LMFDB - Embedded newform 6048.2.a.bp.1.1

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			1.1
Root			$2.87063$ 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)

$f(q)$	$=$	$q-4.24049 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-4.24049 q^{5} +1.00000 q^{7} +5.74125 q^{11} +6.06174 q^{13} -2.56098 q^{17} +7.74125 q^{19} +1.67951 q^{23} +12.9817 q^{25} -5.24049 q^{29} +5.24049 q^{31} -4.24049 q^{35} -7.98174 q^{37} -2.24049 q^{41} -5.92000 q^{43} +6.62271 q^{47} +1.00000 q^{49} -2.85826 q^{53} -24.3457 q^{55} +7.48098 q^{59} -12.2222 q^{61} -25.7047 q^{65} +8.16049 q^{67} +2.32049 q^{71} -1.00152 q^{73} +5.74125 q^{77} -7.59951 q^{79} -13.9817 q^{83} +10.8598 q^{85} +11.8030 q^{89} +6.06174 q^{91} -32.8267 q^{95} +3.61777 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.24049	−1.89640	−0.948202	−	0.317668i	$-0.897100\pi$
$5$	−4.24049	−1.89640	−0.948202	+	0.317668i	$0.897100\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.74125	1.73105	0.865526	−	0.500864i	$-0.166984\pi$
$11$	5.74125	1.73105	0.865526	+	0.500864i	$0.166984\pi$
$12$	0	0
$13$	6.06174	1.68122	0.840612	−	0.541638i	$-0.182196\pi$
$13$	6.06174	1.68122	0.840612	+	0.541638i	$0.182196\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.56098	−0.621128	−0.310564	−	0.950552i	$-0.600518\pi$
$17$	−2.56098	−0.621128	−0.310564	+	0.950552i	$0.600518\pi$
$18$	0	0
$19$	7.74125	1.77596	0.887982	−	0.459878i	$-0.152107\pi$
$19$	7.74125	1.77596	0.887982	+	0.459878i	$0.152107\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.67951	0.350203	0.175101	−	0.984550i	$-0.443975\pi$
$23$	1.67951	0.350203	0.175101	+	0.984550i	$0.443975\pi$
$24$	0	0
$25$	12.9817	2.59635
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.24049	−0.973134	−0.486567	−	0.873643i	$-0.661751\pi$
$29$	−5.24049	−0.973134	−0.486567	+	0.873643i	$0.661751\pi$
$30$	0	0
$31$	5.24049	0.941219	0.470610	−	0.882341i	$-0.344034\pi$
$31$	5.24049	0.941219	0.470610	+	0.882341i	$0.344034\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.24049	−0.716773
$36$	0	0
$37$	−7.98174	−1.31219	−0.656095	−	0.754678i	$-0.727793\pi$
$37$	−7.98174	−1.31219	−0.656095	+	0.754678i	$0.727793\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.24049	−0.349905	−0.174953	−	0.984577i	$-0.555977\pi$
$41$	−2.24049	−0.349905	−0.174953	+	0.984577i	$0.555977\pi$
$42$	0	0
$43$	−5.92000	−0.902792	−0.451396	−	0.892324i	$-0.649074\pi$
$43$	−5.92000	−0.902792	−0.451396	+	0.892324i	$0.649074\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.62271	0.966022	0.483011	−	0.875614i	$-0.339543\pi$
$47$	6.62271	0.966022	0.483011	+	0.875614i	$0.339543\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.85826	−0.392613	−0.196306	−	0.980543i	$-0.562895\pi$
$53$	−2.85826	−0.392613	−0.196306	+	0.980543i	$0.562895\pi$
$54$	0	0
$55$	−24.3457	−3.28277
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.48098	0.973940	0.486970	−	0.873419i	$-0.338102\pi$
$59$	7.48098	0.973940	0.486970	+	0.873419i	$0.338102\pi$
$60$	0	0
$61$	−12.2222	−1.56490	−0.782448	−	0.622716i	$-0.786029\pi$
$61$	−12.2222	−1.56490	−0.782448	+	0.622716i	$0.786029\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−25.7047	−3.18828
$66$	0	0
$67$	8.16049	0.996962	0.498481	−	0.866901i	$-0.333891\pi$
$67$	8.16049	0.996962	0.498481	+	0.866901i	$0.333891\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.32049	0.275391	0.137696	−	0.990475i	$-0.456030\pi$
$71$	2.32049	0.275391	0.137696	+	0.990475i	$0.456030\pi$
$72$	0	0
$73$	−1.00152	−0.117220	−0.0586098	−	0.998281i	$-0.518667\pi$
$73$	−1.00152	−0.117220	−0.0586098	+	0.998281i	$0.518667\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.74125	0.654276
$78$	0	0
$79$	−7.59951	−0.855012	−0.427506	−	0.904013i	$-0.640608\pi$
$79$	−7.59951	−0.855012	−0.427506	+	0.904013i	$0.640608\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.9817	−1.53470	−0.767348	−	0.641231i	$-0.778424\pi$
$83$	−13.9817	−1.53470	−0.767348	+	0.641231i	$0.778424\pi$
$84$	0	0
$85$	10.8598	1.17791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.8030	1.25111	0.625557	−	0.780178i	$-0.284872\pi$
$89$	11.8030	1.25111	0.625557	+	0.780178i	$0.284872\pi$
$90$	0	0
$91$	6.06174	0.635443
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−32.8267	−3.36795
