LMFDB - Embedded newform 588.6.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("588.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.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:	$94.3056860500$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7081}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1770 $ x^2 - x - 1770
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$42.5743$ of defining polynomial
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -65.5743 q^{5} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -65.5743 q^{5} +81.0000 q^{9} -245.574 q^{11} -434.872 q^{13} +590.169 q^{15} +1102.30 q^{17} +2877.29 q^{19} -4229.43 q^{23} +1174.99 q^{25} -729.000 q^{27} +4969.60 q^{29} +8782.65 q^{31} +2210.17 q^{33} -2440.06 q^{37} +3913.85 q^{39} +3668.55 q^{41} -7198.06 q^{43} -5311.52 q^{45} -3273.32 q^{47} -9920.68 q^{51} -3021.67 q^{53} +16103.4 q^{55} -25895.6 q^{57} +51487.5 q^{59} +13313.3 q^{61} +28516.4 q^{65} +30895.7 q^{67} +38064.9 q^{69} -41882.8 q^{71} -34617.3 q^{73} -10574.9 q^{75} +77543.5 q^{79} +6561.00 q^{81} -100908. q^{83} -72282.4 q^{85} -44726.4 q^{87} -40783.7 q^{89} -79043.8 q^{93} -188676. q^{95} -140147. q^{97} -19891.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9	$ 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9} - 407 q^{11} - 449 q^{13} + 423 q^{15} + 1868 q^{17} + 1463 q^{19} - 44 q^{23} - 1605 q^{25} - 1458 q^{27} + 767 q^{29} + 11170 q^{31} + 3663 q^{33} - 3113 q^{37} + 4041 q^{39} + 7842 q^{41} - 12629 q^{43} - 3807 q^{45} - 9576 q^{47} - 16812 q^{51} + 13395 q^{53} + 13105 q^{55} - 13167 q^{57} + 47521 q^{59} + 63652 q^{61} + 28254 q^{65} + 44541 q^{67} + 396 q^{69} - 125840 q^{71} - 6039 q^{73} + 14445 q^{75} + 17588 q^{79} + 13122 q^{81} - 39325 q^{83} - 58060 q^{85} - 6903 q^{87} - 83082 q^{89} - 100530 q^{93} - 214946 q^{95} - 184785 q^{97} - 32967 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9 - 407 * q^11 - 449 * q^13 + 423 * q^15 + 1868 * q^17 + 1463 * q^19 - 44 * q^23 - 1605 * q^25 - 1458 * q^27 + 767 * q^29 + 11170 * q^31 + 3663 * q^33 - 3113 * q^37 + 4041 * q^39 + 7842 * q^41 - 12629 * q^43 - 3807 * q^45 - 9576 * q^47 - 16812 * q^51 + 13395 * q^53 + 13105 * q^55 - 13167 * q^57 + 47521 * q^59 + 63652 * q^61 + 28254 * q^65 + 44541 * q^67 + 396 * q^69 - 125840 * q^71 - 6039 * q^73 + 14445 * q^75 + 17588 * q^79 + 13122 * q^81 - 39325 * q^83 - 58060 * q^85 - 6903 * q^87 - 83082 * q^89 - 100530 * q^93 - 214946 * q^95 - 184785 * q^97 - 32967 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−65.5743	−1.17303	−0.586515	−	0.809939i	$-0.699500\pi$
$5$	−65.5743	−1.17303	−0.586515	+	0.809939i	$0.699500\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−245.574	−0.611929	−0.305965	−	0.952043i	$-0.598979\pi$
$11$	−245.574	−0.611929	−0.305965	+	0.952043i	$0.598979\pi$
$12$	0	0
$13$	−434.872	−0.713679	−0.356839	−	0.934166i	$-0.616146\pi$
$13$	−434.872	−0.713679	−0.356839	+	0.934166i	$0.616146\pi$
$14$	0	0
$15$	590.169	0.677249
$16$	0	0
$17$	1102.30	0.925074	0.462537	−	0.886600i	$-0.346939\pi$
$17$	1102.30	0.925074	0.462537	+	0.886600i	$0.346939\pi$
$18$	0	0
$19$	2877.29	1.82852	0.914260	−	0.405127i	$-0.132773\pi$
$19$	2877.29	1.82852	0.914260	+	0.405127i	$0.132773\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4229.43	−1.66710	−0.833552	−	0.552441i	$-0.813696\pi$
$23$	−4229.43	−1.66710	−0.833552	+	0.552441i	$0.813696\pi$
$24$	0	0
$25$	1174.99	0.375998
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	4969.60	1.09730	0.548652	−	0.836051i	$-0.315141\pi$
$29$	4969.60	1.09730	0.548652	+	0.836051i	$0.315141\pi$
$30$	0	0
$31$	8782.65	1.64143	0.820713	−	0.571341i	$-0.193576\pi$
$31$	8782.65	1.64143	0.820713	+	0.571341i	$0.193576\pi$
$32$	0	0
$33$	2210.17	0.353298
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2440.06	−0.293019	−0.146510	−	0.989209i	$-0.546804\pi$
$37$	−2440.06	−0.293019	−0.146510	+	0.989209i	$0.546804\pi$
$38$	0	0
$39$	3913.85	0.412043
$40$	0	0
$41$	3668.55	0.340828	0.170414	−	0.985373i	$-0.445489\pi$
$41$	3668.55	0.340828	0.170414	+	0.985373i	$0.445489\pi$
$42$	0	0
$43$	−7198.06	−0.593669	−0.296835	−	0.954929i	$-0.595931\pi$
$43$	−7198.06	−0.593669	−0.296835	+	0.954929i	$0.595931\pi$
$44$	0	0
$45$	−5311.52	−0.391010
$46$	0	0
$47$	−3273.32	−0.216145	−0.108072	−	0.994143i	$-0.534468\pi$
$47$	−3273.32	−0.216145	−0.108072	+	0.994143i	$0.534468\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9920.68	−0.534092
$52$	0	0
$53$	−3021.67	−0.147760	−0.0738801	−	0.997267i	$-0.523538\pi$
$53$	−3021.67	−0.147760	−0.0738801	+	0.997267i	$0.523538\pi$
$54$	0	0
$55$	16103.4	0.717811
$56$	0	0
$57$	−25895.6	−1.05570
$58$	0	0
$59$	51487.5	1.92562	0.962812	−	0.270171i	$-0.0870801\pi$
$59$	51487.5	1.92562	0.962812	+	0.270171i	$0.0870801\pi$
$60$	0	0
$61$	13313.3	0.458101	0.229050	−	0.973415i	$-0.426438\pi$
