LMFDB - Embedded newform 6024.2.a.r.1.7

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			1.7
Root			$-1.06426$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.06426 q^{5} +1.22980 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.06426 q^{5} +1.22980 q^{7} +1.00000 q^{9} +5.43020 q^{11} +6.43432 q^{13} -1.06426 q^{15} -1.04698 q^{17} +5.23300 q^{19} +1.22980 q^{21} +9.10972 q^{23} -3.86735 q^{25} +1.00000 q^{27} -1.65181 q^{29} -8.04078 q^{31} +5.43020 q^{33} -1.30882 q^{35} +10.1160 q^{37} +6.43432 q^{39} -5.53665 q^{41} +2.48979 q^{43} -1.06426 q^{45} -0.378348 q^{47} -5.48760 q^{49} -1.04698 q^{51} -3.42049 q^{53} -5.77914 q^{55} +5.23300 q^{57} +7.95133 q^{59} -4.23652 q^{61} +1.22980 q^{63} -6.84779 q^{65} +12.2238 q^{67} +9.10972 q^{69} -1.48748 q^{71} +3.17701 q^{73} -3.86735 q^{75} +6.67803 q^{77} -16.0456 q^{79} +1.00000 q^{81} -10.2098 q^{83} +1.11426 q^{85} -1.65181 q^{87} -14.2105 q^{89} +7.91290 q^{91} -8.04078 q^{93} -5.56928 q^{95} -1.46901 q^{97} +5.43020 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.06426	−0.475952	−0.237976	−	0.971271i	$-0.576484\pi$
$5$	−1.06426	−0.475952	−0.237976	+	0.971271i	$0.576484\pi$
$6$	0	0
$7$	1.22980	0.464819	0.232409	−	0.972618i	$-0.425339\pi$
$7$	1.22980	0.464819	0.232409	+	0.972618i	$0.425339\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.43020	1.63727	0.818633	−	0.574317i	$-0.194732\pi$
$11$	5.43020	1.63727	0.818633	+	0.574317i	$0.194732\pi$
$12$	0	0
$13$	6.43432	1.78456	0.892280	−	0.451483i	$-0.149105\pi$
$13$	6.43432	1.78456	0.892280	+	0.451483i	$0.149105\pi$
$14$	0	0
$15$	−1.06426	−0.274791
$16$	0	0
$17$	−1.04698	−0.253931	−0.126965	−	0.991907i	$-0.540524\pi$
$17$	−1.04698	−0.253931	−0.126965	+	0.991907i	$0.540524\pi$
$18$	0	0
$19$	5.23300	1.20053	0.600267	−	0.799800i	$-0.295061\pi$
$19$	5.23300	1.20053	0.600267	+	0.799800i	$0.295061\pi$
$20$	0	0
$21$	1.22980	0.268363
$22$	0	0
$23$	9.10972	1.89951	0.949754	−	0.312998i	$-0.101333\pi$
$23$	9.10972	1.89951	0.949754	+	0.312998i	$0.101333\pi$
$24$	0	0
$25$	−3.86735	−0.773470
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.65181	−0.306733	−0.153367	−	0.988169i	$-0.549012\pi$
$29$	−1.65181	−0.306733	−0.153367	+	0.988169i	$0.549012\pi$
$30$	0	0
$31$	−8.04078	−1.44417	−0.722083	−	0.691806i	$-0.756815\pi$
$31$	−8.04078	−1.44417	−0.722083	+	0.691806i	$0.756815\pi$
$32$	0	0
$33$	5.43020	0.945276
$34$	0	0
$35$	−1.30882	−0.221231
$36$	0	0
$37$	10.1160	1.66306	0.831528	−	0.555484i	$-0.187467\pi$
$37$	10.1160	1.66306	0.831528	+	0.555484i	$0.187467\pi$
$38$	0	0
$39$	6.43432	1.03032
$40$	0	0
$41$	−5.53665	−0.864679	−0.432340	−	0.901711i	$-0.642312\pi$
$41$	−5.53665	−0.864679	−0.432340	+	0.901711i	$0.642312\pi$
$42$	0	0
$43$	2.48979	0.379690	0.189845	−	0.981814i	$-0.439201\pi$
$43$	2.48979	0.379690	0.189845	+	0.981814i	$0.439201\pi$
$44$	0	0
$45$	−1.06426	−0.158651
$46$	0	0
$47$	−0.378348	−0.0551877	−0.0275938	−	0.999619i	$-0.508785\pi$
$47$	−0.378348	−0.0551877	−0.0275938	+	0.999619i	$0.508785\pi$
$48$	0	0
$49$	−5.48760	−0.783943
$50$	0	0
$51$	−1.04698	−0.146607
$52$	0	0
$53$	−3.42049	−0.469840	−0.234920	−	0.972015i	$-0.575483\pi$
$53$	−3.42049	−0.469840	−0.234920	+	0.972015i	$0.575483\pi$
$54$	0	0
$55$	−5.77914	−0.779259
$56$	0	0
$57$	5.23300	0.693128
$58$	0	0
$59$	7.95133	1.03518	0.517588	−	0.855630i	$-0.326830\pi$
$59$	7.95133	1.03518	0.517588	+	0.855630i	$0.326830\pi$
$60$	0	0
$61$	−4.23652	−0.542430	−0.271215	−	0.962519i	$-0.587425\pi$
$61$	−4.23652	−0.542430	−0.271215	+	0.962519i	$0.587425\pi$
$62$	0	0
$63$	1.22980	0.154940
$64$	0	0
$65$	−6.84779	−0.849364
$66$	0	0
$67$	12.2238	1.49338	0.746690	−	0.665172i	$-0.231642\pi$
$67$	12.2238	1.49338	0.746690	+	0.665172i	$0.231642\pi$
$68$	0	0
$69$	9.10972	1.09668
$70$	0	0
$71$	−1.48748	−0.176531	−0.0882655	−	0.996097i	$-0.528132\pi$
$71$	−1.48748	−0.176531	−0.0882655	+	0.996097i	$0.528132\pi$
$72$	0	0
$73$	3.17701	0.371841	0.185920	−	0.982565i	$-0.440473\pi$
$73$	3.17701	0.371841	0.185920	+	0.982565i	$0.440473\pi$
$74$	0	0
$75$	−3.86735	−0.446563
$76$	0	0
$77$	6.67803	0.761032
$78$	0	0
$79$	−16.0456	−1.80527	−0.902634	−	0.430408i	$-0.858370\pi$
$79$	−16.0456	−1.80527	−0.902634	+	0.430408i	$0.858370\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.2098	−1.12067	−0.560333	−	0.828267i	$-0.689327\pi$
$83$	−10.2098	−1.12067	−0.560333	+	0.828267i	$0.689327\pi$
$84$	0	0
$85$	1.11426	0.120859
