LMFDB - Embedded newform 4864.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1216)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.04547$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04547 q^{5} -0.418627 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-3.04547 q^{5} -0.418627 q^{7} -3.00000 q^{9} +5.27492 q^{11} -2.62685 q^{13} +3.27492 q^{17} -1.00000 q^{19} -3.46410 q^{23} +4.27492 q^{25} -2.62685 q^{29} +6.09095 q^{31} +1.27492 q^{35} -9.55505 q^{37} -4.54983 q^{41} +2.72508 q^{43} +9.13642 q^{45} -0.418627 q^{47} -6.82475 q^{49} -10.3923 q^{53} -16.0646 q^{55} -6.54983 q^{59} +3.04547 q^{61} +1.25588 q^{63} +8.00000 q^{65} +6.54983 q^{67} +11.3446 q^{71} +3.27492 q^{73} -2.20822 q^{77} +13.0192 q^{79} +9.00000 q^{81} -17.0997 q^{83} -9.97368 q^{85} +10.0000 q^{89} +1.09967 q^{91} +3.04547 q^{95} +16.5498 q^{97} -15.8248 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 6 q^{11} - 2 q^{17} - 4 q^{19} + 2 q^{25} - 10 q^{35} + 12 q^{41} + 26 q^{43} + 18 q^{49} + 4 q^{59} + 32 q^{65} - 4 q^{67} - 2 q^{73} + 36 q^{81} - 8 q^{83} + 40 q^{89} - 56 q^{91} + 36 q^{97} - 18 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 6 * q^11 - 2 * q^17 - 4 * q^19 + 2 * q^25 - 10 * q^35 + 12 * q^41 + 26 * q^43 + 18 * q^49 + 4 * q^59 + 32 * q^65 - 4 * q^67 - 2 * q^73 + 36 * q^81 - 8 * q^83 + 40 * q^89 - 56 * q^91 + 36 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−3.04547	−1.36198	−0.680989	−	0.732294i	$-0.738450\pi$
$5$	−3.04547	−1.36198	−0.680989	+	0.732294i	$0.738450\pi$
$6$	0	0
$7$	−0.418627	−0.158226	−0.0791130	−	0.996866i	$-0.525209\pi$
$7$	−0.418627	−0.158226	−0.0791130	+	0.996866i	$0.525209\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	5.27492	1.59045	0.795224	−	0.606316i	$-0.207353\pi$
$11$	5.27492	1.59045	0.795224	+	0.606316i	$0.207353\pi$
$12$	0	0
$13$	−2.62685	−0.728557	−0.364278	−	0.931290i	$-0.618684\pi$
$13$	−2.62685	−0.728557	−0.364278	+	0.931290i	$0.618684\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.27492	0.794284	0.397142	−	0.917757i	$-0.370002\pi$
$17$	3.27492	0.794284	0.397142	+	0.917757i	$0.370002\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	4.27492	0.854983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.62685	−0.487793	−0.243897	−	0.969801i	$-0.578426\pi$
$29$	−2.62685	−0.487793	−0.243897	+	0.969801i	$0.578426\pi$
$30$	0	0
$31$	6.09095	1.09397	0.546983	−	0.837143i	$-0.315776\pi$
$31$	6.09095	1.09397	0.546983	+	0.837143i	$0.315776\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.27492	0.215500
$36$	0	0
$37$	−9.55505	−1.57084	−0.785420	−	0.618963i	$-0.787553\pi$
$37$	−9.55505	−1.57084	−0.785420	+	0.618963i	$0.787553\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.54983	−0.710565	−0.355282	−	0.934759i	$-0.615615\pi$
$41$	−4.54983	−0.710565	−0.355282	+	0.934759i	$0.615615\pi$
$42$	0	0
$43$	2.72508	0.415571	0.207786	−	0.978174i	$-0.433374\pi$
$43$	2.72508	0.415571	0.207786	+	0.978174i	$0.433374\pi$
$44$	0	0
$45$	9.13642	1.36198
$46$	0	0
$47$	−0.418627	−0.0610630	−0.0305315	−	0.999534i	$-0.509720\pi$
$47$	−0.418627	−0.0610630	−0.0305315	+	0.999534i	$0.509720\pi$
$48$	0	0
$49$	−6.82475	−0.974965
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	−16.0646	−2.16615
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.54983	−0.852716	−0.426358	−	0.904555i	$-0.640204\pi$
$59$	−6.54983	−0.852716	−0.426358	+	0.904555i	$0.640204\pi$
$60$	0	0
$61$	3.04547	0.389933	0.194967	−	0.980810i	$-0.437540\pi$
$61$	3.04547	0.389933	0.194967	+	0.980810i	$0.437540\pi$
$62$	0	0
$63$	1.25588	0.158226
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	6.54983	0.800190	0.400095	−	0.916474i	$-0.368977\pi$
$67$	6.54983	0.800190	0.400095	+	0.916474i	$0.368977\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.3446	1.34636	0.673181	−	0.739478i	$-0.264928\pi$
$71$	11.3446	1.34636	0.673181	+	0.739478i	$0.264928\pi$
$72$	0	0
$73$	3.27492	0.383300	0.191650	−	0.981463i	$-0.438616\pi$
$73$	3.27492	0.383300	0.191650	+	0.981463i	$0.438616\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.20822	−0.251650
$78$	0	0
$79$	13.0192	1.46477	0.732385	−	0.680891i	$-0.238407\pi$
$79$	13.0192	1.46477	0.732385	+	0.680891i	$0.238407\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−17.0997	−1.87693	−0.938466	−	0.345371i	$-0.887753\pi$