$61$	13313.3	0.458101	0.229050	+	0.973415i	$0.426438\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	28516.4	0.837166
$66$	0	0
$67$	30895.7	0.840837	0.420418	−	0.907330i	$-0.361883\pi$
$67$	30895.7	0.840837	0.420418	+	0.907330i	$0.361883\pi$
$68$	0	0
$69$	38064.9	0.962503
$70$	0	0
$71$	−41882.8	−0.986030	−0.493015	−	0.870021i	$-0.664105\pi$
$71$	−41882.8	−0.986030	−0.493015	+	0.870021i	$0.664105\pi$
$72$	0	0
$73$	−34617.3	−0.760302	−0.380151	−	0.924924i	$-0.624128\pi$
$73$	−34617.3	−0.760302	−0.380151	+	0.924924i	$0.624128\pi$
$74$	0	0
$75$	−10574.9	−0.217083
$76$	0	0
$77$	0	0
$78$	0	0
$79$	77543.5	1.39790	0.698952	−	0.715168i	$-0.253650\pi$
$79$	77543.5	1.39790	0.698952	+	0.715168i	$0.253650\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−100908.	−1.60779	−0.803897	−	0.594768i	$-0.797244\pi$
$83$	−100908.	−1.60779	−0.803897	+	0.594768i	$0.797244\pi$
$84$	0	0
$85$	−72282.4	−1.08514
$86$	0	0
$87$	−44726.4	−0.633528
$88$	0	0
$89$	−40783.7	−0.545772	−0.272886	−	0.962046i	$-0.587978\pi$
$89$	−40783.7	−0.545772	−0.272886	+	0.962046i	$0.587978\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−79043.8	−0.947678
$94$	0	0
$95$	−188676.	−2.14491
$96$	0	0
$97$	−140147.	−1.51236	−0.756178	−	0.654366i	$-0.772936\pi$
$97$	−140147.	−1.51236	−0.756178	+	0.654366i	$0.772936\pi$
$98$	0	0
$99$	−19891.5	−0.203976
$100$	0	0
$101$	−8576.45	−0.0836574	−0.0418287	−	0.999125i	$-0.513318\pi$
$101$	−8576.45	−0.0836574	−0.0418287	+	0.999125i	$0.513318\pi$
$102$	0	0
$103$	35260.1	0.327485	0.163742	−	0.986503i	$-0.447643\pi$
$103$	35260.1	0.327485	0.163742	+	0.986503i	$0.447643\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−192629.	−1.62653	−0.813264	−	0.581895i	$-0.802311\pi$
$107$	−192629.	−1.62653	−0.813264	+	0.581895i	$0.802311\pi$
$108$	0	0
$109$	123292.	0.993961	0.496980	−	0.867762i	$-0.334442\pi$
$109$	123292.	0.993961	0.496980	+	0.867762i	$0.334442\pi$
$110$	0	0
$111$	21960.6	0.169175
$112$	0	0
$113$	28400.9	0.209236	0.104618	−	0.994512i	$-0.466638\pi$
$113$	28400.9	0.209236	0.104618	+	0.994512i	$0.466638\pi$
$114$	0	0
$115$	277342.	1.95556
$116$	0	0
$117$	−35224.6	−0.237893
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−100744.	−0.625542
$122$	0	0
$123$	−33017.0	−0.196777
$124$	0	0
$125$	127870.	0.731973
$126$	0	0
$127$	−47198.8	−0.259670	−0.129835	−	0.991536i	$-0.541445\pi$
$127$	−47198.8	−0.259670	−0.129835	+	0.991536i	$0.541445\pi$
$128$	0	0
$129$	64782.6	0.342755
$130$	0	0
$131$	−79809.3	−0.406327	−0.203163	−	0.979145i	$-0.565122\pi$
$131$	−79809.3	−0.406327	−0.203163	+	0.979145i	$0.565122\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	47803.7	0.225750
$136$	0	0
$137$	−86064.3	−0.391761	−0.195881	−	0.980628i	$-0.562757\pi$
$137$	−86064.3	−0.391761	−0.195881	+	0.980628i	$0.562757\pi$
$138$	0	0
$139$	−270587.	−1.18787	−0.593935	−	0.804513i	$-0.702427\pi$
$139$	−270587.	−1.18787	−0.593935	+	0.804513i	$0.702427\pi$
$140$	0	0
$141$	29459.9	0.124791
$142$	0	0
$143$	106793.	0.436721
$144$	0	0
$145$	−325878.	−1.28717
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−172548.	−0.636713	−0.318356	−	0.947971i	$-0.603131\pi$
$149$	−172548.	−0.636713	−0.318356	+	0.947971i	$0.603131\pi$
$150$	0	0
$151$	−17899.5	−0.0638849	−0.0319425	−	0.999490i	$-0.510169\pi$
$151$	−17899.5	−0.0638849	−0.0319425	+	0.999490i	$0.510169\pi$
$152$	0	0
$153$	89286.1	0.308358
$154$	0	0
$155$	−575916.	−1.92544
$156$	0	0
$157$	179019.	0.579628	0.289814	−	0.957083i	$-0.406407\pi$
$157$	179019.	0.579628	0.289814	+	0.957083i	$0.406407\pi$
$158$	0	0
$159$	27195.1	0.0853094
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−237289.	−0.699534	−0.349767	−	0.936837i	$-0.613739\pi$
$163$	−237289.	−0.699534	−0.349767	+	0.936837i	$0.613739\pi$
$164$	0	0
$165$	−144930.	−0.414428
$166$	0	0
$167$	94040.0	0.260928	0.130464	−	0.991453i	$-0.458353\pi$
$167$	94040.0	0.260928	0.130464	+	0.991453i	$0.458353\pi$
$168$	0	0
$169$	−182180.	−0.490663
$170$	0	0
$171$	233061.	0.609507
$172$	0	0
$173$	−488713.	−1.24148	−0.620739	−	0.784017i	$-0.713167\pi$
$173$	−488713.	−1.24148	−0.620739	+	0.784017i	$0.713167\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−463387.	−1.11176
$178$	0	0
$179$	812996.	1.89651	0.948257	−	0.317505i	$-0.102845\pi$
$179$	812996.	1.89651	0.948257	+	0.317505i	$0.102845\pi$
$180$	0	0
$181$	332961.	0.755434	0.377717	−	0.925921i	$-0.376709\pi$
$181$	332961.	0.755434	0.377717	+	0.925921i	$0.376709\pi$
$182$	0	0
$183$	−119820.	−0.264484
$184$	0	0
$185$	160005.	0.343720
$186$	0	0
$187$	−270696.	−0.566080
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−144288.	−0.286185	−0.143092	−	0.989709i	$-0.545705\pi$
$191$	−144288.	−0.286185	−0.143092	+	0.989709i	$0.545705\pi$
$192$	0	0
$193$	−894619.	−1.72880	−0.864400	−	0.502804i	$-0.832302\pi$
$193$	−894619.	−1.72880	−0.864400	+	0.502804i	$0.832302\pi$
$194$	0	0
$195$	−256648.	−0.483338
$196$	0	0
$197$	−599462.	−1.10052	−0.550258	−	0.834995i	$-0.685470\pi$