$86$	0	0
$87$	−1.65181	−0.177092
$88$	0	0
$89$	−14.2105	−1.50631	−0.753153	−	0.657846i	$-0.771468\pi$
$89$	−14.2105	−1.50631	−0.753153	+	0.657846i	$0.771468\pi$
$90$	0	0
$91$	7.91290	0.829497
$92$	0	0
$93$	−8.04078	−0.833790
$94$	0	0
$95$	−5.56928	−0.571396
$96$	0	0
$97$	−1.46901	−0.149155	−0.0745775	−	0.997215i	$-0.523761\pi$
$97$	−1.46901	−0.149155	−0.0745775	+	0.997215i	$0.523761\pi$
$98$	0	0
$99$	5.43020	0.545755
$100$	0	0
$101$	−4.09907	−0.407873	−0.203937	−	0.978984i	$-0.565374\pi$
$101$	−4.09907	−0.407873	−0.203937	+	0.978984i	$0.565374\pi$
$102$	0	0
$103$	11.9985	1.18224	0.591121	−	0.806583i	$-0.298685\pi$
$103$	11.9985	1.18224	0.591121	+	0.806583i	$0.298685\pi$
$104$	0	0
$105$	−1.30882	−0.127728
$106$	0	0
$107$	6.44653	0.623209	0.311605	−	0.950212i	$-0.399134\pi$
$107$	6.44653	0.623209	0.311605	+	0.950212i	$0.399134\pi$
$108$	0	0
$109$	3.37203	0.322981	0.161491	−	0.986874i	$-0.448370\pi$
$109$	3.37203	0.322981	0.161491	+	0.986874i	$0.448370\pi$
$110$	0	0
$111$	10.1160	0.960165
$112$	0	0
$113$	−8.75356	−0.823466	−0.411733	−	0.911305i	$-0.635076\pi$
$113$	−8.75356	−0.823466	−0.411733	+	0.911305i	$0.635076\pi$
$114$	0	0
$115$	−9.69511	−0.904074
$116$	0	0
$117$	6.43432	0.594853
$118$	0	0
$119$	−1.28758	−0.118032
$120$	0	0
$121$	18.4870	1.68064
$122$	0	0
$123$	−5.53665	−0.499223
$124$	0	0
$125$	9.43717	0.844086
$126$	0	0
$127$	−15.8099	−1.40290	−0.701449	−	0.712719i	$-0.747463\pi$
$127$	−15.8099	−1.40290	−0.701449	+	0.712719i	$0.747463\pi$
$128$	0	0
$129$	2.48979	0.219214
$130$	0	0
$131$	−14.8438	−1.29691	−0.648454	−	0.761254i	$-0.724584\pi$
$131$	−14.8438	−1.29691	−0.648454	+	0.761254i	$0.724584\pi$
$132$	0	0
$133$	6.43552	0.558031
$134$	0	0
$135$	−1.06426	−0.0915970
$136$	0	0
$137$	−7.38983	−0.631356	−0.315678	−	0.948866i	$-0.602232\pi$
$137$	−7.38983	−0.631356	−0.315678	+	0.948866i	$0.602232\pi$
$138$	0	0
$139$	−4.91837	−0.417171	−0.208585	−	0.978004i	$-0.566886\pi$
$139$	−4.91837	−0.417171	−0.208585	+	0.978004i	$0.566886\pi$
$140$	0	0
$141$	−0.378348	−0.0318626
$142$	0	0
$143$	34.9396	2.92180
$144$	0	0
$145$	1.75795	0.145990
$146$	0	0
$147$	−5.48760	−0.452610
$148$	0	0
$149$	−22.9315	−1.87862	−0.939310	−	0.343069i	$-0.888534\pi$
$149$	−22.9315	−1.87862	−0.939310	+	0.343069i	$0.888534\pi$
$150$	0	0
$151$	−3.13910	−0.255456	−0.127728	−	0.991809i	$-0.540769\pi$
$151$	−3.13910	−0.255456	−0.127728	+	0.991809i	$0.540769\pi$
$152$	0	0
$153$	−1.04698	−0.0846436
$154$	0	0
$155$	8.55748	0.687353
$156$	0	0
$157$	10.8500	0.865927	0.432963	−	0.901412i	$-0.357468\pi$
$157$	10.8500	0.865927	0.432963	+	0.901412i	$0.357468\pi$
$158$	0	0
$159$	−3.42049	−0.271262
$160$	0	0
$161$	11.2031	0.882927
$162$	0	0
$163$	19.6731	1.54091	0.770457	−	0.637492i	$-0.220028\pi$
$163$	19.6731	1.54091	0.770457	+	0.637492i	$0.220028\pi$
$164$	0	0
$165$	−5.77914	−0.449906
$166$	0	0
$167$	5.88714	0.455561	0.227780	−	0.973713i	$-0.426853\pi$
$167$	5.88714	0.455561	0.227780	+	0.973713i	$0.426853\pi$
$168$	0	0
$169$	28.4005	2.18465
$170$	0	0
$171$	5.23300	0.400178
$172$	0	0
$173$	9.23927	0.702449	0.351224	−	0.936291i	$-0.385765\pi$
$173$	9.23927	0.702449	0.351224	+	0.936291i	$0.385765\pi$
$174$	0	0
$175$	−4.75605	−0.359523
$176$	0	0
$177$	7.95133	0.597659
$178$	0	0
$179$	−4.43698	−0.331636	−0.165818	−	0.986156i	$-0.553026\pi$
$179$	−4.43698	−0.331636	−0.165818	+	0.986156i	$0.553026\pi$
$180$	0	0
$181$	−6.79423	−0.505011	−0.252506	−	0.967595i	$-0.581255\pi$
$181$	−6.79423	−0.505011	−0.252506	+	0.967595i	$0.581255\pi$
$182$	0	0
$183$	−4.23652	−0.313172
$184$	0	0
$185$	−10.7660	−0.791534
$186$	0	0
$187$	−5.68533	−0.415752
$188$	0	0
$189$	1.22980	0.0894544
$190$	0	0
$191$	−13.0977	−0.947716	−0.473858	−	0.880601i	$-0.657139\pi$
$191$	−13.0977	−0.947716	−0.473858	+	0.880601i	$0.657139\pi$
$192$	0	0
$193$	−18.7404	−1.34896	−0.674482	−	0.738292i	$-0.735633\pi$
$193$	−18.7404	−1.34896	−0.674482	+	0.738292i	$0.735633\pi$
$194$	0	0
$195$	−6.84779	−0.490381
$196$	0	0
$197$	3.32543	0.236927	0.118464	−	0.992958i	$-0.462203\pi$
$197$	3.32543	0.236927	0.118464	+	0.992958i	$0.462203\pi$
$198$	0	0
$199$	18.4904	1.31075	0.655374	−	0.755305i	$-0.272511\pi$
$199$	18.4904	1.31075	0.655374	+	0.755305i	$0.272511\pi$
$200$	0	0
$201$	12.2238	0.862203
$202$	0	0
$203$	−2.03139	−0.142575
$204$	0	0
$205$	5.89244	0.411546
$206$	0	0
$207$	9.10972	0.633169
$208$	0	0
$209$	28.4162	1.96559
$210$	0	0
