LMFDB - Embedded newform 6048.2.a.bp.1.4

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:	$4$
Coefficient field:	4.4.39528.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 2x + 12 $ x^4 - 9x^2 - 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.12265$ 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)

$f(q)$	$=$	$q+2.73965 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.73965 q^{5} +1.00000 q^{7} +2.24531 q^{11} -0.458685 q^{13} +7.44364 q^{17} +4.24531 q^{19} +4.70399 q^{23} +2.50566 q^{25} +1.73965 q^{29} -1.73965 q^{31} +2.73965 q^{35} +2.49434 q^{37} +4.73965 q^{41} -1.96435 q^{43} -9.90233 q^{47} +1.00000 q^{49} -5.42303 q^{53} +6.15135 q^{55} -6.47929 q^{59} +5.23398 q^{61} -1.25664 q^{65} -2.77530 q^{67} -0.703994 q^{71} -7.96991 q^{73} +2.24531 q^{77} -6.66834 q^{79} -3.50566 q^{83} +20.3929 q^{85} +1.78662 q^{89} -0.458685 q^{91} +11.6306 q^{95} +13.1627 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 4 * q^7	$ 4 q - 2 q^{5} + 4 q^{7} + 8 q^{13} - 2 q^{17} + 8 q^{19} + 14 q^{25} - 6 q^{29} + 6 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} - 2 q^{43} + 2 q^{47} + 4 q^{49} - 6 q^{53} - 12 q^{55} + 4 q^{61} - 4 q^{65} - 4 q^{67} + 16 q^{71} + 12 q^{73} - 2 q^{79} - 18 q^{83} + 22 q^{85} + 8 q^{89} + 8 q^{91} - 16 q^{95} + 24 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 4 * q^7 + 8 * q^13 - 2 * q^17 + 8 * q^19 + 14 * q^25 - 6 * q^29 + 6 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^41 - 2 * q^43 + 2 * q^47 + 4 * q^49 - 6 * q^53 - 12 * q^55 + 4 * q^61 - 4 * q^65 - 4 * q^67 + 16 * q^71 + 12 * q^73 - 2 * q^79 - 18 * q^83 + 22 * q^85 + 8 * q^89 + 8 * q^91 - 16 * q^95 + 24 * 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
$5$	2.73965	1.22521	0.612604	−	0.790390i	$-0.290122\pi$
$5$	2.73965	1.22521	0.612604	+	0.790390i	$0.290122\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.24531	0.676986	0.338493	−	0.940969i	$-0.390083\pi$
$11$	2.24531	0.676986	0.338493	+	0.940969i	$0.390083\pi$
$12$	0	0
$13$	−0.458685	−0.127216	−0.0636082	−	0.997975i	$-0.520261\pi$
$13$	−0.458685	−0.127216	−0.0636082	+	0.997975i	$0.520261\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.44364	1.80535	0.902674	−	0.430325i	$-0.141601\pi$
$17$	7.44364	1.80535	0.902674	+	0.430325i	$0.141601\pi$
$18$	0	0
$19$	4.24531	0.973941	0.486970	−	0.873418i	$-0.338102\pi$
$19$	4.24531	0.973941	0.486970	+	0.873418i	$0.338102\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.70399	0.980851	0.490425	−	0.871483i	$-0.336841\pi$
$23$	4.70399	0.980851	0.490425	+	0.871483i	$0.336841\pi$
$24$	0	0
$25$	2.50566	0.501133
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.73965	0.323044	0.161522	−	0.986869i	$-0.448360\pi$
$29$	1.73965	0.323044	0.161522	+	0.986869i	$0.448360\pi$
$30$	0	0
$31$	−1.73965	−0.312450	−0.156225	−	0.987722i	$-0.549932\pi$
$31$	−1.73965	−0.312450	−0.156225	+	0.987722i	$0.549932\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73965	0.463085
$36$	0	0
$37$	2.49434	0.410066	0.205033	−	0.978755i	$-0.434270\pi$
$37$	2.49434	0.410066	0.205033	+	0.978755i	$0.434270\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.73965	0.740208	0.370104	−	0.928990i	$-0.379322\pi$
$41$	4.73965	0.740208	0.370104	+	0.928990i	$0.379322\pi$
$42$	0	0
$43$	−1.96435	−0.299560	−0.149780	−	0.988719i	$-0.547857\pi$
$43$	−1.96435	−0.299560	−0.149780	+	0.988719i	$0.547857\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.90233	−1.44440	−0.722201	−	0.691683i	$-0.756870\pi$
$47$	−9.90233	−1.44440	−0.722201	+	0.691683i	$0.756870\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.42303	−0.744911	−0.372455	−	0.928050i	$-0.621484\pi$
$53$	−5.42303	−0.744911	−0.372455	+	0.928050i	$0.621484\pi$
$54$	0	0
$55$	6.15135	0.829448
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.47929	−0.843532	−0.421766	−	0.906705i	$-0.638590\pi$
$59$	−6.47929	−0.843532	−0.421766	+	0.906705i	$0.638590\pi$
$60$	0	0
$61$	5.23398	0.670143	0.335071	−	0.942193i	$-0.391240\pi$
$61$	5.23398	0.670143	0.335071	+	0.942193i	$0.391240\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.25664	−0.155866
$66$	0	0
$67$	−2.77530	−0.339057	−0.169528	−	0.985525i	$-0.554224\pi$
$67$	−2.77530	−0.339057	−0.169528	+	0.985525i	$0.554224\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.703994	−0.0835488	−0.0417744	−	0.999127i	$-0.513301\pi$
$71$	−0.703994	−0.0835488	−0.0417744	+	0.999127i	$0.513301\pi$
$72$	0	0
$73$	−7.96991	−0.932808	−0.466404	−	0.884572i	$-0.654451\pi$
