0.786000\pi$
$168$	0	0
$169$	−12.5623	−0.966327
$170$	0	0
$171$	−0.297725	−0.0227676
$172$	0	0
$173$	−3.86566	−0.293901	−0.146950	−	0.989144i	$-0.546946\pi$
$173$	−3.86566	−0.293901	−0.146950	+	0.989144i	$0.546946\pi$
$174$	0	0
$175$	4.73057	0.357597
$176$	0	0
$177$	−14.2430	−1.07057
$178$	0	0
$179$	22.3748	1.67237	0.836187	−	0.548444i	$-0.184780\pi$
$179$	22.3748	1.67237	0.836187	+	0.548444i	$0.184780\pi$
$180$	0	0
$181$	5.75474	0.427746	0.213873	−	0.976861i	$-0.431392\pi$
$181$	5.75474	0.427746	0.213873	+	0.976861i	$0.431392\pi$
$182$	0	0
$183$	−6.19607	−0.458027
$184$	0	0
$185$	0.824074	0.0605871
$186$	0	0
$187$	8.01671	0.586240
$188$	0	0
$189$	0.953396	0.0693493
$190$	0	0
$191$	−20.9778	−1.51790	−0.758950	−	0.651149i	$-0.774287\pi$
$191$	−20.9778	−1.51790	−0.758950	+	0.651149i	$0.774287\pi$
$192$	0	0
$193$	7.72664	0.556176	0.278088	−	0.960556i	$-0.410299\pi$
$193$	7.72664	0.556176	0.278088	+	0.960556i	$0.410299\pi$
$194$	0	0
$195$	0.129299	0.00925926
$196$	0	0
$197$	−15.8930	−1.13233	−0.566164	−	0.824292i	$-0.691573\pi$
$197$	−15.8930	−1.13233	−0.566164	+	0.824292i	$0.691573\pi$
$198$	0	0
$199$	−20.7484	−1.47081	−0.735406	−	0.677626i	$-0.763009\pi$
$199$	−20.7484	−1.47081	−0.735406	+	0.677626i	$0.763009\pi$
$200$	0	0
$201$	6.86617	0.484302
$202$	0	0
$203$	7.07134	0.496311
$204$	0	0
$205$	−0.346977	−0.0242339
$206$	0	0
$207$	−2.03768	−0.141629
$208$	0	0
$209$	−0.830092	−0.0574187
$210$	0	0
$211$	−8.40606	−0.578697	−0.289349	−	0.957224i	$-0.593439\pi$
$211$	−8.40606	−0.578697	−0.289349	+	0.957224i	$0.593439\pi$
$212$	0	0
$213$	6.57408	0.450449
$214$	0	0
$215$	0.134613	0.00918051
$216$	0	0
$217$	−0.169274	−0.0114911
$218$	0	0
$219$	−6.65772	−0.449887
$220$	0	0
$221$	1.90237	0.127967
$222$	0	0
$223$	−17.4068	−1.16565	−0.582823	−	0.812599i	$-0.698052\pi$
$223$	−17.4068	−1.16565	−0.582823	+	0.812599i	$0.698052\pi$
$224$	0	0
$225$	−4.96181	−0.330787
$226$	0	0
$227$	−11.4617	−0.760742	−0.380371	−	0.924834i	$-0.624204\pi$
$227$	−11.4617	−0.760742	−0.380371	+	0.924834i	$0.624204\pi$
$228$	0	0
$229$	8.74504	0.577888	0.288944	−	0.957346i	$-0.406696\pi$
$229$	8.74504	0.577888	0.288944	+	0.957346i	$0.406696\pi$
$230$	0	0
$231$	2.65818	0.174895
$232$	0	0
$233$	−23.8506	−1.56251	−0.781254	−	0.624214i	$-0.785419\pi$
$233$	−23.8506	−1.56251	−0.781254	+	0.624214i	$0.785419\pi$
$234$	0	0
$235$	0.133266	0.00869330
$236$	0	0
$237$	−9.77040	−0.634656
$238$	0	0
$239$	18.7533	1.21305	0.606524	−	0.795065i	$-0.292563\pi$
$239$	18.7533	1.21305	0.606524	+	0.795065i	$0.292563\pi$
$240$	0	0
$241$	−22.1454	−1.42651	−0.713254	−	0.700906i	$-0.752779\pi$
$241$	−22.1454	−1.42651	−0.713254	+	0.700906i	$0.752779\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.19035	0.0760487
$246$	0	0
$247$	−0.196982	−0.0125336
$248$	0	0
$249$	−2.12152	−0.134446
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−5.68128	−0.357179
$254$	0	0
$255$	0.561913	0.0351883
$256$	0	0
$257$	−25.3496	−1.58126	−0.790631	−	0.612293i	$-0.790247\pi$
$257$	−25.3496	−1.58126	−0.790631	+	0.612293i	$0.790247\pi$
$258$	0	0
$259$	4.02027	0.249808
$260$	0	0
$261$	−7.41700	−0.459101
$262$	0	0
$263$	20.7228	1.27782	0.638911	−	0.769281i	$-0.279385\pi$
$263$	20.7228	1.27782	0.638911	+	0.769281i	$0.279385\pi$
$264$	0	0
$265$	0.548701	0.0337064
$266$	0	0
$267$	7.16560	0.438528
$268$	0	0
$269$	−11.0721	−0.675076	−0.337538	−	0.941312i	$-0.609594\pi$
$269$	−11.0721	−0.675076	−0.337538	+	0.941312i	$0.609594\pi$
$270$	0	0
$271$	10.6975	0.649825	0.324913	−	0.945744i	$-0.394665\pi$
