LMFDB - Embedded newform 4864.2.a.bs.1.9

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 23x^{8} + 44x^{7} + 167x^{6} - 266x^{5} - 491x^{4} + 460x^{3} + 546x^{2} + 56x - 8 $ x^10 - 2x^9 - 23x^8 + 44x^7 + 167x^6 - 266x^5 - 491x^4 + 460x^3 + 546x^2 + 56*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.719482$ 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)

$f(q)$	$=$	$q+2.46871 q^{3} -3.22672 q^{5} +4.04213 q^{7} +3.09455 q^{9} +O(q^{10})$	$q+2.46871 q^{3} -3.22672 q^{5} +4.04213 q^{7} +3.09455 q^{9} -6.41171 q^{11} -0.815411 q^{13} -7.96585 q^{15} -2.62028 q^{17} -1.00000 q^{19} +9.97886 q^{21} +6.16717 q^{23} +5.41171 q^{25} +0.233424 q^{27} -3.85818 q^{29} +7.43670 q^{31} -15.8287 q^{33} -13.0428 q^{35} -5.13742 q^{37} -2.01302 q^{39} -11.5007 q^{41} +0.0837292 q^{43} -9.98525 q^{45} +3.99564 q^{47} +9.33880 q^{49} -6.46871 q^{51} -4.22608 q^{53} +20.6888 q^{55} -2.46871 q^{57} -8.53655 q^{59} -5.72239 q^{61} +12.5086 q^{63} +2.63110 q^{65} +11.4278 q^{67} +15.2250 q^{69} -1.10168 q^{71} -7.25458 q^{73} +13.3600 q^{75} -25.9169 q^{77} -13.9906 q^{79} -8.70740 q^{81} +6.51708 q^{83} +8.45489 q^{85} -9.52474 q^{87} -15.3867 q^{89} -3.29599 q^{91} +18.3591 q^{93} +3.22672 q^{95} -13.2654 q^{97} -19.8414 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{3} + 14 q^{9}+O(q^{10}) $ 10 * q - 4 * q^3 + 14 * q^9	$ 10 q - 4 q^{3} + 14 q^{9} - 20 q^{11} + 4 q^{17} - 10 q^{19} + 10 q^{25} - 28 q^{27} - 8 q^{33} - 36 q^{35} - 12 q^{41} + 4 q^{43} + 26 q^{49} - 36 q^{51} + 4 q^{57} - 52 q^{59} - 24 q^{65} - 12 q^{67} + 12 q^{73} + 12 q^{75} + 34 q^{81} - 16 q^{83} - 20 q^{89} - 60 q^{91} - 28 q^{97} - 60 q^{99}+O(q^{100}) $ 10 * q - 4 * q^3 + 14 * q^9 - 20 * q^11 + 4 * q^17 - 10 * q^19 + 10 * q^25 - 28 * q^27 - 8 * q^33 - 36 * q^35 - 12 * q^41 + 4 * q^43 + 26 * q^49 - 36 * q^51 + 4 * q^57 - 52 * q^59 - 24 * q^65 - 12 * q^67 + 12 * q^73 + 12 * q^75 + 34 * q^81 - 16 * q^83 - 20 * q^89 - 60 * q^91 - 28 * q^97 - 60 * 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
$3$	2.46871	1.42531	0.712657	−	0.701513i	$-0.247492\pi$
$3$	2.46871	1.42531	0.712657	+	0.701513i	$0.247492\pi$
$4$	0	0
$5$	−3.22672	−1.44303	−0.721516	−	0.692398i	$-0.756554\pi$
$5$	−3.22672	−1.44303	−0.721516	+	0.692398i	$0.756554\pi$
$6$	0	0
$7$	4.04213	1.52778	0.763890	−	0.645346i	$-0.223287\pi$
$7$	4.04213	1.52778	0.763890	+	0.645346i	$0.223287\pi$
$8$	0	0
$9$	3.09455	1.03152
$10$	0	0
$11$	−6.41171	−1.93320	−0.966601	−	0.256286i	$-0.917501\pi$
$11$	−6.41171	−1.93320	−0.966601	+	0.256286i	$0.917501\pi$
$12$	0	0
$13$	−0.815411	−0.226154	−0.113077	−	0.993586i	$-0.536071\pi$
$13$	−0.815411	−0.226154	−0.113077	+	0.993586i	$0.536071\pi$
$14$	0	0
$15$	−7.96585	−2.05677
$16$	0	0
$17$	−2.62028	−0.635510	−0.317755	−	0.948173i	$-0.602929\pi$
$17$	−2.62028	−0.635510	−0.317755	+	0.948173i	$0.602929\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	9.97886	2.17757
$22$	0	0
$23$	6.16717	1.28594	0.642972	−	0.765890i	$-0.277701\pi$
$23$	6.16717	1.28594	0.642972	+	0.765890i	$0.277701\pi$
$24$	0	0
$25$	5.41171	1.08234
$26$	0	0
$27$	0.233424	0.0449225
$28$	0	0
$29$	−3.85818	−0.716446	−0.358223	−	0.933636i	$-0.616617\pi$
$29$	−3.85818	−0.716446	−0.358223	+	0.933636i	$0.616617\pi$
$30$	0	0
$31$	7.43670	1.33567	0.667835	−	0.744309i	$-0.267221\pi$
$31$	7.43670	1.33567	0.667835	+	0.744309i	$0.267221\pi$
$32$	0	0
$33$	−15.8287	−2.75542
$34$	0	0
$35$	−13.0428	−2.20464
$36$	0	0
$37$	−5.13742	−0.844586	−0.422293	−	0.906459i	$-0.638775\pi$
$37$	−5.13742	−0.844586	−0.422293	+	0.906459i	$0.638775\pi$
$38$	0	0
$39$	−2.01302	−0.322341
$40$	0	0
$41$	−11.5007	−1.79611	−0.898054	−	0.439886i	$-0.855019\pi$
$41$	−11.5007	−1.79611	−0.898054	+	0.439886i	$0.855019\pi$
$42$	0	0
$43$	0.0837292	0.0127686	0.00638429	−	0.999980i	$-0.497968\pi$
$43$	0.0837292	0.0127686	0.00638429	+	0.999980i	$0.497968\pi$
$44$	0	0
$45$	−9.98525	−1.48851
$46$	0	0
$47$	3.99564	0.582824	0.291412	−	0.956598i	$-0.405875\pi$
$47$	3.99564	0.582824	0.291412	+	0.956598i	$0.405875\pi$
$48$	0	0
$49$	9.33880	1.33411
$50$	0	0
$51$	−6.46871	−0.905801
$52$	0	0
$53$	−4.22608	−0.580497	−0.290248	−	0.956951i	$-0.593738\pi$
$53$	−4.22608	−0.580497	−0.290248	+	0.956951i	$0.593738\pi$
$54$	0	0
$55$	20.6888	2.78967
$56$	0	0
$57$	−2.46871	−0.326989
$58$	0	0
$59$	−8.53655	−1.11136	−0.555682	−	0.831395i	$-0.687543\pi$
$59$	−8.53655	−1.11136	−0.555682	+	0.831395i	$0.687543\pi$
$60$	0	0
$61$	−5.72239	−0.732677	−0.366339	−	0.930482i	$-0.619389\pi$
$61$	−5.72239	−0.732677	−0.366339	+	0.930482i	$0.619389\pi$
$62$	0	0
$63$	12.5086	1.57593
$64$	0	0
$65$	2.63110	0.326348
$66$	0	0
$67$	11.4278	1.39613	0.698063	−	0.716036i	$-0.254045\pi$
