LMFDB - Embedded newform 6024.2.a.o.1.4

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} - 3 x^{13} - 43 x^{12} + 119 x^{11} + 679 x^{10} - 1667 x^{9} - 4890 x^{8} + 9662 x^{7} + \cdots - 416 $ x^14 - 3x^13 - 43x^12 + 119x^11 + 679x^10 - 1667x^9 - 4890x^8 + 9662x^7 + 16575x^6 - 20277x^5 - 25196x^4 + 8040x^3 + 10776x^2 + 912*x - 416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.85865$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.85865 q^{5} +3.46115 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.85865 q^{5} +3.46115 q^{7} +1.00000 q^{9} +1.97377 q^{11} -6.43848 q^{13} +2.85865 q^{15} +0.787244 q^{17} -0.821750 q^{19} -3.46115 q^{21} +4.89600 q^{23} +3.17188 q^{25} -1.00000 q^{27} -3.09565 q^{29} +0.755108 q^{31} -1.97377 q^{33} -9.89423 q^{35} +4.19022 q^{37} +6.43848 q^{39} -5.65621 q^{41} +9.67366 q^{43} -2.85865 q^{45} +0.444653 q^{47} +4.97958 q^{49} -0.787244 q^{51} -11.9564 q^{53} -5.64231 q^{55} +0.821750 q^{57} -7.73418 q^{59} -5.93505 q^{61} +3.46115 q^{63} +18.4054 q^{65} +2.06056 q^{67} -4.89600 q^{69} +9.58830 q^{71} -8.89436 q^{73} -3.17188 q^{75} +6.83152 q^{77} +15.9288 q^{79} +1.00000 q^{81} +12.2935 q^{83} -2.25045 q^{85} +3.09565 q^{87} -8.86836 q^{89} -22.2846 q^{91} -0.755108 q^{93} +2.34910 q^{95} +1.21494 q^{97} +1.97377 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * 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
$5$	−2.85865	−1.27843	−0.639214	−	0.769029i	$-0.720740\pi$
$5$	−2.85865	−1.27843	−0.639214	+	0.769029i	$0.720740\pi$
$6$	0	0
$7$	3.46115	1.30819	0.654097	−	0.756411i	$-0.273049\pi$
$7$	3.46115	1.30819	0.654097	+	0.756411i	$0.273049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.97377	0.595114	0.297557	−	0.954704i	$-0.403828\pi$
$11$	1.97377	0.595114	0.297557	+	0.954704i	$0.403828\pi$
$12$	0	0
$13$	−6.43848	−1.78571	−0.892857	−	0.450340i	$-0.851303\pi$
$13$	−6.43848	−1.78571	−0.892857	+	0.450340i	$0.851303\pi$
$14$	0	0
$15$	2.85865	0.738100
$16$	0	0
$17$	0.787244	0.190935	0.0954673	−	0.995433i	$-0.469565\pi$
$17$	0.787244	0.190935	0.0954673	+	0.995433i	$0.469565\pi$
$18$	0	0
$19$	−0.821750	−0.188522	−0.0942612	−	0.995547i	$-0.530049\pi$
$19$	−0.821750	−0.188522	−0.0942612	+	0.995547i	$0.530049\pi$
$20$	0	0
$21$	−3.46115	−0.755286
$22$	0	0
$23$	4.89600	1.02089	0.510444	−	0.859911i	$-0.329481\pi$
$23$	4.89600	1.02089	0.510444	+	0.859911i	$0.329481\pi$
$24$	0	0
$25$	3.17188	0.634376
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.09565	−0.574847	−0.287424	−	0.957804i	$-0.592799\pi$
$29$	−3.09565	−0.574847	−0.287424	+	0.957804i	$0.592799\pi$
$30$	0	0
$31$	0.755108	0.135621	0.0678107	−	0.997698i	$-0.478399\pi$
$31$	0.755108	0.135621	0.0678107	+	0.997698i	$0.478399\pi$
$32$	0	0
$33$	−1.97377	−0.343589
$34$	0	0
$35$	−9.89423	−1.67243
$36$	0	0
$37$	4.19022	0.688867	0.344434	−	0.938811i	$-0.388071\pi$
$37$	4.19022	0.688867	0.344434	+	0.938811i	$0.388071\pi$
$38$	0	0
$39$	6.43848	1.03098
$40$	0	0
$41$	−5.65621	−0.883351	−0.441676	−	0.897175i	$-0.645616\pi$
$41$	−5.65621	−0.883351	−0.441676	+	0.897175i	$0.645616\pi$
$42$	0	0
$43$	9.67366	1.47522	0.737609	−	0.675228i	$-0.235955\pi$
$43$	9.67366	1.47522	0.737609	+	0.675228i	$0.235955\pi$
$44$	0	0
$45$	−2.85865	−0.426142
$46$	0	0
$47$	0.444653	0.0648593	0.0324296	−	0.999474i	$-0.489676\pi$
$47$	0.444653	0.0648593	0.0324296	+	0.999474i	$0.489676\pi$
$48$	0	0
$49$	4.97958	0.711369
$50$	0	0
$51$	−0.787244	−0.110236
$52$	0	0
$53$	−11.9564	−1.64234	−0.821168	−	0.570686i	$-0.806677\pi$
$53$	−11.9564	−1.64234	−0.821168	+	0.570686i	$0.806677\pi$
$54$	0	0
$55$	−5.64231	−0.760809
$56$	0	0
$57$	0.821750	0.108843
$58$	0	0
$59$	−7.73418	−1.00690	−0.503452	−	0.864023i	$-0.667937\pi$
$59$	−7.73418	−1.00690	−0.503452	+	0.864023i	$0.667937\pi$
$60$	0	0
$61$	−5.93505	−0.759905	−0.379953	−	0.925006i	$-0.624060\pi$
$61$	−5.93505	−0.759905	−0.379953	+	0.925006i	$0.624060\pi$
$62$	0	0
$63$	3.46115	0.436064
$64$	0	0
$65$	18.4054	2.28291
$66$	0	0
$67$	2.06056	0.251738	0.125869	−	0.992047i	$-0.459828\pi$
$67$	2.06056	0.251738	0.125869	+	0.992047i	$0.459828\pi$
$68$	0	0
$69$	−4.89600	−0.589409
$70$	0	0
$71$	9.58830	1.13792	0.568961	−	0.822365i	$-0.307346\pi$
$71$	9.58830	1.13792	0.568961	+	0.822365i	$0.307346\pi$
$72$	0	0
$73$	−8.89436	−1.04101	−0.520503	−	0.853860i	$-0.674256\pi$
$73$	−8.89436	−1.04101	−0.520503	+	0.853860i	$0.674256\pi$
$74$	0	0
$75$	−3.17188	−0.366257
$76$	0	0
$77$	6.83152	0.778524
$78$	0	0
$79$	15.9288	1.79214	0.896068	−	0.443918i	$-0.146412\pi$