$271$	10.6975	0.649825	0.324913	+	0.945744i	$0.394665\pi$
$272$	0	0
$273$	0.630788	0.0381770
$274$	0	0
$275$	−13.8341	−0.834227
$276$	0	0
$277$	17.9104	1.07613	0.538066	−	0.842903i	$-0.319155\pi$
$277$	17.9104	1.07613	0.538066	+	0.842903i	$0.319155\pi$
$278$	0	0
$279$	0.177549	0.0106296
$280$	0	0
$281$	−28.7184	−1.71320	−0.856599	−	0.515983i	$-0.827427\pi$
$281$	−28.7184	−1.71320	−0.856599	+	0.515983i	$0.827427\pi$
$282$	0	0
$283$	−10.8434	−0.644572	−0.322286	−	0.946642i	$-0.604451\pi$
$283$	−10.8434	−0.644572	−0.322286	+	0.946642i	$0.604451\pi$
$284$	0	0
$285$	−0.0581834	−0.00344649
$286$	0	0
$287$	−1.69274	−0.0999193
$288$	0	0
$289$	−8.73257	−0.513680
$290$	0	0
$291$	3.02576	0.177373
$292$	0	0
$293$	−5.30242	−0.309771	−0.154885	−	0.987932i	$-0.549501\pi$
$293$	−5.30242	−0.309771	−0.154885	+	0.987932i	$0.549501\pi$
$294$	0	0
$295$	−2.78346	−0.162059
$296$	0	0
$297$	−2.78811	−0.161783
$298$	0	0
$299$	−1.34817	−0.0779669
$300$	0	0
$301$	0.656713	0.0378523
$302$	0	0
$303$	5.40469	0.310491
$304$	0	0
$305$	−1.21088	−0.0693346
$306$	0	0
$307$	31.5081	1.79826	0.899130	−	0.437682i	$-0.144200\pi$
$307$	31.5081	1.79826	0.899130	+	0.437682i	$0.144200\pi$
$308$	0	0
$309$	−12.0964	−0.688140
$310$	0	0
$311$	2.14459	0.121608	0.0608042	−	0.998150i	$-0.480633\pi$
$311$	2.14459	0.121608	0.0608042	+	0.998150i	$0.480633\pi$
$312$	0	0
$313$	23.4893	1.32769	0.663846	−	0.747870i	$-0.268923\pi$
$313$	23.4893	1.32769	0.663846	+	0.747870i	$0.268923\pi$
$314$	0	0
$315$	0.186319	0.0104979
$316$	0	0
$317$	−28.9105	−1.62378	−0.811889	−	0.583812i	$-0.801561\pi$
$317$	−28.9105	−1.62378	−0.811889	+	0.583812i	$0.801561\pi$
$318$	0	0
$319$	−20.6794	−1.15783
$320$	0	0
$321$	−15.3618	−0.857410
$322$	0	0
$323$	−0.856053	−0.0476321
$324$	0	0
$325$	−3.28284	−0.182099
$326$	0	0
$327$	7.73808	0.427917
$328$	0	0
$329$	0.650142	0.0358435
$330$	0	0
$331$	3.26849	0.179652	0.0898262	−	0.995957i	$-0.471369\pi$
$331$	3.26849	0.179652	0.0898262	+	0.995957i	$0.471369\pi$
$332$	0	0
$333$	−4.21680	−0.231079
$334$	0	0
$335$	1.34183	0.0733121
$336$	0	0
$337$	33.9912	1.85162	0.925809	−	0.377991i	$-0.123385\pi$
$337$	33.9912	1.85162	0.925809	+	0.377991i	$0.123385\pi$
$338$	0	0
$339$	−5.61883	−0.305173
$340$	0	0
$341$	0.495027	0.0268072
$342$	0	0
$343$	12.4809	0.673907
$344$	0	0
$345$	−0.398217	−0.0214393
$346$	0	0
$347$	−34.0890	−1.82999	−0.914997	−	0.403462i	$-0.867807\pi$
$347$	−34.0890	−1.82999	−0.914997	+	0.403462i	$0.867807\pi$
$348$	0	0
$349$	−6.66384	−0.356707	−0.178354	−	0.983966i	$-0.557077\pi$
$349$	−6.66384	−0.356707	−0.178354	+	0.983966i	$0.557077\pi$
$350$	0	0
$351$	−0.661622	−0.0353148
$352$	0	0
$353$	29.5421	1.57237	0.786184	−	0.617993i	$-0.212054\pi$
$353$	29.5421	1.57237	0.786184	+	0.617993i	$0.212054\pi$
$354$	0	0
$355$	1.28475	0.0681875
$356$	0	0
$357$	2.74131	0.145086
$358$	0	0
$359$	−5.68669	−0.300132	−0.150066	−	0.988676i	$-0.547949\pi$
$359$	−5.68669	−0.300132	−0.150066	+	0.988676i	$0.547949\pi$
$360$	0	0
$361$	−18.9114	−0.995335
$362$	0	0
$363$	3.22642	0.169343
$364$	0	0
$365$	−1.30109	−0.0681024
$366$	0	0
$367$	−6.00137	−0.313269	−0.156635	−	0.987657i	$-0.550064\pi$
$367$	−6.00137	−0.313269	−0.156635	+	0.987657i	$0.550064\pi$
$368$	0	0
$369$	1.77549	0.0924281
$370$	0	0
$371$	2.67686	0.138976
$372$	0	0
$373$	12.8333	0.664482	0.332241	−	0.943195i	$-0.392195\pi$
$373$	12.8333	0.664482	0.332241	+	0.943195i	$0.392195\pi$
$374$	0	0
$375$	−1.94680	−0.100532
$376$	0	0
$377$	−4.90725	−0.252736
$378$	0	0
$379$	22.2012	1.14040	0.570200	−	0.821506i	$-0.306866\pi$