$96$	0	0
$97$	3.61777	0.367329	0.183665	−	0.982989i	$-0.441204\pi$
$97$	3.61777	0.367329	0.183665	+	0.982989i	$0.441204\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.48250	0.943544	0.471772	−	0.881721i	$-0.343615\pi$
$101$	9.48250	0.943544	0.471772	+	0.881721i	$0.343615\pi$
$102$	0	0
$103$	20.1037	1.98088	0.990438	−	0.137961i	$-0.0440548\pi$
$103$	20.1037	1.98088	0.990438	+	0.137961i	$0.0440548\pi$
$104$	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
$109$	−1.98174	−0.189816	−0.0949080	−	0.995486i	$-0.530256\pi$
$109$	−1.98174	−0.189816	−0.0949080	+	0.995486i	$0.530256\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.74125	−0.163803	−0.0819015	−	0.996640i	$-0.526099\pi$
$113$	−1.74125	−0.163803	−0.0819015	+	0.996640i	$0.526099\pi$
$114$	0	0
$115$	−7.12195	−0.664125
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.56098	−0.234764
$120$	0	0
$121$	21.9620	1.99654
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−33.8465	−3.02732
$126$	0	0
$127$	11.7230	1.04025	0.520123	−	0.854091i	$-0.325886\pi$
$127$	11.7230	1.04025	0.520123	+	0.854091i	$0.325886\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.50229	−0.655478	−0.327739	−	0.944768i	$-0.606287\pi$
$131$	−7.50229	−0.655478	−0.327739	+	0.944768i	$0.606287\pi$
$132$	0	0
$133$	7.74125	0.671252
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.7413	1.00312	0.501561	−	0.865122i	$-0.332759\pi$
$137$	11.7413	1.00312	0.501561	+	0.865122i	$0.332759\pi$
$138$	0	0
$139$	−0.902774	−0.0765723	−0.0382861	−	0.999267i	$-0.512190\pi$
$139$	−0.902774	−0.0765723	−0.0382861	+	0.999267i	$0.512190\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	34.8020	2.91029
$144$	0	0
$145$	22.2222	1.84546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.24201	−0.675212	−0.337606	−	0.941288i	$-0.609617\pi$
$149$	−8.24201	−0.675212	−0.337606	+	0.941288i	$0.609617\pi$
$150$	0	0
$151$	−13.3624	−1.08742	−0.543710	−	0.839273i	$-0.682981\pi$
$151$	−13.3624	−1.08742	−0.543710	+	0.839273i	$0.682981\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−22.2222	−1.78493
$156$	0	0
$157$	−15.0237	−1.19902	−0.599510	−	0.800367i	$-0.704638\pi$
$157$	−15.0237	−1.19902	−0.599510	+	0.800367i	$0.704638\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.67951	0.132364
$162$	0	0
$163$	1.08152	0.0847115	0.0423557	−	0.999103i	$-0.486514\pi$
$163$	1.08152	0.0847115	0.0423557	+	0.999103i	$0.486514\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.6242	1.20904	0.604520	−	0.796590i	$-0.293365\pi$
$167$	15.6242	1.20904	0.604520	+	0.796590i	$0.293365\pi$
$168$	0	0
$169$	23.7447	1.82651
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.9635	0.909566	0.454783	−	0.890602i	$-0.349717\pi$
$173$	11.9635	0.909566	0.454783	+	0.890602i	$0.349717\pi$
$174$	0	0
$175$	12.9817	0.981327
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.22070	−0.240727	−0.120363	−	0.992730i	$-0.538406\pi$
$179$	−3.22070	−0.240727	−0.120363	+	0.992730i	$0.538406\pi$
$180$	0	0
$181$	1.19854	0.0890865	0.0445433	−	0.999007i	$-0.485817\pi$
$181$	1.19854	0.0890865	0.0445433	+	0.999007i	$0.485817\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	33.8465	2.48844
$186$	0	0
$187$	−14.7032	−1.07520
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.0015	−1.37490	−0.687451	−	0.726231i	$-0.741270\pi$
$191$	−19.0015	−1.37490	−0.687451	+	0.726231i	$0.741270\pi$
$192$	0	0
$193$	12.0015	0.863889	0.431944	−	0.901900i	$-0.357828\pi$
$193$	12.0015	0.863889	0.431944	+	0.901900i	$0.357828\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.09875	0.149530	0.0747649	−	0.997201i	$-0.476179\pi$
$197$	2.09875	0.149530	0.0747649	+	0.997201i	$0.476179\pi$
$198$	0	0
$199$	−11.8598	−0.840718	−0.420359	−	0.907358i	$-0.638096\pi$
$199$	−11.8598	−0.840718	−0.420359	+	0.907358i	$0.638096\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.24049	−0.367810
$204$	0	0
$205$	9.50076	0.663562
$206$	0	0
$207$	0	0
$208$	0	0
$209$	44.4445	3.07429
$210$	0	0
$211$	4.92494	0.339047	0.169523	−	0.985526i	$-0.445777\pi$
$211$	4.92494	0.339047	0.169523	+	0.985526i	$0.445777\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	25.1037	1.71206
$216$	0	0
$217$	5.24049	0.355748
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.5240	−1.04425
$222$	0	0