$73$	−7.96991	−0.932808	−0.466404	+	0.884572i	$0.654451\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.24531	0.255877
$78$	0	0
$79$	−6.66834	−0.750247	−0.375124	−	0.926975i	$-0.622400\pi$
$79$	−6.66834	−0.750247	−0.375124	+	0.926975i	$0.622400\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.50566	−0.384796	−0.192398	−	0.981317i	$-0.561627\pi$
$83$	−3.50566	−0.384796	−0.192398	+	0.981317i	$0.561627\pi$
$84$	0	0
$85$	20.3929	2.21193
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.78662	0.189382	0.0946909	−	0.995507i	$-0.469814\pi$
$89$	1.78662	0.189382	0.0946909	+	0.995507i	$0.469814\pi$
$90$	0	0
$91$	−0.458685	−0.0480833
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.6306	1.19328
$96$	0	0
$97$	13.1627	1.33647	0.668234	−	0.743951i	$-0.267051\pi$
$97$	13.1627	1.33647	0.668234	+	0.743951i	$0.267051\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.49062	0.247826	0.123913	−	0.992293i	$-0.460456\pi$
$101$	2.49062	0.247826	0.123913	+	0.992293i	$0.460456\pi$
$102$	0	0
$103$	−10.3816	−1.02293	−0.511466	−	0.859304i	$-0.670897\pi$
$103$	−10.3816	−1.02293	−0.511466	+	0.859304i	$0.670897\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	8.49434	0.813610	0.406805	−	0.913515i	$-0.366643\pi$
$109$	8.49434	0.813610	0.406805	+	0.913515i	$0.366643\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.75469	0.165067	0.0825337	−	0.996588i	$-0.473699\pi$
$113$	1.75469	0.165067	0.0825337	+	0.996588i	$0.473699\pi$
$114$	0	0
$115$	12.8873	1.20175
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.44364	0.682357
$120$	0	0
$121$	−5.95859	−0.541690
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.83360	−0.611216
$126$	0	0
$127$	−2.24903	−0.199569	−0.0997845	−	0.995009i	$-0.531815\pi$
$127$	−2.24903	−0.199569	−0.0997845	+	0.995009i	$0.531815\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.9549	−1.56872	−0.784362	−	0.620303i	$-0.787010\pi$
$131$	−17.9549	−1.56872	−0.784362	+	0.620303i	$0.787010\pi$
$132$	0	0
$133$	4.24531	0.368115
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.24531	0.704444	0.352222	−	0.935916i	$-0.385426\pi$
$137$	8.24531	0.704444	0.352222	+	0.935916i	$0.385426\pi$
$138$	0	0
$139$	−12.2865	−1.04213	−0.521065	−	0.853517i	$-0.674465\pi$
$139$	−12.2865	−1.04213	−0.521065	+	0.853517i	$0.674465\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.02989	−0.0861237
$144$	0	0
$145$	4.76602	0.395796
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.23027	−0.674250	−0.337125	−	0.941460i	$-0.609454\pi$
$149$	−8.23027	−0.674250	−0.337125	+	0.941460i	$0.609454\pi$
$150$	0	0
$151$	13.6269	1.10894	0.554472	−	0.832202i	$-0.312920\pi$
$151$	13.6269	1.10894	0.554472	+	0.832202i	$0.312920\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.76602	−0.382816
$156$	0	0
$157$	19.4173	1.54967	0.774833	−	0.632165i	$-0.217834\pi$
$157$	19.4173	1.54967	0.774833	+	0.632165i	$0.217834\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.70399	0.370727
$162$	0	0
$163$	12.0056	0.940348	0.470174	−	0.882574i	$-0.344191\pi$
$163$	12.0056	0.940348	0.470174	+	0.882574i	$0.344191\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.06759	0.469524	0.234762	−	0.972053i	$-0.424569\pi$
$167$	6.06759	0.469524	0.234762	+	0.972053i	$0.424569\pi$
$168$	0	0
$169$	−12.7896	−0.983816
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.98867	−0.683396	−0.341698	−	0.939810i	$-0.611002\pi$
$173$	−8.98867	−0.683396	−0.341698	+	0.939810i	$0.611002\pi$
$174$	0	0
$175$	2.50566	0.189410
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.2039	1.58485	0.792427	−	0.609967i	$-0.208817\pi$
$179$	21.2039	1.58485	0.792427	+	0.609967i	$0.208817\pi$
$180$	0	0
$181$	18.1833	1.35155	0.675777	−	0.737107i	$-0.263808\pi$
$181$	18.1833	1.35155	0.675777	+	0.737107i	$0.263808\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.83360	0.502416
$186$	0	0
$187$	16.7133	1.22220
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.9699	−1.87912	−0.939558	−	0.342389i	$-0.888764\pi$
$191$	−25.9699	−1.87912	−0.939558	+	0.342389i	$0.888764\pi$
$192$	0	0
$193$	18.9699	1.36548	0.682742	−	0.730659i	$-0.260787\pi$
$193$	18.9699	1.36548	0.682742	+	0.730659i	$0.260787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.31661	−0.165052	−0.0825259	−	0.996589i	$-0.526299\pi$
$197$	−2.31661	−0.165052	−0.0825259	+	0.996589i	$0.526299\pi$
$198$	0	0
$199$	−21.3929	−1.51651	−0.758253	−	0.651961i	$-0.773947\pi$
$199$	−21.3929	−1.51651	−0.758253	+	0.651961i	$0.773947\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.73965	0.122099
