LMFDB - Embedded newform 6012.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.161121.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 3x^2 + 5*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.544588$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.42860 q^{5} +1.88401 q^{7} +O(q^{10})$	$q+4.42860 q^{5} +1.88401 q^{7} -1.84527 q^{11} -5.16786 q^{13} -0.0891752 q^{17} +6.45461 q^{19} +3.21151 q^{23} +14.6125 q^{25} -3.02095 q^{29} -10.4636 q^{31} +8.34353 q^{35} -1.51367 q^{37} +7.82628 q^{41} -7.61344 q^{43} +11.5711 q^{47} -3.45050 q^{49} +2.84132 q^{53} -8.17196 q^{55} +5.92036 q^{59} +4.29853 q^{61} -22.8864 q^{65} +13.3364 q^{67} +11.9240 q^{71} -0.174243 q^{73} -3.47651 q^{77} +2.45661 q^{79} +11.2161 q^{83} -0.394921 q^{85} +18.3673 q^{89} -9.73630 q^{91} +28.5849 q^{95} -3.13202 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 7 * q^5 - 2 * q^7	$ 5 q + 7 q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 7 q^{17} + 2 q^{19} + 13 q^{23} + 2 q^{25} + 11 q^{29} - 12 q^{31} + 12 q^{35} - 7 q^{37} + 12 q^{41} + 19 q^{47} - 9 q^{49} + 21 q^{53} - q^{55} + 7 q^{59} - 6 q^{61} - 14 q^{65} + 10 q^{67} + 35 q^{71} - 8 q^{73} + 6 q^{77} + 11 q^{83} + 5 q^{85} + 32 q^{89} + 5 q^{91} + 19 q^{95} + 11 q^{97}+O(q^{100}) $ 5 * q + 7 * q^5 - 2 * q^7 + 5 * q^11 - 8 * q^13 + 7 * q^17 + 2 * q^19 + 13 * q^23 + 2 * q^25 + 11 * q^29 - 12 * q^31 + 12 * q^35 - 7 * q^37 + 12 * q^41 + 19 * q^47 - 9 * q^49 + 21 * q^53 - q^55 + 7 * q^59 - 6 * q^61 - 14 * q^65 + 10 * q^67 + 35 * q^71 - 8 * q^73 + 6 * q^77 + 11 * q^83 + 5 * q^85 + 32 * q^89 + 5 * q^91 + 19 * q^95 + 11 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.42860	1.98053	0.990265	−	0.139196i	$-0.0444518\pi$
$5$	4.42860	1.98053	0.990265	+	0.139196i	$0.0444518\pi$
$6$	0	0
$7$	1.88401	0.712089	0.356045	−	0.934469i	$-0.384125\pi$
$7$	1.88401	0.712089	0.356045	+	0.934469i	$0.384125\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.84527	−0.556370	−0.278185	−	0.960528i	$-0.589733\pi$
$11$	−1.84527	−0.556370	−0.278185	+	0.960528i	$0.589733\pi$
$12$	0	0
$13$	−5.16786	−1.43331	−0.716653	−	0.697430i	$-0.754327\pi$
$13$	−5.16786	−1.43331	−0.716653	+	0.697430i	$0.754327\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.0891752	−0.0216282	−0.0108141	−	0.999942i	$-0.503442\pi$
$17$	−0.0891752	−0.0216282	−0.0108141	+	0.999942i	$0.503442\pi$
$18$	0	0
$19$	6.45461	1.48079	0.740394	−	0.672173i	$-0.234639\pi$
$19$	6.45461	1.48079	0.740394	+	0.672173i	$0.234639\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.21151	0.669646	0.334823	−	0.942281i	$-0.391324\pi$
$23$	3.21151	0.669646	0.334823	+	0.942281i	$0.391324\pi$
$24$	0	0
$25$	14.6125	2.92250
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.02095	−0.560976	−0.280488	−	0.959858i	$-0.590496\pi$
$29$	−3.02095	−0.560976	−0.280488	+	0.959858i	$0.590496\pi$
$30$	0	0
$31$	−10.4636	−1.87932	−0.939661	−	0.342106i	$-0.888860\pi$
$31$	−10.4636	−1.87932	−0.939661	+	0.342106i	$0.888860\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.34353	1.41031
$36$	0	0
$37$	−1.51367	−0.248845	−0.124423	−	0.992229i	$-0.539708\pi$
$37$	−1.51367	−0.248845	−0.124423	+	0.992229i	$0.539708\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.82628	1.22226	0.611130	−	0.791531i	$-0.290715\pi$
$41$	7.82628	1.22226	0.611130	+	0.791531i	$0.290715\pi$
$42$	0	0
$43$	−7.61344	−1.16104	−0.580520	−	0.814246i	$-0.697151\pi$
$43$	−7.61344	−1.16104	−0.580520	+	0.814246i	$0.697151\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.5711	1.68782	0.843910	−	0.536484i	$-0.180248\pi$
$47$	11.5711	1.68782	0.843910	+	0.536484i	$0.180248\pi$
$48$	0	0
$49$	−3.45050	−0.492929
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.84132	0.390285	0.195142	−	0.980775i	$-0.437483\pi$
$53$	2.84132	0.390285	0.195142	+	0.980775i	$0.437483\pi$
$54$	0	0
$55$	−8.17196	−1.10191
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.92036	0.770766	0.385383	−	0.922757i	$-0.374069\pi$
$59$	5.92036	0.770766	0.385383	+	0.922757i	$0.374069\pi$
$60$	0	0
$61$	4.29853	0.550370	0.275185	−	0.961391i	$-0.411261\pi$
$61$	4.29853	0.550370	0.275185	+	0.961391i	$0.411261\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.8864	−2.83870
$66$	0	0
$67$	13.3364	1.62930	0.814649	−	0.579954i	$-0.196929\pi$
$67$	13.3364	1.62930	0.814649	+	0.579954i	$0.196929\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.9240	1.41511	0.707556	−	0.706657i	$-0.249798\pi$
$71$	11.9240	1.41511	0.707556	+	0.706657i	$0.249798\pi$
$72$	0	0