$83$	−17.0997	−1.87693	−0.938466	+	0.345371i	$0.887753\pi$
$84$	0	0
$85$	−9.97368	−1.08180
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	1.09967	0.115277
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.04547	0.312459
$96$	0	0
$97$	16.5498	1.68038	0.840191	−	0.542291i	$-0.182443\pi$
$97$	16.5498	1.68038	0.840191	+	0.542291i	$0.182443\pi$
$98$	0	0
$99$	−15.8248	−1.59045
$100$	0	0
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	−6.09095	−0.600159	−0.300080	−	0.953914i	$-0.597013\pi$
$103$	−6.09095	−0.600159	−0.300080	+	0.953914i	$0.597013\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.45017	0.140193	0.0700964	−	0.997540i	$-0.477669\pi$
$107$	1.45017	0.140193	0.0700964	+	0.997540i	$0.477669\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.5498	1.18059	0.590295	−	0.807188i	$-0.299012\pi$
$113$	12.5498	1.18059	0.590295	+	0.807188i	$0.299012\pi$
$114$	0	0
$115$	10.5498	0.983777
$116$	0	0
$117$	7.88054	0.728557
$118$	0	0
$119$	−1.37097	−0.125676
$120$	0	0
$121$	16.8248	1.52952
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.20822	0.197509
$126$	0	0
$127$	−17.4356	−1.54716	−0.773579	−	0.633699i	$-0.781536\pi$
$127$	−17.4356	−1.54716	−0.773579	+	0.633699i	$0.781536\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.27492	−0.460872	−0.230436	−	0.973088i	$-0.574015\pi$
$131$	−5.27492	−0.460872	−0.230436	+	0.973088i	$0.574015\pi$
$132$	0	0
$133$	0.418627	0.0362995
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.7251	1.08718	0.543589	−	0.839352i	$-0.317065\pi$
$137$	12.7251	1.08718	0.543589	+	0.839352i	$0.317065\pi$
$138$	0	0
$139$	2.72508	0.231139	0.115569	−	0.993299i	$-0.463131\pi$
$139$	2.72508	0.231139	0.115569	+	0.993299i	$0.463131\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.0646	1.31607	0.658033	−	0.752989i	$-0.271389\pi$
$149$	16.0646	1.31607	0.658033	+	0.752989i	$0.271389\pi$
$150$	0	0
$151$	0.837253	0.0681347	0.0340674	−	0.999420i	$-0.489154\pi$
$151$	0.837253	0.0681347	0.0340674	+	0.999420i	$0.489154\pi$
$152$	0	0
$153$	−9.82475	−0.794284
$154$	0	0
$155$	−18.5498	−1.48996
$156$	0	0
$157$	12.1819	0.972221	0.486111	−	0.873897i	$-0.338415\pi$
$157$	12.1819	0.972221	0.486111	+	0.873897i	$0.338415\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.45017	0.114289
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.1101	1.47878	0.739392	−	0.673275i	$-0.235113\pi$
$167$	19.1101	1.47878	0.739392	+	0.673275i	$0.235113\pi$
$168$	0	0
$169$	−6.09967	−0.469205
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	−8.71780	−0.662802	−0.331401	−	0.943490i	$-0.607521\pi$
$173$	−8.71780	−0.662802	−0.331401	+	0.943490i	$0.607521\pi$
$174$	0	0
$175$	−1.78959	−0.135281
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.0997	1.57706	0.788532	−	0.614994i	$-0.210842\pi$
$179$	21.0997	1.57706	0.788532	+	0.614994i	$0.210842\pi$
$180$	0	0
$181$	−3.46410	−0.257485	−0.128742	−	0.991678i	$-0.541094\pi$
$181$	−3.46410	−0.257485	−0.128742	+	0.991678i	$0.541094\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	29.0997	2.13945
$186$	0	0
$187$	17.2749	1.26327
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7633	0.851161	0.425580	−	0.904921i	$-0.360070\pi$
$191$	11.7633	0.851161	0.425580	+	0.904921i	$0.360070\pi$
$192$	0	0
$193$	11.0997	0.798972	0.399486	−	0.916739i	$-0.369189\pi$
$193$	11.0997	0.798972	0.399486	+	0.916739i	$0.369189\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.25370	−0.374310	−0.187155	−	0.982330i	$-0.559927\pi$
$197$	−5.25370	−0.374310	−0.187155	+	0.982330i	$0.559927\pi$
$198$	0	0
$199$	14.2750	1.01193	0.505965	−	0.862554i	$-0.331136\pi$
$199$	14.2750	1.01193	0.505965	+	0.862554i	$0.331136\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.09967	0.0771816
$204$	0	0
$205$	13.8564	0.967773
$206$	0	0
$207$	10.3923	0.722315
$208$	0	0
$209$	−5.27492	−0.364874
$210$	0	0
$211$	26.5498	1.82777	0.913883	−	0.405978i	$-0.133069\pi$
$211$	26.5498	1.82777	0.913883	+	0.405978i	$0.133069\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.29917	−0.565999
$216$	0	0
$217$	−2.54983	−0.173094
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.60271	−0.578681
$222$	0	0
$223$	17.4356	1.16757	0.583787	−	0.811907i	$-0.301570\pi$