$197$	−599462.	−1.10052	−0.550258	+	0.834995i	$0.685470\pi$
$198$	0	0
$199$	778259.	1.39313	0.696564	−	0.717494i	$-0.254711\pi$
$199$	778259.	1.39313	0.696564	+	0.717494i	$0.254711\pi$
$200$	0	0
$201$	−278062.	−0.485457
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−240563.	−0.399801
$206$	0	0
$207$	−342584.	−0.555701
$208$	0	0
$209$	−706589.	−1.11893
$210$	0	0
$211$	−810532.	−1.25333	−0.626663	−	0.779291i	$-0.715580\pi$
$211$	−810532.	−1.25333	−0.626663	+	0.779291i	$0.715580\pi$
$212$	0	0
$213$	376945.	0.569285
$214$	0	0
$215$	472008.	0.696391
$216$	0	0
$217$	0	0
$218$	0	0
$219$	311556.	0.438961
$220$	0	0
$221$	−479358.	−0.660206
$222$	0	0
$223$	−220486.	−0.296905	−0.148453	−	0.988920i	$-0.547429\pi$
$223$	−220486.	−0.296905	−0.148453	+	0.988920i	$0.547429\pi$
$224$	0	0
$225$	95174.5	0.125333
$226$	0	0
$227$	−1.34378e6	−1.73087	−0.865433	−	0.501025i	$-0.832956\pi$
$227$	−1.34378e6	−1.73087	−0.865433	+	0.501025i	$0.832956\pi$
$228$	0	0
$229$	1.04258e6	1.31378	0.656889	−	0.753987i	$-0.271872\pi$
$229$	1.04258e6	1.31378	0.656889	+	0.753987i	$0.271872\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	632511.	0.763270	0.381635	−	0.924313i	$-0.375361\pi$
$233$	632511.	0.763270	0.381635	+	0.924313i	$0.375361\pi$
$234$	0	0
$235$	214646.	0.253544
$236$	0	0
$237$	−697891.	−0.807081
$238$	0	0
$239$	−684919.	−0.775612	−0.387806	−	0.921741i	$-0.626767\pi$
$239$	−684919.	−0.775612	−0.387806	+	0.921741i	$0.626767\pi$
$240$	0	0
$241$	7732.88	0.00857628	0.00428814	−	0.999991i	$-0.498635\pi$
$241$	7732.88	0.00857628	0.00428814	+	0.999991i	$0.498635\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.25125e6	−1.30498
$248$	0	0
$249$	908172.	0.928261
$250$	0	0
$251$	−1.09614e6	−1.09820	−0.549098	−	0.835758i	$-0.685029\pi$
$251$	−1.09614e6	−1.09820	−0.549098	+	0.835758i	$0.685029\pi$
$252$	0	0
$253$	1.03864e6	1.02015
$254$	0	0
$255$	650542.	0.626505
$256$	0	0
$257$	−714920.	−0.675188	−0.337594	−	0.941292i	$-0.609613\pi$
$257$	−714920.	−0.675188	−0.337594	+	0.941292i	$0.609613\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	402538.	0.365768
$262$	0	0
$263$	−596513.	−0.531778	−0.265889	−	0.964004i	$-0.585665\pi$
$263$	−596513.	−0.531778	−0.265889	+	0.964004i	$0.585665\pi$
$264$	0	0
$265$	198144.	0.173327
$266$	0	0
$267$	367053.	0.315102
$268$	0	0
$269$	2.03308e6	1.71306	0.856531	−	0.516096i	$-0.172615\pi$
$269$	2.03308e6	1.71306	0.856531	+	0.516096i	$0.172615\pi$
$270$	0	0
$271$	−1.79455e6	−1.48434	−0.742170	−	0.670212i	$-0.766203\pi$
$271$	−1.79455e6	−1.48434	−0.742170	+	0.670212i	$0.766203\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−288548.	−0.230084
$276$	0	0
$277$	423580.	0.331693	0.165846	−	0.986152i	$-0.446964\pi$
$277$	423580.	0.331693	0.165846	+	0.986152i	$0.446964\pi$
$278$	0	0
$279$	711395.	0.547142
$280$	0	0
$281$	−1.63799e6	−1.23750	−0.618749	−	0.785589i	$-0.712360\pi$
$281$	−1.63799e6	−1.23750	−0.618749	+	0.785589i	$0.712360\pi$
$282$	0	0
$283$	294035.	0.218240	0.109120	−	0.994029i	$-0.465197\pi$
$283$	294035.	0.218240	0.109120	+	0.994029i	$0.465197\pi$
$284$	0	0
$285$	1.69809e6	1.23836
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−204798.	−0.144238
$290$	0	0
$291$	1.26132e6	0.873159
$292$	0	0
$293$	356961.	0.242913	0.121457	−	0.992597i	$-0.461243\pi$
$293$	356961.	0.242913	0.121457	+	0.992597i	$0.461243\pi$
$294$	0	0
$295$	−3.37626e6	−2.25881
$296$	0	0
$297$	179024.	0.117766
$298$	0	0
$299$	1.83926e6	1.18978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	77188.1	0.0482996
$304$	0	0
$305$	−873010.	−0.537365
$306$	0	0
$307$	2.04097e6	1.23592	0.617960	−	0.786210i	$-0.287959\pi$
$307$	2.04097e6	1.23592	0.617960	+	0.786210i	$0.287959\pi$
$308$	0	0
$309$	−317341.	−0.189073
$310$	0	0
$311$	−2.04674e6	−1.19994	−0.599972	−	0.800021i	$-0.704822\pi$
$311$	−2.04674e6	−1.19994	−0.599972	+	0.800021i	$0.704822\pi$
$312$	0	0
$313$	−638545.	−0.368409	−0.184205	−	0.982888i	$-0.558971\pi$
$313$	−638545.	−0.368409	−0.184205	+	0.982888i	$0.558971\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.49909e6	1.95572	0.977860	−	0.209260i	$-0.0671056\pi$
$317$	3.49909e6	1.95572	0.977860	+	0.209260i	$0.0671056\pi$
$318$	0	0
$319$	−1.22041e6	−0.671472
$320$	0	0
$321$	1.73366e6	0.939076
$322$	0	0
$323$	3.17163e6	1.69152
$324$	0	0
$325$	−510972.	−0.268342
$326$	0	0
$327$	−1.10963e6	−0.573864
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.94179e6	−1.47585	−0.737923	−	0.674885i	$-0.764193\pi$
$331$	−2.94179e6	−1.47585	−0.737923	+	0.674885i	$0.764193\pi$
$332$	0	0
$333$	−197645.	−0.0976731
$334$	0	0
$335$	−2.02597e6	−0.986326
$336$	0	0
$337$	2.77854e6	1.33273	0.666364	−	0.745627i	$-0.267850\pi$
$337$	2.77854e6	1.33273	0.666364	+	0.745627i	$0.267850\pi$
$338$	0	0
$339$	−255608.	−0.120802
$340$	0	0
$341$	−2.15679e6	−1.00444
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.49608e6	−1.12904
$346$	0	0
