LMFDB - Embedded newform 6048.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, 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("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+0.414214 q^{5} -1.00000 q^{7} -0.414214 q^{11} -2.82843 q^{13} +2.82843 q^{17} +1.82843 q^{19} -3.24264 q^{23} -4.82843 q^{25} -2.82843 q^{29} +6.65685 q^{31} -0.414214 q^{35} +5.82843 q^{37} -12.0711 q^{41} +0.828427 q^{43} +11.6569 q^{47} +1.00000 q^{49} +4.00000 q^{53} -0.171573 q^{55} +7.65685 q^{59} +11.3137 q^{61} -1.17157 q^{65} -12.4853 q^{67} +8.41421 q^{71} -8.82843 q^{73} +0.414214 q^{77} -0.828427 q^{79} +5.17157 q^{83} +1.17157 q^{85} +7.24264 q^{89} +2.82843 q^{91} +0.757359 q^{95} +1.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} - 2 q^{19} + 2 q^{23} - 4 q^{25} + 2 q^{31} + 2 q^{35} + 6 q^{37} - 10 q^{41} - 4 q^{43} + 12 q^{47} + 2 q^{49} + 8 q^{53} - 6 q^{55} + 4 q^{59} - 8 q^{65} - 8 q^{67} + 14 q^{71} - 12 q^{73} - 2 q^{77} + 4 q^{79} + 16 q^{83} + 8 q^{85} + 6 q^{89} + 10 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^11 - 2 * q^19 + 2 * q^23 - 4 * q^25 + 2 * q^31 + 2 * q^35 + 6 * q^37 - 10 * q^41 - 4 * q^43 + 12 * q^47 + 2 * q^49 + 8 * q^53 - 6 * q^55 + 4 * q^59 - 8 * q^65 - 8 * q^67 + 14 * q^71 - 12 * q^73 - 2 * q^77 + 4 * q^79 + 16 * q^83 + 8 * q^85 + 6 * q^89 + 10 * q^95 - 8 * 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
$5$	0.414214	0.185242	0.0926210	−	0.995701i	$-0.470476\pi$
$5$	0.414214	0.185242	0.0926210	+	0.995701i	$0.470476\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	1.82843	0.419470	0.209735	−	0.977758i	$-0.432740\pi$
$19$	1.82843	0.419470	0.209735	+	0.977758i	$0.432740\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.24264	−0.676137	−0.338069	−	0.941121i	$-0.609774\pi$
$23$	−3.24264	−0.676137	−0.338069	+	0.941121i	$0.609774\pi$
$24$	0	0
$25$	−4.82843	−0.965685
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	6.65685	1.19561	0.597803	−	0.801643i	$-0.296040\pi$
$31$	6.65685	1.19561	0.597803	+	0.801643i	$0.296040\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.414214	−0.0700149
$36$	0	0
$37$	5.82843	0.958188	0.479094	−	0.877764i	$-0.340965\pi$
$37$	5.82843	0.958188	0.479094	+	0.877764i	$0.340965\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0711	−1.88518	−0.942592	−	0.333946i	$-0.891620\pi$
$41$	−12.0711	−1.88518	−0.942592	+	0.333946i	$0.891620\pi$
$42$	0	0
$43$	0.828427	0.126334	0.0631670	−	0.998003i	$-0.479880\pi$
$43$	0.828427	0.126334	0.0631670	+	0.998003i	$0.479880\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	−0.171573	−0.0231349
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	0	0
$61$	11.3137	1.44857	0.724286	−	0.689500i	$-0.242170\pi$
$61$	11.3137	1.44857	0.724286	+	0.689500i	$0.242170\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.17157	−0.145316
$66$	0	0
$67$	−12.4853	−1.52532	−0.762660	−	0.646800i	$-0.776107\pi$
$67$	−12.4853	−1.52532	−0.762660	+	0.646800i	$0.776107\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.41421	0.998583	0.499292	−	0.866434i	$-0.333594\pi$
$71$	8.41421	0.998583	0.499292	+	0.866434i	$0.333594\pi$
$72$	0	0
$73$	−8.82843	−1.03329	−0.516645	−	0.856200i	$-0.672819\pi$
$73$	−8.82843	−1.03329	−0.516645	+	0.856200i	$0.672819\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.414214	0.0472040
$78$	0	0
$79$	−0.828427	−0.0932053	−0.0466027	−	0.998914i	$-0.514839\pi$
$79$	−0.828427	−0.0932053	−0.0466027	+	0.998914i	$0.514839\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.17157	0.567654	0.283827	−	0.958876i	$-0.408396\pi$
$83$	5.17157	0.567654	0.283827	+	0.958876i	$0.408396\pi$
$84$	0	0
$85$	1.17157	0.127075
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.24264	0.767718	0.383859	−	0.923392i	$-0.374595\pi$
$89$	7.24264	0.767718	0.383859	+	0.923392i	$0.374595\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.757359	0.0777034
$96$	0	0
$97$	1.65685	0.168228	0.0841140	−	0.996456i	$-0.473194\pi$
$97$	1.65685	0.168228	0.0841140	+	0.996456i	$0.473194\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.48528	−0.446302	−0.223151	−	0.974784i	$-0.571634\pi$
$101$	−4.48528	−0.446302	−0.223151	+	0.974784i	$0.571634\pi$
