LMFDB - Embedded newform 6024.2.a.r.1.12

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			1.12
Root			$1.32210$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.32210 q^{5} -1.93279 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.32210 q^{5} -1.93279 q^{7} +1.00000 q^{9} +1.66666 q^{11} +1.77110 q^{13} +1.32210 q^{15} +4.78319 q^{17} -3.38032 q^{19} -1.93279 q^{21} +0.812005 q^{23} -3.25206 q^{25} +1.00000 q^{27} +5.26360 q^{29} -10.3494 q^{31} +1.66666 q^{33} -2.55533 q^{35} -0.350697 q^{37} +1.77110 q^{39} +8.91181 q^{41} +6.65092 q^{43} +1.32210 q^{45} +4.47615 q^{47} -3.26433 q^{49} +4.78319 q^{51} +9.84257 q^{53} +2.20348 q^{55} -3.38032 q^{57} -1.43691 q^{59} +14.4233 q^{61} -1.93279 q^{63} +2.34156 q^{65} +4.16434 q^{67} +0.812005 q^{69} +12.4409 q^{71} -8.86018 q^{73} -3.25206 q^{75} -3.22129 q^{77} +0.706903 q^{79} +1.00000 q^{81} -5.78094 q^{83} +6.32384 q^{85} +5.26360 q^{87} -2.66382 q^{89} -3.42315 q^{91} -10.3494 q^{93} -4.46911 q^{95} +6.34447 q^{97} +1.66666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.32210	0.591260	0.295630	−	0.955303i	$-0.404470\pi$
$5$	1.32210	0.591260	0.295630	+	0.955303i	$0.404470\pi$
$6$	0	0
$7$	−1.93279	−0.730525	−0.365263	−	0.930904i	$-0.619021\pi$
$7$	−1.93279	−0.730525	−0.365263	+	0.930904i	$0.619021\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.66666	0.502516	0.251258	−	0.967920i	$-0.419156\pi$
$11$	1.66666	0.502516	0.251258	+	0.967920i	$0.419156\pi$
$12$	0	0
$13$	1.77110	0.491214	0.245607	−	0.969370i	$-0.421013\pi$
$13$	1.77110	0.491214	0.245607	+	0.969370i	$0.421013\pi$
$14$	0	0
$15$	1.32210	0.341364
$16$	0	0
$17$	4.78319	1.16009	0.580047	−	0.814583i	$-0.303034\pi$
$17$	4.78319	1.16009	0.580047	+	0.814583i	$0.303034\pi$
$18$	0	0
$19$	−3.38032	−0.775498	−0.387749	−	0.921765i	$-0.626747\pi$
$19$	−3.38032	−0.775498	−0.387749	+	0.921765i	$0.626747\pi$
$20$	0	0
$21$	−1.93279	−0.421769
$22$	0	0
$23$	0.812005	0.169315	0.0846574	−	0.996410i	$-0.473020\pi$
$23$	0.812005	0.169315	0.0846574	+	0.996410i	$0.473020\pi$
$24$	0	0
$25$	−3.25206	−0.650412
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.26360	0.977427	0.488713	−	0.872444i	$-0.337466\pi$
$29$	5.26360	0.977427	0.488713	+	0.872444i	$0.337466\pi$
$30$	0	0
$31$	−10.3494	−1.85881	−0.929404	−	0.369064i	$-0.879678\pi$
$31$	−10.3494	−1.85881	−0.929404	+	0.369064i	$0.879678\pi$
$32$	0	0
$33$	1.66666	0.290128
$34$	0	0
$35$	−2.55533	−0.431930
$36$	0	0
$37$	−0.350697	−0.0576543	−0.0288271	−	0.999584i	$-0.509177\pi$
$37$	−0.350697	−0.0576543	−0.0288271	+	0.999584i	$0.509177\pi$
$38$	0	0
$39$	1.77110	0.283602
$40$	0	0
$41$	8.91181	1.39179	0.695895	−	0.718143i	$-0.255008\pi$
$41$	8.91181	1.39179	0.695895	+	0.718143i	$0.255008\pi$
$42$	0	0
$43$	6.65092	1.01426	0.507128	−	0.861871i	$-0.330707\pi$
$43$	6.65092	1.01426	0.507128	+	0.861871i	$0.330707\pi$
$44$	0	0
$45$	1.32210	0.197087
$46$	0	0
$47$	4.47615	0.652913	0.326457	−	0.945212i	$-0.394145\pi$
$47$	4.47615	0.652913	0.326457	+	0.945212i	$0.394145\pi$
$48$	0	0
$49$	−3.26433	−0.466333
$50$	0	0
$51$	4.78319	0.669781
$52$	0	0
$53$	9.84257	1.35198	0.675990	−	0.736910i	$-0.263716\pi$
$53$	9.84257	1.35198	0.675990	+	0.736910i	$0.263716\pi$
$54$	0	0
$55$	2.20348	0.297117
$56$	0	0
$57$	−3.38032	−0.447734
$58$	0	0
$59$	−1.43691	−0.187070	−0.0935350	−	0.995616i	$-0.529817\pi$
$59$	−1.43691	−0.187070	−0.0935350	+	0.995616i	$0.529817\pi$
$60$	0	0
$61$	14.4233	1.84671	0.923356	−	0.383945i	$-0.125435\pi$
$61$	14.4233	1.84671	0.923356	+	0.383945i	$0.125435\pi$
$62$	0	0
$63$	−1.93279	−0.243508
$64$	0	0
$65$	2.34156	0.290435
$66$	0	0
$67$	4.16434	0.508755	0.254378	−	0.967105i	$-0.418129\pi$
$67$	4.16434	0.508755	0.254378	+	0.967105i	$0.418129\pi$
$68$	0	0
$69$	0.812005	0.0977540
$70$	0	0
$71$	12.4409	1.47647	0.738234	−	0.674545i	$-0.235660\pi$
$71$	12.4409	1.47647	0.738234	+	0.674545i	$0.235660\pi$
$72$	0	0
$73$	−8.86018	−1.03701	−0.518503	−	0.855076i	$-0.673510\pi$
$73$	−8.86018	−1.03701	−0.518503	+	0.855076i	$0.673510\pi$
$74$	0	0
$75$	−3.25206	−0.375515
$76$	0	0
$77$	−3.22129	−0.367100
$78$	0	0
$79$	0.706903	0.0795328	0.0397664	−	0.999209i	$-0.487339\pi$
$79$	0.706903	0.0795328	0.0397664	+	0.999209i	$0.487339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.78094	−0.634541	−0.317270	−	0.948335i	$-0.602766\pi$
$83$	−5.78094	−0.634541	−0.317270	+	0.948335i	$0.602766\pi$
$84$	0	0
$85$	6.32384	0.685917
$86$	0	0