$73$	−0.174243	−0.0203936	−0.0101968	−	0.999948i	$-0.503246\pi$
$73$	−0.174243	−0.0203936	−0.0101968	+	0.999948i	$0.503246\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.47651	−0.396185
$78$	0	0
$79$	2.45661	0.276390	0.138195	−	0.990405i	$-0.455870\pi$
$79$	2.45661	0.276390	0.138195	+	0.990405i	$0.455870\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.2161	1.23113	0.615565	−	0.788086i	$-0.288928\pi$
$83$	11.2161	1.23113	0.615565	+	0.788086i	$0.288928\pi$
$84$	0	0
$85$	−0.394921	−0.0428352
$86$	0	0
$87$	0	0
$88$	0	0
$89$	18.3673	1.94693	0.973465	−	0.228835i	$-0.0734918\pi$
$89$	18.3673	1.94693	0.973465	+	0.228835i	$0.0734918\pi$
$90$	0	0
$91$	−9.73630	−1.02064
$92$	0	0
$93$	0	0
$94$	0	0
$95$	28.5849	2.93275
$96$	0	0
$97$	−3.13202	−0.318009	−0.159004	−	0.987278i	$-0.550828\pi$
$97$	−3.13202	−0.318009	−0.159004	+	0.987278i	$0.550828\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.76183	−0.175309	−0.0876544	−	0.996151i	$-0.527937\pi$
$101$	−1.76183	−0.175309	−0.0876544	+	0.996151i	$0.527937\pi$
$102$	0	0
$103$	9.99153	0.984494	0.492247	−	0.870455i	$-0.336176\pi$
$103$	9.99153	0.984494	0.492247	+	0.870455i	$0.336176\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.65486	−0.353329	−0.176664	−	0.984271i	$-0.556531\pi$
$107$	−3.65486	−0.353329	−0.176664	+	0.984271i	$0.556531\pi$
$108$	0	0
$109$	8.58240	0.822045	0.411022	−	0.911625i	$-0.365172\pi$
$109$	8.58240	0.822045	0.411022	+	0.911625i	$0.365172\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.92953	−0.651876	−0.325938	−	0.945391i	$-0.605680\pi$
$113$	−6.92953	−0.651876	−0.325938	+	0.945391i	$0.605680\pi$
$114$	0	0
$115$	14.2225	1.32625
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.168007	−0.0154012
$120$	0	0
$121$	−7.59498	−0.690452
$122$	0	0
$123$	0	0
$124$	0	0
$125$	42.5699	3.80756
$126$	0	0
$127$	−12.7672	−1.13291	−0.566454	−	0.824094i	$-0.691685\pi$
$127$	−12.7672	−1.13291	−0.566454	+	0.824094i	$0.691685\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.75251	−0.153117	−0.0765587	−	0.997065i	$-0.524393\pi$
$131$	−1.75251	−0.153117	−0.0765587	+	0.997065i	$0.524393\pi$
$132$	0	0
$133$	12.1606	1.05445
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.7440	−1.00335	−0.501677	−	0.865055i	$-0.667284\pi$
$137$	−11.7440	−1.00335	−0.501677	+	0.865055i	$0.667284\pi$
$138$	0	0
$139$	−12.8695	−1.09158	−0.545789	−	0.837923i	$-0.683770\pi$
$139$	−12.8695	−1.09158	−0.545789	+	0.837923i	$0.683770\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.53609	0.797448
$144$	0	0
$145$	−13.3786	−1.11103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.74986	0.716816	0.358408	−	0.933565i	$-0.383320\pi$
$149$	8.74986	0.716816	0.358408	+	0.933565i	$0.383320\pi$
$150$	0	0
$151$	0.761158	0.0619422	0.0309711	−	0.999520i	$-0.490140\pi$
$151$	0.761158	0.0619422	0.0309711	+	0.999520i	$0.490140\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−46.3392	−3.72205
$156$	0	0
$157$	22.5093	1.79643	0.898217	−	0.439552i	$-0.144863\pi$
$157$	22.5093	1.79643	0.898217	+	0.439552i	$0.144863\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.05052	0.476848
$162$	0	0
$163$	18.9531	1.48452	0.742260	−	0.670112i	$-0.233754\pi$
$163$	18.9531	1.48452	0.742260	+	0.670112i	$0.233754\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	13.7067	1.05436
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.76587	0.590428	0.295214	−	0.955431i	$-0.404609\pi$
$173$	7.76587	0.590428	0.295214	+	0.955431i	$0.404609\pi$
$174$	0	0
$175$	27.5301	2.08108
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.63802	0.346662	0.173331	−	0.984864i	$-0.444547\pi$
$179$	4.63802	0.346662	0.173331	+	0.984864i	$0.444547\pi$
$180$	0	0
$181$	−11.9653	−0.889375	−0.444688	−	0.895686i	$-0.646685\pi$
$181$	−11.9653	−0.889375	−0.444688	+	0.895686i	$0.646685\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.70342	−0.492846
$186$	0	0
$187$	0.164552	0.0120333
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.49863	−0.180794	−0.0903972	−	0.995906i	$-0.528814\pi$
$191$	−2.49863	−0.180794	−0.0903972	+	0.995906i	$0.528814\pi$
$192$	0	0
$193$	−19.6727	−1.41607	−0.708036	−	0.706176i	$-0.750419\pi$
$193$	−19.6727	−1.41607	−0.708036	+	0.706176i	$0.750419\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.0553	−1.50012	−0.750062	−	0.661367i	$-0.769977\pi$
$197$	−21.0553	−1.50012	−0.750062	+	0.661367i	$0.769977\pi$
$198$	0	0
$199$	−1.30427	−0.0924572	−0.0462286	−	0.998931i	$-0.514720\pi$
$199$	−1.30427	−0.0924572	−0.0462286	+	0.998931i	$0.514720\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.69150	−0.399465