$347$	−172004.	−0.0766857	−0.0383429	−	0.999265i	$-0.512208\pi$
$347$	−172004.	−0.0766857	−0.0383429	+	0.999265i	$0.512208\pi$
$348$	0	0
$349$	3.88321e6	1.70658	0.853290	−	0.521436i	$-0.174603\pi$
$349$	3.88321e6	1.70658	0.853290	+	0.521436i	$0.174603\pi$
$350$	0	0
$351$	317021.	0.137348
$352$	0	0
$353$	770935.	0.329292	0.164646	−	0.986353i	$-0.447352\pi$
$353$	770935.	0.329292	0.164646	+	0.986353i	$0.447352\pi$
$354$	0	0
$355$	2.74644e6	1.15664
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.14033e6	1.28600	0.642998	−	0.765868i	$-0.277691\pi$
$359$	3.14033e6	1.28600	0.642998	+	0.765868i	$0.277691\pi$
$360$	0	0
$361$	5.80271e6	2.34349
$362$	0	0
$363$	906698.	0.361157
$364$	0	0
$365$	2.27001e6	0.891857
$366$	0	0
$367$	365684.	0.141723	0.0708615	−	0.997486i	$-0.477425\pi$
$367$	365684.	0.141723	0.0708615	+	0.997486i	$0.477425\pi$
$368$	0	0
$369$	297153.	0.113609
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−154260.	−0.0574091	−0.0287045	−	0.999588i	$-0.509138\pi$
$373$	−154260.	−0.0574091	−0.0287045	+	0.999588i	$0.509138\pi$
$374$	0	0
$375$	−1.15083e6	−0.422605
$376$	0	0
$377$	−2.16114e6	−0.783122
$378$	0	0
$379$	4.06013e6	1.45192	0.725959	−	0.687738i	$-0.241396\pi$
$379$	4.06013e6	1.45192	0.725959	+	0.687738i	$0.241396\pi$
$380$	0	0
$381$	424789.	0.149920
$382$	0	0
$383$	−2.04303e6	−0.711669	−0.355834	−	0.934549i	$-0.615803\pi$
$383$	−2.04303e6	−0.711669	−0.355834	+	0.934549i	$0.615803\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−583043.	−0.197890
$388$	0	0
$389$	−2.12366e6	−0.711560	−0.355780	−	0.934570i	$-0.615785\pi$
$389$	−2.12366e6	−0.711560	−0.355780	+	0.934570i	$0.615785\pi$
$390$	0	0
$391$	−4.66209e6	−1.54219
$392$	0	0
$393$	718284.	0.234593
$394$	0	0
$395$	−5.08486e6	−1.63978
$396$	0	0
$397$	3.66364e6	1.16664	0.583319	−	0.812243i	$-0.301754\pi$
$397$	3.66364e6	1.16664	0.583319	+	0.812243i	$0.301754\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−64089.7	−0.0199034	−0.00995170	−	0.999950i	$-0.503168\pi$
$401$	−64089.7	−0.0199034	−0.00995170	+	0.999950i	$0.503168\pi$
$402$	0	0
$403$	−3.81933e6	−1.17145
$404$	0	0
$405$	−430233.	−0.130337
$406$	0	0
$407$	599216.	0.179307
$408$	0	0
$409$	−2.02574e6	−0.598790	−0.299395	−	0.954129i	$-0.596785\pi$
$409$	−2.02574e6	−0.598790	−0.299395	+	0.954129i	$0.596785\pi$
$410$	0	0
$411$	774579.	0.226184
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.61698e6	1.88599
$416$	0	0
$417$	2.43528e6	0.685818
$418$	0	0
$419$	−2.38986e6	−0.665025	−0.332513	−	0.943099i	$-0.607896\pi$
$419$	−2.38986e6	−0.665025	−0.332513	+	0.943099i	$0.607896\pi$
$420$	0	0
$421$	3.46875e6	0.953822	0.476911	−	0.878952i	$-0.341756\pi$
$421$	3.46875e6	0.953822	0.476911	+	0.878952i	$0.341756\pi$
$422$	0	0
$423$	−265139.	−0.0720482
$424$	0	0
$425$	1.29519e6	0.347826
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−961140.	−0.252141
$430$	0	0
$431$	−1.60440e6	−0.416025	−0.208013	−	0.978126i	$-0.566700\pi$
$431$	−1.60440e6	−0.416025	−0.208013	+	0.978126i	$0.566700\pi$
$432$	0	0
$433$	−741661.	−0.190102	−0.0950508	−	0.995472i	$-0.530301\pi$
$433$	−741661.	−0.190102	−0.0950508	+	0.995472i	$0.530301\pi$
$434$	0	0
$435$	2.93291e6	0.743147
$436$	0	0
$437$	−1.21693e7	−3.04833
$438$	0	0
$439$	−2.86787e6	−0.710230	−0.355115	−	0.934823i	$-0.615558\pi$
$439$	−2.86787e6	−0.710230	−0.355115	+	0.934823i	$0.615558\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−178298.	−0.0431655	−0.0215827	−	0.999767i	$-0.506871\pi$
$443$	−178298.	−0.0431655	−0.0215827	+	0.999767i	$0.506871\pi$
$444$	0	0
$445$	2.67436e6	0.640207
$446$	0	0
$447$	1.55293e6	0.367606
$448$	0	0
$449$	−2.77890e6	−0.650515	−0.325258	−	0.945625i	$-0.605451\pi$
$449$	−2.77890e6	−0.650515	−0.325258	+	0.945625i	$0.605451\pi$
$450$	0	0
$451$	−900903.	−0.208563
$452$	0	0
$453$	161095.	0.0368840
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.41339e6	1.43647	0.718236	−	0.695800i	$-0.244950\pi$
$457$	6.41339e6	1.43647	0.718236	+	0.695800i	$0.244950\pi$
$458$	0	0
$459$	−803575.	−0.178031
$460$	0	0
$461$	6.88393e6	1.50864	0.754318	−	0.656510i	$-0.227968\pi$
$461$	6.88393e6	1.50864	0.754318	+	0.656510i	$0.227968\pi$
$462$	0	0
$463$	6.70530e6	1.45367	0.726835	−	0.686812i	$-0.240991\pi$
$463$	6.70530e6	1.45367	0.726835	+	0.686812i	$0.240991\pi$
$464$	0	0
$465$	5.18325e6	1.11165
$466$	0	0
$467$	3.25673e6	0.691017	0.345509	−	0.938416i	$-0.387706\pi$
$467$	3.25673e6	0.691017	0.345509	+	0.938416i	$0.387706\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.61117e6	−0.334648
$472$	0	0
$473$	1.76766e6	0.363283
$474$	0	0
$475$	3.38080e6	0.687520
$476$	0	0
$477$	−244755.	−0.0492534
$478$	0	0
$479$	−9.35832e6	−1.86363	−0.931814	−	0.362936i	$-0.881774\pi$
$479$	−9.35832e6	−1.86363	−0.931814	+	0.362936i	$0.881774\pi$
$480$	0	0
$481$	1.06111e6	0.209122
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.19004e6	1.77404
$486$	0	0
$487$	−2.62940e6	−0.502382	−0.251191	−	0.967938i	$-0.580822\pi$