$87$	5.26360	0.564317
$88$	0	0
$89$	−2.66382	−0.282364	−0.141182	−	0.989984i	$-0.545090\pi$
$89$	−2.66382	−0.282364	−0.141182	+	0.989984i	$0.545090\pi$
$90$	0	0
$91$	−3.42315	−0.358844
$92$	0	0
$93$	−10.3494	−1.07318
$94$	0	0
$95$	−4.46911	−0.458521
$96$	0	0
$97$	6.34447	0.644184	0.322092	−	0.946708i	$-0.395614\pi$
$97$	6.34447	0.644184	0.322092	+	0.946708i	$0.395614\pi$
$98$	0	0
$99$	1.66666	0.167505
$100$	0	0
$101$	−14.4190	−1.43474	−0.717372	−	0.696690i	$-0.754655\pi$
$101$	−14.4190	−1.43474	−0.717372	+	0.696690i	$0.754655\pi$
$102$	0	0
$103$	−4.74022	−0.467067	−0.233534	−	0.972349i	$-0.575029\pi$
$103$	−4.74022	−0.467067	−0.233534	+	0.972349i	$0.575029\pi$
$104$	0	0
$105$	−2.55533	−0.249375
$106$	0	0
$107$	−5.17693	−0.500472	−0.250236	−	0.968185i	$-0.580508\pi$
$107$	−5.17693	−0.500472	−0.250236	+	0.968185i	$0.580508\pi$
$108$	0	0
$109$	11.1143	1.06456	0.532279	−	0.846569i	$-0.321336\pi$
$109$	11.1143	1.06456	0.532279	+	0.846569i	$0.321336\pi$
$110$	0	0
$111$	−0.350697	−0.0332867
$112$	0	0
$113$	5.50478	0.517846	0.258923	−	0.965898i	$-0.416632\pi$
$113$	5.50478	0.517846	0.258923	+	0.965898i	$0.416632\pi$
$114$	0	0
$115$	1.07355	0.100109
$116$	0	0
$117$	1.77110	0.163738
$118$	0	0
$119$	−9.24490	−0.847479
$120$	0	0
$121$	−8.22226	−0.747478
$122$	0	0
$123$	8.91181	0.803551
$124$	0	0
$125$	−10.9100	−0.975822
$126$	0	0
$127$	−8.97864	−0.796726	−0.398363	−	0.917228i	$-0.630421\pi$
$127$	−8.97864	−0.796726	−0.398363	+	0.917228i	$0.630421\pi$
$128$	0	0
$129$	6.65092	0.585581
$130$	0	0
$131$	10.5104	0.918300	0.459150	−	0.888359i	$-0.348154\pi$
$131$	10.5104	0.918300	0.459150	+	0.888359i	$0.348154\pi$
$132$	0	0
$133$	6.53344	0.566521
$134$	0	0
$135$	1.32210	0.113788
$136$	0	0
$137$	8.40948	0.718470	0.359235	−	0.933247i	$-0.383038\pi$
$137$	8.40948	0.718470	0.359235	+	0.933247i	$0.383038\pi$
$138$	0	0
$139$	15.2946	1.29727	0.648637	−	0.761098i	$-0.275339\pi$
$139$	15.2946	1.29727	0.648637	+	0.761098i	$0.275339\pi$
$140$	0	0
$141$	4.47615	0.376960
$142$	0	0
$143$	2.95181	0.246842
$144$	0	0
$145$	6.95899	0.577913
$146$	0	0
$147$	−3.26433	−0.269237
$148$	0	0
$149$	4.55732	0.373350	0.186675	−	0.982422i	$-0.440229\pi$
$149$	4.55732	0.373350	0.186675	+	0.982422i	$0.440229\pi$
$150$	0	0
$151$	−0.418269	−0.0340382	−0.0170191	−	0.999855i	$-0.505418\pi$
$151$	−0.418269	−0.0340382	−0.0170191	+	0.999855i	$0.505418\pi$
$152$	0	0
$153$	4.78319	0.386698
$154$	0	0
$155$	−13.6829	−1.09904
$156$	0	0
$157$	18.2998	1.46048	0.730242	−	0.683188i	$-0.239407\pi$
$157$	18.2998	1.46048	0.730242	+	0.683188i	$0.239407\pi$
$158$	0	0
$159$	9.84257	0.780566
$160$	0	0
$161$	−1.56943	−0.123689
$162$	0	0
$163$	−21.3788	−1.67452	−0.837260	−	0.546805i	$-0.815844\pi$
$163$	−21.3788	−1.67452	−0.837260	+	0.546805i	$0.815844\pi$
$164$	0	0
$165$	2.20348	0.171541
$166$	0	0
$167$	8.15518	0.631067	0.315534	−	0.948914i	$-0.397817\pi$
$167$	8.15518	0.631067	0.315534	+	0.948914i	$0.397817\pi$
$168$	0	0
$169$	−9.86322	−0.758709
$170$	0	0
$171$	−3.38032	−0.258499
$172$	0	0
$173$	2.72500	0.207178	0.103589	−	0.994620i	$-0.466967\pi$
$173$	2.72500	0.207178	0.103589	+	0.994620i	$0.466967\pi$
$174$	0	0
$175$	6.28554	0.475142
$176$	0	0
$177$	−1.43691	−0.108005
$178$	0	0
$179$	0.0333956	0.00249610	0.00124805	−	0.999999i	$-0.499603\pi$
$179$	0.0333956	0.00249610	0.00124805	+	0.999999i	$0.499603\pi$
$180$	0	0
$181$	−4.95224	−0.368097	−0.184048	−	0.982917i	$-0.558920\pi$
$181$	−4.95224	−0.368097	−0.184048	+	0.982917i	$0.558920\pi$
$182$	0	0
$183$	14.4233	1.06620
$184$	0	0
$185$	−0.463656	−0.0340887
$186$	0	0
$187$	7.97193	0.582966
$188$	0	0
$189$	−1.93279	−0.140590
$190$	0	0
$191$	12.9827	0.939393	0.469696	−	0.882828i	$-0.344363\pi$
$191$	12.9827	0.939393	0.469696	+	0.882828i	$0.344363\pi$
$192$	0	0
$193$	6.50283	0.468084	0.234042	−	0.972226i	$-0.424805\pi$
$193$	6.50283	0.468084	0.234042	+	0.972226i	$0.424805\pi$
$194$	0	0
$195$	2.34156	0.167683
$196$	0	0
$197$	13.7286	0.978119	0.489060	−	0.872250i	$-0.337340\pi$
$197$	13.7286	0.978119	0.489060	+	0.872250i	$0.337340\pi$
$198$	0	0
$199$	12.6346	0.895641	0.447821	−	0.894123i	$-0.352200\pi$
$199$	12.6346	0.895641	0.447821	+	0.894123i	$0.352200\pi$
$200$	0	0
$201$	4.16434	0.293730
$202$	0	0
$203$	−10.1734	−0.714035
$204$	0	0
$205$	11.7823	0.822910
$206$	0	0
$207$	0.812005	0.0564383
$208$	0	0
$209$	−5.63383	−0.389700
$210$	0	0
$211$	−16.8850	−1.16241	−0.581205	−	0.813757i	$-0.697419\pi$