$379$	22.2012	1.14040	0.570200	+	0.821506i	$0.306866\pi$
$380$	0	0
$381$	0.186938	0.00957712
$382$	0	0
$383$	11.6281	0.594169	0.297085	−	0.954851i	$-0.403986\pi$
$383$	11.6281	0.594169	0.297085	+	0.954851i	$0.403986\pi$
$384$	0	0
$385$	0.519478	0.0264751
$386$	0	0
$387$	−0.688815	−0.0350144
$388$	0	0
$389$	−28.1400	−1.42675	−0.713377	−	0.700781i	$-0.752835\pi$
$389$	−28.1400	−1.42675	−0.713377	+	0.700781i	$0.752835\pi$
$390$	0	0
$391$	−5.85897	−0.296301
$392$	0	0
$393$	18.2430	0.920239
$394$	0	0
$395$	−1.90940	−0.0960721
$396$	0	0
$397$	15.8759	0.796787	0.398393	−	0.917215i	$-0.369568\pi$
$397$	15.8759	0.796787	0.398393	+	0.917215i	$0.369568\pi$
$398$	0	0
$399$	−0.283850	−0.0142103
$400$	0	0
$401$	−17.4417	−0.870998	−0.435499	−	0.900189i	$-0.643428\pi$
$401$	−17.4417	−0.870998	−0.435499	+	0.900189i	$0.643428\pi$
$402$	0	0
$403$	0.117470	0.00585161
$404$	0	0
$405$	−0.195427	−0.00971082
$406$	0	0
$407$	−11.7569	−0.582768
$408$	0	0
$409$	5.19905	0.257076	0.128538	−	0.991705i	$-0.458971\pi$
$409$	5.19905	0.257076	0.128538	+	0.991705i	$0.458971\pi$
$410$	0	0
$411$	−6.84740	−0.337757
$412$	0	0
$413$	−13.5792	−0.668189
$414$	0	0
$415$	−0.414601	−0.0203520
$416$	0	0
$417$	−4.35161	−0.213099
$418$	0	0
$419$	35.0844	1.71399	0.856993	−	0.515328i	$-0.172330\pi$
$419$	35.0844	1.71399	0.856993	+	0.515328i	$0.172330\pi$
$420$	0	0
$421$	−31.2816	−1.52457	−0.762287	−	0.647239i	$-0.775923\pi$
$421$	−31.2816	−1.52457	−0.762287	+	0.647239i	$0.775923\pi$
$422$	0	0
$423$	−0.681923	−0.0331562
$424$	0	0
$425$	−14.2668	−0.692039
$426$	0	0
$427$	−5.90731	−0.285875
$428$	0	0
$429$	−1.84468	−0.0890619
$430$	0	0
$431$	−28.6718	−1.38107	−0.690535	−	0.723299i	$-0.742625\pi$
$431$	−28.6718	−1.38107	−0.690535	+	0.723299i	$0.742625\pi$
$432$	0	0
$433$	−15.7856	−0.758608	−0.379304	−	0.925272i	$-0.623837\pi$
$433$	−15.7856	−0.758608	−0.379304	+	0.925272i	$0.623837\pi$
$434$	0	0
$435$	−1.44948	−0.0694972
$436$	0	0
$437$	0.606669	0.0290209
$438$	0	0
$439$	6.32729	0.301985	0.150993	−	0.988535i	$-0.451753\pi$
$439$	6.32729	0.301985	0.150993	+	0.988535i	$0.451753\pi$
$440$	0	0
$441$	−6.09104	−0.290049
$442$	0	0
$443$	25.8861	1.22989	0.614943	−	0.788572i	$-0.289179\pi$
$443$	25.8861	1.22989	0.614943	+	0.788572i	$0.289179\pi$
$444$	0	0
$445$	1.40035	0.0663829
$446$	0	0
$447$	20.6293	0.975732
$448$	0	0
$449$	8.94102	0.421953	0.210976	−	0.977491i	$-0.432336\pi$
$449$	8.94102	0.421953	0.210976	+	0.977491i	$0.432336\pi$
$450$	0	0
$451$	4.95026	0.233098
$452$	0	0
$453$	9.22216	0.433295
$454$	0	0
$455$	0.123273	0.00577911
$456$	0	0
$457$	−19.0125	−0.889366	−0.444683	−	0.895688i	$-0.646684\pi$
$457$	−19.0125	−0.889366	−0.444683	+	0.895688i	$0.646684\pi$
$458$	0	0
$459$	−2.87531	−0.134208
$460$	0	0
$461$	−41.9761	−1.95502	−0.977511	−	0.210884i	$-0.932366\pi$
$461$	−41.9761	−1.95502	−0.977511	+	0.210884i	$0.932366\pi$
$462$	0	0
$463$	−8.18337	−0.380314	−0.190157	−	0.981754i	$-0.560900\pi$
$463$	−8.18337	−0.380314	−0.190157	+	0.981754i	$0.560900\pi$
$464$	0	0
$465$	0.0346977	0.00160907
$466$	0	0
$467$	−12.9719	−0.600267	−0.300134	−	0.953897i	$-0.597031\pi$
$467$	−12.9719	−0.600267	−0.300134	+	0.953897i	$0.597031\pi$
$468$	0	0
$469$	6.54618	0.302274
$470$	0	0
$471$	4.91211	0.226338
$472$	0	0
$473$	−1.92050	−0.0883045
$474$	0	0
$475$	1.47726	0.0677811
$476$	0	0
$477$	−2.80771	−0.128556
$478$	0	0
$479$	−9.77889	−0.446809	−0.223404	−	0.974726i	$-0.571717\pi$
$479$	−9.77889	−0.446809	−0.223404	+	0.974726i	$0.571717\pi$
$480$	0	0
$481$	−2.78993	−0.127210