$487$	−2.62940e6	−0.502382	−0.251191	+	0.967938i	$0.580822\pi$
$488$	0	0
$489$	2.13560e6	0.403876
$490$	0	0
$491$	−4.81856e6	−0.902015	−0.451008	−	0.892520i	$-0.648935\pi$
$491$	−4.81856e6	−0.902015	−0.451008	+	0.892520i	$0.648935\pi$
$492$	0	0
$493$	5.47798e6	1.01509
$494$	0	0
$495$	1.30437e6	0.239270
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.19098e6	−0.753467	−0.376733	−	0.926322i	$-0.622953\pi$
$499$	−4.19098e6	−0.753467	−0.376733	+	0.926322i	$0.622953\pi$
$500$	0	0
$501$	−846360.	−0.150647
$502$	0	0
$503$	1.75338e6	0.308999	0.154500	−	0.987993i	$-0.450623\pi$
$503$	1.75338e6	0.308999	0.154500	+	0.987993i	$0.450623\pi$
$504$	0	0
$505$	562395.	0.0981326
$506$	0	0
$507$	1.63962e6	0.283284
$508$	0	0
$509$	−5.61846e6	−0.961221	−0.480610	−	0.876934i	$-0.659585\pi$
$509$	−5.61846e6	−0.961221	−0.480610	+	0.876934i	$0.659585\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.09755e6	−0.351899
$514$	0	0
$515$	−2.31216e6	−0.384149
$516$	0	0
$517$	803844.	0.132265
$518$	0	0
$519$	4.39842e6	0.716768
$520$	0	0
$521$	−1.10718e7	−1.78700	−0.893498	−	0.449068i	$-0.851756\pi$
$521$	−1.10718e7	−1.78700	−0.893498	+	0.449068i	$0.851756\pi$
$522$	0	0
$523$	7.59788e6	1.21461	0.607307	−	0.794467i	$-0.292250\pi$
$523$	7.59788e6	1.21461	0.607307	+	0.794467i	$0.292250\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.68109e6	1.51844
$528$	0	0
$529$	1.14518e7	1.77924
$530$	0	0
$531$	4.17049e6	0.641875
$532$	0	0
$533$	−1.59535e6	−0.243242
$534$	0	0
$535$	1.26315e7	1.90796
$536$	0	0
$537$	−7.31697e6	−1.09495
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.35079e6	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$541$	−3.35079e6	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$542$	0	0
$543$	−2.99665e6	−0.436150
$544$	0	0
$545$	−8.08480e6	−1.16595
$546$	0	0
$547$	−1.00856e7	−1.44123	−0.720615	−	0.693335i	$-0.756140\pi$
$547$	−1.00856e7	−1.44123	−0.720615	+	0.693335i	$0.756140\pi$
$548$	0	0
$549$	1.07838e6	0.152700
$550$	0	0
$551$	1.42990e7	2.00644
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.44005e6	−0.198447
$556$	0	0
$557$	1.35667e7	1.85284	0.926419	−	0.376495i	$-0.122871\pi$
$557$	1.35667e7	1.85284	0.926419	+	0.376495i	$0.122871\pi$
$558$	0	0
$559$	3.13023e6	0.423689
$560$	0	0
$561$	2.43626e6	0.326826
$562$	0	0
$563$	3.96988e6	0.527845	0.263922	−	0.964544i	$-0.414984\pi$
$563$	3.96988e6	0.527845	0.263922	+	0.964544i	$0.414984\pi$
$564$	0	0
$565$	−1.86237e6	−0.245440
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.17449e6	−0.540534	−0.270267	−	0.962785i	$-0.587112\pi$
$569$	−4.17449e6	−0.540534	−0.270267	+	0.962785i	$0.587112\pi$
$570$	0	0
$571$	3.85666e6	0.495018	0.247509	−	0.968886i	$-0.420388\pi$
$571$	3.85666e6	0.495018	0.247509	+	0.968886i	$0.420388\pi$
$572$	0	0
$573$	1.29859e6	0.165229
$574$	0	0
$575$	−4.96956e6	−0.626828
$576$	0	0
$577$	2.24575e6	0.280817	0.140408	−	0.990094i	$-0.455158\pi$
$577$	2.24575e6	0.280817	0.140408	+	0.990094i	$0.455158\pi$
$578$	0	0
$579$	8.05157e6	0.998123
$580$	0	0
$581$	0	0
$582$	0	0
$583$	742045.	0.0904188
$584$	0	0
$585$	2.30983e6	0.279055
$586$	0	0
$587$	−6.40082e6	−0.766726	−0.383363	−	0.923598i	$-0.625234\pi$
$587$	−6.40082e6	−0.766726	−0.383363	+	0.923598i	$0.625234\pi$
$588$	0	0
$589$	2.52702e7	3.00138
$590$	0	0
$591$	5.39516e6	0.635383
$592$	0	0
$593$	−1.11206e7	−1.29865	−0.649325	−	0.760511i	$-0.724949\pi$
$593$	−1.11206e7	−1.29865	−0.649325	+	0.760511i	$0.724949\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.00433e6	−0.804323
$598$	0	0
$599$	−1.31341e7	−1.49567	−0.747833	−	0.663887i	$-0.768905\pi$
$599$	−1.31341e7	−1.49567	−0.747833	+	0.663887i	$0.768905\pi$
$600$	0	0
$601$	7.88546e6	0.890514	0.445257	−	0.895403i	$-0.353112\pi$
$601$	7.88546e6	0.890514	0.445257	+	0.895403i	$0.353112\pi$
$602$	0	0
$603$	2.50255e6	0.280279
$604$	0	0
$605$	6.60624e6	0.733780
$606$	0	0
$607$	7.65934e6	0.843762	0.421881	−	0.906651i	$-0.361370\pi$
$607$	7.65934e6	0.843762	0.421881	+	0.906651i	$0.361370\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.42348e6	0.154258
$612$	0	0
$613$	−1.52413e7	−1.63822	−0.819108	−	0.573639i	$-0.805531\pi$
$613$	−1.52413e7	−1.63822	−0.819108	+	0.573639i	$0.805531\pi$
$614$	0	0
$615$	2.16507e6	0.230825
$616$	0	0
$617$	1.35844e7	1.43657	0.718285	−	0.695749i	$-0.244927\pi$
$617$	1.35844e7	1.43657	0.718285	+	0.695749i	$0.244927\pi$
$618$	0	0
$619$	−6.37417e6	−0.668647	−0.334324	−	0.942458i	$-0.608508\pi$
$619$	−6.37417e6	−0.668647	−0.334324	+	0.942458i	$0.608508\pi$
$620$	0	0
$621$	3.08326e6	0.320834
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.20569e7	−1.23462
$626$	0	0
$627$	6.35930e6	0.646012
$628$	0	0
$629$	−2.68967e6	−0.271065
$630$	0	0
$631$	−1.42736e7	−1.42712	−0.713561	−	0.700593i	$-0.752919\pi$
$631$	−1.42736e7	−1.42712	−0.713561	+	0.700593i	$0.752919\pi$
$632$	0	0
$633$	7.29479e6	0.723608
$634$	0	0