$102$	0	0
$103$	−4.65685	−0.458853	−0.229427	−	0.973326i	$-0.573685\pi$
$103$	−4.65685	−0.458853	−0.229427	+	0.973326i	$0.573685\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.3137	1.67378	0.836890	−	0.547372i	$-0.184372\pi$
$107$	17.3137	1.67378	0.836890	+	0.547372i	$0.184372\pi$
$108$	0	0
$109$	−4.31371	−0.413178	−0.206589	−	0.978428i	$-0.566236\pi$
$109$	−4.31371	−0.413178	−0.206589	+	0.978428i	$0.566236\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.1421	1.51852	0.759262	−	0.650785i	$-0.225560\pi$
$113$	16.1421	1.51852	0.759262	+	0.650785i	$0.225560\pi$
$114$	0	0
$115$	−1.34315	−0.125249
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.82843	−0.259281
$120$	0	0
$121$	−10.8284	−0.984402
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.07107	−0.364127
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	−1.82843	−0.158545
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	9.65685	0.819084	0.409542	−	0.912291i	$-0.365689\pi$
$139$	9.65685	0.819084	0.409542	+	0.912291i	$0.365689\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.17157	0.0979718
$144$	0	0
$145$	−1.17157	−0.0972938
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.82843	−0.723253	−0.361626	−	0.932323i	$-0.617778\pi$
$149$	−8.82843	−0.723253	−0.361626	+	0.932323i	$0.617778\pi$
$150$	0	0
$151$	17.3137	1.40897	0.704485	−	0.709719i	$-0.251178\pi$
$151$	17.3137	1.40897	0.704485	+	0.709719i	$0.251178\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.75736	0.221476
$156$	0	0
$157$	14.1421	1.12867	0.564333	−	0.825547i	$-0.309134\pi$
$157$	14.1421	1.12867	0.564333	+	0.825547i	$0.309134\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.24264	0.255556
$162$	0	0
$163$	−0.485281	−0.0380102	−0.0190051	−	0.999819i	$-0.506050\pi$
$163$	−0.485281	−0.0380102	−0.0190051	+	0.999819i	$0.506050\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.4853	1.43043	0.715217	−	0.698902i	$-0.246328\pi$
$167$	18.4853	1.43043	0.715217	+	0.698902i	$0.246328\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.757359	−0.0575810	−0.0287905	−	0.999585i	$-0.509166\pi$
$173$	−0.757359	−0.0575810	−0.0287905	+	0.999585i	$0.509166\pi$
$174$	0	0
$175$	4.82843	0.364995
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.34315	0.623596	0.311798	−	0.950148i	$-0.399069\pi$
$179$	8.34315	0.623596	0.311798	+	0.950148i	$0.399069\pi$
$180$	0	0
$181$	8.34315	0.620141	0.310071	−	0.950714i	$-0.399647\pi$
$181$	8.34315	0.620141	0.310071	+	0.950714i	$0.399647\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.41421	0.177497
$186$	0	0
$187$	−1.17157	−0.0856739
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.58579	0.693603	0.346802	−	0.937939i	$-0.387268\pi$
$191$	9.58579	0.693603	0.346802	+	0.937939i	$0.387268\pi$
$192$	0	0
$193$	7.65685	0.551152	0.275576	−	0.961279i	$-0.411131\pi$
$193$	7.65685	0.551152	0.275576	+	0.961279i	$0.411131\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.1716	1.36592	0.682959	−	0.730457i	$-0.260693\pi$
$197$	19.1716	1.36592	0.682959	+	0.730457i	$0.260693\pi$
$198$	0	0
$199$	−13.4853	−0.955946	−0.477973	−	0.878374i	$-0.658628\pi$
$199$	−13.4853	−0.955946	−0.477973	+	0.878374i	$0.658628\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.82843	0.198517
$204$	0	0
$205$	−5.00000	−0.349215
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.757359	−0.0523876
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.343146	0.0234023
$216$	0	0
$217$	−6.65685	−0.451897
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−14.7990	−0.991014	−0.495507	−	0.868604i	$-0.665018\pi$
$223$	−14.7990	−0.991014	−0.495507	+	0.868604i	$0.665018\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.1716	1.00697	0.503486	−	0.864003i	$-0.332050\pi$
$227$	15.1716	1.00697	0.503486	+	0.864003i	$0.332050\pi$
$228$	0	0
$229$	−3.31371	−0.218976	−0.109488	−	0.993988i	$-0.534921\pi$
$229$	−3.31371	−0.218976	−0.109488	+	0.993988i	$0.534921\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.17157	0.469825	0.234913	−	0.972016i	$-0.424520\pi$
$233$	7.17157	0.469825	0.234913	+	0.972016i	$0.424520\pi$
$234$	0	0
$235$	4.82843	0.314972