$204$	0	0
$205$	12.9850	0.906909
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.53203	0.659344
$210$	0	0
$211$	−25.1007	−1.72800	−0.864000	−	0.503491i	$-0.832049\pi$
$211$	−25.1007	−1.72800	−0.864000	+	0.503491i	$0.832049\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.38162	−0.367023
$216$	0	0
$217$	−1.73965	−0.118095
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.41429	−0.229670
$222$	0	0
$223$	25.1025	1.68099	0.840494	−	0.541821i	$-0.182265\pi$
$223$	25.1025	1.68099	0.840494	+	0.541821i	$0.182265\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.4529	−1.35751	−0.678754	−	0.734366i	$-0.737480\pi$
$227$	−20.4529	−1.35751	−0.678754	+	0.734366i	$0.737480\pi$
$228$	0	0
$229$	−2.88728	−0.190797	−0.0953985	−	0.995439i	$-0.530413\pi$
$229$	−2.88728	−0.190797	−0.0953985	+	0.995439i	$0.530413\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.6119	−1.08828	−0.544140	−	0.838994i	$-0.683144\pi$
$233$	−16.6119	−1.08828	−0.544140	+	0.838994i	$0.683144\pi$
$234$	0	0
$235$	−27.1289	−1.76969
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.7658	1.01981	0.509903	−	0.860232i	$-0.329681\pi$
$239$	15.7658	1.01981	0.509903	+	0.860232i	$0.329681\pi$
$240$	0	0
$241$	1.82599	0.117623	0.0588113	−	0.998269i	$-0.481269\pi$
$241$	1.82599	0.117623	0.0588113	+	0.998269i	$0.481269\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.73965	0.175030
$246$	0	0
$247$	−1.94726	−0.123901
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.4529	1.22786	0.613929	−	0.789361i	$-0.289588\pi$
$251$	19.4529	1.22786	0.613929	+	0.789361i	$0.289588\pi$
$252$	0	0
$253$	10.5619	0.664022
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.43808	0.463975	0.231987	−	0.972719i	$-0.425477\pi$
$257$	7.43808	0.463975	0.231987	+	0.972719i	$0.425477\pi$
$258$	0	0
$259$	2.49434	0.154991
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.90789	0.487621	0.243811	−	0.969823i	$-0.421602\pi$
$263$	7.90789	0.487621	0.243811	+	0.969823i	$0.421602\pi$
$264$	0	0
$265$	−14.8572	−0.912670
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−28.7096	−1.75045	−0.875226	−	0.483714i	$-0.839288\pi$
$269$	−28.7096	−1.75045	−0.875226	+	0.483714i	$0.839288\pi$
$270$	0	0
$271$	21.3929	1.29953	0.649764	−	0.760136i	$-0.274868\pi$
$271$	21.3929	1.29953	0.649764	+	0.760136i	$0.274868\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.62599	0.339260
$276$	0	0
$277$	−1.47557	−0.0886587	−0.0443293	−	0.999017i	$-0.514115\pi$
$277$	−1.47557	−0.0886587	−0.0443293	+	0.999017i	$0.514115\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.96991	−0.475445	−0.237723	−	0.971333i	$-0.576401\pi$
$281$	−7.96991	−0.475445	−0.237723	+	0.971333i	$0.576401\pi$
$282$	0	0
$283$	−12.3166	−0.732147	−0.366073	−	0.930586i	$-0.619298\pi$
$283$	−12.3166	−0.732147	−0.366073	+	0.930586i	$0.619298\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.73965	0.279772
$288$	0	0
$289$	38.4078	2.25928
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−33.0349	−1.92992	−0.964960	−	0.262396i	$-0.915487\pi$
$293$	−33.0349	−1.92992	−0.964960	+	0.262396i	$0.915487\pi$
$294$	0	0
$295$	−17.7510	−1.03350
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.15765	−0.124780
$300$	0	0
$301$	−1.96435	−0.113223
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.3393	0.821064
$306$	0	0
$307$	−10.9285	−0.623722	−0.311861	−	0.950128i	$-0.600952\pi$
$307$	−10.9285	−0.623722	−0.311861	+	0.950128i	$0.600952\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.4342	−1.15871	−0.579357	−	0.815074i	$-0.696696\pi$
$311$	−20.4342	−1.15871	−0.579357	+	0.815074i	$0.696696\pi$
$312$	0	0
$313$	33.3066	1.88260	0.941300	−	0.337571i	$-0.109605\pi$
$313$	33.3066	1.88260	0.941300	+	0.337571i	$0.109605\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.9699	−1.34628	−0.673142	−	0.739513i	$-0.735056\pi$
$317$	−23.9699	−1.34628	−0.673142	+	0.739513i	$0.735056\pi$
$318$	0	0
$319$	3.90604	0.218697
$320$	0	0
$321$	0	0
$322$	0	0
$323$	31.6006	1.75830
$324$	0	0
$325$	−1.14931	−0.0637523
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.90233	−0.545933
$330$	0	0
$331$	18.9530	1.04175	0.520876	−	0.853632i	$-0.325605\pi$
$331$	18.9530	1.04175	0.520876	+	0.853632i	$0.325605\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.60334	−0.415415
$336$	0	0
$337$	23.7933	1.29611	0.648053	−	0.761596i	$-0.275584\pi$
$337$	23.7933	1.29611	0.648053	+	0.761596i	$0.275584\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.90604	−0.211524