$211$	−16.8850	−1.16241	−0.581205	+	0.813757i	$0.697419\pi$
$212$	0	0
$213$	12.4409	0.852439
$214$	0	0
$215$	8.79316	0.599689
$216$	0	0
$217$	20.0032	1.35791
$218$	0	0
$219$	−8.86018	−0.598715
$220$	0	0
$221$	8.47149	0.569854
$222$	0	0
$223$	0.967596	0.0647950	0.0323975	−	0.999475i	$-0.489686\pi$
$223$	0.967596	0.0647950	0.0323975	+	0.999475i	$0.489686\pi$
$224$	0	0
$225$	−3.25206	−0.216804
$226$	0	0
$227$	17.5749	1.16649	0.583243	−	0.812297i	$-0.301783\pi$
$227$	17.5749	1.16649	0.583243	+	0.812297i	$0.301783\pi$
$228$	0	0
$229$	15.0409	0.993929	0.496964	−	0.867771i	$-0.334448\pi$
$229$	15.0409	0.993929	0.496964	+	0.867771i	$0.334448\pi$
$230$	0	0
$231$	−3.22129	−0.211946
$232$	0	0
$233$	−21.9683	−1.43919	−0.719596	−	0.694393i	$-0.755673\pi$
$233$	−21.9683	−1.43919	−0.719596	+	0.694393i	$0.755673\pi$
$234$	0	0
$235$	5.91790	0.386041
$236$	0	0
$237$	0.706903	0.0459183
$238$	0	0
$239$	2.03201	0.131440	0.0657199	−	0.997838i	$-0.479066\pi$
$239$	2.03201	0.131440	0.0657199	+	0.997838i	$0.479066\pi$
$240$	0	0
$241$	18.3591	1.18262	0.591308	−	0.806446i	$-0.298612\pi$
$241$	18.3591	1.18262	0.591308	+	0.806446i	$0.298612\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.31576	−0.275724
$246$	0	0
$247$	−5.98687	−0.380935
$248$	0	0
$249$	−5.78094	−0.366352
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	1.35333	0.0850833
$254$	0	0
$255$	6.32384	0.396015
$256$	0	0
$257$	−23.1750	−1.44561	−0.722807	−	0.691050i	$-0.757148\pi$
$257$	−23.1750	−1.44561	−0.722807	+	0.691050i	$0.757148\pi$
$258$	0	0
$259$	0.677824	0.0421179
$260$	0	0
$261$	5.26360	0.325809
$262$	0	0
$263$	22.5592	1.39106	0.695529	−	0.718498i	$-0.255170\pi$
$263$	22.5592	1.39106	0.695529	+	0.718498i	$0.255170\pi$
$264$	0	0
$265$	13.0128	0.799372
$266$	0	0
$267$	−2.66382	−0.163023
$268$	0	0
$269$	−0.155016	−0.00945150	−0.00472575	−	0.999989i	$-0.501504\pi$
$269$	−0.155016	−0.00945150	−0.00472575	+	0.999989i	$0.501504\pi$
$270$	0	0
$271$	−15.3346	−0.931512	−0.465756	−	0.884913i	$-0.654218\pi$
$271$	−15.3346	−0.931512	−0.465756	+	0.884913i	$0.654218\pi$
$272$	0	0
$273$	−3.42315	−0.207179
$274$	0	0
$275$	−5.42006	−0.326842
$276$	0	0
$277$	−12.4922	−0.750582	−0.375291	−	0.926907i	$-0.622457\pi$
$277$	−12.4922	−0.750582	−0.375291	+	0.926907i	$0.622457\pi$
$278$	0	0
$279$	−10.3494	−0.619603
$280$	0	0
$281$	−0.889653	−0.0530723	−0.0265361	−	0.999648i	$-0.508448\pi$
$281$	−0.889653	−0.0530723	−0.0265361	+	0.999648i	$0.508448\pi$
$282$	0	0
$283$	7.94051	0.472014	0.236007	−	0.971751i	$-0.424161\pi$
$283$	7.94051	0.472014	0.236007	+	0.971751i	$0.424161\pi$
$284$	0	0
$285$	−4.46911	−0.264727
$286$	0	0
$287$	−17.2246	−1.01674
$288$	0	0
$289$	5.87893	0.345819
$290$	0	0
$291$	6.34447	0.371920
$292$	0	0
$293$	26.9991	1.57730	0.788651	−	0.614841i	$-0.210780\pi$
$293$	26.9991	1.57730	0.788651	+	0.614841i	$0.210780\pi$
$294$	0	0
$295$	−1.89974	−0.110607
$296$	0	0
$297$	1.66666	0.0967092
$298$	0	0
$299$	1.43814	0.0831697
$300$	0	0
$301$	−12.8548	−0.740940
$302$	0	0
$303$	−14.4190	−0.828350
$304$	0	0
$305$	19.0690	1.09189
$306$	0	0
$307$	10.9588	0.625454	0.312727	−	0.949843i	$-0.398757\pi$
$307$	10.9588	0.625454	0.312727	+	0.949843i	$0.398757\pi$
$308$	0	0
$309$	−4.74022	−0.269661
$310$	0	0
$311$	−4.76672	−0.270296	−0.135148	−	0.990825i	$-0.543151\pi$
$311$	−4.76672	−0.270296	−0.135148	+	0.990825i	$0.543151\pi$
$312$	0	0
$313$	−11.6334	−0.657558	−0.328779	−	0.944407i	$-0.606637\pi$
$313$	−11.6334	−0.657558	−0.328779	+	0.944407i	$0.606637\pi$
$314$	0	0
$315$	−2.55533	−0.143977
$316$	0	0
$317$	17.7913	0.999262	0.499631	−	0.866238i	$-0.333469\pi$
$317$	17.7913	0.999262	0.499631	+	0.866238i	$0.333469\pi$
$318$	0	0
$319$	8.77261	0.491172
$320$	0	0
$321$	−5.17693	−0.288948
$322$	0	0
$323$	−16.1687	−0.899652
$324$	0	0
$325$	−5.75971	−0.319491
$326$	0	0
$327$	11.1143	0.614623
$328$	0	0
$329$	−8.65145	−0.476970
$330$	0	0
$331$	1.55558	0.0855026	0.0427513	−	0.999086i	$-0.486388\pi$
$331$	1.55558	0.0855026	0.0427513	+	0.999086i	$0.486388\pi$
$332$	0	0
$333$	−0.350697	−0.0192181
$334$	0	0
$335$	5.50567	0.300807
$336$	0	0
$337$	7.81798	0.425873	0.212936	−	0.977066i	$-0.431697\pi$
$337$	7.81798	0.425873	0.212936	+	0.977066i	$0.431697\pi$
$338$	0	0
$339$	5.50478	0.298978
$340$	0	0
$341$	−17.2489	−0.934080
$342$	0	0
$343$	19.8388	1.07119
$344$	0	0
$345$	1.07355	0.0577980
$346$	0	0
$347$	13.8120	0.741465	0.370733	−	0.928740i	$-0.379107\pi$