$223$	17.4356	1.16757	0.583787	+	0.811907i	$0.301570\pi$
$224$	0	0
$225$	−12.8248	−0.854983
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	17.7391	1.17224	0.586118	−	0.810226i	$-0.300656\pi$
$229$	17.7391	1.17224	0.586118	+	0.810226i	$0.300656\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.3746	1.33478	0.667392	−	0.744707i	$-0.267411\pi$
$233$	20.3746	1.33478	0.667392	+	0.744707i	$0.267411\pi$
$234$	0	0
$235$	1.27492	0.0831664
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.6197	−1.65720	−0.828600	−	0.559842i	$-0.810862\pi$
$239$	−25.6197	−1.65720	−0.828600	+	0.559842i	$0.810862\pi$
$240$	0	0
$241$	−12.5498	−0.808406	−0.404203	−	0.914669i	$-0.632451\pi$
$241$	−12.5498	−0.808406	−0.404203	+	0.914669i	$0.632451\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.7846	1.32788
$246$	0	0
$247$	2.62685	0.167142
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.7251	−0.676961	−0.338481	−	0.940973i	$-0.609913\pi$
$251$	−10.7251	−0.676961	−0.338481	+	0.940973i	$0.609913\pi$
$252$	0	0
$253$	−18.2728	−1.14880
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.45017	0.464729	0.232364	−	0.972629i	$-0.425354\pi$
$257$	7.45017	0.464729	0.232364	+	0.972629i	$0.425354\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	7.88054	0.487793
$262$	0	0
$263$	−1.25588	−0.0774409	−0.0387204	−	0.999250i	$-0.512328\pi$
$263$	−1.25588	−0.0774409	−0.0387204	+	0.999250i	$0.512328\pi$
$264$	0	0
$265$	31.6495	1.94421
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.13861	−0.313306	−0.156653	−	0.987654i	$-0.550071\pi$
$269$	−5.13861	−0.313306	−0.156653	+	0.987654i	$0.550071\pi$
$270$	0	0
$271$	−27.8279	−1.69042	−0.845212	−	0.534431i	$-0.820526\pi$
$271$	−27.8279	−1.69042	−0.845212	+	0.534431i	$0.820526\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.5498	1.35981
$276$	0	0
$277$	−16.0646	−0.965230	−0.482615	−	0.875833i	$-0.660313\pi$
$277$	−16.0646	−0.965230	−0.482615	+	0.875833i	$0.660313\pi$
$278$	0	0
$279$	−18.2728	−1.09397
$280$	0	0
$281$	−11.0997	−0.662151	−0.331075	−	0.943604i	$-0.607411\pi$
$281$	−11.0997	−0.662151	−0.331075	+	0.943604i	$0.607411\pi$
$282$	0	0
$283$	−26.3746	−1.56781	−0.783903	−	0.620883i	$-0.786774\pi$
$283$	−26.3746	−1.56781	−0.783903	+	0.620883i	$0.786774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.90468	0.112430
$288$	0	0
$289$	−6.27492	−0.369113
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.3397	1.77246	0.886231	−	0.463244i	$-0.153315\pi$
$293$	30.3397	1.77246	0.886231	+	0.463244i	$0.153315\pi$
$294$	0	0
$295$	19.9474	1.16138
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.09967	0.526247
$300$	0	0
$301$	−1.14079	−0.0657542
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.27492	−0.531080
$306$	0	0
$307$	1.45017	0.0827653	0.0413827	−	0.999143i	$-0.486824\pi$
$307$	1.45017	0.0827653	0.0413827	+	0.999143i	$0.486824\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.7824	1.40528	0.702641	−	0.711544i	$-0.252004\pi$
$311$	24.7824	1.40528	0.702641	+	0.711544i	$0.252004\pi$
$312$	0	0
$313$	−15.0997	−0.853484	−0.426742	−	0.904373i	$-0.640339\pi$
$313$	−15.0997	−0.853484	−0.426742	+	0.904373i	$0.640339\pi$
$314$	0	0
$315$	−3.82475	−0.215500
$316$	0	0
$317$	8.71780	0.489640	0.244820	−	0.969569i	$-0.421271\pi$
$317$	8.71780	0.489640	0.244820	+	0.969569i	$0.421271\pi$
$318$	0	0
$319$	−13.8564	−0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.27492	−0.182221
$324$	0	0
$325$	−11.2296	−0.622904
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.175248	0.00966175
$330$	0	0
$331$	9.45017	0.519428	0.259714	−	0.965686i	$-0.416372\pi$
$331$	9.45017	0.519428	0.259714	+	0.965686i	$0.416372\pi$
$332$	0	0
$333$	28.6652	1.57084
$334$	0	0
$335$	−19.9474	−1.08984
$336$	0	0
$337$	33.6495	1.83301	0.916503	−	0.400029i	$-0.131000\pi$
$337$	33.6495	1.83301	0.916503	+	0.400029i	$0.131000\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.1293	1.73990
$342$	0	0
$343$	5.78741	0.312491
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.82475	0.420055	0.210027	−	0.977696i	$-0.432645\pi$
$347$	7.82475	0.420055	0.210027	+	0.977696i	$0.432645\pi$
$348$	0	0
$349$	−12.7156	−0.680651	−0.340326	−	0.940308i	$-0.610537\pi$
$349$	−12.7156	−0.680651	−0.340326	+	0.940308i	$0.610537\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−34.5498	−1.83371