$223$	1.62082	0.108538	0.0542692	−	0.998526i	$-0.482717\pi$
$223$	1.62082	0.108538	0.0542692	+	0.998526i	$0.482717\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.9437	1.19096	0.595482	−	0.803368i	$-0.296961\pi$
$227$	17.9437	1.19096	0.595482	+	0.803368i	$0.296961\pi$
$228$	0	0
$229$	17.1220	1.13145	0.565725	−	0.824594i	$-0.308596\pi$
$229$	17.1220	1.13145	0.565725	+	0.824594i	$0.308596\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.8617	0.908109	0.454054	−	0.890974i	$-0.349977\pi$
$233$	13.8617	0.908109	0.454054	+	0.890974i	$0.349977\pi$
$234$	0	0
$235$	−28.0835	−1.83197
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.57820	−0.619562	−0.309781	−	0.950808i	$-0.600256\pi$
$239$	−9.57820	−0.619562	−0.309781	+	0.950808i	$0.600256\pi$
$240$	0	0
$241$	−9.58125	−0.617183	−0.308591	−	0.951195i	$-0.599858\pi$
$241$	−9.58125	−0.617183	−0.308591	+	0.951195i	$0.599858\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.24049	−0.270915
$246$	0	0
$247$	46.9254	2.98579
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.9437	−1.19572	−0.597858	−	0.801602i	$-0.703981\pi$
$251$	−18.9437	−1.19572	−0.597858	+	0.801602i	$0.703981\pi$
$252$	0	0
$253$	9.64250	0.606219
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.35750	0.521327	0.260663	−	0.965430i	$-0.416059\pi$
$257$	8.35750	0.521327	0.260663	+	0.965430i	$0.416059\pi$
$258$	0	0
$259$	−7.98174	−0.495961
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.5412	−1.20496	−0.602481	−	0.798133i	$-0.705821\pi$
$263$	−19.5412	−1.20496	−0.602481	+	0.798133i	$0.705821\pi$
$264$	0	0
$265$	12.1204	0.744552
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.7610	−0.899996	−0.449998	−	0.893029i	$-0.648575\pi$
$269$	−14.7610	−0.899996	−0.449998	+	0.893029i	$0.648575\pi$
$270$	0	0
$271$	11.8598	0.720431	0.360215	−	0.932869i	$-0.382703\pi$
$271$	11.8598	0.720431	0.360215	+	0.932869i	$0.382703\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	74.5314	4.49441
$276$	0	0
$277$	−4.98326	−0.299415	−0.149708	−	0.988730i	$-0.547833\pi$
$277$	−4.98326	−0.299415	−0.149708	+	0.988730i	$0.547833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.00152	−0.0597459	−0.0298730	−	0.999554i	$-0.509510\pi$
$281$	−1.00152	−0.0597459	−0.0298730	+	0.999554i	$0.509510\pi$
$282$	0	0
$283$	−7.90125	−0.469681	−0.234840	−	0.972034i	$-0.575457\pi$
$283$	−7.90125	−0.469681	−0.234840	+	0.972034i	$0.575457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.24049	−0.132252
$288$	0	0
$289$	−10.4414	−0.614200
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.00341485	0.000199498 0	9.97490e−5	−	1.00000i	$-0.499968\pi$
$293$	0.00341485	0.000199498 0	9.97490e−5		1.00000i	$0.499968\pi$
$294$	0	0
$295$	−31.7230	−1.84698
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.1808	0.588769
$300$	0	0
$301$	−5.92000	−0.341223
$302$	0	0
$303$	0	0
$304$	0	0
$305$	51.8282	2.96767
$306$	0	0
$307$	23.9604	1.36749	0.683747	−	0.729719i	$-0.260349\pi$
$307$	23.9604	1.36749	0.683747	+	0.729719i	$0.260349\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.97869	0.225611	0.112805	−	0.993617i	$-0.464016\pi$
$311$	3.97869	0.225611	0.112805	+	0.993617i	$0.464016\pi$
$312$	0	0
$313$	28.2005	1.59399	0.796995	−	0.603986i	$-0.206422\pi$
$313$	28.2005	1.59399	0.796995	+	0.603986i	$0.206422\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.0015	−0.954901	−0.477450	−	0.878659i	$-0.658439\pi$
$317$	−17.0015	−0.954901	−0.477450	+	0.878659i	$0.658439\pi$
$318$	0	0
$319$	−30.0870	−1.68455
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.8252	−1.10310
$324$	0	0
$325$	78.6919	4.36504
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.62271	0.365122
$330$	0	0
$331$	1.95652	0.107540	0.0537701	−	0.998553i	$-0.482876\pi$
$331$	1.95652	0.107540	0.0537701	+	0.998553i	$0.482876\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−34.6045	−1.89064
$336$	0	0
$337$	−30.2089	−1.64558	−0.822792	−	0.568343i	$-0.807585\pi$
$337$	−30.2089	−1.64558	−0.822792	+	0.568343i	$0.807585\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	30.0870	1.62930
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.9882	1.28775	0.643877	−	0.765129i	$-0.277325\pi$
$347$	23.9882	1.28775	0.643877	+	0.765129i	$0.277325\pi$
$348$	0	0
$349$	29.2855	1.56762	0.783808	−	0.621003i	$-0.213275\pi$