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.34315	0.280935	0.140467	−	0.990085i	$-0.455139\pi$
$239$	4.34315	0.280935	0.140467	+	0.990085i	$0.455139\pi$
$240$	0	0
$241$	4.14214	0.266818	0.133409	−	0.991061i	$-0.457408\pi$
$241$	4.14214	0.266818	0.133409	+	0.991061i	$0.457408\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.414214	0.0264631
$246$	0	0
$247$	−5.17157	−0.329059
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.97056	−0.566217	−0.283108	−	0.959088i	$-0.591366\pi$
$251$	−8.97056	−0.566217	−0.283108	+	0.959088i	$0.591366\pi$
$252$	0	0
$253$	1.34315	0.0844428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−31.7279	−1.97913	−0.989567	−	0.144075i	$-0.953979\pi$
$257$	−31.7279	−1.97913	−0.989567	+	0.144075i	$0.953979\pi$
$258$	0	0
$259$	−5.82843	−0.362161
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.3848	1.44197	0.720984	−	0.692952i	$-0.243690\pi$
$263$	23.3848	1.44197	0.720984	+	0.692952i	$0.243690\pi$
$264$	0	0
$265$	1.65685	0.101780
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.4142	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$269$	18.4142	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	8.17157	0.490982	0.245491	−	0.969399i	$-0.421051\pi$
$277$	8.17157	0.490982	0.245491	+	0.969399i	$0.421051\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−7.31371	−0.434755	−0.217377	−	0.976088i	$-0.569750\pi$
$283$	−7.31371	−0.434755	−0.217377	+	0.976088i	$0.569750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0711	0.712533
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.8284	−0.866286	−0.433143	−	0.901325i	$-0.642595\pi$
$293$	−14.8284	−0.866286	−0.433143	+	0.901325i	$0.642595\pi$
$294$	0	0
$295$	3.17157	0.184656
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.17157	0.530406
$300$	0	0
$301$	−0.828427	−0.0477497
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.68629	0.268336
$306$	0	0
$307$	8.17157	0.466376	0.233188	−	0.972432i	$-0.425084\pi$
$307$	8.17157	0.466376	0.233188	+	0.972432i	$0.425084\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.4853	0.934795	0.467397	−	0.884047i	$-0.345192\pi$
$311$	16.4853	0.934795	0.467397	+	0.884047i	$0.345192\pi$
$312$	0	0
$313$	−13.1716	−0.744501	−0.372251	−	0.928132i	$-0.621414\pi$
$313$	−13.1716	−0.744501	−0.372251	+	0.928132i	$0.621414\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.2843	1.25161	0.625805	−	0.779980i	$-0.284771\pi$
$317$	22.2843	1.25161	0.625805	+	0.779980i	$0.284771\pi$
$318$	0	0
$319$	1.17157	0.0655955
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.17157	0.287754
$324$	0	0
$325$	13.6569	0.757546
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.6569	−0.642663
$330$	0	0
$331$	−10.6274	−0.584136	−0.292068	−	0.956398i	$-0.594343\pi$
$331$	−10.6274	−0.584136	−0.292068	+	0.956398i	$0.594343\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.17157	−0.282553
$336$	0	0
$337$	−19.6274	−1.06917	−0.534587	−	0.845114i	$-0.679533\pi$
$337$	−19.6274	−1.06917	−0.534587	+	0.845114i	$0.679533\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.75736	−0.149319
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.92893	0.210916	0.105458	−	0.994424i	$-0.466369\pi$
$347$	3.92893	0.210916	0.105458	+	0.994424i	$0.466369\pi$
$348$	0	0
$349$	8.34315	0.446598	0.223299	−	0.974750i	$-0.428317\pi$
$349$	8.34315	0.446598	0.223299	+	0.974750i	$0.428317\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.41421	0.128496	0.0642478	−	0.997934i	$-0.479535\pi$
$353$	2.41421	0.128496	0.0642478	+	0.997934i	$0.479535\pi$
$354$	0	0
$355$	3.48528	0.184980
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.31371	−0.0693349	−0.0346674	−	0.999399i	$-0.511037\pi$
$359$	−1.31371	−0.0693349	−0.0346674	+	0.999399i	$0.511037\pi$
$360$	0	0
$361$	−15.6569	−0.824045
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.65685	−0.191408
$366$	0	0
$367$	19.4853	1.01712	0.508562	−	0.861026i	$-0.330177\pi$
$367$	19.4853	1.01712	0.508562	+	0.861026i	$0.330177\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	26.3137	1.36247	0.681236	−	0.732064i	$-0.261443\pi$
$373$	26.3137	1.36247	0.681236	+	0.732064i	$0.261443\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−2.48528	−0.127660	−0.0638302	−	0.997961i	$-0.520332\pi$