$67$	11.4278	1.39613	0.698063	+	0.716036i	$0.254045\pi$
$68$	0	0
$69$	15.2250	1.83287
$70$	0	0
$71$	−1.10168	−0.130745	−0.0653725	−	0.997861i	$-0.520824\pi$
$71$	−1.10168	−0.130745	−0.0653725	+	0.997861i	$0.520824\pi$
$72$	0	0
$73$	−7.25458	−0.849085	−0.424542	−	0.905408i	$-0.639565\pi$
$73$	−7.25458	−0.849085	−0.424542	+	0.905408i	$0.639565\pi$
$74$	0	0
$75$	13.3600	1.54268
$76$	0	0
$77$	−25.9169	−2.95351
$78$	0	0
$79$	−13.9906	−1.57407	−0.787033	−	0.616911i	$-0.788384\pi$
$79$	−13.9906	−1.57407	−0.787033	+	0.616911i	$0.788384\pi$
$80$	0	0
$81$	−8.70740	−0.967489
$82$	0	0
$83$	6.51708	0.715343	0.357671	−	0.933848i	$-0.383571\pi$
$83$	6.51708	0.715343	0.357671	+	0.933848i	$0.383571\pi$
$84$	0	0
$85$	8.45489	0.917062
$86$	0	0
$87$	−9.52474	−1.02116
$88$	0	0
$89$	−15.3867	−1.63098	−0.815492	−	0.578768i	$-0.803534\pi$
$89$	−15.3867	−1.63098	−0.815492	+	0.578768i	$0.803534\pi$
$90$	0	0
$91$	−3.29599	−0.345514
$92$	0	0
$93$	18.3591	1.90375
$94$	0	0
$95$	3.22672	0.331054
$96$	0	0
$97$	−13.2654	−1.34690	−0.673449	−	0.739234i	$-0.735188\pi$
$97$	−13.2654	−1.34690	−0.673449	+	0.739234i	$0.735188\pi$
$98$	0	0
$99$	−19.8414	−1.99413
$100$	0	0
$101$	−2.76311	−0.274940	−0.137470	−	0.990506i	$-0.543897\pi$
$101$	−2.76311	−0.274940	−0.137470	+	0.990506i	$0.543897\pi$
$102$	0	0
$103$	0.529145	0.0521382	0.0260691	−	0.999660i	$-0.491701\pi$
$103$	0.529145	0.0521382	0.0260691	+	0.999660i	$0.491701\pi$
$104$	0	0
$105$	−32.1990	−3.14230
$106$	0	0
$107$	−0.211777	−0.0204732	−0.0102366	−	0.999948i	$-0.503258\pi$
$107$	−0.211777	−0.0204732	−0.0102366	+	0.999948i	$0.503258\pi$
$108$	0	0
$109$	−9.73908	−0.932835	−0.466417	−	0.884565i	$-0.654456\pi$
$109$	−9.73908	−0.932835	−0.466417	+	0.884565i	$0.654456\pi$
$110$	0	0
$111$	−12.6828	−1.20380
$112$	0	0
$113$	−6.09269	−0.573152	−0.286576	−	0.958058i	$-0.592517\pi$
$113$	−6.09269	−0.573152	−0.286576	+	0.958058i	$0.592517\pi$
$114$	0	0
$115$	−19.8997	−1.85566
$116$	0	0
$117$	−2.52333	−0.233282
$118$	0	0
$119$	−10.5915	−0.970921
$120$	0	0
$121$	30.1100	2.73727
$122$	0	0
$123$	−28.3919	−2.56002
$124$	0	0
$125$	−1.32846	−0.118821
$126$	0	0
$127$	−6.64012	−0.589215	−0.294608	−	0.955618i	$-0.595189\pi$
$127$	−6.64012	−0.589215	−0.294608	+	0.955618i	$0.595189\pi$
$128$	0	0
$129$	0.206704	0.0181992
$130$	0	0
$131$	−3.61315	−0.315682	−0.157841	−	0.987465i	$-0.550453\pi$
$131$	−3.61315	−0.315682	−0.157841	+	0.987465i	$0.550453\pi$
$132$	0	0
$133$	−4.04213	−0.350497
$134$	0	0
$135$	−0.753194	−0.0648246
$136$	0	0
$137$	−11.7468	−1.00360	−0.501799	−	0.864984i	$-0.667328\pi$
$137$	−11.7468	−1.00360	−0.501799	+	0.864984i	$0.667328\pi$
$138$	0	0
$139$	20.4757	1.73672	0.868362	−	0.495931i	$-0.165173\pi$
$139$	20.4757	1.73672	0.868362	+	0.495931i	$0.165173\pi$
$140$	0	0
$141$	9.86410	0.830707
$142$	0	0
$143$	5.22817	0.437202
$144$	0	0
$145$	12.4493	1.03385
$146$	0	0
$147$	23.0548	1.90153
$148$	0	0
$149$	20.4213	1.67298	0.836490	−	0.547982i	$-0.184604\pi$
$149$	20.4213	1.67298	0.836490	+	0.547982i	$0.184604\pi$
$150$	0	0
$151$	−13.1686	−1.07164	−0.535822	−	0.844331i	$-0.679998\pi$
$151$	−13.1686	−1.07164	−0.535822	+	0.844331i	$0.679998\pi$
$152$	0	0
$153$	−8.10858	−0.655540
$154$	0	0
$155$	−23.9961	−1.92742
$156$	0	0
$157$	15.0668	1.20246	0.601232	−	0.799074i	$-0.294677\pi$
$157$	15.0668	1.20246	0.601232	+	0.799074i	$0.294677\pi$
$158$	0	0
$159$	−10.4330	−0.827389
$160$	0	0
$161$	24.9285	1.96464
$162$	0	0
$163$	−11.8860	−0.930982	−0.465491	−	0.885053i	$-0.654122\pi$
$163$	−11.8860	−0.930982	−0.465491	+	0.885053i	$0.654122\pi$
$164$	0	0
$165$	51.0747	3.97616
$166$	0	0
$167$	−19.2419	−1.48898	−0.744491	−	0.667632i	$-0.767308\pi$
$167$	−19.2419	−1.48898	−0.744491	+	0.667632i	$0.767308\pi$
$168$	0	0
$169$	−12.3351	−0.948854
$170$	0	0
$171$	−3.09455	−0.236646
$172$	0	0
$173$	−1.53489	−0.116696	−0.0583478	−	0.998296i	$-0.518583\pi$
$173$	−1.53489	−0.116696	−0.0583478	+	0.998296i	$0.518583\pi$
$174$	0	0
$175$	21.8748	1.65358
$176$	0	0
$177$	−21.0743	−1.58404
$178$	0	0
$179$	3.82867	0.286169	0.143084	−	0.989711i	$-0.454298\pi$
$179$	3.82867	0.286169	0.143084	+	0.989711i	$0.454298\pi$
$180$	0	0
$181$	−5.89663	−0.438293	−0.219147	−	0.975692i	$-0.570327\pi$
$181$	−5.89663	−0.438293	−0.219147	+	0.975692i	$0.570327\pi$
$182$	0	0
$183$	−14.1270	−1.04429
$184$	0	0
$185$	16.5770	1.21877
$186$	0	0
$187$	16.8004	1.22857
$188$	0	0
$189$	0.943531	0.0686318
$190$	0	0
$191$	19.8427	1.43577	0.717885	−	0.696162i	$-0.245110\pi$
$191$	19.8427	1.43577	0.717885	+	0.696162i	$0.245110\pi$
$192$	0	0
$193$	−3.62263	−0.260763	−0.130381	−	0.991464i	$-0.541620\pi$
$193$	−3.62263	−0.260763	−0.130381	+	0.991464i	$0.541620\pi$
$194$	0	0
$195$	6.49544	0.465148
$196$	0	0