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.58943	−0.300056	−0.150028	−	0.988682i	$-0.547936\pi$
$347$	−5.58943	−0.300056	−0.150028	+	0.988682i	$0.547936\pi$
$348$	0	0
$349$	12.2772	0.657186	0.328593	−	0.944472i	$-0.393426\pi$
$349$	12.2772	0.657186	0.328593	+	0.944472i	$0.393426\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.92293	−0.474920	−0.237460	−	0.971397i	$-0.576315\pi$
$353$	−8.92293	−0.474920	−0.237460	+	0.971397i	$0.576315\pi$
$354$	0	0
$355$	−1.92870	−0.102365
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.81651	−0.359762	−0.179881	−	0.983688i	$-0.557571\pi$
$359$	−6.81651	−0.359762	−0.179881	+	0.983688i	$0.557571\pi$
$360$	0	0
$361$	−0.977349	−0.0514394
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.8347	−1.14288
$366$	0	0
$367$	−17.4230	−0.909475	−0.454737	−	0.890626i	$-0.650267\pi$
$367$	−17.4230	−0.909475	−0.454737	+	0.890626i	$0.650267\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.42303	−0.281550
$372$	0	0
$373$	−27.3214	−1.41465	−0.707325	−	0.706888i	$-0.750098\pi$
$373$	−27.3214	−1.41465	−0.707325	+	0.706888i	$0.750098\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.797950	−0.0410965
$378$	0	0
$379$	13.5968	0.698423	0.349211	−	0.937044i	$-0.386450\pi$
$379$	13.5968	0.698423	0.349211	+	0.937044i	$0.386450\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.5582	0.743889	0.371945	−	0.928255i	$-0.378691\pi$
$383$	14.5582	0.743889	0.371945	+	0.928255i	$0.378691\pi$
$384$	0	0
$385$	6.15135	0.313502
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.79591	−0.395268	−0.197634	−	0.980276i	$-0.563326\pi$
$389$	−7.79591	−0.395268	−0.197634	+	0.980276i	$0.563326\pi$
$390$	0	0
$391$	35.0148	1.77078
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.2689	−0.919208
$396$	0	0
$397$	−6.53183	−0.327823	−0.163912	−	0.986475i	$-0.552411\pi$
$397$	−6.53183	−0.327823	−0.163912	+	0.986475i	$0.552411\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.3254	1.41450	0.707250	−	0.706963i	$-0.249935\pi$
$401$	28.3254	1.41450	0.707250	+	0.706963i	$0.249935\pi$
$402$	0	0
$403$	0.797950	0.0397487
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.60056	0.277609
$408$	0	0
$409$	36.8572	1.82247	0.911235	−	0.411886i	$-0.135130\pi$
$409$	36.8572	1.82247	0.911235	+	0.411886i	$0.135130\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.47929	−0.318825
$414$	0	0
$415$	−9.60428	−0.471455
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.3254	1.04181	0.520906	−	0.853614i	$-0.325594\pi$
$419$	21.3254	1.04181	0.520906	+	0.853614i	$0.325594\pi$
$420$	0	0
$421$	−20.0111	−0.975283	−0.487641	−	0.873044i	$-0.662143\pi$
$421$	−20.0111	−0.975283	−0.487641	+	0.873044i	$0.662143\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.6513	0.904719
$426$	0	0
$427$	5.23398	0.253290
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.50938	0.361714	0.180857	−	0.983509i	$-0.442113\pi$
$431$	7.50938	0.361714	0.180857	+	0.983509i	$0.442113\pi$
$432$	0	0
$433$	11.2752	0.541852	0.270926	−	0.962600i	$-0.412670\pi$
$433$	11.2752	0.541852	0.270926	+	0.962600i	$0.412670\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.9699	0.955290
$438$	0	0
$439$	−24.8421	−1.18565	−0.592826	−	0.805331i	$-0.701988\pi$
$439$	−24.8421	−1.18565	−0.592826	+	0.805331i	$0.701988\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.1851	−1.52916	−0.764581	−	0.644528i	$-0.777054\pi$
$443$	−32.1851	−1.52916	−0.764581	+	0.644528i	$0.777054\pi$
$444$	0	0
$445$	4.89472	0.232032
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.2340	0.907708	0.453854	−	0.891076i	$-0.350049\pi$
$449$	19.2340	0.907708	0.453854	+	0.891076i	$0.350049\pi$
$450$	0	0
$451$	10.6420	0.501111
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.25664	−0.0589120
$456$	0	0
$457$	17.8197	0.833570	0.416785	−	0.909005i	$-0.363157\pi$
$457$	17.8197	0.833570	0.416785	+	0.909005i	$0.363157\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.69843	−0.218828	−0.109414	−	0.993996i	$-0.534897\pi$
$461$	−4.69843	−0.218828	−0.109414	+	0.993996i	$0.534897\pi$
$462$	0	0
$463$	−12.2902	−0.571176	−0.285588	−	0.958352i	$-0.592189\pi$
$463$	−12.2902	−0.571176	−0.285588	+	0.958352i	$0.592189\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.2839	1.35510	0.677550	−	0.735477i	$-0.263042\pi$
$467$	29.2839	1.35510	0.677550	+	0.735477i	$0.263042\pi$
$468$	0	0
$469$	−2.77530	−0.128151
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.41057	−0.202798
$474$	0	0
$475$	10.6373	0.488073
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−37.3289	−1.70560	−0.852800	−	0.522238i	$-0.825097\pi$