$482$	0	0
$483$	−1.94272	−0.0883966
$484$	0	0
$485$	0.591315	0.0268502
$486$	0	0
$487$	25.9546	1.17612	0.588058	−	0.808819i	$-0.299893\pi$
$487$	25.9546	1.17612	0.588058	+	0.808819i	$0.299893\pi$
$488$	0	0
$489$	10.0923	0.456392
$490$	0	0
$491$	26.1753	1.18127	0.590636	−	0.806938i	$-0.298877\pi$
$491$	26.1753	1.18127	0.590636	+	0.806938i	$0.298877\pi$
$492$	0	0
$493$	−21.3262	−0.960484
$494$	0	0
$495$	−0.544872	−0.0244902
$496$	0	0
$497$	6.26770	0.281145
$498$	0	0
$499$	−27.0715	−1.21189	−0.605943	−	0.795508i	$-0.707204\pi$
$499$	−27.0715	−1.21189	−0.605943	+	0.795508i	$0.707204\pi$
$500$	0	0
$501$	20.2214	0.903427
$502$	0	0
$503$	−41.2786	−1.84052	−0.920261	−	0.391306i	$-0.872023\pi$
$503$	−41.2786	−1.84052	−0.920261	+	0.391306i	$0.872023\pi$
$504$	0	0
$505$	1.05622	0.0470012
$506$	0	0
$507$	12.5623	0.557909
$508$	0	0
$509$	14.6571	0.649663	0.324831	−	0.945772i	$-0.394692\pi$
$509$	14.6571	0.649663	0.324831	+	0.945772i	$0.394692\pi$
$510$	0	0
$511$	−6.34744	−0.280794
$512$	0	0
$513$	0.297725	0.0131449
$514$	0	0
$515$	−2.36396	−0.104168
$516$	0	0
$517$	−1.90128	−0.0836181
$518$	0	0
$519$	3.86566	0.169684
$520$	0	0
$521$	1.86670	0.0817814	0.0408907	−	0.999164i	$-0.486980\pi$
$521$	1.86670	0.0817814	0.0408907	+	0.999164i	$0.486980\pi$
$522$	0	0
$523$	12.9855	0.567815	0.283908	−	0.958852i	$-0.408369\pi$
$523$	12.9855	0.567815	0.283908	+	0.958852i	$0.408369\pi$
$524$	0	0
$525$	−4.73057	−0.206459
$526$	0	0
$527$	0.510509	0.0222381
$528$	0	0
$529$	−18.8479	−0.819472
$530$	0	0
$531$	14.2430	0.618093
$532$	0	0
$533$	1.17470	0.0508819
$534$	0	0
$535$	−3.00209	−0.129792
$536$	0	0
$537$	−22.3748	−0.965546
$538$	0	0
$539$	−16.9825	−0.731488
$540$	0	0
$541$	22.6986	0.975888	0.487944	−	0.872875i	$-0.337747\pi$
$541$	22.6986	0.975888	0.487944	+	0.872875i	$0.337747\pi$
$542$	0	0
$543$	−5.75474	−0.246959
$544$	0	0
$545$	1.51223	0.0647766
$546$	0	0
$547$	−32.2735	−1.37992	−0.689958	−	0.723849i	$-0.742371\pi$
$547$	−32.2735	−1.37992	−0.689958	+	0.723849i	$0.742371\pi$
$548$	0	0
$549$	6.19607	0.264442
$550$	0	0
$551$	2.20823	0.0940737
$552$	0	0
$553$	−9.31506	−0.396117
$554$	0	0
$555$	−0.824074	−0.0349800
$556$	0	0
$557$	15.7645	0.667963	0.333981	−	0.942580i	$-0.391608\pi$
$557$	15.7645	0.667963	0.333981	+	0.942580i	$0.391608\pi$
$558$	0	0
$559$	−0.455735	−0.0192755
$560$	0	0
$561$	−8.01671	−0.338466
$562$	0	0
$563$	−1.92743	−0.0812315	−0.0406158	−	0.999175i	$-0.512932\pi$
$563$	−1.92743	−0.0812315	−0.0406158	+	0.999175i	$0.512932\pi$
$564$	0	0
$565$	−1.09807	−0.0461961
$566$	0	0
$567$	−0.953396	−0.0400389
$568$	0	0
$569$	20.8910	0.875795	0.437898	−	0.899025i	$-0.355723\pi$
$569$	20.8910	0.875795	0.437898	+	0.899025i	$0.355723\pi$
$570$	0	0
$571$	17.0971	0.715492	0.357746	−	0.933819i	$-0.383545\pi$
$571$	17.0971	0.715492	0.357746	+	0.933819i	$0.383545\pi$
$572$	0	0
$573$	20.9778	0.876360
$574$	0	0
$575$	10.1106	0.421640
$576$	0	0
$577$	−5.15823	−0.214740	−0.107370	−	0.994219i	$-0.534243\pi$
$577$	−5.15823	−0.214740	−0.107370	+	0.994219i	$0.534243\pi$
$578$	0	0
$579$	−7.72664	−0.321108
$580$	0	0
$581$	−2.02265	−0.0839136
$582$	0	0
$583$	−7.82822	−0.324212
$584$	0	0
$585$	−0.129299	−0.00534584
$586$	0	0
$587$	−19.0700	−0.787103	−0.393552	−	0.919302i	$-0.628754\pi$
$587$	−19.0700	−0.787103	−0.393552	+	0.919302i	$0.628754\pi$
$588$	0	0
$589$	−0.0528608	−0.00217809
$590$	0	0
$591$	15.8930	0.653750
$592$	0	0
$593$	−41.2931	−1.69570	−0.847851	−	0.530234i	$-0.822104\pi$
$593$	−41.2931	−1.69570	−0.847851	+	0.530234i	$0.822104\pi$