$635$	3.09503e6	0.304600
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.39251e6	−0.328677
$640$	0	0
$641$	−9.26960e6	−0.891079	−0.445539	−	0.895262i	$-0.646988\pi$
$641$	−9.26960e6	−0.891079	−0.445539	+	0.895262i	$0.646988\pi$
$642$	0	0
$643$	−1.17375e7	−1.11956	−0.559780	−	0.828641i	$-0.689115\pi$
$643$	−1.17375e7	−1.11956	−0.559780	+	0.828641i	$0.689115\pi$
$644$	0	0
$645$	−4.24807e6	−0.402062
$646$	0	0
$647$	−1.02545e7	−0.963059	−0.481529	−	0.876430i	$-0.659918\pi$
$647$	−1.02545e7	−0.963059	−0.481529	+	0.876430i	$0.659918\pi$
$648$	0	0
$649$	−1.26440e7	−1.17835
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.28977e6	0.760781	0.380391	−	0.924826i	$-0.375790\pi$
$653$	8.28977e6	0.760781	0.380391	+	0.924826i	$0.375790\pi$
$654$	0	0
$655$	5.23344e6	0.476633
$656$	0	0
$657$	−2.80400e6	−0.253434
$658$	0	0
$659$	−2.06731e7	−1.85435	−0.927174	−	0.374631i	$-0.877770\pi$
$659$	−2.06731e7	−1.85435	−0.927174	+	0.374631i	$0.877770\pi$
$660$	0	0
$661$	−194342.	−0.0173007	−0.00865035	−	0.999963i	$-0.502754\pi$
$661$	−194342.	−0.0173007	−0.00865035	+	0.999963i	$0.502754\pi$
$662$	0	0
$663$	4.31422e6	0.381170
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.10186e7	−1.82932
$668$	0	0
$669$	1.98437e6	0.171418
$670$	0	0
$671$	−3.26940e6	−0.280325
$672$	0	0
$673$	−1.14437e7	−0.973929	−0.486965	−	0.873422i	$-0.661896\pi$
$673$	−1.14437e7	−0.973929	−0.486965	+	0.873422i	$0.661896\pi$
$674$	0	0
$675$	−856571.	−0.0723609
$676$	0	0
$677$	−9.03738e6	−0.757828	−0.378914	−	0.925432i	$-0.623702\pi$
$677$	−9.03738e6	−0.757828	−0.378914	+	0.925432i	$0.623702\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.20940e7	0.999316
$682$	0	0
$683$	−1.11842e7	−0.917392	−0.458696	−	0.888593i	$-0.651683\pi$
$683$	−1.11842e7	−0.917392	−0.458696	+	0.888593i	$0.651683\pi$
$684$	0	0
$685$	5.64361e6	0.459548
$686$	0	0
$687$	−9.38325e6	−0.758510
$688$	0	0
$689$	1.31404e6	0.105453
$690$	0	0
$691$	−5.66930e6	−0.451684	−0.225842	−	0.974164i	$-0.572513\pi$
$691$	−5.66930e6	−0.451684	−0.225842	+	0.974164i	$0.572513\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.77435e7	1.39341
$696$	0	0
$697$	4.04384e6	0.315291
$698$	0	0
$699$	−5.69260e6	−0.440674
$700$	0	0
$701$	−1.31822e6	−0.101320	−0.0506599	−	0.998716i	$-0.516132\pi$
$701$	−1.31822e6	−0.101320	−0.0506599	+	0.998716i	$0.516132\pi$
$702$	0	0
$703$	−7.02077e6	−0.535792
$704$	0	0
$705$	−1.93181e6	−0.146384
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.26035e6	0.617139	0.308570	−	0.951202i	$-0.400150\pi$
$709$	8.26035e6	0.617139	0.308570	+	0.951202i	$0.400150\pi$
$710$	0	0
$711$	6.28102e6	0.465968
$712$	0	0
$713$	−3.71456e7	−2.73643
$714$	0	0
$715$	−7.00290e6	−0.512287
$716$	0	0
$717$	6.16427e6	0.447800
$718$	0	0
$719$	−2.26743e7	−1.63573	−0.817865	−	0.575410i	$-0.804843\pi$
$719$	−2.26743e7	−1.63573	−0.817865	+	0.575410i	$0.804843\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−69596.0	−0.00495152
$724$	0	0
$725$	5.83925e6	0.412584
$726$	0	0
$727$	1.93477e7	1.35767	0.678833	−	0.734293i	$-0.262486\pi$
$727$	1.93477e7	1.35767	0.678833	+	0.734293i	$0.262486\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.93440e6	−0.549188
$732$	0	0
$733$	−1.48584e7	−1.02144	−0.510720	−	0.859747i	$-0.670621\pi$
$733$	−1.48584e7	−1.02144	−0.510720	+	0.859747i	$0.670621\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.58720e6	−0.514533
$738$	0	0
$739$	−1.76818e7	−1.19101	−0.595504	−	0.803352i	$-0.703048\pi$
$739$	−1.76818e7	−1.19101	−0.595504	+	0.803352i	$0.703048\pi$
$740$	0	0
$741$	1.12613e7	0.753428
$742$	0	0
$743$	−8.28756e6	−0.550750	−0.275375	−	0.961337i	$-0.588802\pi$
$743$	−8.28756e6	−0.550750	−0.275375	+	0.961337i	$0.588802\pi$
$744$	0	0
$745$	1.13147e7	0.746883
$746$	0	0
$747$	−8.17355e6	−0.535932
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.31365e7	−1.49691	−0.748457	−	0.663183i	$-0.769205\pi$
$751$	−2.31365e7	−1.49691	−0.748457	+	0.663183i	$0.769205\pi$
$752$	0	0
$753$	9.86523e6	0.634044
$754$	0	0
$755$	1.17375e6	0.0749389
$756$	0	0
$757$	−1.95475e7	−1.23980	−0.619900	−	0.784681i	$-0.712827\pi$
$757$	−1.95475e7	−1.23980	−0.619900	+	0.784681i	$0.712827\pi$
$758$	0	0
$759$	−9.34776e6	−0.588984
$760$	0	0
$761$	−7.82465e6	−0.489783	−0.244891	−	0.969551i	$-0.578752\pi$
$761$	−7.82465e6	−0.489783	−0.244891	+	0.969551i	$0.578752\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5.85488e6	−0.361713
$766$	0	0
$767$	−2.23905e7	−1.37428
$768$	0	0
$769$	8.27325e6	0.504499	0.252250	−	0.967662i	$-0.418830\pi$
$769$	8.27325e6	0.504499	0.252250	+	0.967662i	$0.418830\pi$
$770$	0	0
$771$	6.43428e6	0.389820
$772$	0	0
$773$	1.88369e7	1.13386	0.566931	−	0.823765i	$-0.308131\pi$
$773$	1.88369e7	1.13386	0.566931	+	0.823765i	$0.308131\pi$
$774$	0	0
$775$	1.03196e7	0.617173
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.05555e7	0.623211
$780$	0	0
$781$	1.02853e7	0.603381
$782$	0	0
$783$	−3.62284e6	−0.211176