$204$	0	0
$205$	34.6594	2.42072
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.9105	−0.823866
$210$	0	0
$211$	−18.2559	−1.25679	−0.628395	−	0.777894i	$-0.716288\pi$
$211$	−18.2559	−1.25679	−0.628395	+	0.777894i	$0.716288\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−33.7169	−2.29947
$216$	0	0
$217$	−19.7136	−1.33825
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.460844	0.0309997
$222$	0	0
$223$	7.16390	0.479730	0.239865	−	0.970806i	$-0.422897\pi$
$223$	7.16390	0.479730	0.239865	+	0.970806i	$0.422897\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.58453	−0.304286	−0.152143	−	0.988359i	$-0.548617\pi$
$227$	−4.58453	−0.304286	−0.152143	+	0.988359i	$0.548617\pi$
$228$	0	0
$229$	−8.62848	−0.570186	−0.285093	−	0.958500i	$-0.592025\pi$
$229$	−8.62848	−0.570186	−0.285093	+	0.958500i	$0.592025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.3435	−0.743139	−0.371570	−	0.928405i	$-0.621180\pi$
$233$	−11.3435	−0.743139	−0.371570	+	0.928405i	$0.621180\pi$
$234$	0	0
$235$	51.2438	3.34278
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.95726	0.514712	0.257356	−	0.966317i	$-0.417149\pi$
$239$	7.95726	0.514712	0.257356	+	0.966317i	$0.417149\pi$
$240$	0	0
$241$	−24.9868	−1.60954	−0.804770	−	0.593586i	$-0.797712\pi$
$241$	−24.9868	−1.60954	−0.804770	+	0.593586i	$0.797712\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−15.2809	−0.976260
$246$	0	0
$247$	−33.3565	−2.12242
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.7487	0.804689	0.402344	−	0.915488i	$-0.368195\pi$
$251$	12.7487	0.804689	0.402344	+	0.915488i	$0.368195\pi$
$252$	0	0
$253$	−5.92610	−0.372571
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.1781	−0.884404	−0.442202	−	0.896916i	$-0.645803\pi$
$257$	−14.1781	−0.884404	−0.442202	+	0.896916i	$0.645803\pi$
$258$	0	0
$259$	−2.85177	−0.177200
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.65792	0.472208	0.236104	−	0.971728i	$-0.424129\pi$
$263$	7.65792	0.472208	0.236104	+	0.971728i	$0.424129\pi$
$264$	0	0
$265$	12.5830	0.772970
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.2060	−1.17101	−0.585506	−	0.810668i	$-0.699104\pi$
$269$	−19.2060	−1.17101	−0.585506	+	0.810668i	$0.699104\pi$
$270$	0	0
$271$	−0.299771	−0.0182098	−0.00910489	−	0.999959i	$-0.502898\pi$
$271$	−0.299771	−0.0182098	−0.00910489	+	0.999959i	$0.502898\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−26.9640	−1.62599
$276$	0	0
$277$	−11.4831	−0.689955	−0.344978	−	0.938611i	$-0.612113\pi$
$277$	−11.4831	−0.689955	−0.344978	+	0.938611i	$0.612113\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.8670	−0.648274	−0.324137	−	0.946010i	$-0.605074\pi$
$281$	−10.8670	−0.648274	−0.324137	+	0.946010i	$0.605074\pi$
$282$	0	0
$283$	9.59764	0.570521	0.285260	−	0.958450i	$-0.407920\pi$
$283$	9.59764	0.570521	0.285260	+	0.958450i	$0.407920\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.7448	0.870358
$288$	0	0
$289$	−16.9920	−0.999532
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−31.9508	−1.86659	−0.933294	−	0.359114i	$-0.883079\pi$
$293$	−31.9508	−1.86659	−0.933294	+	0.359114i	$0.883079\pi$
$294$	0	0
$295$	26.2189	1.52652
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.5966	−0.959807
$300$	0	0
$301$	−14.3438	−0.826764
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.0365	1.09002
$306$	0	0
$307$	11.5965	0.661847	0.330924	−	0.943658i	$-0.392640\pi$
$307$	11.5965	0.661847	0.330924	+	0.943658i	$0.392640\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.68667	−0.322461	−0.161231	−	0.986917i	$-0.551546\pi$
$311$	−5.68667	−0.322461	−0.161231	+	0.986917i	$0.551546\pi$
$312$	0	0
$313$	6.50704	0.367800	0.183900	−	0.982945i	$-0.441128\pi$
$313$	6.50704	0.367800	0.183900	+	0.982945i	$0.441128\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.1953	−1.30278	−0.651390	−	0.758743i	$-0.725814\pi$
$317$	−23.1953	−1.30278	−0.651390	+	0.758743i	$0.725814\pi$
$318$	0	0
$319$	5.57446	0.312110
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.575591	−0.0320267
$324$	0	0
$325$	−75.5152	−4.18883
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.8001	1.20188
$330$	0	0
$331$	−19.7232	−1.08409	−0.542044	−	0.840350i	$-0.682349\pi$
$331$	−19.7232	−1.08409	−0.542044	+	0.840350i	$0.682349\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	59.0615	3.22687
$336$	0	0
$337$	−11.8567	−0.645874	−0.322937	−	0.946420i	$-0.604670\pi$
$337$	−11.8567	−0.645874	−0.322937	+	0.946420i	$0.604670\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.3082	1.04560