$349$	29.2855	1.56762	0.783808	+	0.621003i	$0.213275\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.0420	0.800602	0.400301	−	0.916384i	$-0.368906\pi$
$353$	15.0420	0.800602	0.400301	+	0.916384i	$0.368906\pi$
$354$	0	0
$355$	−9.84000	−0.522253
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.9990	1.00273	0.501363	−	0.865237i	$-0.332832\pi$
$359$	18.9990	1.00273	0.501363	+	0.865237i	$0.332832\pi$
$360$	0	0
$361$	40.9270	2.15405
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.24695	0.222296
$366$	0	0
$367$	−14.8583	−0.775595	−0.387797	−	0.921745i	$-0.626764\pi$
$367$	−14.8583	−0.775595	−0.387797	+	0.921745i	$0.626764\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.85826	−0.148394
$372$	0	0
$373$	17.1006	0.885438	0.442719	−	0.896660i	$-0.354014\pi$
$373$	17.1006	0.885438	0.442719	+	0.896660i	$0.354014\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−31.7665	−1.63606
$378$	0	0
$379$	−20.3609	−1.04587	−0.522935	−	0.852373i	$-0.675163\pi$
$379$	−20.3609	−1.04587	−0.522935	+	0.852373i	$0.675163\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.1067	1.58948	0.794740	−	0.606950i	$-0.207607\pi$
$383$	31.1067	1.58948	0.794740	+	0.606950i	$0.207607\pi$
$384$	0	0
$385$	−24.3457	−1.24077
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.5797	0.536413	0.268207	−	0.963361i	$-0.413569\pi$
$389$	10.5797	0.536413	0.268207	+	0.963361i	$0.413569\pi$
$390$	0	0
$391$	−4.30119	−0.217521
$392$	0	0
$393$	0	0
$394$	0	0
$395$	32.2256	1.62145
$396$	0	0
$397$	1.35598	0.0680545	0.0340272	−	0.999421i	$-0.489167\pi$
$397$	1.35598	0.0680545	0.0340272	+	0.999421i	$0.489167\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.23555	0.461201	0.230601	−	0.973048i	$-0.425931\pi$
$401$	9.23555	0.461201	0.230601	+	0.973048i	$0.425931\pi$
$402$	0	0
$403$	31.7665	1.58240
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−45.8252	−2.27147
$408$	0	0
$409$	9.87957	0.488513	0.244257	−	0.969711i	$-0.421456\pi$
$409$	9.87957	0.488513	0.244257	+	0.969711i	$0.421456\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.48098	0.368115
$414$	0	0
$415$	59.2894	2.91040
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.23555	0.109214	0.0546069	−	0.998508i	$-0.482609\pi$
$419$	2.23555	0.109214	0.0546069	+	0.998508i	$0.482609\pi$
$420$	0	0
$421$	1.83695	0.0895276	0.0447638	−	0.998998i	$-0.485746\pi$
$421$	1.83695	0.0895276	0.0447638	+	0.998998i	$0.485746\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−33.2459	−1.61266
$426$	0	0
$427$	−12.2222	−0.591475
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.517500	0.0249271	0.0124635	−	0.999922i	$-0.496033\pi$
$431$	0.517500	0.0249271	0.0124635	+	0.999922i	$0.496033\pi$
$432$	0	0
$433$	−21.0607	−1.01211	−0.506056	−	0.862500i	$-0.668897\pi$
$433$	−21.0607	−1.01211	−0.506056	+	0.862500i	$0.668897\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.0015	0.621947
$438$	0	0
$439$	5.61967	0.268212	0.134106	−	0.990967i	$-0.457184\pi$
$439$	5.61967	0.268212	0.134106	+	0.990967i	$0.457184\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.7443	−1.03310	−0.516551	−	0.856256i	$-0.672784\pi$
$443$	−21.7443	−1.03310	−0.516551	+	0.856256i	$0.672784\pi$
$444$	0	0
$445$	−50.0504	−2.37262
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.77777	0.0838983	0.0419492	−	0.999120i	$-0.486643\pi$
$449$	1.77777	0.0838983	0.0419492	+	0.999120i	$0.486643\pi$
$450$	0	0
$451$	−12.8632	−0.605704
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25.7047	−1.20506
$456$	0	0
$457$	−11.7462	−0.549464	−0.274732	−	0.961521i	$-0.588589\pi$
$457$	−11.7462	−0.549464	−0.274732	+	0.961521i	$0.588589\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.5980	−0.586747	−0.293373	−	0.955998i	$-0.594778\pi$
$461$	−12.5980	−0.586747	−0.293373	+	0.955998i	$0.594778\pi$
$462$	0	0
$463$	16.5615	0.769677	0.384838	−	0.922984i	$-0.374257\pi$
$463$	16.5615	0.769677	0.384838	+	0.922984i	$0.374257\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.7264	−0.820280	−0.410140	−	0.912023i	$-0.634520\pi$
$467$	−17.7264	−0.820280	−0.410140	+	0.912023i	$0.634520\pi$
$468$	0	0
$469$	8.16049	0.376816
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−33.9882	−1.56278
$474$	0	0
$475$	100.495	4.61102
$476$	0	0
$477$	0	0
$478$	0	0