$211$	3.43034	0.236154	0.118077	−	0.993004i	$-0.462327\pi$
$211$	3.43034	0.236154	0.118077	+	0.993004i	$0.462327\pi$
$212$	0	0
$213$	−1.48748	−0.101920
$214$	0	0
$215$	−2.64979	−0.180714
$216$	0	0
$217$	−9.88851	−0.671276
$218$	0	0
$219$	3.17701	0.214682
$220$	0	0
$221$	−6.73663	−0.453155
$222$	0	0
$223$	−0.0605067	−0.00405183	−0.00202591	−	0.999998i	$-0.500645\pi$
$223$	−0.0605067	−0.00405183	−0.00202591	+	0.999998i	$0.500645\pi$
$224$	0	0
$225$	−3.86735	−0.257823
$226$	0	0
$227$	6.38089	0.423515	0.211757	−	0.977322i	$-0.432081\pi$
$227$	6.38089	0.423515	0.211757	+	0.977322i	$0.432081\pi$
$228$	0	0
$229$	25.4876	1.68427	0.842133	−	0.539270i	$-0.181300\pi$
$229$	25.4876	1.68427	0.842133	+	0.539270i	$0.181300\pi$
$230$	0	0
$231$	6.67803	0.439382
$232$	0	0
$233$	8.91059	0.583752	0.291876	−	0.956456i	$-0.405720\pi$
$233$	8.91059	0.583752	0.291876	+	0.956456i	$0.405720\pi$
$234$	0	0
$235$	0.402660	0.0262667
$236$	0	0
$237$	−16.0456	−1.04227
$238$	0	0
$239$	−27.9565	−1.80836	−0.904178	−	0.427156i	$-0.859516\pi$
$239$	−27.9565	−1.80836	−0.904178	+	0.427156i	$0.859516\pi$
$240$	0	0
$241$	−26.2747	−1.69250	−0.846252	−	0.532783i	$-0.821146\pi$
$241$	−26.2747	−1.69250	−0.846252	+	0.532783i	$0.821146\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.84024	0.373119
$246$	0	0
$247$	33.6708	2.14242
$248$	0	0
$249$	−10.2098	−0.647017
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	49.4676	3.11000
$254$	0	0
$255$	1.11426	0.0697779
$256$	0	0
$257$	27.8655	1.73820	0.869100	−	0.494636i	$-0.164699\pi$
$257$	27.8655	1.73820	0.869100	+	0.494636i	$0.164699\pi$
$258$	0	0
$259$	12.4406	0.773019
$260$	0	0
$261$	−1.65181	−0.102244
$262$	0	0
$263$	28.0309	1.72846	0.864230	−	0.503097i	$-0.167806\pi$
$263$	28.0309	1.72846	0.864230	+	0.503097i	$0.167806\pi$
$264$	0	0
$265$	3.64029	0.223621
$266$	0	0
$267$	−14.2105	−0.869666
$268$	0	0
$269$	14.3015	0.871978	0.435989	−	0.899952i	$-0.356399\pi$
$269$	14.3015	0.871978	0.435989	+	0.899952i	$0.356399\pi$
$270$	0	0
$271$	6.46714	0.392850	0.196425	−	0.980519i	$-0.437067\pi$
$271$	6.46714	0.392850	0.196425	+	0.980519i	$0.437067\pi$
$272$	0	0
$273$	7.91290	0.478910
$274$	0	0
$275$	−21.0005	−1.26638
$276$	0	0
$277$	32.4102	1.94734	0.973670	−	0.227964i	$-0.0732069\pi$
$277$	32.4102	1.94734	0.973670	+	0.227964i	$0.0732069\pi$
$278$	0	0
$279$	−8.04078	−0.481389
$280$	0	0
$281$	−9.46110	−0.564402	−0.282201	−	0.959355i	$-0.591064\pi$
$281$	−9.46110	−0.564402	−0.282201	+	0.959355i	$0.591064\pi$
$282$	0	0
$283$	21.8918	1.30133	0.650665	−	0.759365i	$-0.274490\pi$
$283$	21.8918	1.30133	0.650665	+	0.759365i	$0.274490\pi$
$284$	0	0
$285$	−5.56928	−0.329896
$286$	0	0
$287$	−6.80894	−0.401919
$288$	0	0
$289$	−15.9038	−0.935519
$290$	0	0
$291$	−1.46901	−0.0861146
$292$	0	0
$293$	2.62878	0.153575	0.0767875	−	0.997047i	$-0.475534\pi$
$293$	2.62878	0.153575	0.0767875	+	0.997047i	$0.475534\pi$
$294$	0	0
$295$	−8.46229	−0.492694
$296$	0	0
$297$	5.43020	0.315092
$298$	0	0
$299$	58.6148	3.38978
$300$	0	0
$301$	3.06194	0.176487
$302$	0	0
$303$	−4.09907	−0.235486
$304$	0	0
$305$	4.50876	0.258171
$306$	0	0
$307$	2.11725	0.120838	0.0604188	−	0.998173i	$-0.480756\pi$
$307$	2.11725	0.120838	0.0604188	+	0.998173i	$0.480756\pi$
$308$	0	0
$309$	11.9985	0.682568
$310$	0	0
$311$	1.80626	0.102423	0.0512117	−	0.998688i	$-0.483692\pi$
$311$	1.80626	0.102423	0.0512117	+	0.998688i	$0.483692\pi$
$312$	0	0
$313$	−20.7830	−1.17472	−0.587361	−	0.809325i	$-0.699833\pi$
$313$	−20.7830	−1.17472	−0.587361	+	0.809325i	$0.699833\pi$
$314$	0	0
$315$	−1.30882	−0.0737438
$316$	0	0
$317$	−21.9652	−1.23369	−0.616844	−	0.787085i	$-0.711589\pi$
$317$	−21.9652	−1.23369	−0.616844	+	0.787085i	$0.711589\pi$
$318$	0	0
$319$	−8.96964	−0.502204
$320$	0	0
$321$	6.44653	0.359810
$322$	0	0
$323$	−5.47887	−0.304853
$324$	0	0
$325$	−24.8838	−1.38030
$326$	0	0
$327$	3.37203	0.186473
$328$	0	0
$329$	−0.465290	−0.0256523
$330$	0	0
$331$	−16.2829	−0.894990	−0.447495	−	0.894286i	$-0.647684\pi$
$331$	−16.2829	−0.894990	−0.447495	+	0.894286i	$0.647684\pi$
$332$	0	0
$333$	10.1160	0.554352
$334$	0	0
$335$	−13.0093	−0.710777
$336$	0	0
$337$	0.667707	0.0363723	0.0181862	−	0.999835i	$-0.494211\pi$
$337$	0.667707	0.0363723	0.0181862	+	0.999835i	$0.494211\pi$
$338$	0	0
$339$	−8.75356	−0.475428
$340$	0	0
$341$	−43.6630	−2.36448
$342$	0	0
$343$	−15.3572	−0.829211
$344$	0	0
$345$	−9.69511	−0.521967
$346$	0	0
$347$	−17.7746	−0.954191	−0.477096	−	0.878851i	$-0.658310\pi$