$342$	0	0
$343$	−19.6889	−1.06310
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.4703	0.937857	0.468928	−	0.883236i	$-0.344640\pi$
$347$	17.4703	0.937857	0.468928	+	0.883236i	$0.344640\pi$
$348$	0	0
$349$	30.2137	1.61730	0.808650	−	0.588290i	$-0.200199\pi$
$349$	30.2137	1.61730	0.808650	+	0.588290i	$0.200199\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	23.6819	1.26046	0.630229	−	0.776409i	$-0.282961\pi$
$353$	23.6819	1.26046	0.630229	+	0.776409i	$0.282961\pi$
$354$	0	0
$355$	52.8064	2.80267
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.7084	−1.14573	−0.572863	−	0.819651i	$-0.694167\pi$
$359$	−21.7084	−1.14573	−0.572863	+	0.819651i	$0.694167\pi$
$360$	0	0
$361$	22.6620	1.19274
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.771653	−0.0403902
$366$	0	0
$367$	30.4853	1.59132	0.795661	−	0.605742i	$-0.207124\pi$
$367$	30.4853	1.59132	0.795661	+	0.605742i	$0.207124\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.35307	0.277918
$372$	0	0
$373$	28.3119	1.46593	0.732967	−	0.680264i	$-0.238135\pi$
$373$	28.3119	1.46593	0.732967	+	0.680264i	$0.238135\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.6118	0.804049
$378$	0	0
$379$	22.0469	1.13247	0.566237	−	0.824243i	$-0.308399\pi$
$379$	22.0469	1.13247	0.566237	+	0.824243i	$0.308399\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.3210	−1.03835	−0.519177	−	0.854667i	$-0.673761\pi$
$383$	−20.3210	−1.03835	−0.519177	+	0.854667i	$0.673761\pi$
$384$	0	0
$385$	−15.3961	−0.784656
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.81405	0.294784	0.147392	−	0.989078i	$-0.452912\pi$
$389$	5.81405	0.294784	0.147392	+	0.989078i	$0.452912\pi$
$390$	0	0
$391$	−0.286387	−0.0144832
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.8793	0.547399
$396$	0	0
$397$	−25.3084	−1.27019	−0.635096	−	0.772433i	$-0.719039\pi$
$397$	−25.3084	−1.27019	−0.635096	+	0.772433i	$0.719039\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.2771	1.06253	0.531265	−	0.847206i	$-0.321717\pi$
$401$	21.2771	1.06253	0.531265	+	0.847206i	$0.321717\pi$
$402$	0	0
$403$	54.0745	2.69364
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.79312	0.138450
$408$	0	0
$409$	−39.4470	−1.95053	−0.975263	−	0.221047i	$-0.929053\pi$
$409$	−39.4470	−1.95053	−0.975263	+	0.221047i	$0.929053\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.1540	0.548854
$414$	0	0
$415$	49.6718	2.43829
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.0752	−0.736474	−0.368237	−	0.929732i	$-0.620038\pi$
$419$	−15.0752	−0.736474	−0.368237	+	0.929732i	$0.620038\pi$
$420$	0	0
$421$	−20.2233	−0.985624	−0.492812	−	0.870136i	$-0.664031\pi$
$421$	−20.2233	−0.985624	−0.492812	+	0.870136i	$0.664031\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.30307	−0.0632082
$426$	0	0
$427$	8.09848	0.391913
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.4054	−1.56091	−0.780456	−	0.625211i	$-0.785013\pi$
$431$	−32.4054	−1.56091	−0.780456	+	0.625211i	$0.785013\pi$
$432$	0	0
$433$	−12.6597	−0.608387	−0.304194	−	0.952610i	$-0.598387\pi$
$433$	−12.6597	−0.608387	−0.304194	+	0.952610i	$0.598387\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.7290	0.991604
$438$	0	0
$439$	33.4985	1.59879	0.799397	−	0.600803i	$-0.205152\pi$
$439$	33.4985	1.59879	0.799397	+	0.600803i	$0.205152\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.7875	1.22520	0.612601	−	0.790392i	$-0.290123\pi$
$443$	25.7875	1.22520	0.612601	+	0.790392i	$0.290123\pi$
$444$	0	0
$445$	81.3414	3.85595
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.134437	−0.00634446	−0.00317223	−	0.999995i	$-0.501010\pi$
$449$	−0.134437	−0.00634446	−0.00317223	+	0.999995i	$0.501010\pi$
$450$	0	0
$451$	−14.4416	−0.680028
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−43.1182	−2.02141
$456$	0	0
$457$	−10.7011	−0.500578	−0.250289	−	0.968171i	$-0.580526\pi$
$457$	−10.7011	−0.500578	−0.250289	+	0.968171i	$0.580526\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.23604	0.290441	0.145221	−	0.989399i	$-0.453611\pi$
$461$	6.23604	0.290441	0.145221	+	0.989399i	$0.453611\pi$
$462$	0	0
$463$	−25.0329	−1.16338	−0.581689	−	0.813411i	$-0.697608\pi$
$463$	−25.0329	−1.16338	−0.581689	+	0.813411i	$0.697608\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.25511	0.289452	0.144726	−	0.989472i	$-0.453770\pi$
$467$	6.25511	0.289452	0.144726	+	0.989472i	$0.453770\pi$
$468$	0	0
$469$	25.1259	1.16021
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.0489	0.645967
$474$	0	0
$475$	94.3179	4.32760