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.6005	0.665030	0.332515	−	0.943098i	$-0.392103\pi$
$359$	12.6005	0.665030	0.332515	+	0.943098i	$0.392103\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.97368	−0.522046
$366$	0	0
$367$	15.6460	0.816715	0.408357	−	0.912822i	$-0.366102\pi$
$367$	15.6460	0.816715	0.408357	+	0.912822i	$0.366102\pi$
$368$	0	0
$369$	13.6495	0.710565
$370$	0	0
$371$	4.35050	0.225867
$372$	0	0
$373$	0.952341	0.0493104	0.0246552	−	0.999696i	$-0.492151\pi$
$373$	0.952341	0.0493104	0.0246552	+	0.999696i	$0.492151\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.90033	0.355385
$378$	0	0
$379$	23.6495	1.21479	0.607397	−	0.794399i	$-0.292214\pi$
$379$	23.6495	1.21479	0.607397	+	0.794399i	$0.292214\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.1293	−1.64173	−0.820864	−	0.571124i	$-0.806508\pi$
$383$	−32.1293	−1.64173	−0.820864	+	0.571124i	$0.806508\pi$
$384$	0	0
$385$	6.72508	0.342742
$386$	0	0
$387$	−8.17525	−0.415571
$388$	0	0
$389$	−4.71998	−0.239313	−0.119656	−	0.992815i	$-0.538179\pi$
$389$	−4.71998	−0.239313	−0.119656	+	0.992815i	$0.538179\pi$
$390$	0	0
$391$	−11.3446	−0.573723
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−39.6495	−1.99498
$396$	0	0
$397$	−32.6630	−1.63931	−0.819654	−	0.572859i	$-0.805834\pi$
$397$	−32.6630	−1.63931	−0.819654	+	0.572859i	$0.805834\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.0997	1.75279	0.876397	−	0.481590i	$-0.159940\pi$
$401$	35.0997	1.75279	0.876397	+	0.481590i	$0.159940\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	−27.4093	−1.36198
$406$	0	0
$407$	−50.4021	−2.49834
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.74194	0.134922
$414$	0	0
$415$	52.0766	2.55634
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−24.2487	−1.18181	−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487	−1.18181	−0.590905	+	0.806741i	$0.701229\pi$
$422$	0	0
$423$	1.25588	0.0610630
$424$	0	0
$425$	14.0000	0.679100
$426$	0	0
$427$	−1.27492	−0.0616976
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.67451	0.0806582	0.0403291	−	0.999186i	$-0.487159\pi$
$431$	1.67451	0.0806582	0.0403291	+	0.999186i	$0.487159\pi$
$432$	0	0
$433$	−17.6495	−0.848181	−0.424091	−	0.905620i	$-0.639406\pi$
$433$	−17.6495	−0.848181	−0.424091	+	0.905620i	$0.639406\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46410	0.165710
$438$	0	0
$439$	31.2920	1.49349	0.746743	−	0.665113i	$-0.231617\pi$
$439$	31.2920	1.49349	0.746743	+	0.665113i	$0.231617\pi$
$440$	0	0
$441$	20.4743	0.974965
$442$	0	0
$443$	−10.3746	−0.492911	−0.246456	−	0.969154i	$-0.579266\pi$
$443$	−10.3746	−0.492911	−0.246456	+	0.969154i	$0.579266\pi$
$444$	0	0
$445$	−30.4547	−1.44369
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.5498	−0.969807	−0.484903	−	0.874568i	$-0.661145\pi$
$449$	−20.5498	−0.969807	−0.484903	+	0.874568i	$0.661145\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.34901	−0.157004
$456$	0	0
$457$	15.2749	0.714530	0.357265	−	0.934003i	$-0.383709\pi$
$457$	15.2749	0.714530	0.357265	+	0.934003i	$0.383709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.7156	−0.592225	−0.296113	−	0.955153i	$-0.595690\pi$
$461$	−12.7156	−0.592225	−0.296113	+	0.955153i	$0.595690\pi$
$462$	0	0
$463$	−2.93039	−0.136187	−0.0680933	−	0.997679i	$-0.521692\pi$
$463$	−2.93039	−0.136187	−0.0680933	+	0.997679i	$0.521692\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.17525	0.378305	0.189153	−	0.981948i	$-0.439426\pi$
$467$	8.17525	0.378305	0.189153	+	0.981948i	$0.439426\pi$
$468$	0	0
$469$	−2.74194	−0.126611
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.3746	0.660944
$474$	0	0
$475$	−4.27492	−0.196147
$476$	0	0
$477$	31.1769	1.42749
$478$	0	0
$479$	−17.3205	−0.791394	−0.395697	−	0.918381i	$-0.629497\pi$
$479$	−17.3205	−0.791394	−0.395697	+	0.918381i	$0.629497\pi$
$480$	0	0
$481$	25.0997	1.14445
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−50.4021	−2.28864
$486$	0	0
$487$	−39.0575	−1.76986	−0.884931	−	0.465722i	$-0.845795\pi$
$487$	−39.0575	−1.76986	−0.884931	+	0.465722i	$0.845795\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−8.60271	−0.387447
$494$	0	0
$495$	48.1939	2.16615
$496$	0	0