$479$	−37.3289	−1.70560	−0.852800	+	0.522238i	$0.825097\pi$
$480$	0	0
$481$	−1.14412	−0.0521672
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.0611	1.63745
$486$	0	0
$487$	18.9285	0.857732	0.428866	−	0.903368i	$-0.358913\pi$
$487$	18.9285	0.857732	0.428866	+	0.903368i	$0.358913\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.0273	1.44537	0.722686	−	0.691176i	$-0.242907\pi$
$491$	32.0273	1.44537	0.722686	+	0.691176i	$0.242907\pi$
$492$	0	0
$493$	12.9493	0.583207
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.703994	−0.0315785
$498$	0	0
$499$	36.5554	1.63645	0.818223	−	0.574901i	$-0.194960\pi$
$499$	36.5554	1.63645	0.818223	+	0.574901i	$0.194960\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.57697	0.427016	0.213508	−	0.976941i	$-0.431511\pi$
$503$	9.57697	0.427016	0.213508	+	0.976941i	$0.431511\pi$
$504$	0	0
$505$	6.82341	0.303638
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.54430	0.201422	0.100711	−	0.994916i	$-0.467888\pi$
$509$	4.54430	0.201422	0.100711	+	0.994916i	$0.467888\pi$
$510$	0	0
$511$	−7.96991	−0.352568
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−28.4420	−1.25330
$516$	0	0
$517$	−22.2338	−0.977841
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.2370	1.41233	0.706164	−	0.708049i	$-0.250424\pi$
$521$	32.2370	1.41233	0.706164	+	0.708049i	$0.250424\pi$
$522$	0	0
$523$	24.5707	1.07440	0.537200	−	0.843455i	$-0.319482\pi$
$523$	24.5707	1.07440	0.537200	+	0.843455i	$0.319482\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.9493	−0.564081
$528$	0	0
$529$	−0.872436	−0.0379320
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.17401	−0.0941666
$534$	0	0
$535$	10.9586	0.473781
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.24531	0.0967123
$540$	0	0
$541$	38.8007	1.66817	0.834087	−	0.551633i	$-0.185995\pi$
$541$	38.8007	1.66817	0.834087	+	0.551633i	$0.185995\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	23.2715	0.996841
$546$	0	0
$547$	−24.8035	−1.06052	−0.530261	−	0.847835i	$-0.677906\pi$
$547$	−24.8035	−1.06052	−0.530261	+	0.847835i	$0.677906\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.38534	0.314626
$552$	0	0
$553$	−6.66834	−0.283567
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.29919	−0.0974197	−0.0487099	−	0.998813i	$-0.515511\pi$
$557$	−2.29919	−0.0974197	−0.0487099	+	0.998813i	$0.515511\pi$
$558$	0	0
$559$	0.901017	0.0381090
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.9021	−1.51309	−0.756547	−	0.653939i	$-0.773115\pi$
$563$	−35.9021	−1.51309	−0.756547	+	0.653939i	$0.773115\pi$
$564$	0	0
$565$	4.80723	0.202242
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.3591	1.10503	0.552516	−	0.833502i	$-0.313668\pi$
$569$	26.3591	1.10503	0.552516	+	0.833502i	$0.313668\pi$
$570$	0	0
$571$	−16.8591	−0.705530	−0.352765	−	0.935712i	$-0.614758\pi$
$571$	−16.8591	−0.705530	−0.352765	+	0.935712i	$0.614758\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.7866	0.491536
$576$	0	0
$577$	3.43808	0.143129	0.0715645	−	0.997436i	$-0.477201\pi$
$577$	3.43808	0.143129	0.0715645	+	0.997436i	$0.477201\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.50566	−0.145439
$582$	0	0
$583$	−12.1764	−0.504294
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.3291	−0.962894	−0.481447	−	0.876475i	$-0.659889\pi$
$587$	−23.3291	−0.962894	−0.481447	+	0.876475i	$0.659889\pi$
$588$	0	0
$589$	−7.38534	−0.304308
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.2604	0.626668	0.313334	−	0.949643i	$-0.398554\pi$
$593$	15.2604	0.626668	0.313334	+	0.949643i	$0.398554\pi$
$594$	0	0
$595$	20.3929	0.836029
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.8366	−0.483630	−0.241815	−	0.970322i	$-0.577743\pi$
$599$	−11.8366	−0.483630	−0.241815	+	0.970322i	$0.577743\pi$
$600$	0	0
$601$	−13.7473	−0.560762	−0.280381	−	0.959889i	$-0.590461\pi$
$601$	−13.7473	−0.560762	−0.280381	+	0.959889i	$0.590461\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16.3244	−0.663682
$606$	0	0
$607$	44.5330	1.80754	0.903769	−	0.428021i	$-0.140789\pi$
$607$	44.5330	1.80754	0.903769	+	0.428021i	$0.140789\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.54205	0.183752
$612$	0	0
$613$	−41.2951	−1.66789	−0.833946	−	0.551846i	$-0.813924\pi$
$613$	−41.2951	−1.66789	−0.833946	+	0.551846i	$0.813924\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.46053	0.340608	0.170304	−	0.985392i	$-0.445525\pi$
$617$	8.46053	0.340608	0.170304	+	0.985392i	$0.445525\pi$
$618$	0	0