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.8726	1.82183	0.910913	−	0.412598i	$-0.135378\pi$
$479$	39.8726	1.82183	0.910913	+	0.412598i	$0.135378\pi$
$480$	0	0
$481$	7.82241	0.356671
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.8705	−0.629826
$486$	0	0
$487$	−3.81602	−0.172921	−0.0864603	−	0.996255i	$-0.527556\pi$
$487$	−3.81602	−0.172921	−0.0864603	+	0.996255i	$0.527556\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.4535	0.742535	0.371267	−	0.928526i	$-0.378923\pi$
$491$	16.4535	0.742535	0.371267	+	0.928526i	$0.378923\pi$
$492$	0	0
$493$	0.269393	0.0121329
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.4649	1.00769
$498$	0	0
$499$	14.3871	0.644057	0.322028	−	0.946730i	$-0.395635\pi$
$499$	14.3871	0.644057	0.322028	+	0.946730i	$0.395635\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−38.7487	−1.72772	−0.863860	−	0.503732i	$-0.831960\pi$
$503$	−38.7487	−1.72772	−0.863860	+	0.503732i	$0.831960\pi$
$504$	0	0
$505$	−7.80244	−0.347204
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.9491	−0.618282	−0.309141	−	0.951016i	$-0.600042\pi$
$509$	−13.9491	−0.618282	−0.309141	+	0.951016i	$0.600042\pi$
$510$	0	0
$511$	−0.328276	−0.0145221
$512$	0	0
$513$	0	0
$514$	0	0
$515$	44.2485	1.94982
$516$	0	0
$517$	−21.3518	−0.939053
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−34.7198	−1.52110	−0.760550	−	0.649279i	$-0.775071\pi$
$521$	−34.7198	−1.52110	−0.760550	+	0.649279i	$0.775071\pi$
$522$	0	0
$523$	−7.18919	−0.314362	−0.157181	−	0.987570i	$-0.550241\pi$
$523$	−7.18919	−0.314362	−0.157181	+	0.987570i	$0.550241\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.933096	0.0406463
$528$	0	0
$529$	−12.6862	−0.551575
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−40.4451	−1.75187
$534$	0	0
$535$	−16.1859	−0.699778
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.36711	0.274251
$540$	0	0
$541$	12.5805	0.540878	0.270439	−	0.962737i	$-0.412831\pi$
$541$	12.5805	0.540878	0.270439	+	0.962737i	$0.412831\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	38.0080	1.62808
$546$	0	0
$547$	1.54235	0.0659462	0.0329731	−	0.999456i	$-0.489502\pi$
$547$	1.54235	0.0659462	0.0329731	+	0.999456i	$0.489502\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.4990	−0.830686
$552$	0	0
$553$	4.62828	0.196815
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.4661	1.12140	0.560702	−	0.828017i	$-0.310531\pi$
$557$	26.4661	1.12140	0.560702	+	0.828017i	$0.310531\pi$
$558$	0	0
$559$	39.3452	1.66412
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−41.2657	−1.73914	−0.869572	−	0.493806i	$-0.835605\pi$
$563$	−41.2657	−1.73914	−0.869572	+	0.493806i	$0.835605\pi$
$564$	0	0
$565$	−30.6881	−1.29106
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.6241	1.03229	0.516147	−	0.856500i	$-0.327366\pi$
$569$	24.6241	1.03229	0.516147	+	0.856500i	$0.327366\pi$
$570$	0	0
$571$	−35.5529	−1.48784	−0.743921	−	0.668268i	$-0.767036\pi$
$571$	−35.5529	−1.48784	−0.743921	+	0.668268i	$0.767036\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	46.9281	1.95704
$576$	0	0
$577$	−13.6068	−0.566459	−0.283229	−	0.959052i	$-0.591406\pi$
$577$	−13.6068	−0.566459	−0.283229	+	0.959052i	$0.591406\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	21.1313	0.876675
$582$	0	0
$583$	−5.24299	−0.217143
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.46542	−0.184308	−0.0921538	−	0.995745i	$-0.529375\pi$
$587$	−4.46542	−0.184308	−0.0921538	+	0.995745i	$0.529375\pi$
$588$	0	0
$589$	−67.5386	−2.78288
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.7574	−1.46838	−0.734191	−	0.678943i	$-0.762438\pi$
$593$	−35.7574	−1.46838	−0.734191	+	0.678943i	$0.762438\pi$
$594$	0	0
$595$	−0.744036	−0.0305025
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.4065	−0.711210	−0.355605	−	0.934636i	$-0.615725\pi$
$599$	−17.4065	−0.711210	−0.355605	+	0.934636i	$0.615725\pi$
$600$	0	0
$601$	34.6763	1.41447	0.707237	−	0.706976i	$-0.249941\pi$
$601$	34.6763	1.41447	0.707237	+	0.706976i	$0.249941\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−33.6351	−1.36746
$606$	0	0
$607$	−22.9382	−0.931034	−0.465517	−	0.885039i	$-0.654132\pi$
$607$	−22.9382	−0.931034	−0.465517	+	0.885039i	$0.654132\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−59.7979	−2.41916
$612$	0	0
$613$	19.1586	0.773810	0.386905	−	0.922120i	$-0.373544\pi$
$613$	19.1586	0.773810	0.386905	+	0.922120i	$0.373544\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.9411	−0.722283	−0.361141	−	0.932511i	$-0.617613\pi$
$617$	−17.9411	−0.722283	−0.361141	+	0.932511i	$0.617613\pi$