$784$	0	0
$785$	−1.17390e7	−0.679920
$786$	0	0
$787$	1.83947e7	1.05866	0.529329	−	0.848417i	$-0.322444\pi$
$787$	1.83947e7	1.05866	0.529329	+	0.848417i	$0.322444\pi$
$788$	0	0
$789$	5.36861e6	0.307022
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.78957e6	−0.326937
$794$	0	0
$795$	−1.78330e6	−0.100070
$796$	0	0
$797$	−3.70208e6	−0.206443	−0.103222	−	0.994658i	$-0.532915\pi$
$797$	−3.70208e6	−0.206443	−0.103222	+	0.994658i	$0.532915\pi$
$798$	0	0
$799$	−3.60818e6	−0.199950
$800$	0	0
$801$	−3.30348e6	−0.181924
$802$	0	0
$803$	8.50113e6	0.465251
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.82977e7	−0.989037
$808$	0	0
$809$	1.21913e7	0.654903	0.327451	−	0.944868i	$-0.393810\pi$
$809$	1.21913e7	0.654903	0.327451	+	0.944868i	$0.393810\pi$
$810$	0	0
$811$	2.52433e6	0.134770	0.0673851	−	0.997727i	$-0.478534\pi$
$811$	2.52433e6	0.134770	0.0673851	+	0.997727i	$0.478534\pi$
$812$	0	0
$813$	1.61510e7	0.856984
$814$	0	0
$815$	1.55601e7	0.820574
$816$	0	0
$817$	−2.07109e7	−1.08554
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.50051e6	−0.491914	−0.245957	−	0.969281i	$-0.579102\pi$
$821$	−9.50051e6	−0.491914	−0.245957	+	0.969281i	$0.579102\pi$
$822$	0	0
$823$	−2.26336e7	−1.16481	−0.582403	−	0.812900i	$-0.697887\pi$
$823$	−2.26336e7	−1.16481	−0.582403	+	0.812900i	$0.697887\pi$
$824$	0	0
$825$	2.59694e6	0.132839
$826$	0	0
$827$	1.09007e7	0.554230	0.277115	−	0.960837i	$-0.410622\pi$
$827$	1.09007e7	0.554230	0.277115	+	0.960837i	$0.410622\pi$
$828$	0	0
$829$	2.36120e7	1.19329	0.596647	−	0.802504i	$-0.296499\pi$
$829$	2.36120e7	1.19329	0.596647	+	0.802504i	$0.296499\pi$
$830$	0	0
$831$	−3.81222e6	−0.191503
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.16661e6	−0.306077
$836$	0	0
$837$	−6.40255e6	−0.315893
$838$	0	0
$839$	−2.67438e7	−1.31165	−0.655826	−	0.754912i	$-0.727679\pi$
$839$	−2.67438e7	−1.31165	−0.655826	+	0.754912i	$0.727679\pi$
$840$	0	0
$841$	4.18580e6	0.204075
$842$	0	0
$843$	1.47419e7	0.714469
$844$	0	0
$845$	1.19463e7	0.575562
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.64632e6	−0.126001
$850$	0	0
$851$	1.03201e7	0.488494
$852$	0	0
$853$	−5.00667e6	−0.235600	−0.117800	−	0.993037i	$-0.537584\pi$
$853$	−5.00667e6	−0.235600	−0.117800	+	0.993037i	$0.537584\pi$
$854$	0	0
$855$	−1.52828e7	−0.714969
$856$	0	0
$857$	−2.43624e7	−1.13310	−0.566549	−	0.824028i	$-0.691722\pi$
$857$	−2.43624e7	−1.13310	−0.566549	+	0.824028i	$0.691722\pi$
$858$	0	0
$859$	2.99153e7	1.38328	0.691641	−	0.722241i	$-0.256888\pi$
$859$	2.99153e7	1.38328	0.691641	+	0.722241i	$0.256888\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.60103e7	−1.18883	−0.594414	−	0.804160i	$-0.702616\pi$
$863$	−2.60103e7	−1.18883	−0.594414	+	0.804160i	$0.702616\pi$
$864$	0	0
$865$	3.20471e7	1.45629
$866$	0	0
$867$	1.84318e6	0.0832759
$868$	0	0
$869$	−1.90427e7	−0.855419
$870$	0	0
$871$	−1.34357e7	−0.600087
$872$	0	0
$873$	−1.13519e7	−0.504119
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.01375e7	−0.884109	−0.442055	−	0.896988i	$-0.645750\pi$
$877$	−2.01375e7	−0.884109	−0.442055	+	0.896988i	$0.645750\pi$
$878$	0	0
$879$	−3.21265e6	−0.140246
$880$	0	0
$881$	3.69393e7	1.60343	0.801713	−	0.597710i	$-0.203922\pi$
$881$	3.69393e7	1.60343	0.801713	+	0.597710i	$0.203922\pi$
$882$	0	0
$883$	2.27031e7	0.979904	0.489952	−	0.871749i	$-0.337014\pi$
$883$	2.27031e7	0.979904	0.489952	+	0.871749i	$0.337014\pi$
$884$	0	0
$885$	3.03863e7	1.30413
$886$	0	0
$887$	1.55979e7	0.665666	0.332833	−	0.942986i	$-0.391995\pi$
$887$	1.55979e7	0.665666	0.332833	+	0.942986i	$0.391995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.61121e6	−0.0679921
$892$	0	0
$893$	−9.41831e6	−0.395225
$894$	0	0
$895$	−5.33117e7	−2.22467
$896$	0	0
$897$	−1.65533e7	−0.686918
$898$	0	0
$899$	4.36463e7	1.80114
$900$	0	0
$901$	−3.33078e6	−0.136689
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.18337e7	−0.886146
$906$	0	0
$907$	−1.93485e7	−0.780961	−0.390480	−	0.920611i	$-0.627691\pi$
$907$	−1.93485e7	−0.780961	−0.390480	+	0.920611i	$0.627691\pi$
$908$	0	0
$909$	−694693.	−0.0278858
$910$	0	0
$911$	−3.84661e7	−1.53561	−0.767807	−	0.640682i	$-0.778652\pi$
$911$	−3.84661e7	−1.53561	−0.767807	+	0.640682i	$0.778652\pi$
$912$	0	0
$913$	2.47804e7	0.983857
$914$	0	0
$915$	7.85709e6	0.310248
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.95838e7	0.764905	0.382453	−	0.923975i	$-0.375080\pi$
$919$	1.95838e7	0.764905	0.382453	+	0.923975i	$0.375080\pi$
$920$	0	0
$921$	−1.83687e7	−0.713558
$922$	0	0
$923$	1.82137e7	0.703709
$924$	0	0
$925$	−2.86706e6	−0.110175
$926$	0	0
$927$	2.85607e6	0.109162
$928$	0	0
$929$	1.28256e7	0.487572	0.243786	−	0.969829i	$-0.421610\pi$
$929$	1.28256e7	0.487572	0.243786	+	0.969829i	$0.421610\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.84206e7	0.692789
$934$	0	0
$935$	1.77507e7	0.664028
$936$	0	0
$937$	−5.60430e6	−0.208532	−0.104266	−	0.994549i	$-0.533249\pi$