$197$	13.1190	0.934690	0.467345	−	0.884075i	$-0.345211\pi$
$197$	13.1190	0.934690	0.467345	+	0.884075i	$0.345211\pi$
$198$	0	0
$199$	15.9156	1.12823	0.564113	−	0.825697i	$-0.309218\pi$
$199$	15.9156	1.12823	0.564113	+	0.825697i	$0.309218\pi$
$200$	0	0
$201$	28.2120	1.98992
$202$	0	0
$203$	−15.5953	−1.09457
$204$	0	0
$205$	37.1095	2.59184
$206$	0	0
$207$	19.0846	1.32647
$208$	0	0
$209$	6.41171	0.443507
$210$	0	0
$211$	−5.00713	−0.344705	−0.172352	−	0.985035i	$-0.555137\pi$
$211$	−5.00713	−0.344705	−0.172352	+	0.985035i	$0.555137\pi$
$212$	0	0
$213$	−2.71973	−0.186352
$214$	0	0
$215$	−0.270170	−0.0184255
$216$	0	0
$217$	30.0601	2.04061
$218$	0	0
$219$	−17.9095	−1.21021
$220$	0	0
$221$	2.13660	0.143723
$222$	0	0
$223$	8.30156	0.555913	0.277957	−	0.960594i	$-0.410343\pi$
$223$	8.30156	0.555913	0.277957	+	0.960594i	$0.410343\pi$
$224$	0	0
$225$	16.7468	1.11645
$226$	0	0
$227$	26.8177	1.77995	0.889976	−	0.456007i	$-0.150721\pi$
$227$	26.8177	1.77995	0.889976	+	0.456007i	$0.150721\pi$
$228$	0	0
$229$	−6.15107	−0.406474	−0.203237	−	0.979130i	$-0.565146\pi$
$229$	−6.15107	−0.406474	−0.203237	+	0.979130i	$0.565146\pi$
$230$	0	0
$231$	−63.9815	−4.20968
$232$	0	0
$233$	7.79905	0.510933	0.255466	−	0.966818i	$-0.417771\pi$
$233$	7.79905	0.510933	0.255466	+	0.966818i	$0.417771\pi$
$234$	0	0
$235$	−12.8928	−0.841034
$236$	0	0
$237$	−34.5388	−2.24354
$238$	0	0
$239$	−27.7964	−1.79800	−0.898999	−	0.437951i	$-0.855704\pi$
$239$	−27.7964	−1.79800	−0.898999	+	0.437951i	$0.855704\pi$
$240$	0	0
$241$	−5.53347	−0.356442	−0.178221	−	0.983990i	$-0.557034\pi$
$241$	−5.53347	−0.356442	−0.178221	+	0.983990i	$0.557034\pi$
$242$	0	0
$243$	−22.1964	−1.42390
$244$	0	0
$245$	−30.1337	−1.92517
$246$	0	0
$247$	0.815411	0.0518833
$248$	0	0
$249$	16.0888	1.01959
$250$	0	0
$251$	14.9928	0.946334	0.473167	−	0.880973i	$-0.343111\pi$
$251$	14.9928	0.946334	0.473167	+	0.880973i	$0.343111\pi$
$252$	0	0
$253$	−39.5421	−2.48599
$254$	0	0
$255$	20.8727	1.30710
$256$	0	0
$257$	13.6003	0.848365	0.424182	−	0.905577i	$-0.360562\pi$
$257$	13.6003	0.848365	0.424182	+	0.905577i	$0.360562\pi$
$258$	0	0
$259$	−20.7661	−1.29034
$260$	0	0
$261$	−11.9393	−0.739026
$262$	0	0
$263$	26.4415	1.63046	0.815228	−	0.579140i	$-0.196612\pi$
$263$	26.4415	1.63046	0.815228	+	0.579140i	$0.196612\pi$
$264$	0	0
$265$	13.6364	0.837675
$266$	0	0
$267$	−37.9853	−2.32466
$268$	0	0
$269$	−19.6429	−1.19765	−0.598825	−	0.800880i	$-0.704365\pi$
$269$	−19.6429	−1.19765	−0.598825	+	0.800880i	$0.704365\pi$
$270$	0	0
$271$	−14.0146	−0.851327	−0.425663	−	0.904882i	$-0.639959\pi$
$271$	−14.0146	−0.851327	−0.425663	+	0.904882i	$0.639959\pi$
$272$	0	0
$273$	−8.13687	−0.492466
$274$	0	0
$275$	−34.6983	−2.09238
$276$	0	0
$277$	−4.92581	−0.295963	−0.147982	−	0.988990i	$-0.547278\pi$
$277$	−4.92581	−0.295963	−0.147982	+	0.988990i	$0.547278\pi$
$278$	0	0
$279$	23.0133	1.37777
$280$	0	0
$281$	4.90466	0.292587	0.146294	−	0.989241i	$-0.453266\pi$
$281$	4.90466	0.292587	0.146294	+	0.989241i	$0.453266\pi$
$282$	0	0
$283$	2.07261	0.123204	0.0616018	−	0.998101i	$-0.480379\pi$
$283$	2.07261	0.123204	0.0616018	+	0.998101i	$0.480379\pi$
$284$	0	0
$285$	7.96585	0.471856
$286$	0	0
$287$	−46.4873	−2.74406
$288$	0	0
$289$	−10.1342	−0.596127
$290$	0	0
$291$	−32.7485	−1.91975
$292$	0	0
$293$	10.4606	0.611117	0.305559	−	0.952173i	$-0.401157\pi$
$293$	10.4606	0.611117	0.305559	+	0.952173i	$0.401157\pi$
$294$	0	0
$295$	27.5450	1.60373
$296$	0	0
$297$	−1.49665	−0.0868443
$298$	0	0
$299$	−5.02878	−0.290822
$300$	0	0
$301$	0.338444	0.0195076
$302$	0	0
$303$	−6.82134	−0.391875
$304$	0	0
$305$	18.4645	1.05728
$306$	0	0
$307$	−5.87873	−0.335517	−0.167758	−	0.985828i	$-0.553653\pi$
$307$	−5.87873	−0.335517	−0.167758	+	0.985828i	$0.553653\pi$
$308$	0	0
$309$	1.30631	0.0743132
$310$	0	0
$311$	−12.6861	−0.719365	−0.359683	−	0.933075i	$-0.617115\pi$
$311$	−12.6861	−0.719365	−0.359683	+	0.933075i	$0.617115\pi$
$312$	0	0
$313$	6.25089	0.353321	0.176661	−	0.984272i	$-0.443471\pi$
$313$	6.25089	0.353321	0.176661	+	0.984272i	$0.443471\pi$
$314$	0	0
$315$	−40.3617	−2.27412
$316$	0	0
$317$	−4.72978	−0.265651	−0.132826	−	0.991139i	$-0.542405\pi$
$317$	−4.72978	−0.265651	−0.132826	+	0.991139i	$0.542405\pi$
$318$	0	0
$319$	24.7375	1.38503
$320$	0	0
$321$	−0.522816	−0.0291808
$322$	0	0
$323$	2.62028	0.145796
$324$	0	0
$325$	−4.41276	−0.244776
$326$	0	0
$327$	−24.0430	−1.32958
$328$	0	0
$329$	16.1509	0.890428
$330$	0	0
$331$	−19.8060	−1.08864	−0.544318	−	0.838879i	$-0.683212\pi$
$331$	−19.8060	−1.08864	−0.544318	+	0.838879i	$0.683212\pi$
$332$	0	0
$333$	−15.8980	−0.871206
$334$	0	0
$335$	−36.8743	−2.01466
$336$	0	0
$337$	9.40107	0.512109	0.256054	−	0.966662i	$-0.417577\pi$