$347$	13.8120	0.741465	0.370733	+	0.928740i	$0.379107\pi$
$348$	0	0
$349$	−21.5641	−1.15430	−0.577151	−	0.816638i	$-0.695835\pi$
$349$	−21.5641	−1.15430	−0.577151	+	0.816638i	$0.695835\pi$
$350$	0	0
$351$	1.77110	0.0945341
$352$	0	0
$353$	17.8197	0.948445	0.474222	−	0.880405i	$-0.342729\pi$
$353$	17.8197	0.948445	0.474222	+	0.880405i	$0.342729\pi$
$354$	0	0
$355$	16.4481	0.872976
$356$	0	0
$357$	−9.24490	−0.489292
$358$	0	0
$359$	−23.0153	−1.21470	−0.607350	−	0.794434i	$-0.707767\pi$
$359$	−23.0153	−1.21470	−0.607350	+	0.794434i	$0.707767\pi$
$360$	0	0
$361$	−7.57344	−0.398602
$362$	0	0
$363$	−8.22226	−0.431557
$364$	0	0
$365$	−11.7140	−0.613140
$366$	0	0
$367$	−8.32524	−0.434574	−0.217287	−	0.976108i	$-0.569721\pi$
$367$	−8.32524	−0.434574	−0.217287	+	0.976108i	$0.569721\pi$
$368$	0	0
$369$	8.91181	0.463930
$370$	0	0
$371$	−19.0236	−0.987656
$372$	0	0
$373$	−6.43795	−0.333344	−0.166672	−	0.986012i	$-0.553302\pi$
$373$	−6.43795	−0.333344	−0.166672	+	0.986012i	$0.553302\pi$
$374$	0	0
$375$	−10.9100	−0.563391
$376$	0	0
$377$	9.32235	0.480125
$378$	0	0
$379$	−1.98622	−0.102025	−0.0510127	−	0.998698i	$-0.516245\pi$
$379$	−1.98622	−0.102025	−0.0510127	+	0.998698i	$0.516245\pi$
$380$	0	0
$381$	−8.97864	−0.459990
$382$	0	0
$383$	−14.9856	−0.765730	−0.382865	−	0.923804i	$-0.625063\pi$
$383$	−14.9856	−0.765730	−0.382865	+	0.923804i	$0.625063\pi$
$384$	0	0
$385$	−4.25886	−0.217052
$386$	0	0
$387$	6.65092	0.338085
$388$	0	0
$389$	−2.08249	−0.105586	−0.0527932	−	0.998605i	$-0.516812\pi$
$389$	−2.08249	−0.105586	−0.0527932	+	0.998605i	$0.516812\pi$
$390$	0	0
$391$	3.88398	0.196421
$392$	0	0
$393$	10.5104	0.530181
$394$	0	0
$395$	0.934594	0.0470245
$396$	0	0
$397$	12.4310	0.623893	0.311946	−	0.950100i	$-0.399019\pi$
$397$	12.4310	0.623893	0.311946	+	0.950100i	$0.399019\pi$
$398$	0	0
$399$	6.53344	0.327081
$400$	0	0
$401$	−25.5185	−1.27433	−0.637167	−	0.770726i	$-0.719894\pi$
$401$	−25.5185	−1.27433	−0.637167	+	0.770726i	$0.719894\pi$
$402$	0	0
$403$	−18.3298	−0.913072
$404$	0	0
$405$	1.32210	0.0656955
$406$	0	0
$407$	−0.584492	−0.0289722
$408$	0	0
$409$	37.1233	1.83563	0.917815	−	0.397009i	$-0.129952\pi$
$409$	37.1233	1.83563	0.917815	+	0.397009i	$0.129952\pi$
$410$	0	0
$411$	8.40948	0.414809
$412$	0	0
$413$	2.77725	0.136659
$414$	0	0
$415$	−7.64296	−0.375178
$416$	0	0
$417$	15.2946	0.748982
$418$	0	0
$419$	38.3606	1.87404	0.937020	−	0.349277i	$-0.113573\pi$
$419$	38.3606	1.87404	0.937020	+	0.349277i	$0.113573\pi$
$420$	0	0
$421$	20.5156	0.999870	0.499935	−	0.866063i	$-0.333357\pi$
$421$	20.5156	0.999870	0.499935	+	0.866063i	$0.333357\pi$
$422$	0	0
$423$	4.47615	0.217638
$424$	0	0
$425$	−15.5552	−0.754539
$426$	0	0
$427$	−27.8772	−1.34907
$428$	0	0
$429$	2.95181	0.142515
$430$	0	0
$431$	−11.2812	−0.543396	−0.271698	−	0.962383i	$-0.587585\pi$
$431$	−11.2812	−0.543396	−0.271698	+	0.962383i	$0.587585\pi$
$432$	0	0
$433$	−10.6046	−0.509625	−0.254812	−	0.966991i	$-0.582014\pi$
$433$	−10.6046	−0.509625	−0.254812	+	0.966991i	$0.582014\pi$
$434$	0	0
$435$	6.95899	0.333658
$436$	0	0
$437$	−2.74484	−0.131303
$438$	0	0
$439$	−27.4088	−1.30815	−0.654075	−	0.756430i	$-0.726942\pi$
$439$	−27.4088	−1.30815	−0.654075	+	0.756430i	$0.726942\pi$
$440$	0	0
$441$	−3.26433	−0.155444
$442$	0	0
$443$	−1.16029	−0.0551271	−0.0275636	−	0.999620i	$-0.508775\pi$
$443$	−1.16029	−0.0551271	−0.0275636	+	0.999620i	$0.508775\pi$
$444$	0	0
$445$	−3.52183	−0.166951
$446$	0	0
$447$	4.55732	0.215554
$448$	0	0
$449$	7.98607	0.376886	0.188443	−	0.982084i	$-0.439656\pi$
$449$	7.98607	0.376886	0.188443	+	0.982084i	$0.439656\pi$
$450$	0	0
$451$	14.8529	0.699396
$452$	0	0
$453$	−0.418269	−0.0196520
$454$	0	0
$455$	−4.52574	−0.212170
$456$	0	0
$457$	2.94489	0.137756	0.0688781	−	0.997625i	$-0.478058\pi$
$457$	2.94489	0.137756	0.0688781	+	0.997625i	$0.478058\pi$
$458$	0	0
$459$	4.78319	0.223260
$460$	0	0
$461$	−17.5853	−0.819029	−0.409514	−	0.912304i	$-0.634302\pi$
$461$	−17.5853	−0.819029	−0.409514	+	0.912304i	$0.634302\pi$
$462$	0	0
$463$	2.16308	0.100527	0.0502634	−	0.998736i	$-0.483994\pi$
$463$	2.16308	0.100527	0.0502634	+	0.998736i	$0.483994\pi$
$464$	0	0
$465$	−13.6829	−0.634530
$466$	0	0
$467$	8.91053	0.412330	0.206165	−	0.978517i	$-0.433902\pi$
$467$	8.91053	0.412330	0.206165	+	0.978517i	$0.433902\pi$
$468$	0	0
$469$	−8.04880	−0.371659
$470$	0	0
$471$	18.2998	0.843211
$472$	0	0
$473$	11.0848	0.509679
$474$	0	0
$475$	10.9930	0.504393
$476$	0	0