$79$	15.9288	1.79214	0.896068	+	0.443918i	$0.146412\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.2935	1.34939	0.674693	−	0.738098i	$-0.264276\pi$
$83$	12.2935	1.34939	0.674693	+	0.738098i	$0.264276\pi$
$84$	0	0
$85$	−2.25045	−0.244096
$86$	0	0
$87$	3.09565	0.331888
$88$	0	0
$89$	−8.86836	−0.940044	−0.470022	−	0.882655i	$-0.655754\pi$
$89$	−8.86836	−0.940044	−0.470022	+	0.882655i	$0.655754\pi$
$90$	0	0
$91$	−22.2846	−2.33606
$92$	0	0
$93$	−0.755108	−0.0783010
$94$	0	0
$95$	2.34910	0.241012
$96$	0	0
$97$	1.21494	0.123358	0.0616791	−	0.998096i	$-0.480354\pi$
$97$	1.21494	0.123358	0.0616791	+	0.998096i	$0.480354\pi$
$98$	0	0
$99$	1.97377	0.198371
$100$	0	0
$101$	8.56339	0.852089	0.426045	−	0.904702i	$-0.359907\pi$
$101$	8.56339	0.852089	0.426045	+	0.904702i	$0.359907\pi$
$102$	0	0
$103$	−2.23832	−0.220548	−0.110274	−	0.993901i	$-0.535173\pi$
$103$	−2.23832	−0.220548	−0.110274	+	0.993901i	$0.535173\pi$
$104$	0	0
$105$	9.89423	0.965578
$106$	0	0
$107$	−16.7135	−1.61576	−0.807880	−	0.589347i	$-0.799385\pi$
$107$	−16.7135	−1.61576	−0.807880	+	0.589347i	$0.799385\pi$
$108$	0	0
$109$	6.91488	0.662325	0.331163	−	0.943574i	$-0.392559\pi$
$109$	6.91488	0.662325	0.331163	+	0.943574i	$0.392559\pi$
$110$	0	0
$111$	−4.19022	−0.397718
$112$	0	0
$113$	−14.7421	−1.38682	−0.693408	−	0.720545i	$-0.743892\pi$
$113$	−14.7421	−1.38682	−0.693408	+	0.720545i	$0.743892\pi$
$114$	0	0
$115$	−13.9960	−1.30513
$116$	0	0
$117$	−6.43848	−0.595238
$118$	0	0
$119$	2.72477	0.249779
$120$	0	0
$121$	−7.10424	−0.645840
$122$	0	0
$123$	5.65621	0.510003
$124$	0	0
$125$	5.22596	0.467424
$126$	0	0
$127$	3.57519	0.317247	0.158623	−	0.987339i	$-0.449294\pi$
$127$	3.57519	0.317247	0.158623	+	0.987339i	$0.449294\pi$
$128$	0	0
$129$	−9.67366	−0.851718
$130$	0	0
$131$	8.90148	0.777726	0.388863	−	0.921296i	$-0.372868\pi$
$131$	8.90148	0.777726	0.388863	+	0.921296i	$0.372868\pi$
$132$	0	0
$133$	−2.84420	−0.246624
$134$	0	0
$135$	2.85865	0.246033
$136$	0	0
$137$	11.8943	1.01620	0.508098	−	0.861299i	$-0.330349\pi$
$137$	11.8943	1.01620	0.508098	+	0.861299i	$0.330349\pi$
$138$	0	0
$139$	2.18070	0.184965	0.0924824	−	0.995714i	$-0.470520\pi$
$139$	2.18070	0.184965	0.0924824	+	0.995714i	$0.470520\pi$
$140$	0	0
$141$	−0.444653	−0.0374465
$142$	0	0
$143$	−12.7081	−1.06270
$144$	0	0
$145$	8.84937	0.734900
$146$	0	0
$147$	−4.97958	−0.410709
$148$	0	0
$149$	−15.3444	−1.25706	−0.628532	−	0.777784i	$-0.716344\pi$
$149$	−15.3444	−1.25706	−0.628532	+	0.777784i	$0.716344\pi$
$150$	0	0
$151$	11.5398	0.939098	0.469549	−	0.882906i	$-0.344417\pi$
$151$	11.5398	0.939098	0.469549	+	0.882906i	$0.344417\pi$
$152$	0	0
$153$	0.787244	0.0636449
$154$	0	0
$155$	−2.15859	−0.173382
$156$	0	0
$157$	3.28043	0.261807	0.130903	−	0.991395i	$-0.458212\pi$
$157$	3.28043	0.261807	0.130903	+	0.991395i	$0.458212\pi$
$158$	0	0
$159$	11.9564	0.948203
$160$	0	0
$161$	16.9458	1.33552
$162$	0	0
$163$	−11.0122	−0.862543	−0.431271	−	0.902222i	$-0.641935\pi$
$163$	−11.0122	−0.862543	−0.431271	+	0.902222i	$0.641935\pi$
$164$	0	0
$165$	5.64231	0.439253
$166$	0	0
$167$	−25.1543	−1.94650	−0.973250	−	0.229749i	$-0.926209\pi$
$167$	−25.1543	−1.94650	−0.973250	+	0.229749i	$0.926209\pi$
$168$	0	0
$169$	28.4541	2.18878
$170$	0	0
$171$	−0.821750	−0.0628408
$172$	0	0
$173$	−11.8121	−0.898055	−0.449028	−	0.893518i	$-0.648230\pi$
$173$	−11.8121	−0.898055	−0.449028	+	0.893518i	$0.648230\pi$
$174$	0	0
$175$	10.9784	0.829886
$176$	0	0
$177$	7.73418	0.581337
$178$	0	0
$179$	−23.3312	−1.74385	−0.871926	−	0.489637i	$-0.837129\pi$
$179$	−23.3312	−1.74385	−0.871926	+	0.489637i	$0.837129\pi$
$180$	0	0
$181$	0.301024	0.0223749	0.0111875	−	0.999937i	$-0.496439\pi$
$181$	0.301024	0.0223749	0.0111875	+	0.999937i	$0.496439\pi$
$182$	0	0
$183$	5.93505	0.438732
$184$	0	0
$185$	−11.9784	−0.880667
$186$	0	0
$187$	1.55384	0.113628
$188$	0	0
$189$	−3.46115	−0.251762
$190$	0	0
$191$	−11.3600	−0.821981	−0.410990	−	0.911640i	$-0.634817\pi$
$191$	−11.3600	−0.821981	−0.410990	+	0.911640i	$0.634817\pi$
$192$	0	0
$193$	7.49915	0.539800	0.269900	−	0.962888i	$-0.413009\pi$
$193$	7.49915	0.539800	0.269900	+	0.962888i	$0.413009\pi$
$194$	0	0
$195$	−18.4054	−1.31804
$196$	0	0
$197$	13.1536	0.937152	0.468576	−	0.883423i	$-0.344767\pi$
$197$	13.1536	0.937152	0.468576	+	0.883423i	$0.344767\pi$
$198$	0	0
$199$	24.0140	1.70231	0.851155	−	0.524915i	$-0.175903\pi$
$199$	24.0140	1.70231	0.851155	+	0.524915i	$0.175903\pi$
$200$	0	0
$201$	−2.06056	−0.145341
$202$	0	0
$203$	−10.7145	−0.752011
$204$	0	0
$205$	16.1691	1.12930