$479$	42.0291	1.92036	0.960180	−	0.279383i	$-0.0901300\pi$
$479$	42.0291	1.92036	0.960180	+	0.279383i	$0.0901300\pi$
$480$	0	0
$481$	−48.3832	−2.20608
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.3411	−0.696605
$486$	0	0
$487$	−15.9604	−0.723236	−0.361618	−	0.932326i	$-0.617776\pi$
$487$	−15.9604	−0.723236	−0.361618	+	0.932326i	$0.617776\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.4311	−1.77950	−0.889751	−	0.456446i	$-0.849122\pi$
$491$	−39.4311	−1.77950	−0.889751	+	0.456446i	$0.849122\pi$
$492$	0	0
$493$	13.4208	0.604441
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.32049	0.104088
$498$	0	0
$499$	−25.3229	−1.13361	−0.566804	−	0.823853i	$-0.691820\pi$
$499$	−25.3229	−1.13361	−0.566804	+	0.823853i	$0.691820\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.1417	0.541373	0.270687	−	0.962668i	$-0.412749\pi$
$503$	12.1417	0.541373	0.270687	+	0.962668i	$0.412749\pi$
$504$	0	0
$505$	−40.2104	−1.78934
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.4859	−1.57289	−0.786443	−	0.617663i	$-0.788080\pi$
$509$	−35.4859	−1.57289	−0.786443	+	0.617663i	$0.788080\pi$
$510$	0	0
$511$	−1.00152	−0.0443048
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−85.2495	−3.75654
$516$	0	0
$517$	38.0227	1.67223
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−31.7699	−1.39186	−0.695932	−	0.718108i	$-0.745008\pi$
$521$	−31.7699	−1.39186	−0.695932	+	0.718108i	$0.745008\pi$
$522$	0	0
$523$	8.97680	0.392528	0.196264	−	0.980551i	$-0.437119\pi$
$523$	8.97680	0.392528	0.196264	+	0.980551i	$0.437119\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.4208	−0.584618
$528$	0	0
$529$	−20.1792	−0.877358
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−13.5813	−0.588269
$534$	0	0
$535$	−16.9620	−0.733329
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.74125	0.247293
$540$	0	0
$541$	−19.5816	−0.841879	−0.420940	−	0.907089i	$-0.638300\pi$
$541$	−19.5816	−0.841879	−0.420940	+	0.907089i	$0.638300\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.40354	0.359968
$546$	0	0
$547$	−44.8480	−1.91756	−0.958781	−	0.284147i	$-0.908290\pi$
$547$	−44.8480	−1.91756	−0.958781	+	0.284147i	$0.908290\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−40.5679	−1.72825
$552$	0	0
$553$	−7.59951	−0.323164
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.57326	−0.0666613	−0.0333306	−	0.999444i	$-0.510611\pi$
$557$	−1.57326	−0.0666613	−0.0333306	+	0.999444i	$0.510611\pi$
$558$	0	0
$559$	−35.8855	−1.51779
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.4231	0.987168	0.493584	−	0.869698i	$-0.335687\pi$
$563$	23.4231	0.987168	0.493584	+	0.869698i	$0.335687\pi$
$564$	0	0
$565$	7.38375	0.310637
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.3255	1.14555	0.572773	−	0.819714i	$-0.305868\pi$
$569$	27.3255	1.14555	0.572773	+	0.819714i	$0.305868\pi$
$570$	0	0
$571$	34.1304	1.42831	0.714157	−	0.699986i	$-0.246810\pi$
$571$	34.1304	1.42831	0.714157	+	0.699986i	$0.246810\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.8030	0.909247
$576$	0	0
$577$	4.35750	0.181405	0.0907026	−	0.995878i	$-0.471089\pi$
$577$	4.35750	0.181405	0.0907026	+	0.995878i	$0.471089\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.9817	−0.580060
$582$	0	0
$583$	−16.4100	−0.679633
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.2287	0.546006	0.273003	−	0.962013i	$-0.411983\pi$
$587$	13.2287	0.546006	0.273003	+	0.962013i	$0.411983\pi$
$588$	0	0
$589$	40.5679	1.67157
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.2405	0.913307	0.456654	−	0.889645i	$-0.349048\pi$
$593$	22.2405	0.913307	0.456654	+	0.889645i	$0.349048\pi$
$594$	0	0
$595$	10.8598	0.445208
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.70119	0.314662	0.157331	−	0.987546i	$-0.449711\pi$
$599$	7.70119	0.314662	0.157331	+	0.987546i	$0.449711\pi$
$600$	0	0
$601$	−45.1872	−1.84323	−0.921613	−	0.388111i	$-0.873128\pi$
$601$	−45.1872	−1.84323	−0.921613	+	0.388111i	$0.873128\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−93.1294	−3.78625
$606$	0	0
$607$	−16.4494	−0.667660	−0.333830	−	0.942633i	$-0.608341\pi$
$607$	−16.4494	−0.667660	−0.333830	+	0.942633i	$0.608341\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.1452	1.62410
$612$	0	0
$613$	27.5634	1.11327	0.556637	−	0.830756i	$-0.312092\pi$