$379$	−2.48528	−0.127660	−0.0638302	+	0.997961i	$0.520332\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.4558	−0.789757	−0.394878	−	0.918733i	$-0.629213\pi$
$383$	−15.4558	−0.789757	−0.394878	+	0.918733i	$0.629213\pi$
$384$	0	0
$385$	0.171573	0.00874416
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.3137	−0.573628	−0.286814	−	0.957986i	$-0.592596\pi$
$389$	−11.3137	−0.573628	−0.286814	+	0.957986i	$0.592596\pi$
$390$	0	0
$391$	−9.17157	−0.463826
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.343146	−0.0172655
$396$	0	0
$397$	7.65685	0.384286	0.192143	−	0.981367i	$-0.438456\pi$
$397$	7.65685	0.384286	0.192143	+	0.981367i	$0.438456\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.1716	−0.857507	−0.428754	−	0.903421i	$-0.641047\pi$
$401$	−17.1716	−0.857507	−0.428754	+	0.903421i	$0.641047\pi$
$402$	0	0
$403$	−18.8284	−0.937911
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.41421	−0.119668
$408$	0	0
$409$	−17.5147	−0.866047	−0.433024	−	0.901383i	$-0.642553\pi$
$409$	−17.5147	−0.866047	−0.433024	+	0.901383i	$0.642553\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.65685	−0.376769
$414$	0	0
$415$	2.14214	0.105153
$416$	0	0
$417$	0	0
$418$	0	0
$419$	28.1421	1.37483	0.687417	−	0.726263i	$-0.258745\pi$
$419$	28.1421	1.37483	0.687417	+	0.726263i	$0.258745\pi$
$420$	0	0
$421$	27.4853	1.33955	0.669775	−	0.742564i	$-0.266390\pi$
$421$	27.4853	1.33955	0.669775	+	0.742564i	$0.266390\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.6569	−0.662455
$426$	0	0
$427$	−11.3137	−0.547509
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.2132	0.973636	0.486818	−	0.873503i	$-0.338158\pi$
$431$	20.2132	0.973636	0.486818	+	0.873503i	$0.338158\pi$
$432$	0	0
$433$	−29.4558	−1.41556	−0.707779	−	0.706434i	$-0.750303\pi$
$433$	−29.4558	−1.41556	−0.707779	+	0.706434i	$0.750303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.92893	−0.283619
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.0711	0.668537	0.334268	−	0.942478i	$-0.391511\pi$
$443$	14.0711	0.668537	0.334268	+	0.942478i	$0.391511\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.1716	1.09353	0.546767	−	0.837285i	$-0.315858\pi$
$449$	23.1716	1.09353	0.546767	+	0.837285i	$0.315858\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.17157	0.0549242
$456$	0	0
$457$	2.65685	0.124282	0.0621412	−	0.998067i	$-0.480207\pi$
$457$	2.65685	0.124282	0.0621412	+	0.998067i	$0.480207\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.7574	−0.780468	−0.390234	−	0.920716i	$-0.627606\pi$
$461$	−16.7574	−0.780468	−0.390234	+	0.920716i	$0.627606\pi$
$462$	0	0
$463$	−23.6569	−1.09943	−0.549714	−	0.835353i	$-0.685263\pi$
$463$	−23.6569	−1.09943	−0.549714	+	0.835353i	$0.685263\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.6274	1.32472	0.662359	−	0.749186i	$-0.269555\pi$
$467$	28.6274	1.32472	0.662359	+	0.749186i	$0.269555\pi$
$468$	0	0
$469$	12.4853	0.576517
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.343146	−0.0157779
$474$	0	0
$475$	−8.82843	−0.405076
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.02944	0.138419	0.0692093	−	0.997602i	$-0.477952\pi$
$479$	3.02944	0.138419	0.0692093	+	0.997602i	$0.477952\pi$
$480$	0	0
$481$	−16.4853	−0.751664
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.686292	0.0311629
$486$	0	0
$487$	−5.51472	−0.249896	−0.124948	−	0.992163i	$-0.539876\pi$
$487$	−5.51472	−0.249896	−0.124948	+	0.992163i	$0.539876\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.8701	1.16750	0.583750	−	0.811934i	$-0.301585\pi$
$491$	25.8701	1.16750	0.583750	+	0.811934i	$0.301585\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.41421	−0.377429
$498$	0	0
$499$	38.7696	1.73556	0.867782	−	0.496945i	$-0.165545\pi$
$499$	38.7696	1.73556	0.867782	+	0.496945i	$0.165545\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.1421	0.808918	0.404459	−	0.914556i	$-0.367460\pi$
$503$	18.1421	0.808918	0.404459	+	0.914556i	$0.367460\pi$
$504$	0	0
$505$	−1.85786	−0.0826739
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.5147	−0.687678	−0.343839	−	0.939029i	$-0.611727\pi$
$509$	−15.5147	−0.687678	−0.343839	+	0.939029i	$0.611727\pi$