$497$	−4.74917	−0.213029
$498$	0	0
$499$	26.7251	1.19638	0.598190	−	0.801355i	$-0.295887\pi$
$499$	26.7251	1.19638	0.598190	+	0.801355i	$0.295887\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	41.6843	1.85861	0.929306	−	0.369311i	$-0.120406\pi$
$503$	41.6843	1.85861	0.929306	+	0.369311i	$0.120406\pi$
$504$	0	0
$505$	−42.1993	−1.87785
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.30136	0.190654	0.0953271	−	0.995446i	$-0.469610\pi$
$509$	4.30136	0.190654	0.0953271	+	0.995446i	$0.469610\pi$
$510$	0	0
$511$	−1.37097	−0.0606480
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.5498	0.817403
$516$	0	0
$517$	−2.20822	−0.0971175
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27.0997	1.18726	0.593629	−	0.804739i	$-0.297695\pi$
$521$	27.0997	1.18726	0.593629	+	0.804739i	$0.297695\pi$
$522$	0	0
$523$	−5.45017	−0.238319	−0.119160	−	0.992875i	$-0.538020\pi$
$523$	−5.45017	−0.238319	−0.119160	+	0.992875i	$0.538020\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.9474	0.868920
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	19.6495	0.852716
$532$	0	0
$533$	11.9517	0.517687
$534$	0	0
$535$	−4.41644	−0.190939
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−36.0000	−1.55063
$540$	0	0
$541$	29.0838	1.25041	0.625205	−	0.780461i	$-0.285015\pi$
$541$	29.0838	1.25041	0.625205	+	0.780461i	$0.285015\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.5498	0.451905
$546$	0	0
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	0	0
$549$	−9.13642	−0.389933
$550$	0	0
$551$	2.62685	0.111907
$552$	0	0
$553$	−5.45017	−0.231765
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.9210	1.26779	0.633897	−	0.773417i	$-0.281454\pi$
$557$	29.9210	1.26779	0.633897	+	0.773417i	$0.281454\pi$
$558$	0	0
$559$	−7.15838	−0.302767
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	−38.2202	−1.60794
$566$	0	0
$567$	−3.76764	−0.158226
$568$	0	0
$569$	19.4502	0.815393	0.407697	−	0.913117i	$-0.366332\pi$
$569$	19.4502	0.815393	0.407697	+	0.913117i	$0.366332\pi$
$570$	0	0
$571$	41.0997	1.71997	0.859984	−	0.510321i	$-0.170474\pi$
$571$	41.0997	1.71997	0.859984	+	0.510321i	$0.170474\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14.8087	−0.617567
$576$	0	0
$577$	23.2749	0.968947	0.484474	−	0.874806i	$-0.339011\pi$
$577$	23.2749	0.968947	0.484474	+	0.874806i	$0.339011\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.15838	0.296980
$582$	0	0
$583$	−54.8185	−2.27035
$584$	0	0
$585$	−24.0000	−0.992278
$586$	0	0
$587$	37.2749	1.53850	0.769250	−	0.638948i	$-0.220630\pi$
$587$	37.2749	1.53850	0.769250	+	0.638948i	$0.220630\pi$
$588$	0	0
$589$	−6.09095	−0.250973
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.0997	0.784329	0.392165	−	0.919895i	$-0.371726\pi$
$593$	19.0997	0.784329	0.392165	+	0.919895i	$0.371726\pi$
$594$	0	0
$595$	4.17525	0.171168
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.43996	0.385706	0.192853	−	0.981228i	$-0.438226\pi$
$599$	9.43996	0.385706	0.192853	+	0.981228i	$0.438226\pi$
$600$	0	0
$601$	−11.0997	−0.452765	−0.226382	−	0.974038i	$-0.572690\pi$
$601$	−11.0997	−0.452765	−0.226382	+	0.974038i	$0.572690\pi$
$602$	0	0
$603$	−19.6495	−0.800190
$604$	0	0
$605$	−51.2394	−2.08318
$606$	0	0
$607$	29.3873	1.19279	0.596397	−	0.802689i	$-0.296598\pi$
$607$	29.3873	1.19279	0.596397	+	0.802689i	$0.296598\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.09967	0.0444878
$612$	0	0
$613$	−29.9210	−1.20850	−0.604250	−	0.796795i	$-0.706527\pi$
$613$	−29.9210	−1.20850	−0.604250	+	0.796795i	$0.706527\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.7251	−0.512293	−0.256146	−	0.966638i	$-0.582453\pi$
$617$	−12.7251	−0.512293	−0.256146	+	0.966638i	$0.582453\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.18627	−0.167719
$624$	0	0
$625$	−28.0997	−1.12399
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−31.2920	−1.24769
$630$	0	0
$631$	−13.4378	−0.534950	−0.267475	−	0.963565i	$-0.586189\pi$
$631$	−13.4378	−0.534950	−0.267475	+	0.963565i	$0.586189\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	53.0997	2.10720
$636$	0	0
$637$	17.9276	0.710317
$638$	0	0
$639$	−34.0339	−1.34636
$640$	0	0
$641$	−12.5498	−0.495689	−0.247844	−	0.968800i	$-0.579722\pi$
$641$	−12.5498	−0.495689	−0.247844	+	0.968800i	$0.579722\pi$