$613$	27.5634	1.11327	0.556637	+	0.830756i	$0.312092\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.48402	0.341554	0.170777	−	0.985310i	$-0.445372\pi$
$617$	8.48402	0.341554	0.170777	+	0.985310i	$0.445372\pi$
$618$	0	0
$619$	36.5896	1.47066	0.735330	−	0.677709i	$-0.237027\pi$
$619$	36.5896	1.47066	0.735330	+	0.677709i	$0.237027\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.8030	0.472877
$624$	0	0
$625$	78.6168	3.14467
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.4410	0.815038
$630$	0	0
$631$	−10.7976	−0.429844	−0.214922	−	0.976631i	$-0.568950\pi$
$631$	−10.7976	−0.429844	−0.214922	+	0.976631i	$0.568950\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−49.7112	−1.97273
$636$	0	0
$637$	6.06174	0.240175
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−38.2696	−1.51156	−0.755779	−	0.654826i	$-0.772742\pi$
$641$	−38.2696	−1.51156	−0.755779	+	0.654826i	$0.772742\pi$
$642$	0	0
$643$	35.8499	1.41378	0.706891	−	0.707323i	$-0.250097\pi$
$643$	35.8499	1.41378	0.706891	+	0.707323i	$0.250097\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.4810	1.04107	0.520537	−	0.853839i	$-0.325732\pi$
$647$	26.4810	1.04107	0.520537	+	0.853839i	$0.325732\pi$
$648$	0	0
$649$	42.9502	1.68594
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.00646	0.0393859	0.0196930	−	0.999806i	$-0.493731\pi$
$653$	1.00646	0.0393859	0.0196930	+	0.999806i	$0.493731\pi$
$654$	0	0
$655$	31.8134	1.24305
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.2485	0.438178	0.219089	−	0.975705i	$-0.429691\pi$
$659$	11.2485	0.438178	0.219089	+	0.975705i	$0.429691\pi$
$660$	0	0
$661$	−14.2192	−0.553062	−0.276531	−	0.961005i	$-0.589185\pi$
$661$	−14.2192	−0.553062	−0.276531	+	0.961005i	$0.589185\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−32.8267	−1.27296
$666$	0	0
$667$	−8.80146	−0.340794
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−70.1709	−2.70892
$672$	0	0
$673$	32.2302	1.24238	0.621192	−	0.783659i	$-0.286649\pi$
$673$	32.2302	1.24238	0.621192	+	0.783659i	$0.286649\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−48.5710	−1.86673	−0.933367	−	0.358923i	$-0.883144\pi$
$677$	−48.5710	−1.86673	−0.933367	+	0.358923i	$0.883144\pi$
$678$	0	0
$679$	3.61777	0.138837
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.2222	−0.391143	−0.195571	−	0.980689i	$-0.562656\pi$
$683$	−10.2222	−0.391143	−0.195571	+	0.980689i	$0.562656\pi$
$684$	0	0
$685$	−49.7886	−1.90233
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17.3260	−0.660069
$690$	0	0
$691$	31.2207	1.18769	0.593846	−	0.804579i	$-0.297609\pi$
$691$	31.2207	1.18769	0.593846	+	0.804579i	$0.297609\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.82820	0.145212
$696$	0	0
$697$	5.73784	0.217336
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.3822	−0.694287	−0.347144	−	0.937812i	$-0.612848\pi$
$701$	−18.3822	−0.694287	−0.347144	+	0.937812i	$0.612848\pi$
$702$	0	0
$703$	−61.7886	−2.33040
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.48250	0.356626
$708$	0	0
$709$	46.5466	1.74809	0.874047	−	0.485841i	$-0.161486\pi$
$709$	46.5466	1.74809	0.874047	+	0.485841i	$0.161486\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.80146	0.329617
$714$	0	0
$715$	−147.577	−5.51908
$716$	0	0
$717$	0	0
$718$	0	0
$719$	44.6998	1.66702	0.833510	−	0.552504i	$-0.186328\pi$
$719$	44.6998	1.66702	0.833510	+	0.552504i	$0.186328\pi$
$720$	0	0
$721$	20.1037	0.748701
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−68.0306	−2.52659
$726$	0	0
$727$	22.5285	0.835537	0.417769	−	0.908553i	$-0.362812\pi$
$727$	22.5285	0.835537	0.417769	+	0.908553i	$0.362812\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.1610	0.560749
$732$	0	0
$733$	−8.92189	−0.329538	−0.164769	−	0.986332i	$-0.552688\pi$
$733$	−8.92189	−0.329538	−0.164769	+	0.986332i	$0.552688\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.8514	1.72579
$738$	0	0
$739$	13.6770	0.503115	0.251557	−	0.967842i	$-0.419057\pi$
$739$	13.6770	0.503115	0.251557	+	0.967842i	$0.419057\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.7417	0.944373	0.472186	−	0.881499i	$-0.343465\pi$
$743$	25.7417	0.944373	0.472186	+	0.881499i	$0.343465\pi$
$744$	0	0
$745$	34.9502	1.28047