$347$	−17.7746	−0.954191	−0.477096	+	0.878851i	$0.658310\pi$
$348$	0	0
$349$	19.4872	1.04313	0.521563	−	0.853213i	$-0.325349\pi$
$349$	19.4872	1.04313	0.521563	+	0.853213i	$0.325349\pi$
$350$	0	0
$351$	6.43432	0.343439
$352$	0	0
$353$	−3.75872	−0.200057	−0.100028	−	0.994985i	$-0.531893\pi$
$353$	−3.75872	−0.200057	−0.100028	+	0.994985i	$0.531893\pi$
$354$	0	0
$355$	1.58306	0.0840202
$356$	0	0
$357$	−1.28758	−0.0681457
$358$	0	0
$359$	−5.13439	−0.270983	−0.135491	−	0.990779i	$-0.543261\pi$
$359$	−5.13439	−0.270983	−0.135491	+	0.990779i	$0.543261\pi$
$360$	0	0
$361$	8.38432	0.441280
$362$	0	0
$363$	18.4870	0.970317
$364$	0	0
$365$	−3.38117	−0.176978
$366$	0	0
$367$	27.7111	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$367$	27.7111	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$368$	0	0
$369$	−5.53665	−0.288226
$370$	0	0
$371$	−4.20650	−0.218391
$372$	0	0
$373$	13.6943	0.709062	0.354531	−	0.935044i	$-0.384641\pi$
$373$	13.6943	0.709062	0.354531	+	0.935044i	$0.384641\pi$
$374$	0	0
$375$	9.43717	0.487333
$376$	0	0
$377$	−10.6283	−0.547383
$378$	0	0
$379$	−7.96479	−0.409124	−0.204562	−	0.978854i	$-0.565577\pi$
$379$	−7.96479	−0.409124	−0.204562	+	0.978854i	$0.565577\pi$
$380$	0	0
$381$	−15.8099	−0.809964
$382$	0	0
$383$	27.5549	1.40799	0.703994	−	0.710206i	$-0.251398\pi$
$383$	27.5549	1.40799	0.703994	+	0.710206i	$0.251398\pi$
$384$	0	0
$385$	−7.10716	−0.362215
$386$	0	0
$387$	2.48979	0.126563
$388$	0	0
$389$	18.1005	0.917733	0.458867	−	0.888505i	$-0.348256\pi$
$389$	18.1005	0.917733	0.458867	+	0.888505i	$0.348256\pi$
$390$	0	0
$391$	−9.53773	−0.482344
$392$	0	0
$393$	−14.8438	−0.748770
$394$	0	0
$395$	17.0767	0.859221
$396$	0	0
$397$	12.3304	0.618847	0.309423	−	0.950924i	$-0.399864\pi$
$397$	12.3304	0.618847	0.309423	+	0.950924i	$0.399864\pi$
$398$	0	0
$399$	6.43552	0.322179
$400$	0	0
$401$	13.4979	0.674053	0.337026	−	0.941495i	$-0.390579\pi$
$401$	13.4979	0.674053	0.337026	+	0.941495i	$0.390579\pi$
$402$	0	0
$403$	−51.7369	−2.57720
$404$	0	0
$405$	−1.06426	−0.0528835
$406$	0	0
$407$	54.9317	2.72286
$408$	0	0
$409$	21.9428	1.08500	0.542501	−	0.840055i	$-0.317478\pi$
$409$	21.9428	1.08500	0.542501	+	0.840055i	$0.317478\pi$
$410$	0	0
$411$	−7.38983	−0.364514
$412$	0	0
$413$	9.77851	0.481169
$414$	0	0
$415$	10.8658	0.533383
$416$	0	0
$417$	−4.91837	−0.240854
$418$	0	0
$419$	−12.8002	−0.625328	−0.312664	−	0.949864i	$-0.601221\pi$
$419$	−12.8002	−0.625328	−0.312664	+	0.949864i	$0.601221\pi$
$420$	0	0
$421$	12.8071	0.624178	0.312089	−	0.950053i	$-0.398971\pi$
$421$	12.8071	0.624178	0.312089	+	0.950053i	$0.398971\pi$
$422$	0	0
$423$	−0.378348	−0.0183959
$424$	0	0
$425$	4.04905	0.196408
$426$	0	0
$427$	−5.21005	−0.252132
$428$	0	0
$429$	34.9396	1.68690
$430$	0	0
$431$	−13.9106	−0.670049	−0.335025	−	0.942209i	$-0.608745\pi$
$431$	−13.9106	−0.670049	−0.335025	+	0.942209i	$0.608745\pi$
$432$	0	0
$433$	−22.6140	−1.08676	−0.543380	−	0.839487i	$-0.682856\pi$
$433$	−22.6140	−1.08676	−0.543380	+	0.839487i	$0.682856\pi$
$434$	0	0
$435$	1.75795	0.0842875
$436$	0	0
$437$	47.6712	2.28042
$438$	0	0
$439$	31.4507	1.50106	0.750529	−	0.660837i	$-0.229799\pi$
$439$	31.4507	1.50106	0.750529	+	0.660837i	$0.229799\pi$
$440$	0	0
$441$	−5.48760	−0.261314
$442$	0	0
$443$	−33.1138	−1.57328	−0.786642	−	0.617410i	$-0.788182\pi$
$443$	−33.1138	−1.57328	−0.786642	+	0.617410i	$0.788182\pi$
$444$	0	0
$445$	15.1236	0.716929
$446$	0	0
$447$	−22.9315	−1.08462
$448$	0	0
$449$	6.98902	0.329832	0.164916	−	0.986308i	$-0.447265\pi$
$449$	6.98902	0.329832	0.164916	+	0.986308i	$0.447265\pi$
$450$	0	0
$451$	−30.0651	−1.41571
$452$	0	0
$453$	−3.13910	−0.147488
$454$	0	0
$455$	−8.42138	−0.394801
$456$	0	0
$457$	−13.6339	−0.637767	−0.318884	−	0.947794i	$-0.603308\pi$
$457$	−13.6339	−0.637767	−0.318884	+	0.947794i	$0.603308\pi$
$458$	0	0
$459$	−1.04698	−0.0488690
$460$	0	0
$461$	25.8958	1.20609	0.603044	−	0.797708i	$-0.293954\pi$
$461$	25.8958	1.20609	0.603044	+	0.797708i	$0.293954\pi$
$462$	0	0
$463$	−24.1132	−1.12064	−0.560319	−	0.828277i	$-0.689321\pi$
$463$	−24.1132	−1.12064	−0.560319	+	0.828277i	$0.689321\pi$
$464$	0	0
$465$	8.55748	0.396844
$466$	0	0
$467$	−20.7962	−0.962333	−0.481166	−	0.876629i	$-0.659787\pi$
$467$	−20.7962	−0.962333	−0.481166	+	0.876629i	$0.659787\pi$
$468$	0	0
$469$	15.0328	0.694151
$470$	0	0
$471$	10.8500	0.499943
$472$	0	0
$473$	13.5201	0.621653