$206$	0	0
$207$	4.89600	0.340296
$208$	0	0
$209$	−1.62194	−0.112192
$210$	0	0
$211$	−13.6875	−0.942283	−0.471142	−	0.882058i	$-0.656158\pi$
$211$	−13.6875	−0.942283	−0.471142	+	0.882058i	$0.656158\pi$
$212$	0	0
$213$	−9.58830	−0.656979
$214$	0	0
$215$	−27.6536	−1.88596
$216$	0	0
$217$	2.61354	0.177419
$218$	0	0
$219$	8.89436	0.601025
$220$	0	0
$221$	−5.06866	−0.340955
$222$	0	0
$223$	−11.9355	−0.799262	−0.399631	−	0.916676i	$-0.630862\pi$
$223$	−11.9355	−0.799262	−0.399631	+	0.916676i	$0.630862\pi$
$224$	0	0
$225$	3.17188	0.211459
$226$	0	0
$227$	−11.1186	−0.737968	−0.368984	−	0.929436i	$-0.620294\pi$
$227$	−11.1186	−0.737968	−0.368984	+	0.929436i	$0.620294\pi$
$228$	0	0
$229$	1.46013	0.0964878	0.0482439	−	0.998836i	$-0.484638\pi$
$229$	1.46013	0.0964878	0.0482439	+	0.998836i	$0.484638\pi$
$230$	0	0
$231$	−6.83152	−0.449481
$232$	0	0
$233$	−2.70039	−0.176909	−0.0884544	−	0.996080i	$-0.528193\pi$
$233$	−2.70039	−0.176909	−0.0884544	+	0.996080i	$0.528193\pi$
$234$	0	0
$235$	−1.27111	−0.0829179
$236$	0	0
$237$	−15.9288	−1.03469
$238$	0	0
$239$	−14.1710	−0.916646	−0.458323	−	0.888786i	$-0.651550\pi$
$239$	−14.1710	−0.916646	−0.458323	+	0.888786i	$0.651550\pi$
$240$	0	0
$241$	−11.5165	−0.741843	−0.370921	−	0.928664i	$-0.620958\pi$
$241$	−11.5165	−0.741843	−0.370921	+	0.928664i	$0.620958\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−14.2349	−0.909433
$246$	0	0
$247$	5.29083	0.336647
$248$	0	0
$249$	−12.2935	−0.779069
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	9.66358	0.607544
$254$	0	0
$255$	2.25045	0.140929
$256$	0	0
$257$	−12.9032	−0.804876	−0.402438	−	0.915447i	$-0.631837\pi$
$257$	−12.9032	−0.804876	−0.402438	+	0.915447i	$0.631837\pi$
$258$	0	0
$259$	14.5030	0.901172
$260$	0	0
$261$	−3.09565	−0.191616
$262$	0	0
$263$	−0.412392	−0.0254292	−0.0127146	−	0.999919i	$-0.504047\pi$
$263$	−0.412392	−0.0254292	−0.0127146	+	0.999919i	$0.504047\pi$
$264$	0	0
$265$	34.1791	2.09961
$266$	0	0
$267$	8.86836	0.542735
$268$	0	0
$269$	3.59279	0.219056	0.109528	−	0.993984i	$-0.465066\pi$
$269$	3.59279	0.219056	0.109528	+	0.993984i	$0.465066\pi$
$270$	0	0
$271$	−19.4801	−1.18333	−0.591667	−	0.806183i	$-0.701530\pi$
$271$	−19.4801	−1.18333	−0.591667	+	0.806183i	$0.701530\pi$
$272$	0	0
$273$	22.2846	1.34872
$274$	0	0
$275$	6.26055	0.377526
$276$	0	0
$277$	28.0330	1.68434	0.842169	−	0.539213i	$-0.181278\pi$
$277$	28.0330	1.68434	0.842169	+	0.539213i	$0.181278\pi$
$278$	0	0
$279$	0.755108	0.0452071
$280$	0	0
$281$	−29.0257	−1.73153	−0.865765	−	0.500451i	$-0.833168\pi$
$281$	−29.0257	−1.73153	−0.865765	+	0.500451i	$0.833168\pi$
$282$	0	0
$283$	−17.7692	−1.05627	−0.528134	−	0.849161i	$-0.677108\pi$
$283$	−17.7692	−1.05627	−0.528134	+	0.849161i	$0.677108\pi$
$284$	0	0
$285$	−2.34910	−0.139148
$286$	0	0
$287$	−19.5770	−1.15559
$288$	0	0
$289$	−16.3802	−0.963544
$290$	0	0
$291$	−1.21494	−0.0712209
$292$	0	0
$293$	−17.4501	−1.01945	−0.509724	−	0.860338i	$-0.670252\pi$
$293$	−17.4501	−1.01945	−0.509724	+	0.860338i	$0.670252\pi$
$294$	0	0
$295$	22.1093	1.28725
$296$	0	0
$297$	−1.97377	−0.114530
$298$	0	0
$299$	−31.5228	−1.82301
$300$	0	0
$301$	33.4820	1.92987
$302$	0	0
$303$	−8.56339	−0.491954
$304$	0	0
$305$	16.9662	0.971484
$306$	0	0
$307$	−20.4287	−1.16593	−0.582963	−	0.812498i	$-0.698107\pi$
$307$	−20.4287	−1.16593	−0.582963	+	0.812498i	$0.698107\pi$
$308$	0	0
$309$	2.23832	0.127333
$310$	0	0
$311$	9.32624	0.528842	0.264421	−	0.964407i	$-0.414819\pi$
$311$	9.32624	0.528842	0.264421	+	0.964407i	$0.414819\pi$
$312$	0	0
$313$	7.59242	0.429149	0.214575	−	0.976708i	$-0.431163\pi$
$313$	7.59242	0.429149	0.214575	+	0.976708i	$0.431163\pi$
$314$	0	0
$315$	−9.89423	−0.557476
$316$	0	0
$317$	−20.7358	−1.16464	−0.582320	−	0.812959i	$-0.697855\pi$
$317$	−20.7358	−1.16464	−0.582320	+	0.812959i	$0.697855\pi$
$318$	0	0
$319$	−6.11009	−0.342099
$320$	0	0
$321$	16.7135	0.932859
$322$	0	0
$323$	−0.646918	−0.0359955
$324$	0	0
$325$	−20.4221	−1.13281
$326$	0	0
$327$	−6.91488	−0.382394
$328$	0	0
$329$	1.53901	0.0848485
$330$	0	0
$331$	−2.97892	−0.163736	−0.0818682	−	0.996643i	$-0.526089\pi$
$331$	−2.97892	−0.163736	−0.0818682	+	0.996643i	$0.526089\pi$
$332$	0	0
$333$	4.19022	0.229622
$334$	0	0
$335$	−5.89043	−0.321829
$336$	0	0
$337$	22.7268	1.23801	0.619003	−	0.785389i	$-0.287537\pi$
$337$	22.7268	1.23801	0.619003	+	0.785389i	$0.287537\pi$
$338$	0	0
$339$	14.7421	0.800679
$340$	0	0
$341$	1.49041	0.0807101
$342$	0	0
$343$	−6.99297	−0.377585
$344$	0	0
$345$	13.9960	0.753517
$346$	0	0