$619$	−33.9259	−1.36360	−0.681799	−	0.731540i	$-0.738802\pi$
$619$	−33.9259	−1.36360	−0.681799	+	0.731540i	$0.738802\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.78662	0.0715796
$624$	0	0
$625$	−31.2500	−1.25000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.5670	0.740313
$630$	0	0
$631$	−45.6982	−1.81922	−0.909609	−	0.415465i	$-0.863619\pi$
$631$	−45.6982	−1.81922	−0.909609	+	0.415465i	$0.863619\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.16154	−0.244513
$636$	0	0
$637$	−0.458685	−0.0181738
$638$	0	0
$639$	0	0
$640$	0	0
$641$	48.0685	1.89859	0.949296	−	0.314383i	$-0.101798\pi$
$641$	48.0685	1.89859	0.949296	+	0.314383i	$0.101798\pi$
$642$	0	0
$643$	−24.2013	−0.954407	−0.477203	−	0.878793i	$-0.658349\pi$
$643$	−24.2013	−0.954407	−0.477203	+	0.878793i	$0.658349\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.5207	0.492240	0.246120	−	0.969239i	$-0.420844\pi$
$647$	12.5207	0.492240	0.246120	+	0.969239i	$0.420844\pi$
$648$	0	0
$649$	−14.5480	−0.571060
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.0951	−0.708116	−0.354058	−	0.935224i	$-0.615198\pi$
$653$	−18.0951	−0.708116	−0.354058	+	0.935224i	$0.615198\pi$
$654$	0	0
$655$	−49.1900	−1.92201
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.86483	−0.306370	−0.153185	−	0.988198i	$-0.548953\pi$
$659$	−7.86483	−0.306370	−0.153185	+	0.988198i	$0.548953\pi$
$660$	0	0
$661$	17.1738	0.667984	0.333992	−	0.942576i	$-0.391604\pi$
$661$	17.1738	0.667984	0.333992	+	0.942576i	$0.391604\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.6306	0.451017
$666$	0	0
$667$	8.18329	0.316858
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.7519	0.453678
$672$	0	0
$673$	2.64083	0.101797	0.0508983	−	0.998704i	$-0.483792\pi$
$673$	2.64083	0.101797	0.0508983	+	0.998704i	$0.483792\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.5545	−0.559374	−0.279687	−	0.960091i	$-0.590231\pi$
$677$	−14.5545	−0.559374	−0.279687	+	0.960091i	$0.590231\pi$
$678$	0	0
$679$	13.1627	0.505137
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.23398	0.276801	0.138400	−	0.990376i	$-0.455804\pi$
$683$	7.23398	0.276801	0.138400	+	0.990376i	$0.455804\pi$
$684$	0	0
$685$	22.5892	0.863090
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.48746	0.0947649
$690$	0	0
$691$	6.79610	0.258536	0.129268	−	0.991610i	$-0.458737\pi$
$691$	6.79610	0.258536	0.129268	+	0.991610i	$0.458737\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−33.6607	−1.27682
$696$	0	0
$697$	35.2802	1.33633
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.83732	−0.333781	−0.166890	−	0.985975i	$-0.553373\pi$
$701$	−8.83732	−0.333781	−0.166890	+	0.985975i	$0.553373\pi$
$702$	0	0
$703$	10.5892	0.399380
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.49062	0.0936694
$708$	0	0
$709$	−25.8195	−0.969672	−0.484836	−	0.874605i	$-0.661121\pi$
$709$	−25.8195	−0.969672	−0.484836	+	0.874605i	$0.661121\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.18329	−0.306467
$714$	0	0
$715$	−2.82153	−0.105519
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.3216	1.72751	0.863753	−	0.503916i	$-0.168108\pi$
$719$	46.3216	1.72751	0.863753	+	0.503916i	$0.168108\pi$
$720$	0	0
$721$	−10.3816	−0.386632
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.35897	0.161888
$726$	0	0
$727$	31.3240	1.16174	0.580872	−	0.813995i	$-0.302712\pi$
$727$	31.3240	1.16174	0.580872	+	0.813995i	$0.302712\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−14.6219	−0.540811
$732$	0	0
$733$	35.0405	1.29425	0.647125	−	0.762384i	$-0.275971\pi$
$733$	35.0405	1.29425	0.647125	+	0.762384i	$0.275971\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.23140	−0.229537
$738$	0	0
$739$	−16.0824	−0.591602	−0.295801	−	0.955250i	$-0.595586\pi$
$739$	−16.0824	−0.591602	−0.295801	+	0.955250i	$0.595586\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.39871	0.124686	0.0623432	−	0.998055i	$-0.480143\pi$
$743$	3.39871	0.124686	0.0623432	+	0.998055i	$0.480143\pi$
$744$	0	0
$745$	−22.5480	−0.826095
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	14.4041	0.525613	0.262807	−	0.964849i	$-0.415352\pi$
$751$	14.4041	0.525613	0.262807	+	0.964849i	$0.415352\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37.3330	1.35869
$756$	0	0
$757$	−15.2691	−0.554965	−0.277482	−	0.960731i	$-0.589500\pi$
$757$	−15.2691	−0.554965	−0.277482	+	0.960731i	$0.589500\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.7276	−1.18637	−0.593187	−	0.805065i	$-0.702131\pi$
$761$	−32.7276	−1.18637	−0.593187	+	0.805065i	$0.702131\pi$