$594$	0	0
$595$	0.535725	0.0219626
$596$	0	0
$597$	20.7484	0.849174
$598$	0	0
$599$	23.0543	0.941974	0.470987	−	0.882140i	$-0.343898\pi$
$599$	23.0543	0.941974	0.470987	+	0.882140i	$0.343898\pi$
$600$	0	0
$601$	12.3146	0.502323	0.251161	−	0.967945i	$-0.419188\pi$
$601$	12.3146	0.502323	0.251161	+	0.967945i	$0.419188\pi$
$602$	0	0
$603$	−6.86617	−0.279612
$604$	0	0
$605$	0.630527	0.0256346
$606$	0	0
$607$	−27.0500	−1.09793	−0.548964	−	0.835846i	$-0.684977\pi$
$607$	−27.0500	−1.09793	−0.548964	+	0.835846i	$0.684977\pi$
$608$	0	0
$609$	−7.07134	−0.286545
$610$	0	0
$611$	−0.451175	−0.0182526
$612$	0	0
$613$	7.94498	0.320895	0.160447	−	0.987044i	$-0.448706\pi$
$613$	7.94498	0.320895	0.160447	+	0.987044i	$0.448706\pi$
$614$	0	0
$615$	0.346977	0.0139915
$616$	0	0
$617$	−18.8863	−0.760335	−0.380167	−	0.924918i	$-0.624134\pi$
$617$	−18.8863	−0.760335	−0.380167	+	0.924918i	$0.624134\pi$
$618$	0	0
$619$	39.5562	1.58990	0.794949	−	0.606676i	$-0.207498\pi$
$619$	39.5562	1.58990	0.794949	+	0.606676i	$0.207498\pi$
$620$	0	0
$621$	2.03768	0.0817693
$622$	0	0
$623$	6.83166	0.273705
$624$	0	0
$625$	24.4286	0.977143
$626$	0	0
$627$	0.830092	0.0331507
$628$	0	0
$629$	−12.1246	−0.483440
$630$	0	0
$631$	−15.9495	−0.634941	−0.317471	−	0.948268i	$-0.602834\pi$
$631$	−15.9495	−0.634941	−0.317471	+	0.948268i	$0.602834\pi$
$632$	0	0
$633$	8.40606	0.334111
$634$	0	0
$635$	0.0365326	0.00144975
$636$	0	0
$637$	−4.02996	−0.159673
$638$	0	0
$639$	−6.57408	−0.260067
$640$	0	0
$641$	−7.94266	−0.313716	−0.156858	−	0.987621i	$-0.550137\pi$
$641$	−7.94266	−0.313716	−0.156858	+	0.987621i	$0.550137\pi$
$642$	0	0
$643$	14.4678	0.570554	0.285277	−	0.958445i	$-0.407914\pi$
$643$	14.4678	0.570554	0.285277	+	0.958445i	$0.407914\pi$
$644$	0	0
$645$	−0.134613	−0.00530037
$646$	0	0
$647$	−1.16390	−0.0457577	−0.0228788	−	0.999738i	$-0.507283\pi$
$647$	−1.16390	−0.0457577	−0.0228788	+	0.999738i	$0.507283\pi$
$648$	0	0
$649$	39.7111	1.55880
$650$	0	0
$651$	0.169274	0.00663438
$652$	0	0
$653$	46.6219	1.82446	0.912229	−	0.409681i	$-0.134360\pi$
$653$	46.6219	1.82446	0.912229	+	0.409681i	$0.134360\pi$
$654$	0	0
$655$	3.56517	0.139303
$656$	0	0
$657$	6.65772	0.259742
$658$	0	0
$659$	−9.53776	−0.371538	−0.185769	−	0.982593i	$-0.559478\pi$
$659$	−9.53776	−0.371538	−0.185769	+	0.982593i	$0.559478\pi$
$660$	0	0
$661$	34.4560	1.34018	0.670092	−	0.742278i	$-0.266255\pi$
$661$	34.4560	1.34018	0.670092	+	0.742278i	$0.266255\pi$
$662$	0	0
$663$	−1.90237	−0.0738820
$664$	0	0
$665$	−0.0554718	−0.00215110
$666$	0	0
$667$	15.1135	0.585196
$668$	0	0
$669$	17.4068	0.672986
$670$	0	0
$671$	17.2754	0.666908
$672$	0	0
$673$	−5.83360	−0.224869	−0.112434	−	0.993659i	$-0.535865\pi$
$673$	−5.83360	−0.224869	−0.112434	+	0.993659i	$0.535865\pi$
$674$	0	0
$675$	4.96181	0.190980
$676$	0	0
$677$	−2.87755	−0.110593	−0.0552966	−	0.998470i	$-0.517610\pi$
$677$	−2.87755	−0.110593	−0.0552966	+	0.998470i	$0.517610\pi$
$678$	0	0
$679$	2.88475	0.110707
$680$	0	0
$681$	11.4617	0.439215
$682$	0	0
$683$	−24.9927	−0.956320	−0.478160	−	0.878273i	$-0.658696\pi$
$683$	−24.9927	−0.956320	−0.478160	+	0.878273i	$0.658696\pi$
$684$	0	0
$685$	−1.33816	−0.0511286
$686$	0	0
$687$	−8.74504	−0.333644
$688$	0	0
$689$	−1.85764	−0.0707706
$690$	0	0
$691$	2.73895	0.104195	0.0520973	−	0.998642i	$-0.483409\pi$
$691$	2.73895	0.104195	0.0520973	+	0.998642i	$0.483409\pi$
$692$	0	0
$693$	−2.65818	−0.100976
$694$	0	0
$695$	−0.850419	−0.0322582
$696$	0	0
$697$	5.10508	0.193369
$698$	0	0
$699$	23.8506	0.902114
$700$	0	0