$510$	0	0
$511$	8.82843	0.390547
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.92893	−0.0849989
$516$	0	0
$517$	−4.82843	−0.212354
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.7574	0.821775	0.410887	−	0.911686i	$-0.365219\pi$
$521$	18.7574	0.821775	0.410887	+	0.911686i	$0.365219\pi$
$522$	0	0
$523$	21.9706	0.960706	0.480353	−	0.877075i	$-0.340509\pi$
$523$	21.9706	0.960706	0.480353	+	0.877075i	$0.340509\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.8284	0.820179
$528$	0	0
$529$	−12.4853	−0.542838
$530$	0	0
$531$	0	0
$532$	0	0
$533$	34.1421	1.47886
$534$	0	0
$535$	7.17157	0.310054
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.414214	−0.0178414
$540$	0	0
$541$	14.7990	0.636258	0.318129	−	0.948047i	$-0.396945\pi$
$541$	14.7990	0.636258	0.318129	+	0.948047i	$0.396945\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.78680	−0.0765380
$546$	0	0
$547$	16.9706	0.725609	0.362804	−	0.931865i	$-0.381819\pi$
$547$	16.9706	0.725609	0.362804	+	0.931865i	$0.381819\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.17157	−0.220316
$552$	0	0
$553$	0.828427	0.0352283
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.7696	1.81220	0.906102	−	0.423059i	$-0.139044\pi$
$557$	42.7696	1.81220	0.906102	+	0.423059i	$0.139044\pi$
$558$	0	0
$559$	−2.34315	−0.0991045
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.7990	−0.665848	−0.332924	−	0.942954i	$-0.608035\pi$
$563$	−15.7990	−0.665848	−0.332924	+	0.942954i	$0.608035\pi$
$564$	0	0
$565$	6.68629	0.281294
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.7990	−1.24924	−0.624619	−	0.780929i	$-0.714746\pi$
$569$	−29.7990	−1.24924	−0.624619	+	0.780929i	$0.714746\pi$
$570$	0	0
$571$	32.6274	1.36541	0.682707	−	0.730692i	$-0.260802\pi$
$571$	32.6274	1.36541	0.682707	+	0.730692i	$0.260802\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.6569	0.652936
$576$	0	0
$577$	27.9411	1.16320	0.581602	−	0.813473i	$-0.302426\pi$
$577$	27.9411	1.16320	0.581602	+	0.813473i	$0.302426\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.17157	−0.214553
$582$	0	0
$583$	−1.65685	−0.0686199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.4558	−0.803029	−0.401514	−	0.915853i	$-0.631516\pi$
$587$	−19.4558	−0.803029	−0.401514	+	0.915853i	$0.631516\pi$
$588$	0	0
$589$	12.1716	0.501521
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.7574	0.441752	0.220876	−	0.975302i	$-0.429108\pi$
$593$	10.7574	0.441752	0.220876	+	0.975302i	$0.429108\pi$
$594$	0	0
$595$	−1.17157	−0.0480298
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.0416	−0.859738	−0.429869	−	0.902891i	$-0.641440\pi$
$599$	−21.0416	−0.859738	−0.429869	+	0.902891i	$0.641440\pi$
$600$	0	0
$601$	−38.1421	−1.55585	−0.777925	−	0.628357i	$-0.783728\pi$
$601$	−38.1421	−1.55585	−0.777925	+	0.628357i	$0.783728\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.48528	−0.182353
$606$	0	0
$607$	14.3431	0.582170	0.291085	−	0.956697i	$-0.405984\pi$
$607$	14.3431	0.582170	0.291085	+	0.956697i	$0.405984\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.9706	−1.33385
$612$	0	0
$613$	41.9706	1.69518	0.847588	−	0.530656i	$-0.178054\pi$
$613$	41.9706	1.69518	0.847588	+	0.530656i	$0.178054\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.2843	−0.414029	−0.207015	−	0.978338i	$-0.566375\pi$
$617$	−10.2843	−0.414029	−0.207015	+	0.978338i	$0.566375\pi$
$618$	0	0
$619$	−17.6863	−0.710872	−0.355436	−	0.934701i	$-0.615668\pi$
$619$	−17.6863	−0.710872	−0.355436	+	0.934701i	$0.615668\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.24264	−0.290170
$624$	0	0
$625$	22.4558	0.898234
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.4853	0.657311
$630$	0	0
$631$	−32.1421	−1.27956	−0.639779	−	0.768559i	$-0.720974\pi$
$631$	−32.1421	−1.27956	−0.639779	+	0.768559i	$0.720974\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.17157	−0.0464925
$636$	0	0
$637$	−2.82843	−0.112066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22.4853	0.888115	0.444058	−	0.895998i	$-0.353539\pi$
$641$	22.4853	0.888115	0.444058	+	0.895998i	$0.353539\pi$
$642$	0	0
$643$	−13.4853	−0.531808	−0.265904	−	0.964000i	$-0.585670\pi$