$937$	−5.60430e6	−0.208532	−0.104266	+	0.994549i	$0.533249\pi$
$938$	0	0
$939$	5.74690e6	0.212701
$940$	0	0
$941$	−3.44348e7	−1.26772	−0.633861	−	0.773447i	$-0.718531\pi$
$941$	−3.44348e7	−1.26772	−0.633861	+	0.773447i	$0.718531\pi$
$942$	0	0
$943$	−1.55159e7	−0.568196
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.10626e6	−0.257493	−0.128747	−	0.991678i	$-0.541095\pi$
$947$	−7.10626e6	−0.257493	−0.128747	+	0.991678i	$0.541095\pi$
$948$	0	0
$949$	1.50541e7	0.542612
$950$	0	0
$951$	−3.14918e7	−1.12914
$952$	0	0
$953$	3.78058e7	1.34842	0.674212	−	0.738538i	$-0.264483\pi$
$953$	3.78058e7	1.34842	0.674212	+	0.738538i	$0.264483\pi$
$954$	0	0
$955$	9.46158e6	0.335703
$956$	0	0
$957$	1.09837e7	0.387675
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.85058e7	1.69428
$962$	0	0
$963$	−1.56029e7	−0.542176
$964$	0	0
$965$	5.86641e7	2.02793
$966$	0	0
$967$	1.33643e7	0.459601	0.229800	−	0.973238i	$-0.426193\pi$
$967$	1.33643e7	0.459601	0.229800	+	0.973238i	$0.426193\pi$
$968$	0	0
$969$	−2.85447e7	−0.976598
$970$	0	0
$971$	−4.68668e7	−1.59521	−0.797604	−	0.603181i	$-0.793899\pi$
$971$	−4.68668e7	−1.59521	−0.797604	+	0.603181i	$0.793899\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.59874e6	0.154927
$976$	0	0
$977$	4.25939e7	1.42761	0.713807	−	0.700343i	$-0.246969\pi$
$977$	4.25939e7	1.42761	0.713807	+	0.700343i	$0.246969\pi$
$978$	0	0
$979$	1.00154e7	0.333974
$980$	0	0
$981$	9.98667e6	0.331320
$982$	0	0
$983$	−5.20577e7	−1.71831	−0.859154	−	0.511718i	$-0.829009\pi$
$983$	−5.20577e7	−1.71831	−0.859154	+	0.511718i	$0.829009\pi$
$984$	0	0
$985$	3.93093e7	1.29094
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.04437e7	0.989708
$990$	0	0
$991$	2.91535e7	0.942989	0.471495	−	0.881869i	$-0.343715\pi$
$991$	2.91535e7	0.942989	0.471495	+	0.881869i	$0.343715\pi$
$992$	0	0
$993$	2.64761e7	0.852080
$994$	0	0
$995$	−5.10338e7	−1.63418
$996$	0	0
$997$	−8.99304e6	−0.286529	−0.143264	−	0.989684i	$-0.545760\pi$
$997$	−8.99304e6	−0.286529	−0.143264	+	0.989684i	$0.545760\pi$
$998$	0	0
$999$	1.77880e6	0.0563916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.h.1.1		2
7.2	even	3		588.6.i.m.361.2		4
7.3	odd	6		84.6.i.b.37.1	yes	4
7.4	even	3		588.6.i.m.373.2		4
7.5	odd	6		84.6.i.b.25.1	✓	4
7.6	odd	2		588.6.a.l.1.2		2
21.5	even	6		252.6.k.e.109.2		4
21.17	even	6		252.6.k.e.37.2		4
28.3	even	6		336.6.q.g.289.1		4
28.19	even	6		336.6.q.g.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.b.25.1	✓	4	7.5	odd	6
84.6.i.b.37.1	yes	4	7.3	odd	6
252.6.k.e.37.2		4	21.17	even	6
252.6.k.e.109.2		4	21.5	even	6
336.6.q.g.193.1		4	28.19	even	6
336.6.q.g.289.1		4	28.3	even	6
588.6.a.h.1.1		2	1.1	even	1	trivial
588.6.a.l.1.2		2	7.6	odd	2
588.6.i.m.361.2		4	7.2	even	3
588.6.i.m.373.2		4	7.4	even	3

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -65.5743 q^{5} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -65.5743 q^{5} +81.0000 q^{9} -245.574 q^{11} -434.872 q^{13} +590.169 q^{15} +1102.30 q^{17} +2877.29 q^{19} -4229.43 q^{23} +1174.99 q^{25} -729.000 q^{27} +4969.60 q^{29} +8782.65 q^{31} +2210.17 q^{33} -2440.06 q^{37} +3913.85 q^{39} +3668.55 q^{41} -7198.06 q^{43} -5311.52 q^{45} -3273.32 q^{47} -9920.68 q^{51} -3021.67 q^{53} +16103.4 q^{55} -25895.6 q^{57} +51487.5 q^{59} +13313.3 q^{61} +28516.4 q^{65} +30895.7 q^{67} +38064.9 q^{69} -41882.8 q^{71} -34617.3 q^{73} -10574.9 q^{75} +77543.5 q^{79} +6561.00 q^{81} -100908. q^{83} -72282.4 q^{85} -44726.4 q^{87} -40783.7 q^{89} -79043.8 q^{93} -188676. q^{95} -140147. q^{97} -19891.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9	\( 2 q - 18 q^{3} - 47 q^{5} + 162 q^{9} - 407 q^{11} - 449 q^{13} + 423 q^{15} + 1868 q^{17} + 1463 q^{19} - 44 q^{23} - 1605 q^{25} - 1458 q^{27} + 767 q^{29} + 11170 q^{31} + 3663 q^{33} - 3113 q^{37} + 4041 q^{39} + 7842 q^{41} - 12629 q^{43} - 3807 q^{45} - 9576 q^{47} - 16812 q^{51} + 13395 q^{53} + 13105 q^{55} - 13167 q^{57} + 47521 q^{59} + 63652 q^{61} + 28254 q^{65} + 44541 q^{67} + 396 q^{69} - 125840 q^{71} - 6039 q^{73} + 14445 q^{75} + 17588 q^{79} + 13122 q^{81} - 39325 q^{83} - 58060 q^{85} - 6903 q^{87} - 83082 q^{89} - 100530 q^{93} - 214946 q^{95} - 184785 q^{97} - 32967 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 47 * q^5 + 162 * q^9 - 407 * q^11 - 449 * q^13 + 423 * q^15 + 1868 * q^17 + 1463 * q^19 - 44 * q^23 - 1605 * q^25 - 1458 * q^27 + 767 * q^29 + 11170 * q^31 + 3663 * q^33 - 3113 * q^37 + 4041 * q^39 + 7842 * q^41 - 12629 * q^43 - 3807 * q^45 - 9576 * q^47 - 16812 * q^51 + 13395 * q^53 + 13105 * q^55 - 13167 * q^57 + 47521 * q^59 + 63652 * q^61 + 28254 * q^65 + 44541 * q^67 + 396 * q^69 - 125840 * q^71 - 6039 * q^73 + 14445 * q^75 + 17588 * q^79 + 13122 * q^81 - 39325 * q^83 - 58060 * q^85 - 6903 * q^87 - 83082 * q^89 - 100530 * q^93 - 214946 * q^95 - 184785 * q^97 - 32967 * q^99

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