$477$	9.84257	0.450660
$478$	0	0
$479$	3.94537	0.180269	0.0901343	−	0.995930i	$-0.471270\pi$
$479$	3.94537	0.180269	0.0901343	+	0.995930i	$0.471270\pi$
$480$	0	0
$481$	−0.621119	−0.0283206
$482$	0	0
$483$	−1.56943	−0.0714118
$484$	0	0
$485$	8.38801	0.380880
$486$	0	0
$487$	2.57695	0.116773	0.0583865	−	0.998294i	$-0.481404\pi$
$487$	2.57695	0.116773	0.0583865	+	0.998294i	$0.481404\pi$
$488$	0	0
$489$	−21.3788	−0.966785
$490$	0	0
$491$	−30.1904	−1.36247	−0.681237	−	0.732063i	$-0.738557\pi$
$491$	−30.1904	−1.36247	−0.681237	+	0.732063i	$0.738557\pi$
$492$	0	0
$493$	25.1768	1.13391
$494$	0	0
$495$	2.20348	0.0990391
$496$	0	0
$497$	−24.0457	−1.07860
$498$	0	0
$499$	−7.03717	−0.315027	−0.157514	−	0.987517i	$-0.550348\pi$
$499$	−7.03717	−0.315027	−0.157514	+	0.987517i	$0.550348\pi$
$500$	0	0
$501$	8.15518	0.364347
$502$	0	0
$503$	−10.4139	−0.464333	−0.232167	−	0.972676i	$-0.574581\pi$
$503$	−10.4139	−0.464333	−0.232167	+	0.972676i	$0.574581\pi$
$504$	0	0
$505$	−19.0633	−0.848307
$506$	0	0
$507$	−9.86322	−0.438041
$508$	0	0
$509$	−10.1005	−0.447697	−0.223848	−	0.974624i	$-0.571862\pi$
$509$	−10.1005	−0.447697	−0.223848	+	0.974624i	$0.571862\pi$
$510$	0	0
$511$	17.1249	0.757559
$512$	0	0
$513$	−3.38032	−0.149245
$514$	0	0
$515$	−6.26703	−0.276158
$516$	0	0
$517$	7.46020	0.328099
$518$	0	0
$519$	2.72500	0.119614
$520$	0	0
$521$	−33.4691	−1.46631	−0.733153	−	0.680063i	$-0.761952\pi$
$521$	−33.4691	−1.46631	−0.733153	+	0.680063i	$0.761952\pi$
$522$	0	0
$523$	26.9441	1.17819	0.589093	−	0.808066i	$-0.299485\pi$
$523$	26.9441	1.17819	0.589093	+	0.808066i	$0.299485\pi$
$524$	0	0
$525$	6.28554	0.274324
$526$	0	0
$527$	−49.5032	−2.15639
$528$	0	0
$529$	−22.3406	−0.971332
$530$	0	0
$531$	−1.43691	−0.0623566
$532$	0	0
$533$	15.7837	0.683666
$534$	0	0
$535$	−6.84440	−0.295909
$536$	0	0
$537$	0.0333956	0.00144113
$538$	0	0
$539$	−5.44051	−0.234339
$540$	0	0
$541$	28.7103	1.23435	0.617175	−	0.786826i	$-0.288277\pi$
$541$	28.7103	1.23435	0.617175	+	0.786826i	$0.288277\pi$
$542$	0	0
$543$	−4.95224	−0.212521
$544$	0	0
$545$	14.6942	0.629431
$546$	0	0
$547$	−14.8787	−0.636169	−0.318085	−	0.948062i	$-0.603040\pi$
$547$	−14.8787	−0.636169	−0.318085	+	0.948062i	$0.603040\pi$
$548$	0	0
$549$	14.4233	0.615571
$550$	0	0
$551$	−17.7927	−0.757993
$552$	0	0
$553$	−1.36629	−0.0581007
$554$	0	0
$555$	−0.463656	−0.0196811
$556$	0	0
$557$	16.7417	0.709367	0.354683	−	0.934986i	$-0.384589\pi$
$557$	16.7417	0.709367	0.354683	+	0.934986i	$0.384589\pi$
$558$	0	0
$559$	11.7794	0.498216
$560$	0	0
$561$	7.97193	0.336575
$562$	0	0
$563$	−6.64362	−0.279995	−0.139998	−	0.990152i	$-0.544709\pi$
$563$	−6.64362	−0.279995	−0.139998	+	0.990152i	$0.544709\pi$
$564$	0	0
$565$	7.27785	0.306181
$566$	0	0
$567$	−1.93279	−0.0811695
$568$	0	0
$569$	16.3884	0.687039	0.343520	−	0.939146i	$-0.388381\pi$
$569$	16.3884	0.687039	0.343520	+	0.939146i	$0.388381\pi$
$570$	0	0
$571$	−33.8175	−1.41522	−0.707610	−	0.706604i	$-0.750226\pi$
$571$	−33.8175	−1.41522	−0.707610	+	0.706604i	$0.750226\pi$
$572$	0	0
$573$	12.9827	0.542359
$574$	0	0
$575$	−2.64069	−0.110124
$576$	0	0
$577$	30.3725	1.26442	0.632211	−	0.774796i	$-0.282147\pi$
$577$	30.3725	1.26442	0.632211	+	0.774796i	$0.282147\pi$
$578$	0	0
$579$	6.50283	0.270249
$580$	0	0
$581$	11.1733	0.463548
$582$	0	0
$583$	16.4042	0.679391
$584$	0	0
$585$	2.34156	0.0968116
$586$	0	0
$587$	−22.5939	−0.932548	−0.466274	−	0.884640i	$-0.654404\pi$
$587$	−22.5939	−0.932548	−0.466274	+	0.884640i	$0.654404\pi$
$588$	0	0
$589$	34.9843	1.44150
$590$	0	0
$591$	13.7286	0.564717
$592$	0	0
$593$	22.7426	0.933929	0.466964	−	0.884276i	$-0.345348\pi$
$593$	22.7426	0.933929	0.466964	+	0.884276i	$0.345348\pi$
$594$	0	0
$595$	−12.2227	−0.501080
$596$	0	0
$597$	12.6346	0.517099
$598$	0	0
$599$	−34.5642	−1.41225	−0.706127	−	0.708085i	$-0.749559\pi$
$599$	−34.5642	−1.41225	−0.706127	+	0.708085i	$0.749559\pi$
$600$	0	0
$601$	−17.8672	−0.728818	−0.364409	−	0.931239i	$-0.618729\pi$
$601$	−17.8672	−0.728818	−0.364409	+	0.931239i	$0.618729\pi$
$602$	0	0
$603$	4.16434	0.169585
$604$	0	0
$605$	−10.8706	−0.441954
$606$	0	0
$607$	−33.3465	−1.35349	−0.676747	−	0.736216i	$-0.736611\pi$
$607$	−33.3465	−1.35349	−0.676747	+	0.736216i	$0.736611\pi$
$608$	0	0
$609$	−10.1734	−0.412248
$610$	0	0
$611$	7.92769	0.320720
$612$	0	0
$613$	37.8012	1.52677	0.763387	−	0.645941i	$-0.223535\pi$
$613$	37.8012	1.52677	0.763387	+	0.645941i	$0.223535\pi$