$618$	0	0
$619$	−3.33510	−0.134049	−0.0670245	−	0.997751i	$-0.521351\pi$
$619$	−3.33510	−0.134049	−0.0670245	+	0.997751i	$0.521351\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	34.6042	1.38639
$624$	0	0
$625$	115.462	4.61850
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.134982	0.00538207
$630$	0	0
$631$	35.5155	1.41385	0.706925	−	0.707289i	$-0.250082\pi$
$631$	35.5155	1.41385	0.706925	+	0.707289i	$0.250082\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−56.5409	−2.24376
$636$	0	0
$637$	17.8317	0.706517
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.9112	−0.549461	−0.274730	−	0.961521i	$-0.588589\pi$
$641$	−13.9112	−0.549461	−0.274730	+	0.961521i	$0.588589\pi$
$642$	0	0
$643$	8.57110	0.338011	0.169006	−	0.985615i	$-0.445944\pi$
$643$	8.57110	0.338011	0.169006	+	0.985615i	$0.445944\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.5038	1.74963	0.874813	−	0.484461i	$-0.160984\pi$
$647$	44.5038	1.74963	0.874813	+	0.484461i	$0.160984\pi$
$648$	0	0
$649$	−10.9247	−0.428831
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.37382	−0.171161	−0.0855805	−	0.996331i	$-0.527274\pi$
$653$	−4.37382	−0.171161	−0.0855805	+	0.996331i	$0.527274\pi$
$654$	0	0
$655$	−7.76116	−0.303254
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.68235	−0.377171	−0.188585	−	0.982057i	$-0.560390\pi$
$659$	−9.68235	−0.377171	−0.188585	+	0.982057i	$0.560390\pi$
$660$	0	0
$661$	20.8248	0.809992	0.404996	−	0.914318i	$-0.367273\pi$
$661$	20.8248	0.809992	0.404996	+	0.914318i	$0.367273\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	53.8542	2.08838
$666$	0	0
$667$	−9.70179	−0.375655
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.93195	−0.306209
$672$	0	0
$673$	9.75633	0.376079	0.188039	−	0.982161i	$-0.439787\pi$
$673$	9.75633	0.376079	0.188039	+	0.982161i	$0.439787\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.4827	0.825647	0.412823	−	0.910811i	$-0.364543\pi$
$677$	21.4827	0.825647	0.412823	+	0.910811i	$0.364543\pi$
$678$	0	0
$679$	−5.90077	−0.226451
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.4925	0.631068	0.315534	−	0.948914i	$-0.397816\pi$
$683$	16.4925	0.631068	0.315534	+	0.948914i	$0.397816\pi$
$684$	0	0
$685$	−52.0093	−1.98717
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.6835	−0.559397
$690$	0	0
$691$	24.0700	0.915666	0.457833	−	0.889038i	$-0.348626\pi$
$691$	24.0700	0.915666	0.457833	+	0.889038i	$0.348626\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−56.9939	−2.16190
$696$	0	0
$697$	−0.697910	−0.0264352
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.5056	0.585640	0.292820	−	0.956168i	$-0.405406\pi$
$701$	15.5056	0.585640	0.292820	+	0.956168i	$0.405406\pi$
$702$	0	0
$703$	−9.77013	−0.368487
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.31931	−0.124835
$708$	0	0
$709$	−32.9477	−1.23738	−0.618689	−	0.785636i	$-0.712336\pi$
$709$	−32.9477	−1.23738	−0.618689	+	0.785636i	$0.712336\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−33.6040	−1.25848
$714$	0	0
$715$	42.2315	1.57937
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.1464	1.23615	0.618076	−	0.786119i	$-0.287913\pi$
$719$	33.1464	1.23615	0.618076	+	0.786119i	$0.287913\pi$
$720$	0	0
$721$	18.8242	0.701048
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−44.1435	−1.63945
$726$	0	0
$727$	−3.57131	−0.132453	−0.0662263	−	0.997805i	$-0.521096\pi$
$727$	−3.57131	−0.132453	−0.0662263	+	0.997805i	$0.521096\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.678930	0.0251111
$732$	0	0
$733$	−11.4545	−0.423081	−0.211541	−	0.977369i	$-0.567848\pi$
$733$	−11.4545	−0.423081	−0.211541	+	0.977369i	$0.567848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.6092	−0.906493
$738$	0	0
$739$	−1.93622	−0.0712251	−0.0356126	−	0.999366i	$-0.511338\pi$
$739$	−1.93622	−0.0712251	−0.0356126	+	0.999366i	$0.511338\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.0852	1.72739	0.863695	−	0.504015i	$-0.168144\pi$
$743$	47.0852	1.72739	0.863695	+	0.504015i	$0.168144\pi$
$744$	0	0
$745$	38.7496	1.41968
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.88580	−0.251602
$750$	0	0
$751$	−38.9219	−1.42028	−0.710141	−	0.704060i	$-0.751369\pi$
$751$	−38.9219	−1.42028	−0.710141	+	0.704060i	$0.751369\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.37087	0.122678
$756$	0	0
$757$	25.1383	0.913669	0.456834	−	0.889552i	$-0.348983\pi$
$757$	25.1383	0.913669	0.456834	+	0.889552i	$0.348983\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.6330	0.892945	0.446473	−	0.894797i	$-0.352680\pi$