$701$	−28.2624	−1.06746	−0.533728	−	0.845656i	$-0.679209\pi$
$701$	−28.2624	−1.06746	−0.533728	+	0.845656i	$0.679209\pi$
$702$	0	0
$703$	1.25545	0.0473500
$704$	0	0
$705$	−0.133266	−0.00501908
$706$	0	0
$707$	5.15281	0.193791
$708$	0	0
$709$	20.2281	0.759681	0.379840	−	0.925052i	$-0.375979\pi$
$709$	20.2281	0.759681	0.379840	+	0.925052i	$0.375979\pi$
$710$	0	0
$711$	9.77040	0.366419
$712$	0	0
$713$	−0.361788	−0.0135491
$714$	0	0
$715$	−0.360499	−0.0134819
$716$	0	0
$717$	−18.7533	−0.700354
$718$	0	0
$719$	−37.5155	−1.39909	−0.699546	−	0.714587i	$-0.746615\pi$
$719$	−37.5155	−1.39909	−0.699546	+	0.714587i	$0.746615\pi$
$720$	0	0
$721$	−11.5326	−0.429498
$722$	0	0
$723$	22.1454	0.823595
$724$	0	0
$725$	36.8017	1.36678
$726$	0	0
$727$	−16.1806	−0.600104	−0.300052	−	0.953923i	$-0.597004\pi$
$727$	−16.1806	−0.600104	−0.300052	+	0.953923i	$0.597004\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.98056	−0.0732536
$732$	0	0
$733$	−44.6196	−1.64806	−0.824032	−	0.566543i	$-0.808280\pi$
$733$	−44.6196	−1.64806	−0.824032	+	0.566543i	$0.808280\pi$
$734$	0	0
$735$	−1.19035	−0.0439067
$736$	0	0
$737$	−19.1437	−0.705166
$738$	0	0
$739$	9.45004	0.347626	0.173813	−	0.984779i	$-0.444391\pi$
$739$	9.45004	0.347626	0.173813	+	0.984779i	$0.444391\pi$
$740$	0	0
$741$	0.196982	0.00723630
$742$	0	0
$743$	−24.5874	−0.902024	−0.451012	−	0.892518i	$-0.648937\pi$
$743$	−24.5874	−0.902024	−0.451012	+	0.892518i	$0.648937\pi$
$744$	0	0
$745$	4.03151	0.147703
$746$	0	0
$747$	2.12152	0.0776224
$748$	0	0
$749$	−14.6458	−0.535147
$750$	0	0
$751$	−30.3380	−1.10705	−0.553524	−	0.832833i	$-0.686717\pi$
$751$	−30.3380	−1.10705	−0.553524	+	0.832833i	$0.686717\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	1.80225	0.0655907
$756$	0	0
$757$	−14.0340	−0.510075	−0.255037	−	0.966931i	$-0.582088\pi$
$757$	−14.0340	−0.510075	−0.255037	+	0.966931i	$0.582088\pi$
$758$	0	0
$759$	5.68128	0.206218
$760$	0	0
$761$	−27.7370	−1.00546	−0.502732	−	0.864442i	$-0.667672\pi$
$761$	−27.7370	−1.00546	−0.502732	+	0.864442i	$0.667672\pi$
$762$	0	0
$763$	7.37745	0.267082
$764$	0	0
$765$	−0.561913	−0.0203160
$766$	0	0
$767$	9.42348	0.340262
$768$	0	0
$769$	−5.26201	−0.189753	−0.0948764	−	0.995489i	$-0.530246\pi$
$769$	−5.26201	−0.189753	−0.0948764	+	0.995489i	$0.530246\pi$
$770$	0	0
$771$	25.3496	0.912942
$772$	0	0
$773$	−2.45085	−0.0881508	−0.0440754	−	0.999028i	$-0.514034\pi$
$773$	−2.45085	−0.0881508	−0.0440754	+	0.999028i	$0.514034\pi$
$774$	0	0
$775$	−0.880963	−0.0316451
$776$	0	0
$777$	−4.02027	−0.144227
$778$	0	0
$779$	−0.528607	−0.0189393
$780$	0	0
$781$	−18.3293	−0.655874
$782$	0	0
$783$	7.41700	0.265062
$784$	0	0
$785$	0.959956	0.0342623
$786$	0	0
$787$	0.655232	0.0233565	0.0116783	−	0.999932i	$-0.496283\pi$
$787$	0.655232	0.0233565	0.0116783	+	0.999932i	$0.496283\pi$
$788$	0	0
$789$	−20.7228	−0.737751
$790$	0	0
$791$	−5.35697	−0.190472
$792$	0	0
$793$	4.09946	0.145576
$794$	0	0
$795$	−0.548701	−0.0194604
$796$	0	0
$797$	0.147686	0.00523132	0.00261566	−	0.999997i	$-0.499167\pi$
$797$	0.147686	0.00523132	0.00261566	+	0.999997i	$0.499167\pi$
$798$	0	0
$799$	−1.96074	−0.0693661
$800$	0	0
$801$	−7.16560	−0.253184
$802$	0	0
$803$	18.5625	0.655056
$804$	0	0
$805$	−0.379658	−0.0133812
$806$	0	0
$807$	11.0721	0.389755
$808$	0	0
$809$	−39.9449	−1.40439	−0.702193	−	0.711987i	$-0.747796\pi$
$809$	−39.9449	−1.40439	−0.702193	+	0.711987i	$0.747796\pi$
$810$	0	0
$811$	−45.3749	−1.59333	−0.796663	−	0.604423i	$-0.793404\pi$
$811$	−45.3749	−1.59333	−0.796663	+	0.604423i	$0.793404\pi$