$337$	9.40107	0.512109	0.256054	+	0.966662i	$0.417577\pi$
$338$	0	0
$339$	−15.0411	−0.816921
$340$	0	0
$341$	−47.6819	−2.58212
$342$	0	0
$343$	9.45373	0.510454
$344$	0	0
$345$	−49.1267	−2.64489
$346$	0	0
$347$	25.2991	1.35813	0.679063	−	0.734080i	$-0.262386\pi$
$347$	25.2991	1.35813	0.679063	+	0.734080i	$0.262386\pi$
$348$	0	0
$349$	28.9343	1.54882	0.774408	−	0.632687i	$-0.218048\pi$
$349$	28.9343	1.54882	0.774408	+	0.632687i	$0.218048\pi$
$350$	0	0
$351$	−0.190337	−0.0101594
$352$	0	0
$353$	−22.4897	−1.19701	−0.598503	−	0.801121i	$-0.704238\pi$
$353$	−22.4897	−1.19701	−0.598503	+	0.801121i	$0.704238\pi$
$354$	0	0
$355$	3.55480	0.188669
$356$	0	0
$357$	−26.1474	−1.38387
$358$	0	0
$359$	25.7745	1.36033	0.680164	−	0.733060i	$-0.261909\pi$
$359$	25.7745	1.36033	0.680164	+	0.733060i	$0.261909\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	74.3330	3.90147
$364$	0	0
$365$	23.4085	1.22526
$366$	0	0
$367$	−3.91733	−0.204483	−0.102242	−	0.994760i	$-0.532601\pi$
$367$	−3.91733	−0.204483	−0.102242	+	0.994760i	$0.532601\pi$
$368$	0	0
$369$	−35.5895	−1.85272
$370$	0	0
$371$	−17.0824	−0.886871
$372$	0	0
$373$	−7.17543	−0.371530	−0.185765	−	0.982594i	$-0.559476\pi$
$373$	−7.17543	−0.371530	−0.185765	+	0.982594i	$0.559476\pi$
$374$	0	0
$375$	−3.27959	−0.169357
$376$	0	0
$377$	3.14600	0.162027
$378$	0	0
$379$	9.11028	0.467964	0.233982	−	0.972241i	$-0.424824\pi$
$379$	9.11028	0.467964	0.233982	+	0.972241i	$0.424824\pi$
$380$	0	0
$381$	−16.3926	−0.839816
$382$	0	0
$383$	−28.4774	−1.45513	−0.727564	−	0.686040i	$-0.759348\pi$
$383$	−28.4774	−1.45513	−0.727564	+	0.686040i	$0.759348\pi$
$384$	0	0
$385$	83.6266	4.26201
$386$	0	0
$387$	0.259104	0.0131710
$388$	0	0
$389$	−27.5101	−1.39482	−0.697409	−	0.716674i	$-0.745664\pi$
$389$	−27.5101	−1.39482	−0.697409	+	0.716674i	$0.745664\pi$
$390$	0	0
$391$	−16.1597	−0.817231
$392$	0	0
$393$	−8.91983	−0.449946
$394$	0	0
$395$	45.1437	2.27143
$396$	0	0
$397$	32.7251	1.64242	0.821212	−	0.570624i	$-0.193298\pi$
$397$	32.7251	1.64242	0.821212	+	0.570624i	$0.193298\pi$
$398$	0	0
$399$	−9.97886	−0.499568
$400$	0	0
$401$	13.5472	0.676515	0.338257	−	0.941054i	$-0.390163\pi$
$401$	13.5472	0.676515	0.338257	+	0.941054i	$0.390163\pi$
$402$	0	0
$403$	−6.06397	−0.302068
$404$	0	0
$405$	28.0963	1.39612
$406$	0	0
$407$	32.9396	1.63276
$408$	0	0
$409$	27.0909	1.33956	0.669779	−	0.742561i	$-0.266389\pi$
$409$	27.0909	1.33956	0.669779	+	0.742561i	$0.266389\pi$
$410$	0	0
$411$	−28.9995	−1.43044
$412$	0	0
$413$	−34.5058	−1.69792
$414$	0	0
$415$	−21.0288	−1.03226
$416$	0	0
$417$	50.5486	2.47538
$418$	0	0
$419$	−12.0175	−0.587092	−0.293546	−	0.955945i	$-0.594835\pi$
$419$	−12.0175	−0.587092	−0.293546	+	0.955945i	$0.594835\pi$
$420$	0	0
$421$	−21.6253	−1.05395	−0.526977	−	0.849880i	$-0.676675\pi$
$421$	−21.6253	−1.05395	−0.526977	+	0.849880i	$0.676675\pi$
$422$	0	0
$423$	12.3647	0.601194
$424$	0	0
$425$	−14.1802	−0.687839
$426$	0	0
$427$	−23.1306	−1.11937
$428$	0	0
$429$	12.9069	0.623150
$430$	0	0
$431$	−0.466340	−0.0224628	−0.0112314	−	0.999937i	$-0.503575\pi$
$431$	−0.466340	−0.0224628	−0.0112314	+	0.999937i	$0.503575\pi$
$432$	0	0
$433$	−12.1789	−0.585281	−0.292640	−	0.956223i	$-0.594534\pi$
$433$	−12.1789	−0.585281	−0.292640	+	0.956223i	$0.594534\pi$
$434$	0	0
$435$	30.7337	1.47357
$436$	0	0
$437$	−6.16717	−0.295016
$438$	0	0
$439$	22.9016	1.09303	0.546517	−	0.837448i	$-0.315953\pi$
$439$	22.9016	1.09303	0.546517	+	0.837448i	$0.315953\pi$
$440$	0	0
$441$	28.8994	1.37616
$442$	0	0
$443$	−4.15864	−0.197583	−0.0987914	−	0.995108i	$-0.531498\pi$
$443$	−4.15864	−0.197583	−0.0987914	+	0.995108i	$0.531498\pi$
$444$	0	0
$445$	49.6485	2.35356
$446$	0	0
$447$	50.4144	2.38452
$448$	0	0
$449$	−13.4154	−0.633112	−0.316556	−	0.948574i	$-0.602527\pi$
$449$	−13.4154	−0.633112	−0.316556	+	0.948574i	$0.602527\pi$
$450$	0	0
$451$	73.7391	3.47224
$452$	0	0
$453$	−32.5095	−1.52743
$454$	0	0
$455$	10.6352	0.498588
$456$	0	0
$457$	−23.9359	−1.11968	−0.559838	−	0.828602i	$-0.689136\pi$
$457$	−23.9359	−1.11968	−0.559838	+	0.828602i	$0.689136\pi$
$458$	0	0
$459$	−0.611636	−0.0285487
$460$	0	0
$461$	31.0335	1.44537	0.722686	−	0.691176i	$-0.242907\pi$
$461$	31.0335	1.44537	0.722686	+	0.691176i	$0.242907\pi$
$462$	0	0
$463$	31.3472	1.45683	0.728415	−	0.685136i	$-0.240257\pi$
$463$	31.3472	1.45683	0.728415	+	0.685136i	$0.240257\pi$
$464$	0	0
$465$	−59.2396	−2.74717
$466$	0	0
$467$	−7.14257	−0.330519	−0.165259	−	0.986250i	$-0.552846\pi$
$467$	−7.14257	−0.330519	−0.165259	+	0.986250i	$0.552846\pi$
$468$	0	0
$469$	46.1926	2.13298
$470$	0	0
$471$	37.1957	1.71389
$472$	0	0
$473$	−0.536847	−0.0246843
$474$	0	0
$475$	−5.41171	−0.248306
$476$	0	0
$477$	−13.0778	−0.598792
$478$	0	0