$761$	24.6330	0.892945	0.446473	+	0.894797i	$0.352680\pi$
$762$	0	0
$763$	16.1693	0.585369
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−30.5956	−1.10474
$768$	0	0
$769$	5.50988	0.198692	0.0993458	−	0.995053i	$-0.468325\pi$
$769$	5.50988	0.198692	0.0993458	+	0.995053i	$0.468325\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.73543	0.134354	0.0671771	−	0.997741i	$-0.478601\pi$
$773$	3.73543	0.134354	0.0671771	+	0.997741i	$0.478601\pi$
$774$	0	0
$775$	−152.900	−5.49232
$776$	0	0
$777$	0	0
$778$	0	0
$779$	50.5156	1.80991
$780$	0	0
$781$	−22.0029	−0.787326
$782$	0	0
$783$	0	0
$784$	0	0
$785$	99.6845	3.55789
$786$	0	0
$787$	20.4386	0.728557	0.364279	−	0.931290i	$-0.381316\pi$
$787$	20.4386	0.728557	0.364279	+	0.931290i	$0.381316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.0553	−0.464194
$792$	0	0
$793$	−22.2142	−0.788848
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.0001	−0.425065	−0.212533	−	0.977154i	$-0.568171\pi$
$797$	−12.0001	−0.425065	−0.212533	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−1.03186	−0.0365045
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.321526	0.0113464
$804$	0	0
$805$	26.7953	0.944411
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.91890	−0.102623	−0.0513116	−	0.998683i	$-0.516340\pi$
$809$	−2.91890	−0.102623	−0.0513116	+	0.998683i	$0.516340\pi$
$810$	0	0
$811$	−10.1192	−0.355333	−0.177667	−	0.984091i	$-0.556855\pi$
$811$	−10.1192	−0.355333	−0.177667	+	0.984091i	$0.556855\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	83.9356	2.94014
$816$	0	0
$817$	−49.1418	−1.71925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−49.3896	−1.72371	−0.861855	−	0.507155i	$-0.830697\pi$
$821$	−49.3896	−1.72371	−0.861855	+	0.507155i	$0.830697\pi$
$822$	0	0
$823$	29.1619	1.01652	0.508261	−	0.861203i	$-0.330289\pi$
$823$	29.1619	1.01652	0.508261	+	0.861203i	$0.330289\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0347	0.557583	0.278791	−	0.960352i	$-0.410066\pi$
$827$	16.0347	0.557583	0.278791	+	0.960352i	$0.410066\pi$
$828$	0	0
$829$	−25.0995	−0.871743	−0.435871	−	0.900009i	$-0.643560\pi$
$829$	−25.0995	−0.871743	−0.435871	+	0.900009i	$0.643560\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.307699	0.0106611
$834$	0	0
$835$	4.42860	0.153258
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.9491	−1.06848	−0.534241	−	0.845332i	$-0.679402\pi$
$839$	−30.9491	−1.06848	−0.534241	+	0.845332i	$0.679402\pi$
$840$	0	0
$841$	−19.8739	−0.685306
$842$	0	0
$843$	0	0
$844$	0	0
$845$	60.7016	2.08820
$846$	0	0
$847$	−14.3090	−0.491664
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.86115	−0.166638
$852$	0	0
$853$	−38.7371	−1.32633	−0.663166	−	0.748472i	$-0.730788\pi$
$853$	−38.7371	−1.32633	−0.663166	+	0.748472i	$0.730788\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.5386	1.55557	0.777784	−	0.628532i	$-0.216344\pi$
$857$	45.5386	1.55557	0.777784	+	0.628532i	$0.216344\pi$
$858$	0	0
$859$	−20.4128	−0.696477	−0.348238	−	0.937406i	$-0.613220\pi$
$859$	−20.4128	−0.696477	−0.348238	+	0.937406i	$0.613220\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.2563	−0.927815	−0.463907	−	0.885884i	$-0.653553\pi$
$863$	−27.2563	−0.927815	−0.463907	+	0.885884i	$0.653553\pi$
$864$	0	0
$865$	34.3919	1.16936
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.53311	−0.153775
$870$	0	0
$871$	−68.9205	−2.33528
$872$	0	0
$873$	0	0
$874$	0	0
$875$	80.2021	2.71133
$876$	0	0
$877$	−54.1311	−1.82788	−0.913939	−	0.405851i	$-0.866975\pi$
$877$	−54.1311	−1.82788	−0.913939	+	0.405851i	$0.866975\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−49.1257	−1.65509	−0.827543	−	0.561403i	$-0.810262\pi$
$881$	−49.1257	−1.65509	−0.827543	+	0.561403i	$0.810262\pi$
$882$	0	0
$883$	−35.8325	−1.20586	−0.602930	−	0.797794i	$-0.706000\pi$
$883$	−35.8325	−1.20586	−0.602930	+	0.797794i	$0.706000\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.2790	1.08382	0.541911	−	0.840436i	$-0.317701\pi$
$887$	32.2790	1.08382	0.541911	+	0.840436i	$0.317701\pi$
$888$	0	0
$889$	−24.0536	−0.806731
$890$	0	0
$891$	0	0
$892$	0	0
$893$	74.6870	2.49931
$894$	0	0
$895$	20.5399	0.686574
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31.6101	1.05425
$900$	0	0
$901$	−0.253375	−0.00844114
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−52.9896	−1.76143
$906$	0	0
$907$	−12.9281	−0.429269	−0.214635	−	0.976694i	$-0.568856\pi$
$907$	−12.9281	−0.429269	−0.214635	+	0.976694i	$0.568856\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.4084	−1.37192	−0.685961	−	0.727638i	$-0.740618\pi$