$642$	0	0
$643$	−44.9244	−1.77165	−0.885823	−	0.464023i	$-0.846405\pi$
$643$	−44.9244	−1.77165	−0.885823	+	0.464023i	$0.846405\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.6197	−1.00721	−0.503607	−	0.863933i	$-0.667994\pi$
$647$	−25.6197	−1.00721	−0.503607	+	0.863933i	$0.667994\pi$
$648$	0	0
$649$	−34.5498	−1.35620
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.0646	0.628657	0.314329	−	0.949314i	$-0.398221\pi$
$653$	16.0646	0.628657	0.314329	+	0.949314i	$0.398221\pi$
$654$	0	0
$655$	16.0646	0.627697
$656$	0	0
$657$	−9.82475	−0.383300
$658$	0	0
$659$	−13.4502	−0.523944	−0.261972	−	0.965075i	$-0.584373\pi$
$659$	−13.4502	−0.523944	−0.261972	+	0.965075i	$0.584373\pi$
$660$	0	0
$661$	−9.55505	−0.371648	−0.185824	−	0.982583i	$-0.559495\pi$
$661$	−9.55505	−0.371648	−0.185824	+	0.982583i	$0.559495\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.27492	−0.0494392
$666$	0	0
$667$	9.09967	0.352341
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.0646	0.620168
$672$	0	0
$673$	−47.0997	−1.81556	−0.907779	−	0.419448i	$-0.862224\pi$
$673$	−47.0997	−1.81556	−0.907779	+	0.419448i	$0.862224\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.722166	−0.0277551	−0.0138775	−	0.999904i	$-0.504418\pi$
$677$	−0.722166	−0.0277551	−0.0138775	+	0.999904i	$0.504418\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.1993	−0.849434	−0.424717	−	0.905326i	$-0.639626\pi$
$683$	−22.1993	−0.849434	−0.424717	+	0.905326i	$0.639626\pi$
$684$	0	0
$685$	−38.7539	−1.48071
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27.2990	1.04001
$690$	0	0
$691$	23.4743	0.893003	0.446501	−	0.894783i	$-0.352670\pi$
$691$	23.4743	0.893003	0.446501	+	0.894783i	$0.352670\pi$
$692$	0	0
$693$	6.62466	0.251650
$694$	0	0
$695$	−8.29917	−0.314806
$696$	0	0
$697$	−14.9003	−0.564390
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.0148	0.793717	0.396859	−	0.917880i	$-0.370100\pi$
$701$	21.0148	0.793717	0.396859	+	0.917880i	$0.370100\pi$
$702$	0	0
$703$	9.55505	0.360376
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.80066	−0.218156
$708$	0	0
$709$	38.2202	1.43539	0.717695	−	0.696358i	$-0.245197\pi$
$709$	38.2202	1.43539	0.717695	+	0.696358i	$0.245197\pi$
$710$	0	0
$711$	−39.0575	−1.46477
$712$	0	0
$713$	−21.0997	−0.790189
$714$	0	0
$715$	42.1993	1.57817
$716$	0	0
$717$	0	0
$718$	0	0
$719$	51.6580	1.92652	0.963259	−	0.268575i	$-0.0865526\pi$
$719$	51.6580	1.92652	0.963259	+	0.268575i	$0.0865526\pi$
$720$	0	0
$721$	2.54983	0.0949608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−11.2296	−0.417055
$726$	0	0
$727$	10.9260	0.405224	0.202612	−	0.979259i	$-0.435057\pi$
$727$	10.9260	0.405224	0.202612	+	0.979259i	$0.435057\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	8.92442	0.330082
$732$	0	0
$733$	19.1101	0.705848	0.352924	−	0.935652i	$-0.385187\pi$
$733$	19.1101	0.705848	0.352924	+	0.935652i	$0.385187\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.5498	1.27266
$738$	0	0
$739$	7.82475	0.287838	0.143919	−	0.989589i	$-0.454029\pi$
$739$	7.82475	0.287838	0.143919	+	0.989589i	$0.454029\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	52.9139	1.94122	0.970611	−	0.240655i	$-0.0773622\pi$
$743$	52.9139	1.94122	0.970611	+	0.240655i	$0.0773622\pi$
$744$	0	0
$745$	−48.9244	−1.79245
$746$	0	0
$747$	51.2990	1.87693
$748$	0	0
$749$	−0.607078	−0.0221822
$750$	0	0
$751$	−36.3155	−1.32517	−0.662586	−	0.748986i	$-0.730541\pi$
$751$	−36.3155	−1.32517	−0.662586	+	0.748986i	$0.730541\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.54983	−0.0927980
$756$	0	0
$757$	3.04547	0.110690	0.0553448	−	0.998467i	$-0.482374\pi$
$757$	3.04547	0.110690	0.0553448	+	0.998467i	$0.482374\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.27492	−0.118716	−0.0593578	−	0.998237i	$-0.518905\pi$
$761$	−3.27492	−0.118716	−0.0593578	+	0.998237i	$0.518905\pi$
$762$	0	0
$763$	1.45017	0.0524995
$764$	0	0
$765$	29.9210	1.08180
$766$	0	0
$767$	17.2054	0.621252
$768$	0	0
$769$	−13.8248	−0.498533	−0.249267	−	0.968435i	$-0.580190\pi$
$769$	−13.8248	−0.498533	−0.249267	+	0.968435i	$0.580190\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.7561	1.25009	0.625045	−	0.780589i	$-0.285081\pi$
$773$	34.7561	1.25009	0.625045	+	0.780589i	$0.285081\pi$
$774$	0	0