$643$	−13.4853	−0.531808	−0.265904	+	0.964000i	$0.585670\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.3137	−1.30970	−0.654849	−	0.755760i	$-0.727268\pi$
$647$	−33.3137	−1.30970	−0.654849	+	0.755760i	$0.727268\pi$
$648$	0	0
$649$	−3.17157	−0.124495
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−38.9706	−1.52504	−0.762518	−	0.646967i	$-0.776037\pi$
$653$	−38.9706	−1.52504	−0.762518	+	0.646967i	$0.776037\pi$
$654$	0	0
$655$	0.828427	0.0323693
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.1005	−0.666141	−0.333071	−	0.942902i	$-0.608085\pi$
$659$	−17.1005	−0.666141	−0.333071	+	0.942902i	$0.608085\pi$
$660$	0	0
$661$	5.17157	0.201151	0.100575	−	0.994929i	$-0.467932\pi$
$661$	5.17157	0.201151	0.100575	+	0.994929i	$0.467932\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.757359	−0.0293691
$666$	0	0
$667$	9.17157	0.355125
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.68629	−0.180912
$672$	0	0
$673$	−22.2843	−0.858996	−0.429498	−	0.903068i	$-0.641309\pi$
$673$	−22.2843	−0.858996	−0.429498	+	0.903068i	$0.641309\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.0416	−1.65422	−0.827112	−	0.562037i	$-0.810018\pi$
$677$	−43.0416	−1.65422	−0.827112	+	0.562037i	$0.810018\pi$
$678$	0	0
$679$	−1.65685	−0.0635842
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.89949	−0.264002	−0.132001	−	0.991250i	$-0.542140\pi$
$683$	−6.89949	−0.264002	−0.132001	+	0.991250i	$0.542140\pi$
$684$	0	0
$685$	−1.17157	−0.0447635
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.3137	−0.431018
$690$	0	0
$691$	−47.3137	−1.79990	−0.899949	−	0.435995i	$-0.856397\pi$
$691$	−47.3137	−1.79990	−0.899949	+	0.435995i	$0.856397\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−34.1421	−1.29323
$698$	0	0
$699$	0	0
$700$	0	0
$701$	48.4853	1.83126	0.915632	−	0.402018i	$-0.131691\pi$
$701$	48.4853	1.83126	0.915632	+	0.402018i	$0.131691\pi$
$702$	0	0
$703$	10.6569	0.401931
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.48528	0.168686
$708$	0	0
$709$	3.68629	0.138442	0.0692208	−	0.997601i	$-0.477949\pi$
$709$	3.68629	0.138442	0.0692208	+	0.997601i	$0.477949\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.5858	−0.808394
$714$	0	0
$715$	0.485281	0.0181485
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.65685	−0.136378	−0.0681888	−	0.997672i	$-0.521722\pi$
$719$	−3.65685	−0.136378	−0.0681888	+	0.997672i	$0.521722\pi$
$720$	0	0
$721$	4.65685	0.173430
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13.6569	0.507203
$726$	0	0
$727$	38.6274	1.43261	0.716306	−	0.697787i	$-0.245832\pi$
$727$	38.6274	1.43261	0.716306	+	0.697787i	$0.245832\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.34315	0.0866644
$732$	0	0
$733$	4.82843	0.178342	0.0891710	−	0.996016i	$-0.471578\pi$
$733$	4.82843	0.178342	0.0891710	+	0.996016i	$0.471578\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.17157	0.190497
$738$	0	0
$739$	−2.97056	−0.109274	−0.0546370	−	0.998506i	$-0.517400\pi$
$739$	−2.97056	−0.109274	−0.0546370	+	0.998506i	$0.517400\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.89949	−0.106372	−0.0531861	−	0.998585i	$-0.516938\pi$
$743$	−2.89949	−0.106372	−0.0531861	+	0.998585i	$0.516938\pi$
$744$	0	0
$745$	−3.65685	−0.133977
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.3137	−0.632629
$750$	0	0
$751$	−33.5147	−1.22297	−0.611485	−	0.791256i	$-0.709427\pi$
$751$	−33.5147	−1.22297	−0.611485	+	0.791256i	$0.709427\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.17157	0.261000
$756$	0	0
$757$	−26.2843	−0.955318	−0.477659	−	0.878545i	$-0.658515\pi$
$757$	−26.2843	−0.955318	−0.477659	+	0.878545i	$0.658515\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.20101	−0.297286	−0.148643	−	0.988891i	$-0.547491\pi$
$761$	−8.20101	−0.297286	−0.148643	+	0.988891i	$0.547491\pi$
$762$	0	0
$763$	4.31371	0.156167
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.6569	−0.781984
$768$	0	0
$769$	30.1421	1.08695	0.543477	−	0.839424i	$-0.317108\pi$
$769$	30.1421	1.08695	0.543477	+	0.839424i	$0.317108\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.5858	1.27993	0.639966	−	0.768403i	$-0.278948\pi$
$773$	35.5858	1.27993	0.639966	+	0.768403i	$0.278948\pi$