$479$	26.5447	1.21286	0.606430	−	0.795137i	$-0.292601\pi$
$479$	26.5447	1.21286	0.606430	+	0.795137i	$0.292601\pi$
$480$	0	0
$481$	4.18911	0.191007
$482$	0	0
$483$	61.5413	2.80023
$484$	0	0
$485$	42.8037	1.94362
$486$	0	0
$487$	−9.74569	−0.441619	−0.220810	−	0.975317i	$-0.570870\pi$
$487$	−9.74569	−0.441619	−0.220810	+	0.975317i	$0.570870\pi$
$488$	0	0
$489$	−29.3431	−1.32694
$490$	0	0
$491$	−25.6150	−1.15599	−0.577995	−	0.816040i	$-0.696165\pi$
$491$	−25.6150	−1.15599	−0.577995	+	0.816040i	$0.696165\pi$
$492$	0	0
$493$	10.1095	0.455309
$494$	0	0
$495$	64.0225	2.87760
$496$	0	0
$497$	−4.45312	−0.199750
$498$	0	0
$499$	5.48801	0.245677	0.122838	−	0.992427i	$-0.460800\pi$
$499$	5.48801	0.245677	0.122838	+	0.992427i	$0.460800\pi$
$500$	0	0
$501$	−47.5028	−2.12227
$502$	0	0
$503$	9.00013	0.401296	0.200648	−	0.979663i	$-0.435695\pi$
$503$	9.00013	0.401296	0.200648	+	0.979663i	$0.435695\pi$
$504$	0	0
$505$	8.91578	0.396747
$506$	0	0
$507$	−30.4519	−1.35241
$508$	0	0
$509$	−28.2818	−1.25357	−0.626784	−	0.779193i	$-0.715629\pi$
$509$	−28.2818	−1.25357	−0.626784	+	0.779193i	$0.715629\pi$
$510$	0	0
$511$	−29.3240	−1.29722
$512$	0	0
$513$	−0.233424	−0.0103059
$514$	0	0
$515$	−1.70740	−0.0752371
$516$	0	0
$517$	−25.6189	−1.12672
$518$	0	0
$519$	−3.78921	−0.166328
$520$	0	0
$521$	−4.19618	−0.183838	−0.0919190	−	0.995766i	$-0.529300\pi$
$521$	−4.19618	−0.183838	−0.0919190	+	0.995766i	$0.529300\pi$
$522$	0	0
$523$	33.6631	1.47198	0.735992	−	0.676990i	$-0.236716\pi$
$523$	33.6631	1.47198	0.735992	+	0.676990i	$0.236716\pi$
$524$	0	0
$525$	54.0027	2.35687
$526$	0	0
$527$	−19.4862	−0.848833
$528$	0	0
$529$	15.0340	0.653651
$530$	0	0
$531$	−26.4168	−1.14639
$532$	0	0
$533$	9.37779	0.406197
$534$	0	0
$535$	0.683344	0.0295435
$536$	0	0
$537$	9.45191	0.407880
$538$	0	0
$539$	−59.8776	−2.57911
$540$	0	0
$541$	18.3995	0.791056	0.395528	−	0.918454i	$-0.370562\pi$
$541$	18.3995	0.791056	0.395528	+	0.918454i	$0.370562\pi$
$542$	0	0
$543$	−14.5571	−0.624705
$544$	0	0
$545$	31.4253	1.34611
$546$	0	0
$547$	5.83156	0.249340	0.124670	−	0.992198i	$-0.460213\pi$
$547$	5.83156	0.249340	0.124670	+	0.992198i	$0.460213\pi$
$548$	0	0
$549$	−17.7082	−0.755770
$550$	0	0
$551$	3.85818	0.164364
$552$	0	0
$553$	−56.5518	−2.40483
$554$	0	0
$555$	40.9239	1.73712
$556$	0	0
$557$	30.9517	1.31146	0.655732	−	0.754993i	$-0.272360\pi$
$557$	30.9517	1.31146	0.655732	+	0.754993i	$0.272360\pi$
$558$	0	0
$559$	−0.0682737	−0.00288767
$560$	0	0
$561$	41.4755	1.75110
$562$	0	0
$563$	−43.3007	−1.82490	−0.912452	−	0.409183i	$-0.865814\pi$
$563$	−43.3007	−1.82490	−0.912452	+	0.409183i	$0.865814\pi$
$564$	0	0
$565$	19.6594	0.827076
$566$	0	0
$567$	−35.1964	−1.47811
$568$	0	0
$569$	−8.32798	−0.349127	−0.174563	−	0.984646i	$-0.555851\pi$
$569$	−8.32798	−0.349127	−0.174563	+	0.984646i	$0.555851\pi$
$570$	0	0
$571$	−21.5676	−0.902574	−0.451287	−	0.892379i	$-0.649035\pi$
$571$	−21.5676	−0.902574	−0.451287	+	0.892379i	$0.649035\pi$
$572$	0	0
$573$	48.9861	2.04642
$574$	0	0
$575$	33.3749	1.39183
$576$	0	0
$577$	−31.5716	−1.31434	−0.657172	−	0.753741i	$-0.728247\pi$
$577$	−31.5716	−1.31434	−0.657172	+	0.753741i	$0.728247\pi$
$578$	0	0
$579$	−8.94324	−0.371669
$580$	0	0
$581$	26.3429	1.09289
$582$	0	0
$583$	27.0964	1.12222
$584$	0	0
$585$	8.14208	0.336634
$586$	0	0
$587$	−28.9679	−1.19563	−0.597817	−	0.801633i	$-0.703965\pi$
$587$	−28.9679	−1.19563	−0.597817	+	0.801633i	$0.703965\pi$
$588$	0	0
$589$	−7.43670	−0.306424
$590$	0	0
$591$	32.3871	1.33223
$592$	0	0
$593$	23.8431	0.979117	0.489559	−	0.871970i	$-0.337158\pi$
$593$	23.8431	0.979117	0.489559	+	0.871970i	$0.337158\pi$
$594$	0	0
$595$	34.1758	1.40107
$596$	0	0
$597$	39.2911	1.60808
$598$	0	0
$599$	−24.0716	−0.983537	−0.491769	−	0.870726i	$-0.663649\pi$
$599$	−24.0716	−0.983537	−0.491769	+	0.870726i	$0.663649\pi$
$600$	0	0
$601$	−1.72226	−0.0702523	−0.0351262	−	0.999383i	$-0.511183\pi$
$601$	−1.72226	−0.0702523	−0.0351262	+	0.999383i	$0.511183\pi$
$602$	0	0
$603$	35.3639	1.44013
$604$	0	0
$605$	−97.1564	−3.94997
$606$	0	0
$607$	−17.2198	−0.698928	−0.349464	−	0.936950i	$-0.613636\pi$
$607$	−17.2198	−0.698928	−0.349464	+	0.936950i	$0.613636\pi$
$608$	0	0
$609$	−38.5002	−1.56011
$610$	0	0
$611$	−3.25809	−0.131808
$612$	0	0
$613$	−32.5924	−1.31639	−0.658197	−	0.752845i	$-0.728681\pi$
$613$	−32.5924	−1.31639	−0.658197	+	0.752845i	$0.728681\pi$
$614$	0	0
$615$	91.6128	3.69418
$616$	0	0
$617$	6.44448	0.259445	0.129722	−	0.991550i	$-0.458591\pi$
$617$	6.44448	0.259445	0.129722	+	0.991550i	$0.458591\pi$
$618$	0	0
$619$	31.8637	1.28071	0.640356	−	0.768078i	$-0.278787\pi$
$619$	31.8637	1.28071	0.640356	+	0.768078i	$0.278787\pi$
$620$	0	0
$621$	1.43957	0.0577678
$622$	0	0