$347$	−33.9602	−1.82308	−0.911539	−	0.411214i	$-0.865105\pi$
$347$	−33.9602	−1.82308	−0.911539	+	0.411214i	$0.865105\pi$
$348$	0	0
$349$	−24.2883	−1.30012	−0.650062	−	0.759882i	$-0.725257\pi$
$349$	−24.2883	−1.30012	−0.650062	+	0.759882i	$0.725257\pi$
$350$	0	0
$351$	6.43848	0.343661
$352$	0	0
$353$	−1.49292	−0.0794603	−0.0397301	−	0.999210i	$-0.512650\pi$
$353$	−1.49292	−0.0794603	−0.0397301	+	0.999210i	$0.512650\pi$
$354$	0	0
$355$	−27.4096	−1.45475
$356$	0	0
$357$	−2.72477	−0.144210
$358$	0	0
$359$	10.0541	0.530634	0.265317	−	0.964161i	$-0.414523\pi$
$359$	10.0541	0.530634	0.265317	+	0.964161i	$0.414523\pi$
$360$	0	0
$361$	−18.3247	−0.964459
$362$	0	0
$363$	7.10424	0.372876
$364$	0	0
$365$	25.4259	1.33085
$366$	0	0
$367$	−28.4158	−1.48329	−0.741647	−	0.670791i	$-0.765955\pi$
$367$	−28.4158	−1.48329	−0.741647	+	0.670791i	$0.765955\pi$
$368$	0	0
$369$	−5.65621	−0.294450
$370$	0	0
$371$	−41.3829	−2.14849
$372$	0	0
$373$	−24.6509	−1.27638	−0.638188	−	0.769881i	$-0.720316\pi$
$373$	−24.6509	−1.27638	−0.638188	+	0.769881i	$0.720316\pi$
$374$	0	0
$375$	−5.22596	−0.269867
$376$	0	0
$377$	19.9313	1.02651
$378$	0	0
$379$	10.8229	0.555937	0.277968	−	0.960590i	$-0.410339\pi$
$379$	10.8229	0.555937	0.277968	+	0.960590i	$0.410339\pi$
$380$	0	0
$381$	−3.57519	−0.183162
$382$	0	0
$383$	28.3722	1.44975	0.724875	−	0.688881i	$-0.241898\pi$
$383$	28.3722	1.44975	0.724875	+	0.688881i	$0.241898\pi$
$384$	0	0
$385$	−19.5289	−0.995286
$386$	0	0
$387$	9.67366	0.491740
$388$	0	0
$389$	15.2903	0.775250	0.387625	−	0.921817i	$-0.373295\pi$
$389$	15.2903	0.775250	0.387625	+	0.921817i	$0.373295\pi$
$390$	0	0
$391$	3.85435	0.194923
$392$	0	0
$393$	−8.90148	−0.449020
$394$	0	0
$395$	−45.5350	−2.29111
$396$	0	0
$397$	−6.93483	−0.348049	−0.174025	−	0.984741i	$-0.555677\pi$
$397$	−6.93483	−0.348049	−0.174025	+	0.984741i	$0.555677\pi$
$398$	0	0
$399$	2.84420	0.142388
$400$	0	0
$401$	−11.0000	−0.549313	−0.274656	−	0.961542i	$-0.588564\pi$
$401$	−11.0000	−0.549313	−0.274656	+	0.961542i	$0.588564\pi$
$402$	0	0
$403$	−4.86175	−0.242181
$404$	0	0
$405$	−2.85865	−0.142047
$406$	0	0
$407$	8.27052	0.409954
$408$	0	0
$409$	−29.4426	−1.45584	−0.727921	−	0.685661i	$-0.759513\pi$
$409$	−29.4426	−1.45584	−0.727921	+	0.685661i	$0.759513\pi$
$410$	0	0
$411$	−11.8943	−0.586701
$412$	0	0
$413$	−26.7692	−1.31723
$414$	0	0
$415$	−35.1428	−1.72509
$416$	0	0
$417$	−2.18070	−0.106789
$418$	0	0
$419$	−1.79204	−0.0875467	−0.0437734	−	0.999041i	$-0.513938\pi$
$419$	−1.79204	−0.0875467	−0.0437734	+	0.999041i	$0.513938\pi$
$420$	0	0
$421$	−38.8402	−1.89296	−0.946479	−	0.322765i	$-0.895387\pi$
$421$	−38.8402	−1.89296	−0.946479	+	0.322765i	$0.895387\pi$
$422$	0	0
$423$	0.444653	0.0216198
$424$	0	0
$425$	2.49704	0.121124
$426$	0	0
$427$	−20.5421	−0.994103
$428$	0	0
$429$	12.7081	0.613552
$430$	0	0
$431$	−25.0252	−1.20542	−0.602710	−	0.797960i	$-0.705912\pi$
$431$	−25.0252	−1.20542	−0.602710	+	0.797960i	$0.705912\pi$
$432$	0	0
$433$	−29.6033	−1.42264	−0.711322	−	0.702867i	$-0.751903\pi$
$433$	−29.6033	−1.42264	−0.711322	+	0.702867i	$0.751903\pi$
$434$	0	0
$435$	−8.84937	−0.424295
$436$	0	0
$437$	−4.02329	−0.192460
$438$	0	0
$439$	18.0498	0.861472	0.430736	−	0.902478i	$-0.358254\pi$
$439$	18.0498	0.861472	0.430736	+	0.902478i	$0.358254\pi$
$440$	0	0
$441$	4.97958	0.237123
$442$	0	0
$443$	−19.9971	−0.950089	−0.475045	−	0.879962i	$-0.657568\pi$
$443$	−19.9971	−0.950089	−0.475045	+	0.879962i	$0.657568\pi$
$444$	0	0
$445$	25.3515	1.20178
$446$	0	0
$447$	15.3444	0.725766
$448$	0	0
$449$	−19.6482	−0.927257	−0.463629	−	0.886030i	$-0.653453\pi$
$449$	−19.6482	−0.927257	−0.463629	+	0.886030i	$0.653453\pi$
$450$	0	0
$451$	−11.1640	−0.525694
$452$	0	0
$453$	−11.5398	−0.542189
$454$	0	0
$455$	63.7038	2.98648
$456$	0	0
$457$	12.9819	0.607267	0.303633	−	0.952789i	$-0.401800\pi$
$457$	12.9819	0.607267	0.303633	+	0.952789i	$0.401800\pi$
$458$	0	0
$459$	−0.787244	−0.0367454
$460$	0	0
$461$	−9.22485	−0.429644	−0.214822	−	0.976653i	$-0.568917\pi$
$461$	−9.22485	−0.429644	−0.214822	+	0.976653i	$0.568917\pi$
$462$	0	0
$463$	29.6083	1.37601	0.688007	−	0.725704i	$-0.258486\pi$
$463$	29.6083	1.37601	0.688007	+	0.725704i	$0.258486\pi$
$464$	0	0
$465$	2.15859	0.100102
$466$	0	0
$467$	20.3417	0.941303	0.470651	−	0.882319i	$-0.344019\pi$
$467$	20.3417	0.941303	0.470651	+	0.882319i	$0.344019\pi$
$468$	0	0
$469$	7.13193	0.329322
$470$	0	0
$471$	−3.28043	−0.151154
$472$	0	0
$473$	19.0936	0.877923
$474$	0	0
$475$	−2.60649	−0.119594
$476$	0	0
$477$	−11.9564	−0.547445
$478$	0	0