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−47.5329	−1.73450	−0.867251	−	0.497872i	$-0.834115\pi$
$751$	−47.5329	−1.73450	−0.867251	+	0.497872i	$0.834115\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	56.6633	2.06219
$756$	0	0
$757$	−7.57479	−0.275310	−0.137655	−	0.990480i	$-0.543957\pi$
$757$	−7.57479	−0.275310	−0.137655	+	0.990480i	$0.543957\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.2874	0.880417	0.440208	−	0.897896i	$-0.354905\pi$
$761$	24.2874	0.880417	0.440208	+	0.897896i	$0.354905\pi$
$762$	0	0
$763$	−1.98174	−0.0717437
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.3477	1.63741
$768$	0	0
$769$	1.35902	0.0490077	0.0245038	−	0.999700i	$-0.492199\pi$
$769$	1.35902	0.0490077	0.0245038	+	0.999700i	$0.492199\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.2024	0.762599	0.381299	−	0.924452i	$-0.375477\pi$
$773$	21.2024	0.762599	0.381299	+	0.924452i	$0.375477\pi$
$774$	0	0
$775$	68.0306	2.44373
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.3442	−0.621420
$780$	0	0
$781$	13.3225	0.476717
$782$	0	0
$783$	0	0
$784$	0	0
$785$	63.7078	2.27383
$786$	0	0
$787$	−37.1990	−1.32600	−0.663001	−	0.748618i	$-0.730718\pi$
$787$	−37.1990	−1.32600	−0.663001	+	0.748618i	$0.730718\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.74125	−0.0619117
$792$	0	0
$793$	−74.0879	−2.63094
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.2875	1.21453	0.607263	−	0.794501i	$-0.292268\pi$
$797$	34.2875	1.21453	0.607263	+	0.794501i	$0.292268\pi$
$798$	0	0
$799$	−16.9606	−0.600023
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.75000	−0.202913
$804$	0	0
$805$	−7.12195	−0.251016
$806$	0	0
$807$	0	0
$808$	0	0
$809$	11.2424	0.395261	0.197630	−	0.980277i	$-0.436675\pi$
$809$	11.2424	0.395261	0.197630	+	0.980277i	$0.436675\pi$
$810$	0	0
$811$	10.8997	0.382741	0.191371	−	0.981518i	$-0.438707\pi$
$811$	10.8997	0.382741	0.191371	+	0.981518i	$0.438707\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.58619	−0.160647
$816$	0	0
$817$	−45.8282	−1.60333
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.5497	1.45009	0.725047	−	0.688700i	$-0.241818\pi$
$821$	41.5497	1.45009	0.725047	+	0.688700i	$0.241818\pi$
$822$	0	0
$823$	−27.3999	−0.955102	−0.477551	−	0.878604i	$-0.658475\pi$
$823$	−27.3999	−0.955102	−0.477551	+	0.878604i	$0.658475\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.1003	1.15101	0.575505	−	0.817798i	$-0.304806\pi$
$827$	33.1003	1.15101	0.575505	+	0.817798i	$0.304806\pi$
$828$	0	0
$829$	−24.4623	−0.849612	−0.424806	−	0.905284i	$-0.639658\pi$
$829$	−24.4623	−0.849612	−0.424806	+	0.905284i	$0.639658\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.56098	−0.0887325
$834$	0	0
$835$	−66.2544	−2.29283
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.1792	−1.24905	−0.624523	−	0.781006i	$-0.714707\pi$
$839$	−36.1792	−1.24905	−0.624523	+	0.781006i	$0.714707\pi$
$840$	0	0
$841$	−1.53729	−0.0530099
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−100.689	−3.46381
$846$	0	0
$847$	21.9620	0.754622
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.4054	−0.459532
$852$	0	0
$853$	−18.1605	−0.621803	−0.310902	−	0.950442i	$-0.600631\pi$
$853$	−18.1605	−0.621803	−0.310902	+	0.950442i	$0.600631\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.4499	1.21094	0.605472	−	0.795866i	$-0.292984\pi$
$857$	35.4499	1.21094	0.605472	+	0.795866i	$0.292984\pi$
$858$	0	0
$859$	34.2036	1.16701	0.583506	−	0.812109i	$-0.301681\pi$
$859$	34.2036	1.16701	0.583506	+	0.812109i	$0.301681\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.91354	−0.0651376	−0.0325688	−	0.999469i	$-0.510369\pi$
$863$	−1.91354	−0.0651376	−0.0325688	+	0.999469i	$0.510369\pi$
$864$	0	0
$865$	−50.7310	−1.72490
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−43.6307	−1.48007
$870$	0	0
$871$	49.4667	1.67612
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−33.8465	−1.14422
$876$	0	0
$877$	46.6237	1.57437	0.787185	−	0.616717i	$-0.211538\pi$
$877$	46.6237	1.57437	0.787185	+	0.616717i	$0.211538\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19.6795	0.663020	0.331510	−	0.943452i	$-0.392442\pi$
$881$	19.6795	0.663020	0.331510	+	0.943452i	$0.392442\pi$
$882$	0	0
$883$	−1.83305	−0.0616870	−0.0308435	−	0.999524i	$-0.509819\pi$