$623$	−62.1949	−2.49179
$624$	0	0
$625$	−22.7720	−0.910879
$626$	0	0
$627$	15.8287	0.632136
$628$	0	0
$629$	13.4615	0.536743
$630$	0	0
$631$	7.36348	0.293135	0.146568	−	0.989201i	$-0.453177\pi$
$631$	7.36348	0.293135	0.146568	+	0.989201i	$0.453177\pi$
$632$	0	0
$633$	−12.3612	−0.491312
$634$	0	0
$635$	21.4258	0.850257
$636$	0	0
$637$	−7.61496	−0.301716
$638$	0	0
$639$	−3.40920	−0.134866
$640$	0	0
$641$	39.4715	1.55903	0.779514	−	0.626384i	$-0.215466\pi$
$641$	39.4715	1.55903	0.779514	+	0.626384i	$0.215466\pi$
$642$	0	0
$643$	−15.7411	−0.620768	−0.310384	−	0.950611i	$-0.600458\pi$
$643$	−15.7411	−0.620768	−0.310384	+	0.950611i	$0.600458\pi$
$644$	0	0
$645$	−0.666974	−0.0262621
$646$	0	0
$647$	−17.8689	−0.702499	−0.351250	−	0.936282i	$-0.614243\pi$
$647$	−17.8689	−0.702499	−0.351250	+	0.936282i	$0.614243\pi$
$648$	0	0
$649$	54.7338	2.14849
$650$	0	0
$651$	74.2098	2.90851
$652$	0	0
$653$	−1.83114	−0.0716581	−0.0358291	−	0.999358i	$-0.511407\pi$
$653$	−1.83114	−0.0716581	−0.0358291	+	0.999358i	$0.511407\pi$
$654$	0	0
$655$	11.6586	0.455540
$656$	0	0
$657$	−22.4497	−0.875846
$658$	0	0
$659$	−17.9446	−0.699021	−0.349510	−	0.936932i	$-0.613652\pi$
$659$	−17.9446	−0.699021	−0.349510	+	0.936932i	$0.613652\pi$
$660$	0	0
$661$	0.256100	0.00996114	0.00498057	−	0.999988i	$-0.498415\pi$
$661$	0.256100	0.00996114	0.00498057	+	0.999988i	$0.498415\pi$
$662$	0	0
$663$	5.27466	0.204851
$664$	0	0
$665$	13.0428	0.505778
$666$	0	0
$667$	−23.7940	−0.921309
$668$	0	0
$669$	20.4942	0.792350
$670$	0	0
$671$	36.6903	1.41641
$672$	0	0
$673$	−3.46127	−0.133422	−0.0667111	−	0.997772i	$-0.521251\pi$
$673$	−3.46127	−0.133422	−0.0667111	+	0.997772i	$0.521251\pi$
$674$	0	0
$675$	1.26322	0.0486215
$676$	0	0
$677$	13.7402	0.528080	0.264040	−	0.964512i	$-0.414945\pi$
$677$	13.7402	0.528080	0.264040	+	0.964512i	$0.414945\pi$
$678$	0	0
$679$	−53.6205	−2.05777
$680$	0	0
$681$	66.2052	2.53699
$682$	0	0
$683$	5.31127	0.203230	0.101615	−	0.994824i	$-0.467599\pi$
$683$	5.31127	0.203230	0.101615	+	0.994824i	$0.467599\pi$
$684$	0	0
$685$	37.9036	1.44822
$686$	0	0
$687$	−15.1852	−0.579353
$688$	0	0
$689$	3.44599	0.131282
$690$	0	0
$691$	−14.0583	−0.534802	−0.267401	−	0.963585i	$-0.586165\pi$
$691$	−14.0583	−0.534802	−0.267401	+	0.963585i	$0.586165\pi$
$692$	0	0
$693$	−80.2013	−3.04660
$694$	0	0
$695$	−66.0692	−2.50615
$696$	0	0
$697$	30.1350	1.14144
$698$	0	0
$699$	19.2536	0.728239
$700$	0	0
$701$	47.7193	1.80233	0.901167	−	0.433473i	$-0.142712\pi$
$701$	47.7193	1.80233	0.901167	+	0.433473i	$0.142712\pi$
$702$	0	0
$703$	5.13742	0.193761
$704$	0	0
$705$	−31.8287	−1.19874
$706$	0	0
$707$	−11.1689	−0.420048
$708$	0	0
$709$	−12.8747	−0.483519	−0.241759	−	0.970336i	$-0.577724\pi$
$709$	−12.8747	−0.483519	−0.241759	+	0.970336i	$0.577724\pi$
$710$	0	0
$711$	−43.2947	−1.62368
$712$	0	0
$713$	45.8634	1.71760
$714$	0	0
$715$	−16.8698	−0.630896
$716$	0	0
$717$	−68.6213	−2.56271
$718$	0	0
$719$	6.30671	0.235201	0.117600	−	0.993061i	$-0.462480\pi$
$719$	6.30671	0.235201	0.117600	+	0.993061i	$0.462480\pi$
$720$	0	0
$721$	2.13887	0.0796557
$722$	0	0
$723$	−13.6606	−0.508042
$724$	0	0
$725$	−20.8793	−0.775439
$726$	0	0
$727$	24.8092	0.920123	0.460061	−	0.887887i	$-0.347827\pi$
$727$	24.8092	0.920123	0.460061	+	0.887887i	$0.347827\pi$
$728$	0	0
$729$	−28.6743	−1.06201
$730$	0	0
$731$	−0.219394	−0.00811457
$732$	0	0
$733$	20.8961	0.771816	0.385908	−	0.922537i	$-0.373888\pi$
$733$	20.8961	0.771816	0.385908	+	0.922537i	$0.373888\pi$
$734$	0	0
$735$	−74.3914	−2.74397
$736$	0	0
$737$	−73.2716	−2.69900
$738$	0	0
$739$	−9.57101	−0.352075	−0.176038	−	0.984383i	$-0.556328\pi$
$739$	−9.57101	−0.352075	−0.176038	+	0.984383i	$0.556328\pi$
$740$	0	0
$741$	2.01302	0.0739500
$742$	0	0
$743$	10.3553	0.379900	0.189950	−	0.981794i	$-0.439167\pi$
$743$	10.3553	0.379900	0.189950	+	0.981794i	$0.439167\pi$
$744$	0	0
$745$	−65.8939	−2.41416
$746$	0	0
$747$	20.1675	0.737889
$748$	0	0
$749$	−0.856029	−0.0312786
$750$	0	0
$751$	30.4868	1.11248	0.556239	−	0.831022i	$-0.312244\pi$
$751$	30.4868	1.11248	0.556239	+	0.831022i	$0.312244\pi$
$752$	0	0
$753$	37.0128	1.34882
$754$	0	0
$755$	42.4913	1.54642
$756$	0	0
$757$	−13.0841	−0.475549	−0.237774	−	0.971320i	$-0.576418\pi$
$757$	−13.0841	−0.475549	−0.237774	+	0.971320i	$0.576418\pi$
$758$	0	0
$759$	−97.6181	−3.54331
$760$	0	0
$761$	32.3638	1.17319	0.586594	−	0.809881i	$-0.300468\pi$
$761$	32.3638	1.17319	0.586594	+	0.809881i	$0.300468\pi$
$762$	0	0
$763$	−39.3666	−1.42517
$764$	0	0
$765$	26.1641	0.945965
$766$	0	0
$767$	6.96079	0.251340
$768$	0	0
$769$	10.9234	0.393908	0.196954	−	0.980413i	$-0.436895\pi$
$769$	10.9234	0.393908	0.196954	+	0.980413i	$0.436895\pi$
$770$	0	0
$771$	33.5753	1.20919