$774$	0	0
$775$	−32.1421	−1.15458
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−22.0711	−0.790778
$780$	0	0
$781$	−3.48528	−0.124713
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.85786	0.209076
$786$	0	0
$787$	−41.9411	−1.49504	−0.747520	−	0.664239i	$-0.768756\pi$
$787$	−41.9411	−1.49504	−0.747520	+	0.664239i	$0.768756\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.1421	−0.573948
$792$	0	0
$793$	−32.0000	−1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.1005	−0.747418	−0.373709	−	0.927546i	$-0.621914\pi$
$797$	−21.1005	−0.747418	−0.373709	+	0.927546i	$0.621914\pi$
$798$	0	0
$799$	32.9706	1.16641
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.65685	0.129048
$804$	0	0
$805$	1.34315	0.0473397
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31.5147	1.10800	0.553999	−	0.832517i	$-0.313101\pi$
$809$	31.5147	1.10800	0.553999	+	0.832517i	$0.313101\pi$
$810$	0	0
$811$	20.6569	0.725360	0.362680	−	0.931914i	$-0.381862\pi$
$811$	20.6569	0.725360	0.362680	+	0.931914i	$0.381862\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.201010	−0.00704108
$816$	0	0
$817$	1.51472	0.0529933
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.2843	−1.75493	−0.877467	−	0.479638i	$-0.840768\pi$
$821$	−50.2843	−1.75493	−0.877467	+	0.479638i	$0.840768\pi$
$822$	0	0
$823$	3.37258	0.117561	0.0587804	−	0.998271i	$-0.481279\pi$
$823$	3.37258	0.117561	0.0587804	+	0.998271i	$0.481279\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.4142	0.362138	0.181069	−	0.983470i	$-0.442044\pi$
$827$	10.4142	0.362138	0.181069	+	0.983470i	$0.442044\pi$
$828$	0	0
$829$	−20.1421	−0.699565	−0.349783	−	0.936831i	$-0.613745\pi$
$829$	−20.1421	−0.699565	−0.349783	+	0.936831i	$0.613745\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.82843	0.0979992
$834$	0	0
$835$	7.65685	0.264976
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.85786	−0.202236	−0.101118	−	0.994874i	$-0.532242\pi$
$839$	−5.85786	−0.202236	−0.101118	+	0.994874i	$0.532242\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.07107	−0.0712469
$846$	0	0
$847$	10.8284	0.372069
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.8995	−0.647866
$852$	0	0
$853$	44.9706	1.53976	0.769881	−	0.638187i	$-0.220315\pi$
$853$	44.9706	1.53976	0.769881	+	0.638187i	$0.220315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.5563	1.31706	0.658530	−	0.752555i	$-0.271179\pi$
$857$	38.5563	1.31706	0.658530	+	0.752555i	$0.271179\pi$
$858$	0	0
$859$	−16.1127	−0.549758	−0.274879	−	0.961479i	$-0.588638\pi$
$859$	−16.1127	−0.549758	−0.274879	+	0.961479i	$0.588638\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.3431	−0.828650	−0.414325	−	0.910129i	$-0.635982\pi$
$863$	−24.3431	−0.828650	−0.414325	+	0.910129i	$0.635982\pi$
$864$	0	0
$865$	−0.313708	−0.0106664
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.343146	0.0116404
$870$	0	0
$871$	35.3137	1.19656
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.07107	0.137627
$876$	0	0
$877$	−27.9411	−0.943505	−0.471752	−	0.881731i	$-0.656378\pi$
$877$	−27.9411	−0.943505	−0.471752	+	0.881731i	$0.656378\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.27208	−0.211312	−0.105656	−	0.994403i	$-0.533694\pi$
$881$	−6.27208	−0.211312	−0.105656	+	0.994403i	$0.533694\pi$
$882$	0	0
$883$	39.3137	1.32301	0.661506	−	0.749940i	$-0.269918\pi$
$883$	39.3137	1.32301	0.661506	+	0.749940i	$0.269918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.7696	1.36891	0.684454	−	0.729056i	$-0.260041\pi$
$887$	40.7696	1.36891	0.684454	+	0.729056i	$0.260041\pi$
$888$	0	0
$889$	2.82843	0.0948624
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21.3137	0.713236
$894$	0	0
$895$	3.45584	0.115516
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.8284	−0.627963
$900$	0	0
$901$	11.3137	0.376914
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.45584	0.114876
$906$	0	0
$907$	−33.3137	−1.10616	−0.553082	−	0.833127i	$-0.686548\pi$
$907$	−33.3137	−1.10616	−0.553082	+	0.833127i	$0.686548\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42.9706	−1.42368	−0.711839	−	0.702343i	$-0.752137\pi$
$911$	−42.9706	−1.42368	−0.711839	+	0.702343i	$0.752137\pi$
$912$	0	0