$479$	−14.8343	−0.677795	−0.338898	−	0.940823i	$-0.610054\pi$
$479$	−14.8343	−0.677795	−0.338898	+	0.940823i	$0.610054\pi$
$480$	0	0
$481$	−26.9786	−1.23012
$482$	0	0
$483$	−16.9458	−0.771061
$484$	0	0
$485$	−3.47308	−0.157705
$486$	0	0
$487$	21.2044	0.960863	0.480431	−	0.877032i	$-0.340480\pi$
$487$	21.2044	0.960863	0.480431	+	0.877032i	$0.340480\pi$
$488$	0	0
$489$	11.0122	0.497989
$490$	0	0
$491$	−24.9738	−1.12705	−0.563525	−	0.826099i	$-0.690555\pi$
$491$	−24.9738	−1.12705	−0.563525	+	0.826099i	$0.690555\pi$
$492$	0	0
$493$	−2.43703	−0.109758
$494$	0	0
$495$	−5.64231	−0.253603
$496$	0	0
$497$	33.1866	1.48862
$498$	0	0
$499$	31.1777	1.39571	0.697853	−	0.716241i	$-0.254139\pi$
$499$	31.1777	1.39571	0.697853	+	0.716241i	$0.254139\pi$
$500$	0	0
$501$	25.1543	1.12381
$502$	0	0
$503$	−23.7468	−1.05882	−0.529408	−	0.848367i	$-0.677586\pi$
$503$	−23.7468	−1.05882	−0.529408	+	0.848367i	$0.677586\pi$
$504$	0	0
$505$	−24.4797	−1.08933
$506$	0	0
$507$	−28.4541	−1.26369
$508$	0	0
$509$	−2.08465	−0.0924006	−0.0462003	−	0.998932i	$-0.514711\pi$
$509$	−2.08465	−0.0924006	−0.0462003	+	0.998932i	$0.514711\pi$
$510$	0	0
$511$	−30.7847	−1.36184
$512$	0	0
$513$	0.821750	0.0362812
$514$	0	0
$515$	6.39856	0.281954
$516$	0	0
$517$	0.877642	0.0385986
$518$	0	0
$519$	11.8121	0.518492
$520$	0	0
$521$	−9.81549	−0.430025	−0.215012	−	0.976611i	$-0.568979\pi$
$521$	−9.81549	−0.430025	−0.215012	+	0.976611i	$0.568979\pi$
$522$	0	0
$523$	5.73921	0.250958	0.125479	−	0.992096i	$-0.459953\pi$
$523$	5.73921	0.250958	0.125479	+	0.992096i	$0.459953\pi$
$524$	0	0
$525$	−10.9784	−0.479135
$526$	0	0
$527$	0.594454	0.0258948
$528$	0	0
$529$	0.970842	0.0422105
$530$	0	0
$531$	−7.73418	−0.335635
$532$	0	0
$533$	36.4174	1.57741
$534$	0	0
$535$	47.7782	2.06563
$536$	0	0
$537$	23.3312	1.00681
$538$	0	0
$539$	9.82855	0.423345
$540$	0	0
$541$	24.2136	1.04102	0.520511	−	0.853855i	$-0.325741\pi$
$541$	24.2136	1.04102	0.520511	+	0.853855i	$0.325741\pi$
$542$	0	0
$543$	−0.301024	−0.0129182
$544$	0	0
$545$	−19.7672	−0.846734
$546$	0	0
$547$	24.9403	1.06637	0.533186	−	0.845998i	$-0.320995\pi$
$547$	24.9403	1.06637	0.533186	+	0.845998i	$0.320995\pi$
$548$	0	0
$549$	−5.93505	−0.253302
$550$	0	0
$551$	2.54385	0.108372
$552$	0	0
$553$	55.1322	2.34446
$554$	0	0
$555$	11.9784	0.508453
$556$	0	0
$557$	41.8016	1.77119	0.885595	−	0.464457i	$-0.153751\pi$
$557$	41.8016	1.77119	0.885595	+	0.464457i	$0.153751\pi$
$558$	0	0
$559$	−62.2837	−2.63432
$560$	0	0
$561$	−1.55384	−0.0656030
$562$	0	0
$563$	−19.4501	−0.819725	−0.409863	−	0.912147i	$-0.634423\pi$
$563$	−19.4501	−0.819725	−0.409863	+	0.912147i	$0.634423\pi$
$564$	0	0
$565$	42.1424	1.77294
$566$	0	0
$567$	3.46115	0.145355
$568$	0	0
$569$	34.8453	1.46079	0.730396	−	0.683024i	$-0.239336\pi$
$569$	34.8453	1.46079	0.730396	+	0.683024i	$0.239336\pi$
$570$	0	0
$571$	9.55201	0.399739	0.199869	−	0.979823i	$-0.435948\pi$
$571$	9.55201	0.399739	0.199869	+	0.979823i	$0.435948\pi$
$572$	0	0
$573$	11.3600	0.474571
$574$	0	0
$575$	15.5295	0.647626
$576$	0	0
$577$	−21.0700	−0.877155	−0.438577	−	0.898693i	$-0.644517\pi$
$577$	−21.0700	−0.877155	−0.438577	+	0.898693i	$0.644517\pi$
$578$	0	0
$579$	−7.49915	−0.311654
$580$	0	0
$581$	42.5497	1.76526
$582$	0	0
$583$	−23.5991	−0.977377
$584$	0	0
$585$	18.4054	0.760968
$586$	0	0
$587$	−20.0729	−0.828499	−0.414249	−	0.910163i	$-0.635956\pi$
$587$	−20.0729	−0.828499	−0.414249	+	0.910163i	$0.635956\pi$
$588$	0	0
$589$	−0.620510	−0.0255677
$590$	0	0
$591$	−13.1536	−0.541065
$592$	0	0
$593$	17.8472	0.732895	0.366448	−	0.930439i	$-0.380574\pi$
$593$	17.8472	0.732895	0.366448	+	0.930439i	$0.380574\pi$
$594$	0	0
$595$	−7.78917	−0.319325
$596$	0	0
$597$	−24.0140	−0.982829
$598$	0	0
$599$	−31.2348	−1.27622	−0.638110	−	0.769945i	$-0.720284\pi$
$599$	−31.2348	−1.27622	−0.638110	+	0.769945i	$0.720284\pi$
$600$	0	0
$601$	−33.3546	−1.36056	−0.680280	−	0.732952i	$-0.738142\pi$
$601$	−33.3546	−1.36056	−0.680280	+	0.732952i	$0.738142\pi$
$602$	0	0
$603$	2.06056	0.0839126
$604$	0	0
$605$	20.3085	0.825659
$606$	0	0
$607$	5.13488	0.208418	0.104209	−	0.994555i	$-0.466769\pi$
$607$	5.13488	0.208418	0.104209	+	0.994555i	$0.466769\pi$
$608$	0	0
$609$	10.7145	0.434174
$610$	0	0
$611$	−2.86289	−0.115820
$612$	0	0
$613$	31.0561	1.25435	0.627173	−	0.778880i	$-0.284212\pi$
$613$	31.0561	1.25435	0.627173	+	0.778880i	$0.284212\pi$
$614$	0	0
$615$	−16.1691	−0.652002
$616$	0	0
$617$	−4.61070	−0.185620	−0.0928100	−	0.995684i	$-0.529585\pi$
$617$	−4.61070	−0.185620	−0.0928100	+	0.995684i	$0.529585\pi$