$474$	0	0
$475$	−20.2379	−0.928576
$476$	0	0
$477$	−3.42049	−0.156613
$478$	0	0
$479$	−23.6828	−1.08210	−0.541048	−	0.840992i	$-0.681972\pi$
$479$	−23.6828	−1.08210	−0.541048	+	0.840992i	$0.681972\pi$
$480$	0	0
$481$	65.0894	2.96782
$482$	0	0
$483$	11.2031	0.509758
$484$	0	0
$485$	1.56340	0.0709906
$486$	0	0
$487$	−28.8107	−1.30554	−0.652768	−	0.757558i	$-0.726392\pi$
$487$	−28.8107	−1.30554	−0.652768	+	0.757558i	$0.726392\pi$
$488$	0	0
$489$	19.6731	0.889647
$490$	0	0
$491$	−37.9736	−1.71372	−0.856862	−	0.515546i	$-0.827589\pi$
$491$	−37.9736	−1.71372	−0.856862	+	0.515546i	$0.827589\pi$
$492$	0	0
$493$	1.72942	0.0778890
$494$	0	0
$495$	−5.77914	−0.259753
$496$	0	0
$497$	−1.82929	−0.0820549
$498$	0	0
$499$	−34.7142	−1.55402	−0.777010	−	0.629488i	$-0.783265\pi$
$499$	−34.7142	−1.55402	−0.777010	+	0.629488i	$0.783265\pi$
$500$	0	0
$501$	5.88714	0.263018
$502$	0	0
$503$	−30.3257	−1.35216	−0.676079	−	0.736829i	$-0.736322\pi$
$503$	−30.3257	−1.35216	−0.676079	+	0.736829i	$0.736322\pi$
$504$	0	0
$505$	4.36248	0.194128
$506$	0	0
$507$	28.4005	1.26131
$508$	0	0
$509$	6.78075	0.300552	0.150276	−	0.988644i	$-0.451984\pi$
$509$	6.78075	0.300552	0.150276	+	0.988644i	$0.451984\pi$
$510$	0	0
$511$	3.90707	0.172839
$512$	0	0
$513$	5.23300	0.231043
$514$	0	0
$515$	−12.7695	−0.562691
$516$	0	0
$517$	−2.05450	−0.0903569
$518$	0	0
$519$	9.23927	0.405559
$520$	0	0
$521$	−4.16861	−0.182630	−0.0913150	−	0.995822i	$-0.529107\pi$
$521$	−4.16861	−0.182630	−0.0913150	+	0.995822i	$0.529107\pi$
$522$	0	0
$523$	19.8719	0.868936	0.434468	−	0.900687i	$-0.356936\pi$
$523$	19.8719	0.868936	0.434468	+	0.900687i	$0.356936\pi$
$524$	0	0
$525$	−4.75605	−0.207571
$526$	0	0
$527$	8.41856	0.366718
$528$	0	0
$529$	59.9870	2.60813
$530$	0	0
$531$	7.95133	0.345058
$532$	0	0
$533$	−35.6246	−1.54307
$534$	0	0
$535$	−6.86078	−0.296617
$536$	0	0
$537$	−4.43698	−0.191470
$538$	0	0
$539$	−29.7988	−1.28352
$540$	0	0
$541$	−31.8524	−1.36944	−0.684721	−	0.728806i	$-0.740076\pi$
$541$	−31.8524	−1.36944	−0.684721	+	0.728806i	$0.740076\pi$
$542$	0	0
$543$	−6.79423	−0.291568
$544$	0	0
$545$	−3.58871	−0.153724
$546$	0	0
$547$	10.2594	0.438659	0.219330	−	0.975651i	$-0.429613\pi$
$547$	10.2594	0.438659	0.219330	+	0.975651i	$0.429613\pi$
$548$	0	0
$549$	−4.23652	−0.180810
$550$	0	0
$551$	−8.64392	−0.368243
$552$	0	0
$553$	−19.7328	−0.839123
$554$	0	0
$555$	−10.7660	−0.456992
$556$	0	0
$557$	−21.6310	−0.916535	−0.458268	−	0.888814i	$-0.651530\pi$
$557$	−21.6310	−0.916535	−0.458268	+	0.888814i	$0.651530\pi$
$558$	0	0
$559$	16.0201	0.677579
$560$	0	0
$561$	−5.68533	−0.240035
$562$	0	0
$563$	−30.6291	−1.29086	−0.645431	−	0.763819i	$-0.723322\pi$
$563$	−30.6291	−1.29086	−0.645431	+	0.763819i	$0.723322\pi$
$564$	0	0
$565$	9.31607	0.391930
$566$	0	0
$567$	1.22980	0.0516465
$568$	0	0
$569$	−21.9034	−0.918239	−0.459119	−	0.888375i	$-0.651835\pi$
$569$	−21.9034	−0.918239	−0.459119	+	0.888375i	$0.651835\pi$
$570$	0	0
$571$	−7.96131	−0.333170	−0.166585	−	0.986027i	$-0.553274\pi$
$571$	−7.96131	−0.333170	−0.166585	+	0.986027i	$0.553274\pi$
$572$	0	0
$573$	−13.0977	−0.547164
$574$	0	0
$575$	−35.2305	−1.46921
$576$	0	0
$577$	2.54093	0.105780	0.0528902	−	0.998600i	$-0.483157\pi$
$577$	2.54093	0.105780	0.0528902	+	0.998600i	$0.483157\pi$
$578$	0	0
$579$	−18.7404	−0.778824
$580$	0	0
$581$	−12.5559	−0.520907
$582$	0	0
$583$	−18.5739	−0.769253
$584$	0	0
$585$	−6.84779	−0.283121
$586$	0	0
$587$	1.10380	0.0455586	0.0227793	−	0.999741i	$-0.492748\pi$
$587$	1.10380	0.0455586	0.0227793	+	0.999741i	$0.492748\pi$
$588$	0	0
$589$	−42.0774	−1.73377
$590$	0	0
$591$	3.32543	0.136790
$592$	0	0
$593$	−23.9562	−0.983761	−0.491881	−	0.870663i	$-0.663690\pi$
$593$	−23.9562	−0.983761	−0.491881	+	0.870663i	$0.663690\pi$
$594$	0	0
$595$	1.37032	0.0561775
$596$	0	0
$597$	18.4904	0.756760
$598$	0	0
$599$	−8.28504	−0.338517	−0.169259	−	0.985572i	$-0.554137\pi$
$599$	−8.28504	−0.338517	−0.169259	+	0.985572i	$0.554137\pi$
$600$	0	0
$601$	−26.5770	−1.08410	−0.542050	−	0.840346i	$-0.682352\pi$
$601$	−26.5770	−1.08410	−0.542050	+	0.840346i	$0.682352\pi$
$602$	0	0
$603$	12.2238	0.497793
$604$	0	0
$605$	−19.6750	−0.799903
$606$	0	0
$607$	4.25341	0.172641	0.0863203	−	0.996267i	$-0.472489\pi$
$607$	4.25341	0.172641	0.0863203	+	0.996267i	$0.472489\pi$
$608$	0	0
$609$	−2.03139	−0.0823159
$610$	0	0
$611$	−2.43441	−0.0984857
$612$	0	0
$613$	−15.4821	−0.625317	−0.312659	−	0.949866i	$-0.601220\pi$