$913$	−2.14214	−0.0708943
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.00000	−0.0660458
$918$	0	0
$919$	−41.6569	−1.37413	−0.687066	−	0.726595i	$-0.741102\pi$
$919$	−41.6569	−1.37413	−0.687066	+	0.726595i	$0.741102\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−23.7990	−0.783353
$924$	0	0
$925$	−28.1421	−0.925308
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17.8579	−0.585898	−0.292949	−	0.956128i	$-0.594637\pi$
$929$	−17.8579	−0.585898	−0.292949	+	0.956128i	$0.594637\pi$
$930$	0	0
$931$	1.82843	0.0599243
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.485281	−0.0158704
$936$	0	0
$937$	11.3137	0.369603	0.184801	−	0.982776i	$-0.440836\pi$
$937$	11.3137	0.369603	0.184801	+	0.982776i	$0.440836\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29.3848	0.957916	0.478958	−	0.877838i	$-0.341015\pi$
$941$	29.3848	0.957916	0.478958	+	0.877838i	$0.341015\pi$
$942$	0	0
$943$	39.1421	1.27464
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.7279	0.706063	0.353031	−	0.935612i	$-0.385151\pi$
$947$	21.7279	0.706063	0.353031	+	0.935612i	$0.385151\pi$
$948$	0	0
$949$	24.9706	0.810579
$950$	0	0
$951$	0	0
$952$	0	0
$953$	58.8284	1.90564	0.952820	−	0.303536i	$-0.0981674\pi$
$953$	58.8284	1.90564	0.952820	+	0.303536i	$0.0981674\pi$
$954$	0	0
$955$	3.97056	0.128484
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.82843	0.0913347
$960$	0	0
$961$	13.3137	0.429474
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.17157	0.102097
$966$	0	0
$967$	46.2843	1.48840	0.744201	−	0.667956i	$-0.232831\pi$
$967$	46.2843	1.48840	0.744201	+	0.667956i	$0.232831\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.8284	−1.50280	−0.751398	−	0.659849i	$-0.770620\pi$
$971$	−46.8284	−1.50280	−0.751398	+	0.659849i	$0.770620\pi$
$972$	0	0
$973$	−9.65685	−0.309585
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.9706	−1.56671	−0.783354	−	0.621576i	$-0.786493\pi$
$977$	−48.9706	−1.56671	−0.783354	+	0.621576i	$0.786493\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	0	0
$982$	0	0
$983$	53.5980	1.70951	0.854755	−	0.519032i	$-0.173707\pi$
$983$	53.5980	1.70951	0.854755	+	0.519032i	$0.173707\pi$
$984$	0	0
$985$	7.94113	0.253025
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.68629	−0.0854191
$990$	0	0
$991$	−13.4558	−0.427439	−0.213719	−	0.976895i	$-0.568558\pi$
$991$	−13.4558	−0.427439	−0.213719	+	0.976895i	$0.568558\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.58579	−0.177081
$996$	0	0
$997$	45.3137	1.43510	0.717550	−	0.696507i	$-0.245264\pi$
$997$	45.3137	1.43510	0.717550	+	0.696507i	$0.245264\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.z.1.2	✓	2
3.2	odd	2		6048.2.a.bg.1.1	yes	2
4.3	odd	2		6048.2.a.ba.1.2	yes	2
12.11	even	2		6048.2.a.bj.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.z.1.2	✓	2	1.1	even	1	trivial
6048.2.a.ba.1.2	yes	2	4.3	odd	2
6048.2.a.bg.1.1	yes	2	3.2	odd	2
6048.2.a.bj.1.1	yes	2	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+0.414214 q^{5} -1.00000 q^{7} -0.414214 q^{11} -2.82843 q^{13} +2.82843 q^{17} +1.82843 q^{19} -3.24264 q^{23} -4.82843 q^{25} -2.82843 q^{29} +6.65685 q^{31} -0.414214 q^{35} +5.82843 q^{37} -12.0711 q^{41} +0.828427 q^{43} +11.6569 q^{47} +1.00000 q^{49} +4.00000 q^{53} -0.171573 q^{55} +7.65685 q^{59} +11.3137 q^{61} -1.17157 q^{65} -12.4853 q^{67} +8.41421 q^{71} -8.82843 q^{73} +0.414214 q^{77} -0.828427 q^{79} +5.17157 q^{83} +1.17157 q^{85} +7.24264 q^{89} +2.82843 q^{91} +0.757359 q^{95} +1.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} - 2 q^{19} + 2 q^{23} - 4 q^{25} + 2 q^{31} + 2 q^{35} + 6 q^{37} - 10 q^{41} - 4 q^{43} + 12 q^{47} + 2 q^{49} + 8 q^{53} - 6 q^{55} + 4 q^{59} - 8 q^{65} - 8 q^{67} + 14 q^{71} - 12 q^{73} - 2 q^{77} + 4 q^{79} + 16 q^{83} + 8 q^{85} + 6 q^{89} + 10 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^11 - 2 * q^19 + 2 * q^23 - 4 * q^25 + 2 * q^31 + 2 * q^35 + 6 * q^37 - 10 * q^41 - 4 * q^43 + 12 * q^47 + 2 * q^49 + 8 * q^53 - 6 * q^55 + 4 * q^59 - 8 * q^65 - 8 * q^67 + 14 * q^71 - 12 * q^73 - 2 * q^77 + 4 * q^79 + 16 * q^83 + 8 * q^85 + 6 * q^89 + 10 * q^95 - 8 * q^97

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