$618$	0	0
$619$	39.0155	1.56817	0.784084	−	0.620655i	$-0.213133\pi$
$619$	39.0155	1.56817	0.784084	+	0.620655i	$0.213133\pi$
$620$	0	0
$621$	−4.89600	−0.196470
$622$	0	0
$623$	−30.6947	−1.22976
$624$	0	0
$625$	−30.7986	−1.23194
$626$	0	0
$627$	1.62194	0.0647742
$628$	0	0
$629$	3.29872	0.131529
$630$	0	0
$631$	−6.89757	−0.274588	−0.137294	−	0.990530i	$-0.543840\pi$
$631$	−6.89757	−0.274588	−0.137294	+	0.990530i	$0.543840\pi$
$632$	0	0
$633$	13.6875	0.544027
$634$	0	0
$635$	−10.2202	−0.405577
$636$	0	0
$637$	−32.0610	−1.27030
$638$	0	0
$639$	9.58830	0.379307
$640$	0	0
$641$	−11.5659	−0.456824	−0.228412	−	0.973565i	$-0.573353\pi$
$641$	−11.5659	−0.456824	−0.228412	+	0.973565i	$0.573353\pi$
$642$	0	0
$643$	45.8983	1.81005	0.905026	−	0.425356i	$-0.139851\pi$
$643$	45.8983	1.81005	0.905026	+	0.425356i	$0.139851\pi$
$644$	0	0
$645$	27.6536	1.08886
$646$	0	0
$647$	17.3528	0.682207	0.341104	−	0.940026i	$-0.389199\pi$
$647$	17.3528	0.682207	0.341104	+	0.940026i	$0.389199\pi$
$648$	0	0
$649$	−15.2655	−0.599223
$650$	0	0
$651$	−2.61354	−0.102433
$652$	0	0
$653$	−19.9185	−0.779470	−0.389735	−	0.920927i	$-0.627433\pi$
$653$	−19.9185	−0.779470	−0.389735	+	0.920927i	$0.627433\pi$
$654$	0	0
$655$	−25.4462	−0.994266
$656$	0	0
$657$	−8.89436	−0.347002
$658$	0	0
$659$	27.3468	1.06528	0.532640	−	0.846342i	$-0.321200\pi$
$659$	27.3468	1.06528	0.532640	+	0.846342i	$0.321200\pi$
$660$	0	0
$661$	−5.50954	−0.214296	−0.107148	−	0.994243i	$-0.534172\pi$
$661$	−5.50954	−0.214296	−0.107148	+	0.994243i	$0.534172\pi$
$662$	0	0
$663$	5.06866	0.196850
$664$	0	0
$665$	8.13058	0.315290
$666$	0	0
$667$	−15.1563	−0.586854
$668$	0	0
$669$	11.9355	0.461454
$670$	0	0
$671$	−11.7144	−0.452230
$672$	0	0
$673$	29.9655	1.15508	0.577542	−	0.816361i	$-0.304012\pi$
$673$	29.9655	1.15508	0.577542	+	0.816361i	$0.304012\pi$
$674$	0	0
$675$	−3.17188	−0.122086
$676$	0	0
$677$	−24.6128	−0.945948	−0.472974	−	0.881076i	$-0.656819\pi$
$677$	−24.6128	−0.945948	−0.472974	+	0.881076i	$0.656819\pi$
$678$	0	0
$679$	4.20509	0.161376
$680$	0	0
$681$	11.1186	0.426066
$682$	0	0
$683$	4.85611	0.185814	0.0929069	−	0.995675i	$-0.470384\pi$
$683$	4.85611	0.185814	0.0929069	+	0.995675i	$0.470384\pi$
$684$	0	0
$685$	−34.0016	−1.29913
$686$	0	0
$687$	−1.46013	−0.0557073
$688$	0	0
$689$	76.9810	2.93274
$690$	0	0
$691$	8.60305	0.327276	0.163638	−	0.986520i	$-0.447677\pi$
$691$	8.60305	0.327276	0.163638	+	0.986520i	$0.447677\pi$
$692$	0	0
$693$	6.83152	0.259508
$694$	0	0
$695$	−6.23387	−0.236464
$696$	0	0
$697$	−4.45281	−0.168662
$698$	0	0
$699$	2.70039	0.102138
$700$	0	0
$701$	−18.6734	−0.705283	−0.352642	−	0.935758i	$-0.614717\pi$
$701$	−18.6734	−0.705283	−0.352642	+	0.935758i	$0.614717\pi$
$702$	0	0
$703$	−3.44331	−0.129867
$704$	0	0
$705$	1.27111	0.0478726
$706$	0	0
$707$	29.6392	1.11470
$708$	0	0
$709$	−44.1323	−1.65742	−0.828712	−	0.559675i	$-0.810926\pi$
$709$	−44.1323	−1.65742	−0.828712	+	0.559675i	$0.810926\pi$
$710$	0	0
$711$	15.9288	0.597378
$712$	0	0
$713$	3.69701	0.138454
$714$	0	0
$715$	36.3279	1.35859
$716$	0	0
$717$	14.1710	0.529226
$718$	0	0
$719$	21.9981	0.820390	0.410195	−	0.911998i	$-0.365461\pi$
$719$	21.9981	0.820390	0.410195	+	0.911998i	$0.365461\pi$
$720$	0	0
$721$	−7.74716	−0.288519
$722$	0	0
$723$	11.5165	0.428303
$724$	0	0
$725$	−9.81902	−0.364669
$726$	0	0
$727$	−0.933796	−0.0346326	−0.0173163	−	0.999850i	$-0.505512\pi$
$727$	−0.933796	−0.0346326	−0.0173163	+	0.999850i	$0.505512\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.61553	0.281670
$732$	0	0
$733$	−0.500954	−0.0185032	−0.00925158	−	0.999957i	$-0.502945\pi$
$733$	−0.500954	−0.0185032	−0.00925158	+	0.999957i	$0.502945\pi$
$734$	0	0
$735$	14.2349	0.525062
$736$	0	0
$737$	4.06708	0.149813
$738$	0	0
$739$	−13.4313	−0.494078	−0.247039	−	0.969006i	$-0.579458\pi$
$739$	−13.4313	−0.494078	−0.247039	+	0.969006i	$0.579458\pi$
$740$	0	0
$741$	−5.29083	−0.194363
$742$	0	0
$743$	42.5141	1.55969	0.779845	−	0.625972i	$-0.215298\pi$
$743$	42.5141	1.55969	0.779845	+	0.625972i	$0.215298\pi$
$744$	0	0
$745$	43.8643	1.60706
$746$	0	0
$747$	12.2935	0.449796
$748$	0	0
$749$	−57.8482	−2.11373
$750$	0	0
$751$	−8.87531	−0.323865	−0.161932	−	0.986802i	$-0.551773\pi$
$751$	−8.87531	−0.323865	−0.161932	+	0.986802i	$0.551773\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−32.9883	−1.20057
$756$	0	0
$757$	33.3317	1.21146	0.605730	−	0.795670i	$-0.292881\pi$
$757$	33.3317	1.21146	0.605730	+	0.795670i	$0.292881\pi$
$758$	0	0
$759$	−9.66358	−0.350766
$760$	0	0