$613$	−15.4821	−0.625317	−0.312659	+	0.949866i	$0.601220\pi$
$614$	0	0
$615$	5.89244	0.237606
$616$	0	0
$617$	32.7116	1.31692	0.658460	−	0.752616i	$-0.271208\pi$
$617$	32.7116	1.31692	0.658460	+	0.752616i	$0.271208\pi$
$618$	0	0
$619$	18.9797	0.762857	0.381428	−	0.924398i	$-0.375432\pi$
$619$	18.9797	0.762857	0.381428	+	0.924398i	$0.375432\pi$
$620$	0	0
$621$	9.10972	0.365560
$622$	0	0
$623$	−17.4760	−0.700159
$624$	0	0
$625$	9.29314	0.371726
$626$	0	0
$627$	28.4162	1.13484
$628$	0	0
$629$	−10.5913	−0.422301
$630$	0	0
$631$	23.4485	0.933468	0.466734	−	0.884398i	$-0.345431\pi$
$631$	23.4485	0.933468	0.466734	+	0.884398i	$0.345431\pi$
$632$	0	0
$633$	3.43034	0.136344
$634$	0	0
$635$	16.8258	0.667712
$636$	0	0
$637$	−35.3090	−1.39899
$638$	0	0
$639$	−1.48748	−0.0588436
$640$	0	0
$641$	38.2792	1.51194	0.755968	−	0.654608i	$-0.227166\pi$
$641$	38.2792	1.51194	0.755968	+	0.654608i	$0.227166\pi$
$642$	0	0
$643$	−35.8562	−1.41403	−0.707016	−	0.707197i	$-0.749959\pi$
$643$	−35.8562	−1.41403	−0.707016	+	0.707197i	$0.749959\pi$
$644$	0	0
$645$	−2.64979	−0.104335
$646$	0	0
$647$	16.2331	0.638188	0.319094	−	0.947723i	$-0.396621\pi$
$647$	16.2331	0.638188	0.319094	+	0.947723i	$0.396621\pi$
$648$	0	0
$649$	43.1773	1.69486
$650$	0	0
$651$	−9.88851	−0.387561
$652$	0	0
$653$	11.8505	0.463746	0.231873	−	0.972746i	$-0.425515\pi$
$653$	11.8505	0.463746	0.231873	+	0.972746i	$0.425515\pi$
$654$	0	0
$655$	15.7977	0.617266
$656$	0	0
$657$	3.17701	0.123947
$658$	0	0
$659$	16.4022	0.638938	0.319469	−	0.947597i	$-0.396495\pi$
$659$	16.4022	0.638938	0.319469	+	0.947597i	$0.396495\pi$
$660$	0	0
$661$	32.5802	1.26722	0.633611	−	0.773652i	$-0.281572\pi$
$661$	32.5802	1.26722	0.633611	+	0.773652i	$0.281572\pi$
$662$	0	0
$663$	−6.73663	−0.261629
$664$	0	0
$665$	−6.84907	−0.265596
$666$	0	0
$667$	−15.0475	−0.582642
$668$	0	0
$669$	−0.0605067	−0.00233933
$670$	0	0
$671$	−23.0051	−0.888102
$672$	0	0
$673$	7.49413	0.288878	0.144439	−	0.989514i	$-0.453862\pi$
$673$	7.49413	0.288878	0.144439	+	0.989514i	$0.453862\pi$
$674$	0	0
$675$	−3.86735	−0.148854
$676$	0	0
$677$	−16.4272	−0.631350	−0.315675	−	0.948867i	$-0.602231\pi$
$677$	−16.4272	−0.631350	−0.315675	+	0.948867i	$0.602231\pi$
$678$	0	0
$679$	−1.80658	−0.0693300
$680$	0	0
$681$	6.38089	0.244516
$682$	0	0
$683$	−45.6097	−1.74521	−0.872603	−	0.488430i	$-0.837570\pi$
$683$	−45.6097	−1.74521	−0.872603	+	0.488430i	$0.837570\pi$
$684$	0	0
$685$	7.86471	0.300495
$686$	0	0
$687$	25.4876	0.972411
$688$	0	0
$689$	−22.0085	−0.838457
$690$	0	0
$691$	−22.6585	−0.861971	−0.430985	−	0.902359i	$-0.641834\pi$
$691$	−22.6585	−0.861971	−0.430985	+	0.902359i	$0.641834\pi$
$692$	0	0
$693$	6.67803	0.253677
$694$	0	0
$695$	5.23443	0.198553
$696$	0	0
$697$	5.79678	0.219569
$698$	0	0
$699$	8.91059	0.337030
$700$	0	0
$701$	−48.3811	−1.82733	−0.913664	−	0.406470i	$-0.866760\pi$
$701$	−48.3811	−1.82733	−0.913664	+	0.406470i	$0.866760\pi$
$702$	0	0
$703$	52.9369	1.99655
$704$	0	0
$705$	0.402660	0.0151651
$706$	0	0
$707$	−5.04102	−0.189587
$708$	0	0
$709$	20.2892	0.761977	0.380988	−	0.924580i	$-0.375584\pi$
$709$	20.2892	0.761977	0.380988	+	0.924580i	$0.375584\pi$
$710$	0	0
$711$	−16.0456	−0.601756
$712$	0	0
$713$	−73.2492	−2.74320
$714$	0	0
$715$	−37.1849	−1.39063
$716$	0	0
$717$	−27.9565	−1.04405
$718$	0	0
$719$	−21.3427	−0.795949	−0.397975	−	0.917396i	$-0.630287\pi$
$719$	−21.3427	−0.795949	−0.397975	+	0.917396i	$0.630287\pi$
$720$	0	0
$721$	14.7556	0.549529
$722$	0	0
$723$	−26.2747	−0.977167
$724$	0	0
$725$	6.38812	0.237249
$726$	0	0
$727$	−18.3692	−0.681275	−0.340637	−	0.940195i	$-0.610643\pi$
$727$	−18.3692	−0.681275	−0.340637	+	0.940195i	$0.610643\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.60677	−0.0964150
$732$	0	0
$733$	35.3094	1.30418	0.652091	−	0.758140i	$-0.273892\pi$
$733$	35.3094	1.30418	0.652091	+	0.758140i	$0.273892\pi$
$734$	0	0
$735$	5.84024	0.215421
$736$	0	0
$737$	66.3778	2.44506
$738$	0	0
$739$	3.93787	0.144857	0.0724284	−	0.997374i	$-0.476925\pi$
$739$	3.93787	0.144857	0.0724284	+	0.997374i	$0.476925\pi$
$740$	0	0
$741$	33.6708	1.23693
$742$	0	0
$743$	12.6513	0.464130	0.232065	−	0.972700i	$-0.425452\pi$
$743$	12.6513	0.464130	0.232065	+	0.972700i	$0.425452\pi$
$744$	0	0
$745$	24.4051	0.894133
$746$	0	0
$747$	−10.2098	−0.373555
$748$	0	0
$749$	7.92791	0.289679
$750$	0	0
$751$	13.1563	0.480079	0.240039	−	0.970763i	$-0.422840\pi$