$614$	0	0
$615$	11.7823	0.475107
$616$	0	0
$617$	−37.2122	−1.49811	−0.749054	−	0.662509i	$-0.769492\pi$
$617$	−37.2122	−1.49811	−0.749054	+	0.662509i	$0.769492\pi$
$618$	0	0
$619$	−19.6476	−0.789703	−0.394851	−	0.918745i	$-0.629204\pi$
$619$	−19.6476	−0.789703	−0.394851	+	0.918745i	$0.629204\pi$
$620$	0	0
$621$	0.812005	0.0325847
$622$	0	0
$623$	5.14860	0.206274
$624$	0	0
$625$	1.83619	0.0734474
$626$	0	0
$627$	−5.63383	−0.224993
$628$	0	0
$629$	−1.67745	−0.0668844
$630$	0	0
$631$	31.3683	1.24875	0.624377	−	0.781123i	$-0.285353\pi$
$631$	31.3683	1.24875	0.624377	+	0.781123i	$0.285353\pi$
$632$	0	0
$633$	−16.8850	−0.671118
$634$	0	0
$635$	−11.8706	−0.471072
$636$	0	0
$637$	−5.78144	−0.229069
$638$	0	0
$639$	12.4409	0.492156
$640$	0	0
$641$	−39.1187	−1.54509	−0.772547	−	0.634957i	$-0.781018\pi$
$641$	−39.1187	−1.54509	−0.772547	+	0.634957i	$0.781018\pi$
$642$	0	0
$643$	29.1156	1.14821	0.574104	−	0.818783i	$-0.305351\pi$
$643$	29.1156	1.14821	0.574104	+	0.818783i	$0.305351\pi$
$644$	0	0
$645$	8.79316	0.346230
$646$	0	0
$647$	7.10250	0.279228	0.139614	−	0.990206i	$-0.455414\pi$
$647$	7.10250	0.279228	0.139614	+	0.990206i	$0.455414\pi$
$648$	0	0
$649$	−2.39484	−0.0940056
$650$	0	0
$651$	20.0032	0.783988
$652$	0	0
$653$	16.8297	0.658595	0.329298	−	0.944226i	$-0.393188\pi$
$653$	16.8297	0.658595	0.329298	+	0.944226i	$0.393188\pi$
$654$	0	0
$655$	13.8958	0.542954
$656$	0	0
$657$	−8.86018	−0.345668
$658$	0	0
$659$	6.28973	0.245013	0.122507	−	0.992468i	$-0.460907\pi$
$659$	6.28973	0.245013	0.122507	+	0.992468i	$0.460907\pi$
$660$	0	0
$661$	22.2912	0.867026	0.433513	−	0.901147i	$-0.357274\pi$
$661$	22.2912	0.867026	0.433513	+	0.901147i	$0.357274\pi$
$662$	0	0
$663$	8.47149	0.329005
$664$	0	0
$665$	8.63785	0.334961
$666$	0	0
$667$	4.27407	0.165493
$668$	0	0
$669$	0.967596	0.0374094
$670$	0	0
$671$	24.0386	0.928002
$672$	0	0
$673$	−15.4922	−0.597180	−0.298590	−	0.954382i	$-0.596516\pi$
$673$	−15.4922	−0.597180	−0.298590	+	0.954382i	$0.596516\pi$
$674$	0	0
$675$	−3.25206	−0.125172
$676$	0	0
$677$	−26.4012	−1.01468	−0.507340	−	0.861746i	$-0.669371\pi$
$677$	−26.4012	−1.01468	−0.507340	+	0.861746i	$0.669371\pi$
$678$	0	0
$679$	−12.2625	−0.470593
$680$	0	0
$681$	17.5749	0.673472
$682$	0	0
$683$	−36.5118	−1.39708	−0.698542	−	0.715569i	$-0.746167\pi$
$683$	−36.5118	−1.39708	−0.698542	+	0.715569i	$0.746167\pi$
$684$	0	0
$685$	11.1181	0.424803
$686$	0	0
$687$	15.0409	0.573845
$688$	0	0
$689$	17.4321	0.664111
$690$	0	0
$691$	−43.8047	−1.66641	−0.833204	−	0.552965i	$-0.813496\pi$
$691$	−43.8047	−1.66641	−0.833204	+	0.552965i	$0.813496\pi$
$692$	0	0
$693$	−3.22129	−0.122367
$694$	0	0
$695$	20.2210	0.767026
$696$	0	0
$697$	42.6269	1.61461
$698$	0	0
$699$	−21.9683	−0.830918
$700$	0	0
$701$	6.80240	0.256923	0.128461	−	0.991715i	$-0.458996\pi$
$701$	6.80240	0.256923	0.128461	+	0.991715i	$0.458996\pi$
$702$	0	0
$703$	1.18547	0.0447108
$704$	0	0
$705$	5.91790	0.222881
$706$	0	0
$707$	27.8689	1.04812
$708$	0	0
$709$	−11.9384	−0.448355	−0.224177	−	0.974548i	$-0.571970\pi$
$709$	−11.9384	−0.448355	−0.224177	+	0.974548i	$0.571970\pi$
$710$	0	0
$711$	0.706903	0.0265109
$712$	0	0
$713$	−8.40377	−0.314724
$714$	0	0
$715$	3.90258	0.145948
$716$	0	0
$717$	2.03201	0.0758868
$718$	0	0
$719$	48.4513	1.80693	0.903464	−	0.428664i	$-0.141016\pi$
$719$	48.4513	1.80693	0.903464	+	0.428664i	$0.141016\pi$
$720$	0	0
$721$	9.16184	0.341205
$722$	0	0
$723$	18.3591	0.682783
$724$	0	0
$725$	−17.1175	−0.635730
$726$	0	0
$727$	−1.03513	−0.0383907	−0.0191954	−	0.999816i	$-0.506110\pi$
$727$	−1.03513	−0.0383907	−0.0191954	+	0.999816i	$0.506110\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.8126	1.17663
$732$	0	0
$733$	−41.1013	−1.51811	−0.759056	−	0.651025i	$-0.774339\pi$
$733$	−41.1013	−1.51811	−0.759056	+	0.651025i	$0.774339\pi$
$734$	0	0
$735$	−4.31576	−0.159189
$736$	0	0
$737$	6.94053	0.255658
$738$	0	0
$739$	43.9415	1.61641	0.808207	−	0.588899i	$-0.200438\pi$
$739$	43.9415	1.61641	0.808207	+	0.588899i	$0.200438\pi$
$740$	0	0
$741$	−5.98687	−0.219933
$742$	0	0
$743$	−25.6425	−0.940731	−0.470365	−	0.882472i	$-0.655878\pi$
$743$	−25.6425	−0.940731	−0.470365	+	0.882472i	$0.655878\pi$
$744$	0	0
$745$	6.02522	0.220747
$746$	0	0
$747$	−5.78094	−0.211514
$748$	0	0
$749$	10.0059	0.365608
$750$	0	0
$751$	2.65521	0.0968899	0.0484450	−	0.998826i	$-0.484573\pi$
$751$	2.65521	0.0968899	0.0484450	+	0.998826i	$0.484573\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−0.552992	−0.0201254