$812$	0	0
$813$	−10.6975	−0.375177
$814$	0	0
$815$	1.97231	0.0690871
$816$	0	0
$817$	0.205078	0.00717476
$818$	0	0
$819$	−0.630788	−0.0220415
$820$	0	0
$821$	11.8600	0.413916	0.206958	−	0.978350i	$-0.433644\pi$
$821$	11.8600	0.413916	0.206958	+	0.978350i	$0.433644\pi$
$822$	0	0
$823$	47.5309	1.65682	0.828412	−	0.560119i	$-0.189245\pi$
$823$	47.5309	1.65682	0.828412	+	0.560119i	$0.189245\pi$
$824$	0	0
$825$	13.8341	0.481641
$826$	0	0
$827$	22.9039	0.796447	0.398223	−	0.917289i	$-0.369627\pi$
$827$	22.9039	0.796447	0.398223	+	0.917289i	$0.369627\pi$
$828$	0	0
$829$	19.7278	0.685174	0.342587	−	0.939486i	$-0.388697\pi$
$829$	19.7278	0.685174	0.342587	+	0.939486i	$0.388697\pi$
$830$	0	0
$831$	−17.9104	−0.621305
$832$	0	0
$833$	−17.5136	−0.606812
$834$	0	0
$835$	3.95180	0.136758
$836$	0	0
$837$	−0.177549	−0.00613699
$838$	0	0
$839$	−1.74066	−0.0600944	−0.0300472	−	0.999548i	$-0.509566\pi$
$839$	−1.74066	−0.0600944	−0.0300472	+	0.999548i	$0.509566\pi$
$840$	0	0
$841$	26.0119	0.896962
$842$	0	0
$843$	28.7184	0.989115
$844$	0	0
$845$	2.45500	0.0844545
$846$	0	0
$847$	3.07605	0.105694
$848$	0	0
$849$	10.8434	0.372144
$850$	0	0
$851$	8.59248	0.294546
$852$	0	0
$853$	3.49853	0.119788	0.0598938	−	0.998205i	$-0.480924\pi$
$853$	3.49853	0.119788	0.0598938	+	0.998205i	$0.480924\pi$
$854$	0	0
$855$	0.0581834	0.00198983
$856$	0	0
$857$	30.8830	1.05494	0.527472	−	0.849572i	$-0.323140\pi$
$857$	30.8830	1.05494	0.527472	+	0.849572i	$0.323140\pi$
$858$	0	0
$859$	26.9133	0.918269	0.459135	−	0.888367i	$-0.348160\pi$
$859$	26.9133	0.918269	0.459135	+	0.888367i	$0.348160\pi$
$860$	0	0
$861$	1.69274	0.0576884
$862$	0	0
$863$	34.9757	1.19059	0.595293	−	0.803508i	$-0.297036\pi$
$863$	34.9757	1.19059	0.595293	+	0.803508i	$0.297036\pi$
$864$	0	0
$865$	0.755452	0.0256862
$866$	0	0
$867$	8.73257	0.296574
$868$	0	0
$869$	27.2410	0.924088
$870$	0	0
$871$	−4.54281	−0.153927
$872$	0	0
$873$	−3.02576	−0.102407
$874$	0	0
$875$	−1.85607	−0.0627467
$876$	0	0
$877$	8.83450	0.298320	0.149160	−	0.988813i	$-0.452343\pi$
$877$	8.83450	0.298320	0.149160	+	0.988813i	$0.452343\pi$
$878$	0	0
$879$	5.30242	0.178846
$880$	0	0
$881$	−9.47059	−0.319072	−0.159536	−	0.987192i	$-0.551000\pi$
$881$	−9.47059	−0.319072	−0.159536	+	0.987192i	$0.551000\pi$
$882$	0	0
$883$	−34.3079	−1.15455	−0.577276	−	0.816549i	$-0.695885\pi$
$883$	−34.3079	−1.15455	−0.577276	+	0.816549i	$0.695885\pi$
$884$	0	0
$885$	2.78346	0.0935650
$886$	0	0
$887$	−38.7313	−1.30047	−0.650236	−	0.759733i	$-0.725330\pi$
$887$	−38.7313	−1.30047	−0.650236	+	0.759733i	$0.725330\pi$
$888$	0	0
$889$	0.178226	0.00597750
$890$	0	0
$891$	2.78811	0.0934054
$892$	0	0
$893$	0.203026	0.00679399
$894$	0	0
$895$	−4.37264	−0.146161
$896$	0	0
$897$	1.34817	0.0450142
$898$	0	0
$899$	−1.31688	−0.0439204
$900$	0	0
$901$	−8.07305	−0.268952
$902$	0	0
$903$	−0.656713	−0.0218541
$904$	0	0
$905$	−1.12463	−0.0373839
$906$	0	0
$907$	5.43451	0.180450	0.0902250	−	0.995921i	$-0.471241\pi$
$907$	5.43451	0.180450	0.0902250	+	0.995921i	$0.471241\pi$
$908$	0	0
$909$	−5.40469	−0.179262
$910$	0	0
$911$	33.1227	1.09740	0.548702	−	0.836018i	$-0.315122\pi$
$911$	33.1227	1.09740	0.548702	+	0.836018i	$0.315122\pi$
$912$	0	0
$913$	5.91504	0.195759
$914$	0	0
$915$	1.21088	0.0400303
$916$	0	0
$917$	17.3928	0.574362
$918$	0	0
$919$	46.3016	1.52735	0.763675	−	0.645601i	$-0.223393\pi$
$919$	46.3016	1.52735	0.763675	+	0.645601i	$0.223393\pi$
$920$	0	0
$921$	−31.5081	−1.03823
$922$	0	0
$923$	−4.34956	−0.143168
$924$	0	0