$772$	0	0
$773$	−34.9008	−1.25530	−0.627648	−	0.778497i	$-0.715982\pi$
$773$	−34.9008	−1.25530	−0.627648	+	0.778497i	$0.715982\pi$
$774$	0	0
$775$	40.2452	1.44565
$776$	0	0
$777$	−51.2656	−1.83914
$778$	0	0
$779$	11.5007	0.412055
$780$	0	0
$781$	7.06363	0.252756
$782$	0	0
$783$	−0.900592	−0.0321845
$784$	0	0
$785$	−48.6164	−1.73519
$786$	0	0
$787$	5.72739	0.204159	0.102080	−	0.994776i	$-0.467450\pi$
$787$	5.72739	0.204159	0.102080	+	0.994776i	$0.467450\pi$
$788$	0	0
$789$	65.2766	2.32391
$790$	0	0
$791$	−24.6274	−0.875650
$792$	0	0
$793$	4.66610	0.165698
$794$	0	0
$795$	33.6643	1.19395
$796$	0	0
$797$	45.1622	1.59973	0.799863	−	0.600183i	$-0.204906\pi$
$797$	45.1622	1.59973	0.799863	+	0.600183i	$0.204906\pi$
$798$	0	0
$799$	−10.4697	−0.370391
$800$	0	0
$801$	−47.6149	−1.68239
$802$	0	0
$803$	46.5143	1.64145
$804$	0	0
$805$	−80.4372	−2.83504
$806$	0	0
$807$	−48.4928	−1.70703
$808$	0	0
$809$	35.4870	1.24766	0.623828	−	0.781562i	$-0.285577\pi$
$809$	35.4870	1.24766	0.623828	+	0.781562i	$0.285577\pi$
$810$	0	0
$811$	20.2334	0.710492	0.355246	−	0.934773i	$-0.384397\pi$
$811$	20.2334	0.710492	0.355246	+	0.934773i	$0.384397\pi$
$812$	0	0
$813$	−34.5981	−1.21341
$814$	0	0
$815$	38.3527	1.34344
$816$	0	0
$817$	−0.0837292	−0.00292931
$818$	0	0
$819$	−10.1996	−0.356404
$820$	0	0
$821$	52.0305	1.81588	0.907938	−	0.419104i	$-0.137656\pi$
$821$	52.0305	1.81588	0.907938	+	0.419104i	$0.137656\pi$
$822$	0	0
$823$	5.60310	0.195312	0.0976559	−	0.995220i	$-0.468866\pi$
$823$	5.60310	0.195312	0.0976559	+	0.995220i	$0.468866\pi$
$824$	0	0
$825$	−85.6601	−2.98230
$826$	0	0
$827$	−28.3875	−0.987129	−0.493565	−	0.869709i	$-0.664306\pi$
$827$	−28.3875	−0.987129	−0.493565	+	0.869709i	$0.664306\pi$
$828$	0	0
$829$	32.5931	1.13201	0.566003	−	0.824403i	$-0.308489\pi$
$829$	32.5931	1.13201	0.566003	+	0.824403i	$0.308489\pi$
$830$	0	0
$831$	−12.1604	−0.421841
$832$	0	0
$833$	−24.4702	−0.847844
$834$	0	0
$835$	62.0882	2.14865
$836$	0	0
$837$	1.73591	0.0600017
$838$	0	0
$839$	45.0055	1.55376	0.776881	−	0.629647i	$-0.216801\pi$
$839$	45.0055	1.55376	0.776881	+	0.629647i	$0.216801\pi$
$840$	0	0
$841$	−14.1145	−0.486706
$842$	0	0
$843$	12.1082	0.417029
$844$	0	0
$845$	39.8019	1.36923
$846$	0	0
$847$	121.708	4.18195
$848$	0	0
$849$	5.11667	0.175604
$850$	0	0
$851$	−31.6833	−1.08609
$852$	0	0
$853$	−21.6381	−0.740875	−0.370438	−	0.928857i	$-0.620792\pi$
$853$	−21.6381	−0.740875	−0.370438	+	0.928857i	$0.620792\pi$
$854$	0	0
$855$	9.98525	0.341488
$856$	0	0
$857$	−11.4432	−0.390892	−0.195446	−	0.980714i	$-0.562616\pi$
$857$	−11.4432	−0.390892	−0.195446	+	0.980714i	$0.562616\pi$
$858$	0	0
$859$	−7.57422	−0.258429	−0.129215	−	0.991617i	$-0.541246\pi$
$859$	−7.57422	−0.258429	−0.129215	+	0.991617i	$0.541246\pi$
$860$	0	0
$861$	−114.764	−3.91114
$862$	0	0
$863$	−34.6063	−1.17801	−0.589007	−	0.808128i	$-0.700481\pi$
$863$	−34.6063	−1.17801	−0.589007	+	0.808128i	$0.700481\pi$
$864$	0	0
$865$	4.95266	0.168396
$866$	0	0
$867$	−25.0183	−0.849667
$868$	0	0
$869$	89.7036	3.04299
$870$	0	0
$871$	−9.31834	−0.315740
$872$	0	0
$873$	−41.0505	−1.38935
$874$	0	0
$875$	−5.36981	−0.181533
$876$	0	0
$877$	−29.4493	−0.994434	−0.497217	−	0.867626i	$-0.665645\pi$
$877$	−29.4493	−0.994434	−0.497217	+	0.867626i	$0.665645\pi$
$878$	0	0
$879$	25.8243	0.871033
$880$	0	0
$881$	51.5583	1.73704	0.868522	−	0.495651i	$-0.165070\pi$
$881$	51.5583	1.73704	0.868522	+	0.495651i	$0.165070\pi$
$882$	0	0
$883$	−25.2991	−0.851382	−0.425691	−	0.904869i	$-0.639969\pi$
$883$	−25.2991	−0.851382	−0.425691	+	0.904869i	$0.639969\pi$
$884$	0	0
$885$	68.0008	2.28582
$886$	0	0
$887$	−1.92340	−0.0645815	−0.0322908	−	0.999479i	$-0.510280\pi$
$887$	−1.92340	−0.0645815	−0.0322908	+	0.999479i	$0.510280\pi$
$888$	0	0
$889$	−26.8402	−0.900192
$890$	0	0
$891$	55.8293	1.87035
$892$	0	0
$893$	−3.99564	−0.133709
$894$	0	0
$895$	−12.3541	−0.412950
$896$	0	0
$897$	−12.4146	−0.414512
$898$	0	0
$899$	−28.6921	−0.956936
$900$	0	0
$901$	11.0735	0.368912
$902$	0	0
$903$	0.835522	0.0278044
$904$	0	0
$905$	19.0268	0.632471
$906$	0	0
$907$	27.7093	0.920071	0.460036	−	0.887900i	$-0.347837\pi$
$907$	27.7093	0.920071	0.460036	+	0.887900i	$0.347837\pi$
$908$	0	0
$909$	−8.55060	−0.283605
$910$	0	0
$911$	51.4735	1.70539	0.852696	−	0.522407i	$-0.174966\pi$
$911$	51.4735	1.70539	0.852696	+	0.522407i	$0.174966\pi$
$912$	0	0
$913$	−41.7856	−1.38290
$914$	0	0
$915$	45.5837	1.50695
$916$	0	0
$917$	−14.6048	−0.482293
$918$	0	0
$919$	−25.4158	−0.838390	−0.419195	−	0.907896i	$-0.637688\pi$
$919$	−25.4158	−0.838390	−0.419195	+	0.907896i	$0.637688\pi$
$920$	0	0
$921$	−14.5129	−0.478216
$922$	0	0
$923$	0.898319	0.0295685
$924$	0	0
$925$	−27.8022	−0.914131
$926$	0	0