$775$	26.0383	0.935324
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.54983	0.163015
$780$	0	0
$781$	59.8421	2.14132
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−37.0997	−1.32414
$786$	0	0
$787$	−14.1993	−0.506152	−0.253076	−	0.967446i	$-0.581442\pi$
$787$	−14.1993	−0.506152	−0.253076	+	0.967446i	$0.581442\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.25370	−0.186800
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.2487	−0.858933	−0.429467	−	0.903083i	$-0.641298\pi$
$797$	−24.2487	−0.858933	−0.429467	+	0.903083i	$0.641298\pi$
$798$	0	0
$799$	−1.37097	−0.0485014
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	17.2749	0.609619
$804$	0	0
$805$	−4.41644	−0.155659
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.82475	0.204787	0.102394	−	0.994744i	$-0.467350\pi$
$809$	5.82475	0.204787	0.102394	+	0.994744i	$0.467350\pi$
$810$	0	0
$811$	−47.2990	−1.66089	−0.830446	−	0.557099i	$-0.811915\pi$
$811$	−47.2990	−1.66089	−0.830446	+	0.557099i	$0.811915\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.1819	0.426713
$816$	0	0
$817$	−2.72508	−0.0953386
$818$	0	0
$819$	−3.29901	−0.115277
$820$	0	0
$821$	26.5720	0.927370	0.463685	−	0.886000i	$-0.346527\pi$
$821$	26.5720	0.927370	0.463685	+	0.886000i	$0.346527\pi$
$822$	0	0
$823$	21.4334	0.747122	0.373561	−	0.927606i	$-0.378137\pi$
$823$	21.4334	0.747122	0.373561	+	0.927606i	$0.378137\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.6495	−1.65694	−0.828468	−	0.560037i	$-0.810787\pi$
$827$	−47.6495	−1.65694	−0.828468	+	0.560037i	$0.810787\pi$
$828$	0	0
$829$	25.3161	0.879266	0.439633	−	0.898178i	$-0.355109\pi$
$829$	25.3161	0.879266	0.439633	+	0.898178i	$0.355109\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−22.3505	−0.774399
$834$	0	0
$835$	−58.1993	−2.01407
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.6602	1.64541	0.822706	−	0.568467i	$-0.192463\pi$
$839$	47.6602	1.64541	0.822706	+	0.568467i	$0.192463\pi$
$840$	0	0
$841$	−22.0997	−0.762058
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.5764	0.639047
$846$	0	0
$847$	−7.04329	−0.242010
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.0997	1.13464
$852$	0	0
$853$	32.9665	1.12875	0.564376	−	0.825518i	$-0.309117\pi$
$853$	32.9665	1.12875	0.564376	+	0.825518i	$0.309117\pi$
$854$	0	0
$855$	−9.13642	−0.312459
$856$	0	0
$857$	16.1993	0.553359	0.276679	−	0.960962i	$-0.410766\pi$
$857$	16.1993	0.553359	0.276679	+	0.960962i	$0.410766\pi$
$858$	0	0
$859$	−55.4743	−1.89276	−0.946379	−	0.323060i	$-0.895289\pi$
$859$	−55.4743	−1.89276	−0.946379	+	0.323060i	$0.895289\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.7994	−1.42287	−0.711434	−	0.702753i	$-0.751954\pi$
$863$	−41.7994	−1.42287	−0.711434	+	0.702753i	$0.751954\pi$
$864$	0	0
$865$	26.5498	0.902721
$866$	0	0
$867$	0	0
$868$	0	0
$869$	68.6750	2.32964
$870$	0	0
$871$	−17.2054	−0.582983
$872$	0	0
$873$	−49.6495	−1.68038
$874$	0	0
$875$	−0.924421	−0.0312511
$876$	0	0
$877$	31.4071	1.06054	0.530271	−	0.847828i	$-0.322090\pi$
$877$	31.4071	1.06054	0.530271	+	0.847828i	$0.322090\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.82475	0.0614774	0.0307387	−	0.999527i	$-0.490214\pi$
$881$	1.82475	0.0614774	0.0307387	+	0.999527i	$0.490214\pi$
$882$	0	0
$883$	13.6254	0.458532	0.229266	−	0.973364i	$-0.426367\pi$
$883$	13.6254	0.458532	0.229266	+	0.973364i	$0.426367\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.837253	−0.0281122	−0.0140561	−	0.999901i	$-0.504474\pi$
$887$	−0.837253	−0.0281122	−0.0140561	+	0.999901i	$0.504474\pi$
$888$	0	0
$889$	7.29901	0.244801
$890$	0	0
$891$	47.4743	1.59045
$892$	0	0
$893$	0.418627	0.0140088
$894$	0	0
$895$	−64.2585	−2.14793
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.0000	−0.533630
$900$	0	0
$901$	−34.0339	−1.13383
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.5498	0.350688
$906$	0	0
$907$	−23.6495	−0.785269	−0.392634	−	0.919695i	$-0.628436\pi$
$907$	−23.6495	−0.785269	−0.392634	+	0.919695i	$0.628436\pi$
$908$	0	0
$909$	−41.5692	−1.37876
$910$	0	0
$911$	34.0339	1.12759	0.563797	−	0.825913i	$-0.309340\pi$
$911$	34.0339	1.12759	0.563797	+	0.825913i	$0.309340\pi$
$912$	0	0
$913$	−90.1993	−2.98516
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.20822	0.0729219
$918$	0	0