$756$	0	0
$757$	24.8495	0.903170	0.451585	−	0.892228i	$-0.350859\pi$
$757$	24.8495	0.903170	0.451585	+	0.892228i	$0.350859\pi$
$758$	0	0
$759$	1.35333	0.0491229
$760$	0	0
$761$	−37.4431	−1.35731	−0.678656	−	0.734457i	$-0.737437\pi$
$761$	−37.4431	−1.35731	−0.678656	+	0.734457i	$0.737437\pi$
$762$	0	0
$763$	−21.4816	−0.777687
$764$	0	0
$765$	6.32384	0.228639
$766$	0	0
$767$	−2.54491	−0.0918913
$768$	0	0
$769$	26.3670	0.950817	0.475409	−	0.879765i	$-0.342300\pi$
$769$	26.3670	0.950817	0.475409	+	0.879765i	$0.342300\pi$
$770$	0	0
$771$	−23.1750	−0.834626
$772$	0	0
$773$	−25.7856	−0.927444	−0.463722	−	0.885981i	$-0.653486\pi$
$773$	−25.7856	−0.927444	−0.463722	+	0.885981i	$0.653486\pi$
$774$	0	0
$775$	33.6569	1.20899
$776$	0	0
$777$	0.677824	0.0243168
$778$	0	0
$779$	−30.1248	−1.07933
$780$	0	0
$781$	20.7348	0.741948
$782$	0	0
$783$	5.26360	0.188106
$784$	0	0
$785$	24.1941	0.863526
$786$	0	0
$787$	10.6994	0.381392	0.190696	−	0.981649i	$-0.438926\pi$
$787$	10.6994	0.381392	0.190696	+	0.981649i	$0.438926\pi$
$788$	0	0
$789$	22.5592	0.803128
$790$	0	0
$791$	−10.6396	−0.378300
$792$	0	0
$793$	25.5450	0.907130
$794$	0	0
$795$	13.0128	0.461518
$796$	0	0
$797$	38.3024	1.35674	0.678371	−	0.734720i	$-0.262686\pi$
$797$	38.3024	1.35674	0.678371	+	0.734720i	$0.262686\pi$
$798$	0	0
$799$	21.4103	0.757441
$800$	0	0
$801$	−2.66382	−0.0941214
$802$	0	0
$803$	−14.7669	−0.521111
$804$	0	0
$805$	−2.07495	−0.0731322
$806$	0	0
$807$	−0.155016	−0.00545683
$808$	0	0
$809$	−3.40961	−0.119875	−0.0599377	−	0.998202i	$-0.519090\pi$
$809$	−3.40961	−0.119875	−0.0599377	+	0.998202i	$0.519090\pi$
$810$	0	0
$811$	−47.0083	−1.65068	−0.825342	−	0.564633i	$-0.809018\pi$
$811$	−47.0083	−1.65068	−0.825342	+	0.564633i	$0.809018\pi$
$812$	0	0
$813$	−15.3346	−0.537809
$814$	0	0
$815$	−28.2649	−0.990076
$816$	0	0
$817$	−22.4822	−0.786554
$818$	0	0
$819$	−3.42315	−0.119615
$820$	0	0
$821$	−37.3958	−1.30512	−0.652562	−	0.757736i	$-0.726306\pi$
$821$	−37.3958	−1.30512	−0.652562	+	0.757736i	$0.726306\pi$
$822$	0	0
$823$	−45.3275	−1.58002	−0.790010	−	0.613095i	$-0.789924\pi$
$823$	−45.3275	−1.58002	−0.790010	+	0.613095i	$0.789924\pi$
$824$	0	0
$825$	−5.42006	−0.188702
$826$	0	0
$827$	−13.1051	−0.455709	−0.227854	−	0.973695i	$-0.573171\pi$
$827$	−13.1051	−0.455709	−0.227854	+	0.973695i	$0.573171\pi$
$828$	0	0
$829$	−26.7113	−0.927723	−0.463862	−	0.885908i	$-0.653537\pi$
$829$	−26.7113	−0.927723	−0.463862	+	0.885908i	$0.653537\pi$
$830$	0	0
$831$	−12.4922	−0.433349
$832$	0	0
$833$	−15.6139	−0.540990
$834$	0	0
$835$	10.7819	0.373125
$836$	0	0
$837$	−10.3494	−0.357728
$838$	0	0
$839$	−6.43873	−0.222290	−0.111145	−	0.993804i	$-0.535452\pi$
$839$	−6.43873	−0.222290	−0.111145	+	0.993804i	$0.535452\pi$
$840$	0	0
$841$	−1.29448	−0.0446374
$842$	0	0
$843$	−0.889653	−0.0306413
$844$	0	0
$845$	−13.0401	−0.448594
$846$	0	0
$847$	15.8919	0.546052
$848$	0	0
$849$	7.94051	0.272518
$850$	0	0
$851$	−0.284768	−0.00976173
$852$	0	0
$853$	−45.5846	−1.56079	−0.780394	−	0.625289i	$-0.784981\pi$
$853$	−45.5846	−1.56079	−0.780394	+	0.625289i	$0.784981\pi$
$854$	0	0
$855$	−4.46911	−0.152840
$856$	0	0
$857$	−25.5630	−0.873215	−0.436608	−	0.899652i	$-0.643820\pi$
$857$	−25.5630	−0.873215	−0.436608	+	0.899652i	$0.643820\pi$
$858$	0	0
$859$	9.62934	0.328549	0.164274	−	0.986415i	$-0.447472\pi$
$859$	9.62934	0.328549	0.164274	+	0.986415i	$0.447472\pi$
$860$	0	0
$861$	−17.2246	−0.587014
$862$	0	0
$863$	35.2448	1.19975	0.599874	−	0.800094i	$-0.295217\pi$
$863$	35.2448	1.19975	0.599874	+	0.800094i	$0.295217\pi$
$864$	0	0
$865$	3.60271	0.122496
$866$	0	0
$867$	5.87893	0.199659
$868$	0	0
$869$	1.17816	0.0399665
$870$	0	0
$871$	7.37545	0.249908
$872$	0	0
$873$	6.34447	0.214728
$874$	0	0
$875$	21.0868	0.712863
$876$	0	0
$877$	38.3883	1.29628	0.648140	−	0.761522i	$-0.275547\pi$
$877$	38.3883	1.29628	0.648140	+	0.761522i	$0.275547\pi$
$878$	0	0
$879$	26.9991	0.910655
$880$	0	0
$881$	20.1378	0.678459	0.339229	−	0.940704i	$-0.389834\pi$
$881$	20.1378	0.678459	0.339229	+	0.940704i	$0.389834\pi$
$882$	0	0
$883$	−39.6513	−1.33437	−0.667187	−	0.744891i	$-0.732502\pi$
$883$	−39.6513	−1.33437	−0.667187	+	0.744891i	$0.732502\pi$
$884$	0	0
$885$	−1.89974	−0.0638589
$886$	0	0
$887$	−47.0425	−1.57953	−0.789766	−	0.613408i	$-0.789798\pi$
$887$	−47.0425	−1.57953	−0.789766	+	0.613408i	$0.789798\pi$
$888$	0	0