$761$	9.47965	0.343637	0.171819	−	0.985129i	$-0.445036\pi$
$761$	9.47965	0.343637	0.171819	+	0.985129i	$0.445036\pi$
$762$	0	0
$763$	23.9335	0.866449
$764$	0	0
$765$	−2.25045	−0.0813653
$766$	0	0
$767$	49.7964	1.79804
$768$	0	0
$769$	−0.189343	−0.00682789	−0.00341395	−	0.999994i	$-0.501087\pi$
$769$	−0.189343	−0.00682789	−0.00341395	+	0.999994i	$0.501087\pi$
$770$	0	0
$771$	12.9032	0.464696
$772$	0	0
$773$	18.0931	0.650763	0.325381	−	0.945583i	$-0.394507\pi$
$773$	18.0931	0.650763	0.325381	+	0.945583i	$0.394507\pi$
$774$	0	0
$775$	2.39511	0.0860349
$776$	0	0
$777$	−14.5030	−0.520292
$778$	0	0
$779$	4.64799	0.166531
$780$	0	0
$781$	18.9251	0.677193
$782$	0	0
$783$	3.09565	0.110629
$784$	0	0
$785$	−9.37759	−0.334701
$786$	0	0
$787$	−20.8907	−0.744672	−0.372336	−	0.928098i	$-0.621443\pi$
$787$	−20.8907	−0.744672	−0.372336	+	0.928098i	$0.621443\pi$
$788$	0	0
$789$	0.412392	0.0146816
$790$	0	0
$791$	−51.0245	−1.81422
$792$	0	0
$793$	38.2127	1.35697
$794$	0	0
$795$	−34.1791	−1.21221
$796$	0	0
$797$	30.2794	1.07255	0.536276	−	0.844042i	$-0.319830\pi$
$797$	30.2794	1.07255	0.536276	+	0.844042i	$0.319830\pi$
$798$	0	0
$799$	0.350050	0.0123839
$800$	0	0
$801$	−8.86836	−0.313348
$802$	0	0
$803$	−17.5554	−0.619517
$804$	0	0
$805$	−48.4422	−1.70736
$806$	0	0
$807$	−3.59279	−0.126472
$808$	0	0
$809$	29.5352	1.03840	0.519202	−	0.854652i	$-0.326229\pi$
$809$	29.5352	1.03840	0.519202	+	0.854652i	$0.326229\pi$
$810$	0	0
$811$	28.8882	1.01440	0.507202	−	0.861827i	$-0.330680\pi$
$811$	28.8882	1.01440	0.507202	+	0.861827i	$0.330680\pi$
$812$	0	0
$813$	19.4801	0.683198
$814$	0	0
$815$	31.4800	1.10270
$816$	0	0
$817$	−7.94933	−0.278112
$818$	0	0
$819$	−22.2846	−0.778686
$820$	0	0
$821$	28.4936	0.994434	0.497217	−	0.867626i	$-0.334355\pi$
$821$	28.4936	0.994434	0.497217	+	0.867626i	$0.334355\pi$
$822$	0	0
$823$	−18.0661	−0.629744	−0.314872	−	0.949134i	$-0.601962\pi$
$823$	−18.0661	−0.629744	−0.314872	+	0.949134i	$0.601962\pi$
$824$	0	0
$825$	−6.26055	−0.217965
$826$	0	0
$827$	6.67691	0.232179	0.116090	−	0.993239i	$-0.462964\pi$
$827$	6.67691	0.232179	0.116090	+	0.993239i	$0.462964\pi$
$828$	0	0
$829$	−14.5448	−0.505162	−0.252581	−	0.967576i	$-0.581279\pi$
$829$	−14.5448	−0.505162	−0.252581	+	0.967576i	$0.581279\pi$
$830$	0	0
$831$	−28.0330	−0.972454
$832$	0	0
$833$	3.92015	0.135825
$834$	0	0
$835$	71.9074	2.48846
$836$	0	0
$837$	−0.755108	−0.0261003
$838$	0	0
$839$	36.0848	1.24579	0.622893	−	0.782307i	$-0.285957\pi$
$839$	36.0848	1.24579	0.622893	+	0.782307i	$0.285957\pi$
$840$	0	0
$841$	−19.4170	−0.669551
$842$	0	0
$843$	29.0257	0.999699
$844$	0	0
$845$	−81.3403	−2.79819
$846$	0	0
$847$	−24.5889	−0.844883
$848$	0	0
$849$	17.7692	0.609837
$850$	0	0
$851$	20.5153	0.703256
$852$	0	0
$853$	−1.33709	−0.0457811	−0.0228905	−	0.999738i	$-0.507287\pi$
$853$	−1.33709	−0.0457811	−0.0228905	+	0.999738i	$0.507287\pi$
$854$	0	0
$855$	2.34910	0.0803374
$856$	0	0
$857$	15.9900	0.546208	0.273104	−	0.961984i	$-0.411950\pi$
$857$	15.9900	0.546208	0.273104	+	0.961984i	$0.411950\pi$
$858$	0	0
$859$	25.1781	0.859066	0.429533	−	0.903051i	$-0.358678\pi$
$859$	25.1781	0.859066	0.429533	+	0.903051i	$0.358678\pi$
$860$	0	0
$861$	19.5770	0.667182
$862$	0	0
$863$	−10.5553	−0.359305	−0.179653	−	0.983730i	$-0.557497\pi$
$863$	−10.5553	−0.359305	−0.179653	+	0.983730i	$0.557497\pi$
$864$	0	0
$865$	33.7666	1.14810
$866$	0	0
$867$	16.3802	0.556302
$868$	0	0
$869$	31.4399	1.06652
$870$	0	0
$871$	−13.2669	−0.449532
$872$	0	0
$873$	1.21494	0.0411194
$874$	0	0
$875$	18.0878	0.611481
$876$	0	0
$877$	−41.1743	−1.39036	−0.695179	−	0.718837i	$-0.744675\pi$
$877$	−41.1743	−1.39036	−0.695179	+	0.718837i	$0.744675\pi$
$878$	0	0
$879$	17.4501	0.588579
$880$	0	0
$881$	13.8405	0.466299	0.233150	−	0.972441i	$-0.425097\pi$
$881$	13.8405	0.466299	0.233150	+	0.972441i	$0.425097\pi$
$882$	0	0
$883$	−19.7487	−0.664595	−0.332298	−	0.943175i	$-0.607824\pi$
$883$	−19.7487	−0.664595	−0.332298	+	0.943175i	$0.607824\pi$
$884$	0	0
$885$	−22.1093	−0.743196
$886$	0	0
$887$	11.1268	0.373602	0.186801	−	0.982398i	$-0.440188\pi$
$887$	11.1268	0.373602	0.186801	+	0.982398i	$0.440188\pi$
$888$	0	0
$889$	12.3743	0.415020
$890$	0	0
$891$	1.97377	0.0661237
$892$	0	0
$893$	−0.365394	−0.0122274
$894$	0	0
$895$	66.6956	2.22939
$896$	0	0
$897$	31.5228	1.05252
$898$	0	0
$899$	−2.33755	−0.0779616
$900$	0	0
$901$	−9.41259	−0.313579
$902$	0	0
$903$	−33.4820	−1.11421
$904$	0	0
$905$	−0.860523	−0.0286047
$906$	0	0
$907$	30.9956	1.02919	0.514596	−	0.857433i	$-0.327942\pi$