$762$	0	0
$763$	8.49434	0.307516
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.97196	0.107311
$768$	0	0
$769$	7.40799	0.267139	0.133569	−	0.991039i	$-0.457356\pi$
$769$	7.40799	0.267139	0.133569	+	0.991039i	$0.457356\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.6982	−0.492691	−0.246346	−	0.969182i	$-0.579230\pi$
$773$	−13.6982	−0.492691	−0.246346	+	0.969182i	$0.579230\pi$
$774$	0	0
$775$	−4.35897	−0.156579
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.1213	0.720919
$780$	0	0
$781$	−1.58069	−0.0565614
$782$	0	0
$783$	0	0
$784$	0	0
$785$	53.1965	1.89866
$786$	0	0
$787$	−35.3367	−1.25962	−0.629808	−	0.776751i	$-0.716867\pi$
$787$	−35.3367	−1.25962	−0.629808	+	0.776751i	$0.716867\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.75469	0.0623896
$792$	0	0
$793$	−2.40075	−0.0852531
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.40055	0.191297	0.0956487	−	0.995415i	$-0.469507\pi$
$797$	5.40055	0.191297	0.0956487	+	0.995415i	$0.469507\pi$
$798$	0	0
$799$	−73.7094	−2.60765
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.8949	−0.631498
$804$	0	0
$805$	12.8873	0.454217
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−35.7445	−1.25671	−0.628354	−	0.777927i	$-0.716271\pi$
$809$	−35.7445	−1.25671	−0.628354	+	0.777927i	$0.716271\pi$
$810$	0	0
$811$	8.34670	0.293092	0.146546	−	0.989204i	$-0.453184\pi$
$811$	8.34670	0.293092	0.146546	+	0.989204i	$0.453184\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	32.8910	1.15212
$816$	0	0
$817$	−8.33926	−0.291754
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.8797	−0.589105	−0.294552	−	0.955635i	$-0.595171\pi$
$821$	−16.8797	−0.589105	−0.294552	+	0.955635i	$0.595171\pi$
$822$	0	0
$823$	16.3315	0.569279	0.284639	−	0.958635i	$-0.408126\pi$
$823$	16.3315	0.569279	0.284639	+	0.958635i	$0.408126\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.6533	1.23979	0.619893	−	0.784686i	$-0.287176\pi$
$827$	35.6533	1.23979	0.619893	+	0.784686i	$0.287176\pi$
$828$	0	0
$829$	−47.0011	−1.63242	−0.816208	−	0.577758i	$-0.803928\pi$
$829$	−47.0011	−1.63242	−0.816208	+	0.577758i	$0.803928\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.44364	0.257907
$834$	0	0
$835$	16.6230	0.575264
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.8724	−0.582501	−0.291251	−	0.956647i	$-0.594071\pi$
$839$	−16.8724	−0.582501	−0.291251	+	0.956647i	$0.594071\pi$
$840$	0	0
$841$	−25.9736	−0.895642
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−35.0390	−1.20538
$846$	0	0
$847$	−5.95859	−0.204739
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.7333	0.402214
$852$	0	0
$853$	−7.22470	−0.247369	−0.123685	−	0.992322i	$-0.539471\pi$
$853$	−7.22470	−0.247369	−0.123685	+	0.992322i	$0.539471\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.3796	−1.51598	−0.757989	−	0.652268i	$-0.773818\pi$
$857$	−44.3796	−1.51598	−0.757989	+	0.652268i	$0.773818\pi$
$858$	0	0
$859$	53.2464	1.81674	0.908372	−	0.418163i	$-0.137326\pi$
$859$	53.2464	1.81674	0.908372	+	0.418163i	$0.137326\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.0594	−0.580710	−0.290355	−	0.956919i	$-0.593773\pi$
$863$	−17.0594	−0.580710	−0.290355	+	0.956919i	$0.593773\pi$
$864$	0	0
$865$	−24.6258	−0.837302
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.9725	−0.507907
$870$	0	0
$871$	1.27299	0.0431336
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.83360	−0.231018
$876$	0	0
$877$	−7.59553	−0.256483	−0.128241	−	0.991743i	$-0.540933\pi$
$877$	−7.59553	−0.256483	−0.128241	+	0.991743i	$0.540933\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.7040	0.764917	0.382459	−	0.923973i	$-0.375077\pi$
$881$	22.7040	0.764917	0.382459	+	0.923973i	$0.375077\pi$
$882$	0	0
$883$	−31.8704	−1.07252	−0.536262	−	0.844052i	$-0.680164\pi$
$883$	−31.8704	−1.07252	−0.536262	+	0.844052i	$0.680164\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.3029	1.18535	0.592677	−	0.805440i	$-0.298071\pi$
$887$	35.3029	1.18535	0.592677	+	0.805440i	$0.298071\pi$
$888$	0	0
$889$	−2.24903	−0.0754300
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−42.0384	−1.40676
$894$	0	0
$895$	58.0912	1.94177
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.02637	−0.100935
$900$	0	0
$901$	−40.3671	−1.34482
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49.8158	1.65593
$906$	0	0
$907$	15.1890	0.504344	0.252172	−	0.967682i	$-0.418855\pi$
$907$	15.1890	0.504344	0.252172	+	0.967682i	$0.418855\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.1138	−1.46156	−0.730778	−	0.682615i	$-0.760843\pi$