$889$	17.3538	0.582028
$890$	0	0
$891$	1.66666	0.0558351
$892$	0	0
$893$	−15.1308	−0.506333
$894$	0	0
$895$	0.0441522	0.00147584
$896$	0	0
$897$	1.43814	0.0480181
$898$	0	0
$899$	−54.4752	−1.81685
$900$	0	0
$901$	47.0789	1.56843
$902$	0	0
$903$	−12.8548	−0.427782
$904$	0	0
$905$	−6.54734	−0.217641
$906$	0	0
$907$	−9.82104	−0.326102	−0.163051	−	0.986618i	$-0.552134\pi$
$907$	−9.82104	−0.326102	−0.163051	+	0.986618i	$0.552134\pi$
$908$	0	0
$909$	−14.4190	−0.478248
$910$	0	0
$911$	−36.9127	−1.22297	−0.611486	−	0.791255i	$-0.709428\pi$
$911$	−36.9127	−1.22297	−0.611486	+	0.791255i	$0.709428\pi$
$912$	0	0
$913$	−9.63484	−0.318867
$914$	0	0
$915$	19.0690	0.630401
$916$	0	0
$917$	−20.3144	−0.670842
$918$	0	0
$919$	−5.38703	−0.177702	−0.0888509	−	0.996045i	$-0.528319\pi$
$919$	−5.38703	−0.177702	−0.0888509	+	0.996045i	$0.528319\pi$
$920$	0	0
$921$	10.9588	0.361106
$922$	0	0
$923$	22.0341	0.725261
$924$	0	0
$925$	1.14049	0.0374990
$926$	0	0
$927$	−4.74022	−0.155689
$928$	0	0
$929$	−16.7402	−0.549229	−0.274615	−	0.961554i	$-0.588550\pi$
$929$	−16.7402	−0.549229	−0.274615	+	0.961554i	$0.588550\pi$
$930$	0	0
$931$	11.0345	0.361640
$932$	0	0
$933$	−4.76672	−0.156055
$934$	0	0
$935$	10.5397	0.344684
$936$	0	0
$937$	−19.2799	−0.629848	−0.314924	−	0.949117i	$-0.601979\pi$
$937$	−19.2799	−0.629848	−0.314924	+	0.949117i	$0.601979\pi$
$938$	0	0
$939$	−11.6334	−0.379641
$940$	0	0
$941$	43.3895	1.41446	0.707228	−	0.706985i	$-0.249945\pi$
$941$	43.3895	1.41446	0.707228	+	0.706985i	$0.249945\pi$
$942$	0	0
$943$	7.23644	0.235651
$944$	0	0
$945$	−2.55533	−0.0831250
$946$	0	0
$947$	−46.1881	−1.50091	−0.750456	−	0.660921i	$-0.770166\pi$
$947$	−46.1881	−1.50091	−0.750456	+	0.660921i	$0.770166\pi$
$948$	0	0
$949$	−15.6922	−0.509391
$950$	0	0
$951$	17.7913	0.576924
$952$	0	0
$953$	−22.1935	−0.718919	−0.359459	−	0.933161i	$-0.617039\pi$
$953$	−22.1935	−0.718919	−0.359459	+	0.933161i	$0.617039\pi$
$954$	0	0
$955$	17.1643	0.555425
$956$	0	0
$957$	8.77261	0.283578
$958$	0	0
$959$	−16.2537	−0.524861
$960$	0	0
$961$	76.1102	2.45517
$962$	0	0
$963$	−5.17693	−0.166824
$964$	0	0
$965$	8.59738	0.276759
$966$	0	0
$967$	12.5322	0.403008	0.201504	−	0.979488i	$-0.435417\pi$
$967$	12.5322	0.403008	0.201504	+	0.979488i	$0.435417\pi$
$968$	0	0
$969$	−16.1687	−0.519414
$970$	0	0
$971$	−20.0017	−0.641885	−0.320942	−	0.947099i	$-0.604000\pi$
$971$	−20.0017	−0.641885	−0.320942	+	0.947099i	$0.604000\pi$
$972$	0	0
$973$	−29.5613	−0.947692
$974$	0	0
$975$	−5.75971	−0.184458
$976$	0	0
$977$	51.1686	1.63703	0.818514	−	0.574487i	$-0.194798\pi$
$977$	51.1686	1.63703	0.818514	+	0.574487i	$0.194798\pi$
$978$	0	0
$979$	−4.43967	−0.141892
$980$	0	0
$981$	11.1143	0.354853
$982$	0	0
$983$	45.1780	1.44095	0.720477	−	0.693479i	$-0.243923\pi$
$983$	45.1780	1.44095	0.720477	+	0.693479i	$0.243923\pi$
$984$	0	0
$985$	18.1505	0.578323
$986$	0	0
$987$	−8.65145	−0.275379
$988$	0	0
$989$	5.40058	0.171729
$990$	0	0
$991$	−24.7370	−0.785797	−0.392899	−	0.919582i	$-0.628528\pi$
$991$	−24.7370	−0.785797	−0.392899	+	0.919582i	$0.628528\pi$
$992$	0	0
$993$	1.55558	0.0493650
$994$	0	0
$995$	16.7041	0.529557
$996$	0	0
$997$	−53.5696	−1.69657	−0.848284	−	0.529542i	$-0.822364\pi$
$997$	−53.5696	−1.69657	−0.848284	+	0.529542i	$0.822364\pi$
$998$	0	0
$999$	−0.350697	−0.0110956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.12	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.32210 q^{5} -1.93279 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.32210 q^{5} -1.93279 q^{7} +1.00000 q^{9} +1.66666 q^{11} +1.77110 q^{13} +1.32210 q^{15} +4.78319 q^{17} -3.38032 q^{19} -1.93279 q^{21} +0.812005 q^{23} -3.25206 q^{25} +1.00000 q^{27} +5.26360 q^{29} -10.3494 q^{31} +1.66666 q^{33} -2.55533 q^{35} -0.350697 q^{37} +1.77110 q^{39} +8.91181 q^{41} +6.65092 q^{43} +1.32210 q^{45} +4.47615 q^{47} -3.26433 q^{49} +4.78319 q^{51} +9.84257 q^{53} +2.20348 q^{55} -3.38032 q^{57} -1.43691 q^{59} +14.4233 q^{61} -1.93279 q^{63} +2.34156 q^{65} +4.16434 q^{67} +0.812005 q^{69} +12.4409 q^{71} -8.86018 q^{73} -3.25206 q^{75} -3.22129 q^{77} +0.706903 q^{79} +1.00000 q^{81} -5.78094 q^{83} +6.32384 q^{85} +5.26360 q^{87} -2.66382 q^{89} -3.42315 q^{91} -10.3494 q^{93} -4.46911 q^{95} +6.34447 q^{97} +1.66666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 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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