$883$	−1.83305	−0.0616870	−0.0308435	+	0.999524i	$0.509819\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.6648	1.60043	0.800213	−	0.599715i	$-0.204719\pi$
$887$	47.6648	1.60043	0.800213	+	0.599715i	$0.204719\pi$
$888$	0	0
$889$	11.7230	0.393176
$890$	0	0
$891$	0	0
$892$	0	0
$893$	51.2681	1.71562
$894$	0	0
$895$	13.6573	0.456515
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−27.4627	−0.915933
$900$	0	0
$901$	7.31994	0.243863
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.08238	−0.168944
$906$	0	0
$907$	30.0805	0.998806	0.499403	−	0.866370i	$-0.333553\pi$
$907$	30.0805	0.998806	0.499403	+	0.866370i	$0.333553\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.5843	−1.37775	−0.688875	−	0.724880i	$-0.741895\pi$
$911$	−41.5843	−1.37775	−0.688875	+	0.724880i	$0.741895\pi$
$912$	0	0
$913$	−80.2727	−2.65664
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.50229	−0.247747
$918$	0	0
$919$	19.0820	0.629458	0.314729	−	0.949182i	$-0.398086\pi$
$919$	19.0820	0.629458	0.314729	+	0.949182i	$0.398086\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.0662	0.462994
$924$	0	0
$925$	−103.617	−3.40690
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.9684	1.54098	0.770492	−	0.637450i	$-0.220011\pi$
$929$	46.9684	1.54098	0.770492	+	0.637450i	$0.220011\pi$
$930$	0	0
$931$	7.74125	0.253709
$932$	0	0
$933$	0	0
$934$	0	0
$935$	62.3487	2.03902
$936$	0	0
$937$	−10.7397	−0.350852	−0.175426	−	0.984493i	$-0.556130\pi$
$937$	−10.7397	−0.350852	−0.175426	+	0.984493i	$0.556130\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	35.2024	1.14757	0.573783	−	0.819007i	$-0.305475\pi$
$941$	35.2024	1.14757	0.573783	+	0.819007i	$0.305475\pi$
$942$	0	0
$943$	−3.76293	−0.122538
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.7196	0.640800	0.320400	−	0.947282i	$-0.396183\pi$
$947$	19.7196	0.640800	0.320400	+	0.947282i	$0.396183\pi$
$948$	0	0
$949$	−6.07098	−0.197072
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.0870	−1.36333	−0.681665	−	0.731664i	$-0.738744\pi$
$953$	−42.0870	−1.36333	−0.681665	+	0.731664i	$0.738744\pi$
$954$	0	0
$955$	80.5757	2.60737
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.7413	0.379145
$960$	0	0
$961$	−3.53729	−0.114106
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−50.8923	−1.63828
$966$	0	0
$967$	−6.87500	−0.221085	−0.110543	−	0.993871i	$-0.535259\pi$
$967$	−6.87500	−0.221085	−0.110543	+	0.993871i	$0.535259\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.9718	1.15439	0.577195	−	0.816606i	$-0.304147\pi$
$971$	35.9718	1.15439	0.577195	+	0.816606i	$0.304147\pi$
$972$	0	0
$973$	−0.902774	−0.0289416
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.6014	0.339169	0.169584	−	0.985516i	$-0.445757\pi$
$977$	10.6014	0.339169	0.169584	+	0.985516i	$0.445757\pi$
$978$	0	0
$979$	67.7639	2.16574
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.3027	−1.15788	−0.578938	−	0.815371i	$-0.696533\pi$
$983$	−36.3027	−1.15788	−0.578938	+	0.815371i	$0.696533\pi$
$984$	0	0
$985$	−8.89973	−0.283569
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.94271	−0.316160
$990$	0	0
$991$	15.0049	0.476648	0.238324	−	0.971186i	$-0.423402\pi$
$991$	15.0049	0.476648	0.238324	+	0.971186i	$0.423402\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	50.2913	1.59434
$996$	0	0
$997$	−37.6677	−1.19295	−0.596474	−	0.802632i	$-0.703432\pi$
$997$	−37.6677	−1.19295	−0.596474	+	0.802632i	$0.703432\pi$
$998$	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.1.1	yes	4
3.2	odd	2		6048.2.a.bt.1.4	yes	4
4.3	odd	2		6048.2.a.bk.1.1	✓	4
12.11	even	2		6048.2.a.bs.1.4	yes	4

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

\(f(q)\)	\(=\)	\(q-4.24049 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-4.24049 q^{5} +1.00000 q^{7} +5.74125 q^{11} +6.06174 q^{13} -2.56098 q^{17} +7.74125 q^{19} +1.67951 q^{23} +12.9817 q^{25} -5.24049 q^{29} +5.24049 q^{31} -4.24049 q^{35} -7.98174 q^{37} -2.24049 q^{41} -5.92000 q^{43} +6.62271 q^{47} +1.00000 q^{49} -2.85826 q^{53} -24.3457 q^{55} +7.48098 q^{59} -12.2222 q^{61} -25.7047 q^{65} +8.16049 q^{67} +2.32049 q^{71} -1.00152 q^{73} +5.74125 q^{77} -7.59951 q^{79} -13.9817 q^{83} +10.8598 q^{85} +11.8030 q^{89} +6.06174 q^{91} -32.8267 q^{95} +3.61777 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more