$751$	13.1563	0.480079	0.240039	+	0.970763i	$0.422840\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	3.34082	0.121585
$756$	0	0
$757$	16.7342	0.608215	0.304107	−	0.952638i	$-0.401642\pi$
$757$	16.7342	0.608215	0.304107	+	0.952638i	$0.401642\pi$
$758$	0	0
$759$	49.4676	1.79556
$760$	0	0
$761$	−19.9857	−0.724481	−0.362241	−	0.932085i	$-0.617988\pi$
$761$	−19.9857	−0.724481	−0.362241	+	0.932085i	$0.617988\pi$
$762$	0	0
$763$	4.14690	0.150128
$764$	0	0
$765$	1.11426	0.0402863
$766$	0	0
$767$	51.1614	1.84733
$768$	0	0
$769$	7.31569	0.263810	0.131905	−	0.991262i	$-0.457891\pi$
$769$	7.31569	0.263810	0.131905	+	0.991262i	$0.457891\pi$
$770$	0	0
$771$	27.8655	1.00355
$772$	0	0
$773$	22.3226	0.802888	0.401444	−	0.915884i	$-0.368508\pi$
$773$	22.3226	0.802888	0.401444	+	0.915884i	$0.368508\pi$
$774$	0	0
$775$	31.0965	1.11702
$776$	0	0
$777$	12.4406	0.446303
$778$	0	0
$779$	−28.9733	−1.03808
$780$	0	0
$781$	−8.07729	−0.289028
$782$	0	0
$783$	−1.65181	−0.0590308
$784$	0	0
$785$	−11.5473	−0.412139
$786$	0	0
$787$	21.2108	0.756083	0.378042	−	0.925789i	$-0.376598\pi$
$787$	21.2108	0.756083	0.378042	+	0.925789i	$0.376598\pi$
$788$	0	0
$789$	28.0309	0.997927
$790$	0	0
$791$	−10.7651	−0.382762
$792$	0	0
$793$	−27.2591	−0.967999
$794$	0	0
$795$	3.64029	0.129108
$796$	0	0
$797$	45.4158	1.60871	0.804355	−	0.594149i	$-0.202511\pi$
$797$	45.4158	1.60871	0.804355	+	0.594149i	$0.202511\pi$
$798$	0	0
$799$	0.396124	0.0140139
$800$	0	0
$801$	−14.2105	−0.502102
$802$	0	0
$803$	17.2518	0.608802
$804$	0	0
$805$	−11.9230	−0.420231
$806$	0	0
$807$	14.3015	0.503437
$808$	0	0
$809$	−1.44145	−0.0506788	−0.0253394	−	0.999679i	$-0.508067\pi$
$809$	−1.44145	−0.0506788	−0.0253394	+	0.999679i	$0.508067\pi$
$810$	0	0
$811$	18.0279	0.633044	0.316522	−	0.948585i	$-0.397485\pi$
$811$	18.0279	0.633044	0.316522	+	0.948585i	$0.397485\pi$
$812$	0	0
$813$	6.46714	0.226812
$814$	0	0
$815$	−20.9373	−0.733401
$816$	0	0
$817$	13.0291	0.455830
$818$	0	0
$819$	7.91290	0.276499
$820$	0	0
$821$	36.6461	1.27896	0.639478	−	0.768809i	$-0.279150\pi$
$821$	36.6461	1.27896	0.639478	+	0.768809i	$0.279150\pi$
$822$	0	0
$823$	−52.3454	−1.82465	−0.912324	−	0.409470i	$-0.865714\pi$
$823$	−52.3454	−1.82465	−0.912324	+	0.409470i	$0.865714\pi$
$824$	0	0
$825$	−21.0005	−0.731142
$826$	0	0
$827$	8.63182	0.300158	0.150079	−	0.988674i	$-0.452047\pi$
$827$	8.63182	0.300158	0.150079	+	0.988674i	$0.452047\pi$
$828$	0	0
$829$	−10.9599	−0.380653	−0.190326	−	0.981721i	$-0.560955\pi$
$829$	−10.9599	−0.380653	−0.190326	+	0.981721i	$0.560955\pi$
$830$	0	0
$831$	32.4102	1.12430
$832$	0	0
$833$	5.74543	0.199067
$834$	0	0
$835$	−6.26545	−0.216825
$836$	0	0
$837$	−8.04078	−0.277930
$838$	0	0
$839$	−7.29686	−0.251915	−0.125958	−	0.992036i	$-0.540200\pi$
$839$	−7.29686	−0.251915	−0.125958	+	0.992036i	$0.540200\pi$
$840$	0	0
$841$	−26.2715	−0.905915
$842$	0	0
$843$	−9.46110	−0.325858
$844$	0	0
$845$	−30.2255	−1.03979
$846$	0	0
$847$	22.7353	0.781193
$848$	0	0
$849$	21.8918	0.751323
$850$	0	0
$851$	92.1536	3.15899
$852$	0	0
$853$	−20.5960	−0.705195	−0.352598	−	0.935775i	$-0.614702\pi$
$853$	−20.5960	−0.705195	−0.352598	+	0.935775i	$0.614702\pi$
$854$	0	0
$855$	−5.56928	−0.190465
$856$	0	0
$857$	−19.4261	−0.663584	−0.331792	−	0.943353i	$-0.607653\pi$
$857$	−19.4261	−0.663584	−0.331792	+	0.943353i	$0.607653\pi$
$858$	0	0
$859$	21.3158	0.727284	0.363642	−	0.931539i	$-0.381533\pi$
$859$	21.3158	0.727284	0.363642	+	0.931539i	$0.381533\pi$
$860$	0	0
$861$	−6.80894	−0.232048
$862$	0	0
$863$	−9.90040	−0.337013	−0.168507	−	0.985701i	$-0.553894\pi$
$863$	−9.90040	−0.337013	−0.168507	+	0.985701i	$0.553894\pi$
$864$	0	0
$865$	−9.83299	−0.334332
$866$	0	0
$867$	−15.9038	−0.540122
$868$	0	0
$869$	−87.1306	−2.95570
$870$	0	0
$871$	78.6521	2.66502
$872$	0	0
$873$	−1.46901	−0.0497183
$874$	0	0
$875$	11.6058	0.392347
$876$	0	0
$877$	−27.9888	−0.945115	−0.472557	−	0.881300i	$-0.656669\pi$
$877$	−27.9888	−0.945115	−0.472557	+	0.881300i	$0.656669\pi$
$878$	0	0
$879$	2.62878	0.0886666
$880$	0	0
$881$	1.19522	0.0402681	0.0201341	−	0.999797i	$-0.493591\pi$
$881$	1.19522	0.0402681	0.0201341	+	0.999797i	$0.493591\pi$
$882$	0	0
$883$	27.9421	0.940328	0.470164	−	0.882579i	$-0.344195\pi$
$883$	27.9421	0.940328	0.470164	+	0.882579i	$0.344195\pi$
$884$	0	0
$885$	−8.46229	−0.284457
$886$	0	0
$887$	−23.8893	−0.802124	−0.401062	−	0.916051i	$-0.631359\pi$