$911$	−44.1138	−1.46156	−0.730778	+	0.682615i	$0.760843\pi$
$912$	0	0
$913$	−7.87130	−0.260502
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.9549	−0.592922
$918$	0	0
$919$	11.1590	0.368100	0.184050	−	0.982917i	$-0.441079\pi$
$919$	11.1590	0.368100	0.184050	+	0.982917i	$0.441079\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.322912	0.0106288
$924$	0	0
$925$	6.24997	0.205498
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.0536799	−0.00176118	−0.000880591	−	1.00000i	$-0.500280\pi$
$929$	−0.0536799	−0.00176118	−0.000880591		1.00000i	$0.500280\pi$
$930$	0	0
$931$	4.24531	0.139134
$932$	0	0
$933$	0	0
$934$	0	0
$935$	45.7885	1.49744
$936$	0	0
$937$	−0.275398	−0.00899685	−0.00449843	−	0.999990i	$-0.501432\pi$
$937$	−0.275398	−0.00899685	−0.00449843	+	0.999990i	$0.501432\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.301768	0.00983734	0.00491867	−	0.999988i	$-0.498434\pi$
$941$	0.301768	0.00983734	0.00491867	+	0.999988i	$0.498434\pi$
$942$	0	0
$943$	22.2953	0.726034
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.7859	1.26037	0.630186	−	0.776444i	$-0.282979\pi$
$947$	38.7859	1.26037	0.630186	+	0.776444i	$0.282979\pi$
$948$	0	0
$949$	3.65568	0.118668
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.09396	−0.262189	−0.131094	−	0.991370i	$-0.541849\pi$
$953$	−8.09396	−0.262189	−0.131094	+	0.991370i	$0.541849\pi$
$954$	0	0
$955$	−71.1484	−2.30231
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.24531	0.266255
$960$	0	0
$961$	−27.9736	−0.902375
$962$	0	0
$963$	0	0
$964$	0	0
$965$	51.9709	1.67300
$966$	0	0
$967$	−12.9475	−0.416362	−0.208181	−	0.978090i	$-0.566754\pi$
$967$	−12.9475	−0.416362	−0.208181	+	0.978090i	$0.566754\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−44.0886	−1.41487	−0.707435	−	0.706778i	$-0.750148\pi$
$971$	−44.0886	−1.41487	−0.707435	+	0.706778i	$0.750148\pi$
$972$	0	0
$973$	−12.2865	−0.393888
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.3365	−0.970550	−0.485275	−	0.874362i	$-0.661281\pi$
$977$	−30.3365	−0.970550	−0.485275	+	0.874362i	$0.661281\pi$
$978$	0	0
$979$	4.01152	0.128209
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.95507	−0.126147	−0.0630735	−	0.998009i	$-0.520090\pi$
$983$	−3.95507	−0.126147	−0.0630735	+	0.998009i	$0.520090\pi$
$984$	0	0
$985$	−6.34670	−0.202223
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.24028	−0.293824
$990$	0	0
$991$	−11.0650	−0.351491	−0.175746	−	0.984436i	$-0.556234\pi$
$991$	−11.0650	−0.351491	−0.175746	+	0.984436i	$0.556234\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−58.6091	−1.85803
$996$	0	0
$997$	−11.1146	−0.352002	−0.176001	−	0.984390i	$-0.556316\pi$
$997$	−11.1146	−0.352002	−0.176001	+	0.984390i	$0.556316\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.bp.1.4	yes	4
3.2	odd	2		6048.2.a.bt.1.1	yes	4
4.3	odd	2		6048.2.a.bk.1.4	✓	4
12.11	even	2		6048.2.a.bs.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bk.1.4	✓	4	4.3	odd	2
6048.2.a.bp.1.4	yes	4	1.1	even	1	trivial
6048.2.a.bs.1.1	yes	4	12.11	even	2
6048.2.a.bt.1.1	yes	4	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+2.73965 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.73965 q^{5} +1.00000 q^{7} +2.24531 q^{11} -0.458685 q^{13} +7.44364 q^{17} +4.24531 q^{19} +4.70399 q^{23} +2.50566 q^{25} +1.73965 q^{29} -1.73965 q^{31} +2.73965 q^{35} +2.49434 q^{37} +4.73965 q^{41} -1.96435 q^{43} -9.90233 q^{47} +1.00000 q^{49} -5.42303 q^{53} +6.15135 q^{55} -6.47929 q^{59} +5.23398 q^{61} -1.25664 q^{65} -2.77530 q^{67} -0.703994 q^{71} -7.96991 q^{73} +2.24531 q^{77} -6.66834 q^{79} -3.50566 q^{83} +20.3929 q^{85} +1.78662 q^{89} -0.458685 q^{91} +11.6306 q^{95} +13.1627 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 4 * q^7	\( 4 q - 2 q^{5} + 4 q^{7} + 8 q^{13} - 2 q^{17} + 8 q^{19} + 14 q^{25} - 6 q^{29} + 6 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} - 2 q^{43} + 2 q^{47} + 4 q^{49} - 6 q^{53} - 12 q^{55} + 4 q^{61} - 4 q^{65} - 4 q^{67} + 16 q^{71} + 12 q^{73} - 2 q^{79} - 18 q^{83} + 22 q^{85} + 8 q^{89} + 8 q^{91} - 16 q^{95} + 24 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 4 * q^7 + 8 * q^13 - 2 * q^17 + 8 * q^19 + 14 * q^25 - 6 * q^29 + 6 * q^31 - 2 * q^35 + 6 * q^37 + 6 * q^41 - 2 * q^43 + 2 * q^47 + 4 * q^49 - 6 * q^53 - 12 * q^55 + 4 * q^61 - 4 * q^65 - 4 * q^67 + 16 * q^71 + 12 * q^73 - 2 * q^79 - 18 * q^83 + 22 * q^85 + 8 * q^89 + 8 * q^91 - 16 * q^95 + 24 * q^97

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