$927$	1.63747	0.0537815
$928$	0	0
$929$	−41.2485	−1.35332	−0.676661	−	0.736295i	$-0.736574\pi$
$929$	−41.2485	−1.35332	−0.676661	+	0.736295i	$0.736574\pi$
$930$	0	0
$931$	−9.33880	−0.306067
$932$	0	0
$933$	−31.3185	−1.02532
$934$	0	0
$935$	−54.2103	−1.77287
$936$	0	0
$937$	−36.1849	−1.18211	−0.591054	−	0.806632i	$-0.701288\pi$
$937$	−36.1849	−1.18211	−0.591054	+	0.806632i	$0.701288\pi$
$938$	0	0
$939$	15.4317	0.503593
$940$	0	0
$941$	−17.2262	−0.561559	−0.280779	−	0.959772i	$-0.590593\pi$
$941$	−17.2262	−0.561559	−0.280779	+	0.959772i	$0.590593\pi$
$942$	0	0
$943$	−70.9267	−2.30969
$944$	0	0
$945$	−3.04451	−0.0990378
$946$	0	0
$947$	−17.7823	−0.577847	−0.288924	−	0.957352i	$-0.593297\pi$
$947$	−17.7823	−0.577847	−0.288924	+	0.957352i	$0.593297\pi$
$948$	0	0
$949$	5.91546	0.192024
$950$	0	0
$951$	−11.6765	−0.378636
$952$	0	0
$953$	−47.3478	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$953$	−47.3478	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$954$	0	0
$955$	−64.0269	−2.07186
$956$	0	0
$957$	61.0698	1.97411
$958$	0	0
$959$	−47.4821	−1.53328
$960$	0	0
$961$	24.3045	0.784017
$962$	0	0
$963$	−0.655354	−0.0211185
$964$	0	0
$965$	11.6892	0.376289
$966$	0	0
$967$	−25.3504	−0.815213	−0.407606	−	0.913158i	$-0.633636\pi$
$967$	−25.3504	−0.815213	−0.407606	+	0.913158i	$0.633636\pi$
$968$	0	0
$969$	6.46871	0.207805
$970$	0	0
$971$	−45.3901	−1.45664	−0.728319	−	0.685238i	$-0.759698\pi$
$971$	−45.3901	−1.45664	−0.728319	+	0.685238i	$0.759698\pi$
$972$	0	0
$973$	82.7653	2.65333
$974$	0	0
$975$	−10.8939	−0.348883
$976$	0	0
$977$	27.3398	0.874679	0.437339	−	0.899297i	$-0.355921\pi$
$977$	27.3398	0.874679	0.437339	+	0.899297i	$0.355921\pi$
$978$	0	0
$979$	98.6549	3.15302
$980$	0	0
$981$	−30.1381	−0.962236
$982$	0	0
$983$	−4.84750	−0.154611	−0.0773056	−	0.997007i	$-0.524632\pi$
$983$	−4.84750	−0.154611	−0.0773056	+	0.997007i	$0.524632\pi$
$984$	0	0
$985$	−42.3313	−1.34879
$986$	0	0
$987$	39.8720	1.26914
$988$	0	0
$989$	0.516372	0.0164197
$990$	0	0
$991$	28.1022	0.892696	0.446348	−	0.894859i	$-0.352724\pi$
$991$	28.1022	0.892696	0.446348	+	0.894859i	$0.352724\pi$
$992$	0	0
$993$	−48.8954	−1.55165
$994$	0	0
$995$	−51.3551	−1.62807
$996$	0	0
$997$	−6.99262	−0.221458	−0.110729	−	0.993851i	$-0.535319\pi$
$997$	−6.99262	−0.221458	−0.110729	+	0.993851i	$0.535319\pi$
$998$	0	0
$999$	−1.19920	−0.0379409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bs.1.9		10
4.3	odd	2		4864.2.a.bt.1.1		10
8.3	odd	2	inner	4864.2.a.bs.1.10		10
8.5	even	2		4864.2.a.bt.1.2		10
16.3	odd	4		2432.2.c.j.1217.4	yes	20
16.5	even	4		2432.2.c.j.1217.3	✓	20
16.11	odd	4		2432.2.c.j.1217.17	yes	20
16.13	even	4		2432.2.c.j.1217.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.3	✓	20	16.5	even	4
2432.2.c.j.1217.4	yes	20	16.3	odd	4
2432.2.c.j.1217.17	yes	20	16.11	odd	4
2432.2.c.j.1217.18	yes	20	16.13	even	4
4864.2.a.bs.1.9		10	1.1	even	1	trivial
4864.2.a.bs.1.10		10	8.3	odd	2	inner
4864.2.a.bt.1.1		10	4.3	odd	2
4864.2.a.bt.1.2		10	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.46871 q^{3} -3.22672 q^{5} +4.04213 q^{7} +3.09455 q^{9} +O(q^{10})\)	\(q+2.46871 q^{3} -3.22672 q^{5} +4.04213 q^{7} +3.09455 q^{9} -6.41171 q^{11} -0.815411 q^{13} -7.96585 q^{15} -2.62028 q^{17} -1.00000 q^{19} +9.97886 q^{21} +6.16717 q^{23} +5.41171 q^{25} +0.233424 q^{27} -3.85818 q^{29} +7.43670 q^{31} -15.8287 q^{33} -13.0428 q^{35} -5.13742 q^{37} -2.01302 q^{39} -11.5007 q^{41} +0.0837292 q^{43} -9.98525 q^{45} +3.99564 q^{47} +9.33880 q^{49} -6.46871 q^{51} -4.22608 q^{53} +20.6888 q^{55} -2.46871 q^{57} -8.53655 q^{59} -5.72239 q^{61} +12.5086 q^{63} +2.63110 q^{65} +11.4278 q^{67} +15.2250 q^{69} -1.10168 q^{71} -7.25458 q^{73} +13.3600 q^{75} -25.9169 q^{77} -13.9906 q^{79} -8.70740 q^{81} +6.51708 q^{83} +8.45489 q^{85} -9.52474 q^{87} -15.3867 q^{89} -3.29599 q^{91} +18.3591 q^{93} +3.22672 q^{95} -13.2654 q^{97} -19.8414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{3} + 14 q^{9}+O(q^{10}) \) 10 * q - 4 * q^3 + 14 * q^9	\( 10 q - 4 q^{3} + 14 q^{9} - 20 q^{11} + 4 q^{17} - 10 q^{19} + 10 q^{25} - 28 q^{27} - 8 q^{33} - 36 q^{35} - 12 q^{41} + 4 q^{43} + 26 q^{49} - 36 q^{51} + 4 q^{57} - 52 q^{59} - 24 q^{65} - 12 q^{67} + 12 q^{73} + 12 q^{75} + 34 q^{81} - 16 q^{83} - 20 q^{89} - 60 q^{91} - 28 q^{97} - 60 q^{99}+O(q^{100}) \) 10 * q - 4 * q^3 + 14 * q^9 - 20 * q^11 + 4 * q^17 - 10 * q^19 + 10 * q^25 - 28 * q^27 - 8 * q^33 - 36 * q^35 - 12 * q^41 + 4 * q^43 + 26 * q^49 - 36 * q^51 + 4 * q^57 - 52 * q^59 - 24 * q^65 - 12 * q^67 + 12 * q^73 + 12 * q^75 + 34 * q^81 - 16 * q^83 - 20 * q^89 - 60 * q^91 - 28 * q^97 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more