$919$	−0.115088	−0.00379639	−0.00189820	−	0.999998i	$-0.500604\pi$
$919$	−0.115088	−0.00379639	−0.00189820	+	0.999998i	$0.500604\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−29.8007	−0.980901
$924$	0	0
$925$	−40.8471	−1.34304
$926$	0	0
$927$	18.2728	0.600159
$928$	0	0
$929$	−7.09967	−0.232933	−0.116466	−	0.993195i	$-0.537157\pi$
$929$	−7.09967	−0.232933	−0.116466	+	0.993195i	$0.537157\pi$
$930$	0	0
$931$	6.82475	0.223672
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−52.6103	−1.72054
$936$	0	0
$937$	14.9244	0.487560	0.243780	−	0.969831i	$-0.421613\pi$
$937$	14.9244	0.487560	0.243780	+	0.969831i	$0.421613\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.1769	1.01634	0.508169	−	0.861257i	$-0.330322\pi$
$941$	31.1769	1.01634	0.508169	+	0.861257i	$0.330322\pi$
$942$	0	0
$943$	15.7611	0.513252
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.1993	0.981347	0.490673	−	0.871344i	$-0.336751\pi$
$947$	30.1993	0.981347	0.490673	+	0.871344i	$0.336751\pi$
$948$	0	0
$949$	−8.60271	−0.279256
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.2990	1.46738	0.733689	−	0.679485i	$-0.237797\pi$
$953$	45.2990	1.46738	0.733689	+	0.679485i	$0.237797\pi$
$954$	0	0
$955$	−35.8248	−1.15926
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.32706	−0.172020
$960$	0	0
$961$	6.09967	0.196764
$962$	0	0
$963$	−4.35050	−0.140193
$964$	0	0
$965$	−33.8038	−1.08818
$966$	0	0
$967$	−55.5407	−1.78607	−0.893034	−	0.449988i	$-0.851428\pi$
$967$	−55.5407	−1.78607	−0.893034	+	0.449988i	$0.851428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	−1.14079	−0.0365721
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.1993	−0.902177	−0.451088	−	0.892479i	$-0.648964\pi$
$977$	−28.1993	−0.902177	−0.451088	+	0.892479i	$0.648964\pi$
$978$	0	0
$979$	52.7492	1.68587
$980$	0	0
$981$	10.3923	0.331801
$982$	0	0
$983$	22.6893	0.723676	0.361838	−	0.932241i	$-0.382149\pi$
$983$	22.6893	0.723676	0.361838	+	0.932241i	$0.382149\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.43996	−0.300173
$990$	0	0
$991$	47.0531	1.49469	0.747345	−	0.664436i	$-0.231328\pi$
$991$	47.0531	1.49469	0.747345	+	0.664436i	$0.231328\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−43.4743	−1.37823
$996$	0	0
$997$	−42.1029	−1.33341	−0.666707	−	0.745320i	$-0.732297\pi$
$997$	−42.1029	−1.33341	−0.666707	+	0.745320i	$0.732297\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bk.1.1		4
4.3	odd	2		4864.2.a.bj.1.1		4
8.3	odd	2	inner	4864.2.a.bk.1.4		4
8.5	even	2		4864.2.a.bj.1.4		4
16.3	odd	4		1216.2.c.i.609.7	yes	8
16.5	even	4		1216.2.c.i.609.2	yes	8
16.11	odd	4		1216.2.c.i.609.1	✓	8
16.13	even	4		1216.2.c.i.609.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.i.609.1	✓	8	16.11	odd	4
1216.2.c.i.609.2	yes	8	16.5	even	4
1216.2.c.i.609.7	yes	8	16.3	odd	4
1216.2.c.i.609.8	yes	8	16.13	even	4
4864.2.a.bj.1.1		4	4.3	odd	2
4864.2.a.bj.1.4		4	8.5	even	2
4864.2.a.bk.1.1		4	1.1	even	1	trivial
4864.2.a.bk.1.4		4	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q-3.04547 q^{5} -0.418627 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-3.04547 q^{5} -0.418627 q^{7} -3.00000 q^{9} +5.27492 q^{11} -2.62685 q^{13} +3.27492 q^{17} -1.00000 q^{19} -3.46410 q^{23} +4.27492 q^{25} -2.62685 q^{29} +6.09095 q^{31} +1.27492 q^{35} -9.55505 q^{37} -4.54983 q^{41} +2.72508 q^{43} +9.13642 q^{45} -0.418627 q^{47} -6.82475 q^{49} -10.3923 q^{53} -16.0646 q^{55} -6.54983 q^{59} +3.04547 q^{61} +1.25588 q^{63} +8.00000 q^{65} +6.54983 q^{67} +11.3446 q^{71} +3.27492 q^{73} -2.20822 q^{77} +13.0192 q^{79} +9.00000 q^{81} -17.0997 q^{83} -9.97368 q^{85} +10.0000 q^{89} +1.09967 q^{91} +3.04547 q^{95} +16.5498 q^{97} -15.8248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 6 q^{11} - 2 q^{17} - 4 q^{19} + 2 q^{25} - 10 q^{35} + 12 q^{41} + 26 q^{43} + 18 q^{49} + 4 q^{59} + 32 q^{65} - 4 q^{67} - 2 q^{73} + 36 q^{81} - 8 q^{83} + 40 q^{89} - 56 q^{91} + 36 q^{97} - 18 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 6 * q^11 - 2 * q^17 - 4 * q^19 + 2 * q^25 - 10 * q^35 + 12 * q^41 + 26 * q^43 + 18 * q^49 + 4 * q^59 + 32 * q^65 - 4 * q^67 - 2 * q^73 + 36 * q^81 - 8 * q^83 + 40 * q^89 - 56 * q^91 + 36 * q^97 - 18 * q^99

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