$911$	−41.4084	−1.37192	−0.685961	+	0.727638i	$0.740618\pi$
$912$	0	0
$913$	−20.6968	−0.684964
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.30175	−0.109033
$918$	0	0
$919$	−0.204782	−0.00675514	−0.00337757	−	0.999994i	$-0.501075\pi$
$919$	−0.204782	−0.00675514	−0.00337757	+	0.999994i	$0.501075\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−61.6212	−2.02829
$924$	0	0
$925$	−22.1184	−0.727250
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.8315	−0.978741	−0.489371	−	0.872076i	$-0.662773\pi$
$929$	−29.8315	−0.978741	−0.489371	+	0.872076i	$0.662773\pi$
$930$	0	0
$931$	−22.2716	−0.729923
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.728736	0.0238322
$936$	0	0
$937$	27.7370	0.906128	0.453064	−	0.891478i	$-0.350331\pi$
$937$	27.7370	0.906128	0.453064	+	0.891478i	$0.350331\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47.2119	1.53906	0.769532	−	0.638609i	$-0.220490\pi$
$941$	47.2119	1.53906	0.769532	+	0.638609i	$0.220490\pi$
$942$	0	0
$943$	25.1341	0.818481
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.6942	−0.542487	−0.271244	−	0.962511i	$-0.587435\pi$
$947$	−16.6942	−0.542487	−0.271244	+	0.962511i	$0.587435\pi$
$948$	0	0
$949$	0.900464	0.0292303
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−12.0236	−0.389482	−0.194741	−	0.980855i	$-0.562387\pi$
$953$	−12.0236	−0.389482	−0.194741	+	0.980855i	$0.562387\pi$
$954$	0	0
$955$	−11.0654	−0.358069
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.1258	−0.714478
$960$	0	0
$961$	78.4875	2.53185
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−87.1225	−2.80457
$966$	0	0
$967$	−40.3542	−1.29770	−0.648852	−	0.760914i	$-0.724751\pi$
$967$	−40.3542	−1.29770	−0.648852	+	0.760914i	$0.724751\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.5121	−0.947088	−0.473544	−	0.880770i	$-0.657026\pi$
$971$	−29.5121	−0.947088	−0.473544	+	0.880770i	$0.657026\pi$
$972$	0	0
$973$	−24.2463	−0.777301
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.577211	0.0184666	0.00923330	−	0.999957i	$-0.497061\pi$
$977$	0.577211	0.0184666	0.00923330	+	0.999957i	$0.497061\pi$
$978$	0	0
$979$	−33.8926	−1.08321
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41.6804	1.32940	0.664698	−	0.747112i	$-0.268560\pi$
$983$	41.6804	1.32940	0.664698	+	0.747112i	$0.268560\pi$
$984$	0	0
$985$	−93.2453	−2.97104
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.4506	−0.777485
$990$	0	0
$991$	−8.68185	−0.275788	−0.137894	−	0.990447i	$-0.544033\pi$
$991$	−8.68185	−0.275788	−0.137894	+	0.990447i	$0.544033\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.77609	−0.183114
$996$	0	0
$997$	33.2583	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$997$	33.2583	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.f.1.5		5
3.2	odd	2		2004.2.a.b.1.1	✓	5
12.11	even	2		8016.2.a.q.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.b.1.1	✓	5	3.2	odd	2
6012.2.a.f.1.5		5	1.1	even	1	trivial
8016.2.a.q.1.1		5	12.11	even	2

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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+4.42860 q^{5} +1.88401 q^{7} +O(q^{10})\)	\(q+4.42860 q^{5} +1.88401 q^{7} -1.84527 q^{11} -5.16786 q^{13} -0.0891752 q^{17} +6.45461 q^{19} +3.21151 q^{23} +14.6125 q^{25} -3.02095 q^{29} -10.4636 q^{31} +8.34353 q^{35} -1.51367 q^{37} +7.82628 q^{41} -7.61344 q^{43} +11.5711 q^{47} -3.45050 q^{49} +2.84132 q^{53} -8.17196 q^{55} +5.92036 q^{59} +4.29853 q^{61} -22.8864 q^{65} +13.3364 q^{67} +11.9240 q^{71} -0.174243 q^{73} -3.47651 q^{77} +2.45661 q^{79} +11.2161 q^{83} -0.394921 q^{85} +18.3673 q^{89} -9.73630 q^{91} +28.5849 q^{95} -3.13202 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 7 * q^5 - 2 * q^7	\( 5 q + 7 q^{5} - 2 q^{7} + 5 q^{11} - 8 q^{13} + 7 q^{17} + 2 q^{19} + 13 q^{23} + 2 q^{25} + 11 q^{29} - 12 q^{31} + 12 q^{35} - 7 q^{37} + 12 q^{41} + 19 q^{47} - 9 q^{49} + 21 q^{53} - q^{55} + 7 q^{59} - 6 q^{61} - 14 q^{65} + 10 q^{67} + 35 q^{71} - 8 q^{73} + 6 q^{77} + 11 q^{83} + 5 q^{85} + 32 q^{89} + 5 q^{91} + 19 q^{95} + 11 q^{97}+O(q^{100}) \) 5 * q + 7 * q^5 - 2 * q^7 + 5 * q^11 - 8 * q^13 + 7 * q^17 + 2 * q^19 + 13 * q^23 + 2 * q^25 + 11 * q^29 - 12 * q^31 + 12 * q^35 - 7 * q^37 + 12 * q^41 + 19 * q^47 - 9 * q^49 + 21 * q^53 - q^55 + 7 * q^59 - 6 * q^61 - 14 * q^65 + 10 * q^67 + 35 * q^71 - 8 * q^73 + 6 * q^77 + 11 * q^83 + 5 * q^85 + 32 * q^89 + 5 * q^91 + 19 * q^95 + 11 * q^97

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