$907$	30.9956	1.02919	0.514596	+	0.857433i	$0.327942\pi$
$908$	0	0
$909$	8.56339	0.284030
$910$	0	0
$911$	−9.74559	−0.322886	−0.161443	−	0.986882i	$-0.551615\pi$
$911$	−9.74559	−0.322886	−0.161443	+	0.986882i	$0.551615\pi$
$912$	0	0
$913$	24.2645	0.803038
$914$	0	0
$915$	−16.9662	−0.560886
$916$	0	0
$917$	30.8094	1.01742
$918$	0	0
$919$	21.9408	0.723759	0.361880	−	0.932225i	$-0.382135\pi$
$919$	21.9408	0.723759	0.361880	+	0.932225i	$0.382135\pi$
$920$	0	0
$921$	20.4287	0.673148
$922$	0	0
$923$	−61.7341	−2.03200
$924$	0	0
$925$	13.2909	0.437001
$926$	0	0
$927$	−2.23832	−0.0735160
$928$	0	0
$929$	28.5351	0.936206	0.468103	−	0.883674i	$-0.344938\pi$
$929$	28.5351	0.936206	0.468103	+	0.883674i	$0.344938\pi$
$930$	0	0
$931$	−4.09197	−0.134109
$932$	0	0
$933$	−9.32624	−0.305327
$934$	0	0
$935$	−4.44188	−0.145265
$936$	0	0
$937$	−13.9150	−0.454583	−0.227291	−	0.973827i	$-0.572987\pi$
$937$	−13.9150	−0.454583	−0.227291	+	0.973827i	$0.572987\pi$
$938$	0	0
$939$	−7.59242	−0.247769
$940$	0	0
$941$	34.7287	1.13212	0.566062	−	0.824363i	$-0.308466\pi$
$941$	34.7287	1.13212	0.566062	+	0.824363i	$0.308466\pi$
$942$	0	0
$943$	−27.6928	−0.901802
$944$	0	0
$945$	9.89423	0.321859
$946$	0	0
$947$	8.52499	0.277025	0.138513	−	0.990361i	$-0.455768\pi$
$947$	8.52499	0.277025	0.138513	+	0.990361i	$0.455768\pi$
$948$	0	0
$949$	57.2662	1.85894
$950$	0	0
$951$	20.7358	0.672406
$952$	0	0
$953$	−2.65475	−0.0859957	−0.0429978	−	0.999075i	$-0.513691\pi$
$953$	−2.65475	−0.0859957	−0.0429978	+	0.999075i	$0.513691\pi$
$954$	0	0
$955$	32.4743	1.05084
$956$	0	0
$957$	6.11009	0.197511
$958$	0	0
$959$	41.1679	1.32938
$960$	0	0
$961$	−30.4298	−0.981607
$962$	0	0
$963$	−16.7135	−0.538587
$964$	0	0
$965$	−21.4374	−0.690095
$966$	0	0
$967$	0.591523	0.0190221	0.00951105	−	0.999955i	$-0.496972\pi$
$967$	0.591523	0.0190221	0.00951105	+	0.999955i	$0.496972\pi$
$968$	0	0
$969$	0.646918	0.0207820
$970$	0	0
$971$	−26.3921	−0.846963	−0.423481	−	0.905905i	$-0.639192\pi$
$971$	−26.3921	−0.846963	−0.423481	+	0.905905i	$0.639192\pi$
$972$	0	0
$973$	7.54775	0.241970
$974$	0	0
$975$	20.4221	0.654030
$976$	0	0
$977$	−44.6729	−1.42921	−0.714606	−	0.699527i	$-0.753394\pi$
$977$	−44.6729	−1.42921	−0.714606	+	0.699527i	$0.753394\pi$
$978$	0	0
$979$	−17.5041	−0.559433
$980$	0	0
$981$	6.91488	0.220775
$982$	0	0
$983$	34.3542	1.09573	0.547864	−	0.836567i	$-0.315441\pi$
$983$	34.3542	1.09573	0.547864	+	0.836567i	$0.315441\pi$
$984$	0	0
$985$	−37.6014	−1.19808
$986$	0	0
$987$	−1.53901	−0.0489873
$988$	0	0
$989$	47.3623	1.50603
$990$	0	0
$991$	−5.25605	−0.166964	−0.0834819	−	0.996509i	$-0.526604\pi$
$991$	−5.25605	−0.166964	−0.0834819	+	0.996509i	$0.526604\pi$
$992$	0	0
$993$	2.97892	0.0945332
$994$	0	0
$995$	−68.6477	−2.17628
$996$	0	0
$997$	10.9584	0.347055	0.173527	−	0.984829i	$-0.444484\pi$
$997$	10.9584	0.347055	0.173527	+	0.984829i	$0.444484\pi$
$998$	0	0
$999$	−4.19022	−0.132573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.o.1.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.o.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.85865 q^{5} +3.46115 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.85865 q^{5} +3.46115 q^{7} +1.00000 q^{9} +1.97377 q^{11} -6.43848 q^{13} +2.85865 q^{15} +0.787244 q^{17} -0.821750 q^{19} -3.46115 q^{21} +4.89600 q^{23} +3.17188 q^{25} -1.00000 q^{27} -3.09565 q^{29} +0.755108 q^{31} -1.97377 q^{33} -9.89423 q^{35} +4.19022 q^{37} +6.43848 q^{39} -5.65621 q^{41} +9.67366 q^{43} -2.85865 q^{45} +0.444653 q^{47} +4.97958 q^{49} -0.787244 q^{51} -11.9564 q^{53} -5.64231 q^{55} +0.821750 q^{57} -7.73418 q^{59} -5.93505 q^{61} +3.46115 q^{63} +18.4054 q^{65} +2.06056 q^{67} -4.89600 q^{69} +9.58830 q^{71} -8.89436 q^{73} -3.17188 q^{75} +6.83152 q^{77} +15.9288 q^{79} +1.00000 q^{81} +12.2935 q^{83} -2.25045 q^{85} +3.09565 q^{87} -8.86836 q^{89} -22.2846 q^{91} -0.755108 q^{93} +2.34910 q^{95} +1.21494 q^{97} +1.97377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 3 * q^5 + 7 * q^7 + 14 * q^9 - 22 * q^11 + 9 * q^13 + 3 * q^15 - 7 * q^21 - 17 * q^23 + 25 * q^25 - 14 * q^27 - 18 * q^29 - 7 * q^31 + 22 * q^33 - 27 * q^35 + 9 * q^37 - 9 * q^39 - 6 * q^41 - 14 * q^43 - 3 * q^45 - 15 * q^47 + 15 * q^49 - 11 * q^53 + 2 * q^55 - 36 * q^59 + 2 * q^61 + 7 * q^63 + 8 * q^65 + 3 * q^67 + 17 * q^69 - 29 * q^71 + 2 * q^73 - 25 * q^75 + 8 * q^77 + 23 * q^79 + 14 * q^81 - 55 * q^83 + 7 * q^85 + 18 * q^87 + 9 * q^89 - 22 * q^91 + 7 * q^93 - 27 * q^95 + 17 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more