$925$	20.9229	0.687942
$926$	0	0
$927$	12.0964	0.397298
$928$	0	0
$929$	−42.7628	−1.40300	−0.701500	−	0.712669i	$-0.747486\pi$
$929$	−42.7628	−1.40300	−0.701500	+	0.712669i	$0.747486\pi$
$930$	0	0
$931$	1.81345	0.0594336
$932$	0	0
$933$	−2.14459	−0.0702106
$934$	0	0
$935$	−1.56668	−0.0512358
$936$	0	0
$937$	12.1186	0.395898	0.197949	−	0.980212i	$-0.436572\pi$
$937$	12.1186	0.395898	0.197949	+	0.980212i	$0.436572\pi$
$938$	0	0
$939$	−23.4893	−0.766543
$940$	0	0
$941$	−2.76665	−0.0901902	−0.0450951	−	0.998983i	$-0.514359\pi$
$941$	−2.76665	−0.0901902	−0.0450951	+	0.998983i	$0.514359\pi$
$942$	0	0
$943$	−3.61787	−0.117814
$944$	0	0
$945$	−0.186319	−0.00606095
$946$	0	0
$947$	53.1596	1.72746	0.863728	−	0.503959i	$-0.168124\pi$
$947$	53.1596	1.72746	0.863728	+	0.503959i	$0.168124\pi$
$948$	0	0
$949$	4.40489	0.142989
$950$	0	0
$951$	28.9105	0.937489
$952$	0	0
$953$	−25.5609	−0.827998	−0.413999	−	0.910277i	$-0.635868\pi$
$953$	−25.5609	−0.827998	−0.413999	+	0.910277i	$0.635868\pi$
$954$	0	0
$955$	4.09962	0.132661
$956$	0	0
$957$	20.6794	0.668472
$958$	0	0
$959$	−6.52828	−0.210809
$960$	0	0
$961$	−30.9685	−0.998983
$962$	0	0
$963$	15.3618	0.495026
$964$	0	0
$965$	−1.50999	−0.0486083
$966$	0	0
$967$	27.2234	0.875445	0.437722	−	0.899110i	$-0.355785\pi$
$967$	27.2234	0.875445	0.437722	+	0.899110i	$0.355785\pi$
$968$	0	0
$969$	0.856053	0.0275004
$970$	0	0
$971$	31.5699	1.01313	0.506563	−	0.862203i	$-0.330916\pi$
$971$	31.5699	1.01313	0.506563	+	0.862203i	$0.330916\pi$
$972$	0	0
$973$	−4.14880	−0.133005
$974$	0	0
$975$	3.28284	0.105135
$976$	0	0
$977$	−20.3913	−0.652374	−0.326187	−	0.945305i	$-0.605764\pi$
$977$	−20.3913	−0.652374	−0.326187	+	0.945305i	$0.605764\pi$
$978$	0	0
$979$	−19.9785	−0.638516
$980$	0	0
$981$	−7.73808	−0.247058
$982$	0	0
$983$	−49.3299	−1.57338	−0.786690	−	0.617348i	$-0.788207\pi$
$983$	−49.3299	−1.57338	−0.786690	+	0.617348i	$0.788207\pi$
$984$	0	0
$985$	3.10591	0.0989626
$986$	0	0
$987$	−0.650142	−0.0206943
$988$	0	0
$989$	1.40358	0.0446314
$990$	0	0
$991$	49.2570	1.56470	0.782350	−	0.622839i	$-0.214021\pi$
$991$	49.2570	1.56470	0.782350	+	0.622839i	$0.214021\pi$
$992$	0	0
$993$	−3.26849	−0.103722
$994$	0	0
$995$	4.05478	0.128545
$996$	0	0
$997$	10.1880	0.322658	0.161329	−	0.986901i	$-0.448422\pi$
$997$	10.1880	0.322658	0.161329	+	0.986901i	$0.448422\pi$
$998$	0	0
$999$	4.21680	0.133414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.n.1.8	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.n.1.8	✓	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.195427 q^{5} -0.953396 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.195427 q^{5} -0.953396 q^{7} +1.00000 q^{9} +2.78811 q^{11} +0.661622 q^{13} +0.195427 q^{15} +2.87531 q^{17} -0.297725 q^{19} +0.953396 q^{21} -2.03768 q^{23} -4.96181 q^{25} -1.00000 q^{27} -7.41700 q^{29} +0.177549 q^{31} -2.78811 q^{33} +0.186319 q^{35} -4.21680 q^{37} -0.661622 q^{39} +1.77549 q^{41} -0.688815 q^{43} -0.195427 q^{45} -0.681923 q^{47} -6.09104 q^{49} -2.87531 q^{51} -2.80771 q^{53} -0.544872 q^{55} +0.297725 q^{57} +14.2430 q^{59} +6.19607 q^{61} -0.953396 q^{63} -0.129299 q^{65} -6.86617 q^{67} +2.03768 q^{69} -6.57408 q^{71} +6.65772 q^{73} +4.96181 q^{75} -2.65818 q^{77} +9.77040 q^{79} +1.00000 q^{81} +2.12152 q^{83} -0.561913 q^{85} +7.41700 q^{87} -7.16560 q^{89} -0.630788 q^{91} -0.177549 q^{93} +0.0581834 q^{95} -3.02576 q^{97} +2.78811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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