$887$	−23.8893	−0.802124	−0.401062	+	0.916051i	$0.631359\pi$
$888$	0	0
$889$	−19.4429	−0.652094
$890$	0	0
$891$	5.43020	0.181918
$892$	0	0
$893$	−1.97989	−0.0662546
$894$	0	0
$895$	4.72211	0.157843
$896$	0	0
$897$	58.6148	1.95709
$898$	0	0
$899$	13.2818	0.442974
$900$	0	0
$901$	3.58120	0.119307
$902$	0	0
$903$	3.06194	0.101895
$904$	0	0
$905$	7.23083	0.240361
$906$	0	0
$907$	−50.5044	−1.67697	−0.838485	−	0.544925i	$-0.816558\pi$
$907$	−50.5044	−1.67697	−0.838485	+	0.544925i	$0.816558\pi$
$908$	0	0
$909$	−4.09907	−0.135958
$910$	0	0
$911$	37.3000	1.23580	0.617901	−	0.786256i	$-0.287983\pi$
$911$	37.3000	1.23580	0.617901	+	0.786256i	$0.287983\pi$
$912$	0	0
$913$	−55.4410	−1.83483
$914$	0	0
$915$	4.50876	0.149055
$916$	0	0
$917$	−18.2548	−0.602827
$918$	0	0
$919$	35.9679	1.18647	0.593236	−	0.805029i	$-0.297850\pi$
$919$	35.9679	1.18647	0.593236	+	0.805029i	$0.297850\pi$
$920$	0	0
$921$	2.11725	0.0697657
$922$	0	0
$923$	−9.57090	−0.315030
$924$	0	0
$925$	−39.1220	−1.28632
$926$	0	0
$927$	11.9985	0.394081
$928$	0	0
$929$	−43.6455	−1.43196	−0.715981	−	0.698120i	$-0.754020\pi$
$929$	−43.6455	−1.43196	−0.715981	+	0.698120i	$0.754020\pi$
$930$	0	0
$931$	−28.7166	−0.941150
$932$	0	0
$933$	1.80626	0.0591342
$934$	0	0
$935$	6.05067	0.197878
$936$	0	0
$937$	29.2531	0.955656	0.477828	−	0.878454i	$-0.341424\pi$
$937$	29.2531	0.955656	0.477828	+	0.878454i	$0.341424\pi$
$938$	0	0
$939$	−20.7830	−0.678227
$940$	0	0
$941$	20.2861	0.661309	0.330654	−	0.943752i	$-0.392731\pi$
$941$	20.2861	0.661309	0.330654	+	0.943752i	$0.392731\pi$
$942$	0	0
$943$	−50.4373	−1.64246
$944$	0	0
$945$	−1.30882	−0.0425760
$946$	0	0
$947$	19.8798	0.646006	0.323003	−	0.946398i	$-0.395308\pi$
$947$	19.8798	0.646006	0.323003	+	0.946398i	$0.395308\pi$
$948$	0	0
$949$	20.4419	0.663572
$950$	0	0
$951$	−21.9652	−0.712271
$952$	0	0
$953$	55.1034	1.78497	0.892487	−	0.451072i	$-0.148958\pi$
$953$	55.1034	1.78497	0.892487	+	0.451072i	$0.148958\pi$
$954$	0	0
$955$	13.9394	0.451067
$956$	0	0
$957$	−8.96964	−0.289947
$958$	0	0
$959$	−9.08798	−0.293466
$960$	0	0
$961$	33.6541	1.08562
$962$	0	0
$963$	6.44653	0.207736
$964$	0	0
$965$	19.9447	0.642041
$966$	0	0
$967$	−0.547168	−0.0175958	−0.00879788	−	0.999961i	$-0.502800\pi$
$967$	−0.547168	−0.0175958	−0.00879788	+	0.999961i	$0.502800\pi$
$968$	0	0
$969$	−5.47887	−0.176007
$970$	0	0
$971$	−56.9967	−1.82911	−0.914555	−	0.404461i	$-0.867459\pi$
$971$	−56.9967	−1.82911	−0.914555	+	0.404461i	$0.867459\pi$
$972$	0	0
$973$	−6.04859	−0.193909
$974$	0	0
$975$	−24.8838	−0.796918
$976$	0	0
$977$	−43.5761	−1.39412	−0.697061	−	0.717011i	$-0.745510\pi$
$977$	−43.5761	−1.39412	−0.697061	+	0.717011i	$0.745510\pi$
$978$	0	0
$979$	−77.1656	−2.46622
$980$	0	0
$981$	3.37203	0.107660
$982$	0	0
$983$	−2.72737	−0.0869896	−0.0434948	−	0.999054i	$-0.513849\pi$
$983$	−2.72737	−0.0869896	−0.0434948	+	0.999054i	$0.513849\pi$
$984$	0	0
$985$	−3.53912	−0.112766
$986$	0	0
$987$	−0.465290	−0.0148103
$988$	0	0
$989$	22.6813	0.721224
$990$	0	0
$991$	−13.4465	−0.427143	−0.213571	−	0.976927i	$-0.568510\pi$
$991$	−13.4465	−0.427143	−0.213571	+	0.976927i	$0.568510\pi$
$992$	0	0
$993$	−16.2829	−0.516723
$994$	0	0
$995$	−19.6786	−0.623853
$996$	0	0
$997$	33.1155	1.04878	0.524389	−	0.851479i	$-0.324294\pi$
$997$	33.1155	1.04878	0.524389	+	0.851479i	$0.324294\pi$
$998$	0	0
$999$	10.1160	0.320055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.06426 q^{5} +1.22980 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.06426 q^{5} +1.22980 q^{7} +1.00000 q^{9} +5.43020 q^{11} +6.43432 q^{13} -1.06426 q^{15} -1.04698 q^{17} +5.23300 q^{19} +1.22980 q^{21} +9.10972 q^{23} -3.86735 q^{25} +1.00000 q^{27} -1.65181 q^{29} -8.04078 q^{31} +5.43020 q^{33} -1.30882 q^{35} +10.1160 q^{37} +6.43432 q^{39} -5.53665 q^{41} +2.48979 q^{43} -1.06426 q^{45} -0.378348 q^{47} -5.48760 q^{49} -1.04698 q^{51} -3.42049 q^{53} -5.77914 q^{55} +5.23300 q^{57} +7.95133 q^{59} -4.23652 q^{61} +1.22980 q^{63} -6.84779 q^{65} +12.2238 q^{67} +9.10972 q^{69} -1.48748 q^{71} +3.17701 q^{73} -3.86735 q^{75} +6.67803 q^{77} -16.0456 q^{79} +1.00000 q^{81} -10.2098 q^{83} +1.11426 q^{85} -1.65181 q^{87} -14.2105 q^{89} +7.91290 q^{91} -8.04078 q^{93} -5.56928 q^{95} -1.46901 q^{97} +5.43020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 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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