LMFDB - Embedded newform 588.6.a.i.1.2

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:	$\Q(\sqrt{2641})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 660$ x^2 - x - 660
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-25.1953$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +96.3907 q^{5} +81.0000 q^{9} -179.516 q^{11} -56.8747 q^{13} -867.516 q^{15} +1416.89 q^{17} -1292.59 q^{19} -4668.61 q^{23} +6166.16 q^{25} -729.000 q^{27} -5003.10 q^{29} -7444.35 q^{31} +1615.64 q^{33} -9080.90 q^{37} +511.872 q^{39} +6628.36 q^{41} -12924.5 q^{43} +7807.64 q^{45} -5498.00 q^{47} -12752.0 q^{51} +29539.1 q^{53} -17303.7 q^{55} +11633.4 q^{57} +1058.51 q^{59} +9487.86 q^{61} -5482.19 q^{65} +1331.03 q^{67} +42017.5 q^{69} +49811.2 q^{71} -75758.3 q^{73} -55495.4 q^{75} +62370.8 q^{79} +6561.00 q^{81} +53221.5 q^{83} +136575. q^{85} +45027.9 q^{87} -118682. q^{89} +66999.1 q^{93} -124594. q^{95} -136608. q^{97} -14540.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 18 q^{3} + 90 q^{5} + 162 q^{9} + 566 q^{11} - 936 q^{13} - 810 q^{15} - 558 q^{17} - 324 q^{19} - 2862 q^{23} + 3082 q^{25} - 1458 q^{27} - 4456 q^{29} - 1116 q^{31} - 5094 q^{33} - 23712 q^{37}+ \cdots + 45846 q^{99}+O(q^{100})$ 2 * q - 18 * q^3 + 90 * q^5 + 162 * q^9 + 566 * q^11 - 936 * q^13 - 810 * q^15 - 558 * q^17 - 324 * q^19 - 2862 * q^23 + 3082 * q^25 - 1458 * q^27 - 4456 * q^29 - 1116 * q^31 - 5094 * q^33 - 23712 * q^37 + 8424 * q^39 + 21582 * q^41 + 3752 * q^43 + 7290 * q^45 + 2160 * q^47 + 5022 * q^51 + 53528 * q^53 - 22068 * q^55 + 2916 * q^57 - 14328 * q^59 - 17820 * q^61 + 136 * q^65 + 812 * q^67 + 25758 * q^69 + 59846 * q^71 - 141444 * q^73 - 27738 * q^75 + 22988 * q^79 + 13122 * q^81 + 17640 * q^83 + 149196 * q^85 + 40104 * q^87 - 227394 * q^89 + 10044 * q^93 - 130784 * q^95 + 4500 * q^97 + 45846 * q^99

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
$5$	96.3907	1.72429	0.862144	−	0.506663i	$-0.169121\pi$
$5$	96.3907	1.72429	0.862144	+	0.506663i	$0.169121\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−179.516	−0.447323	−0.223662	−	0.974667i	$-0.571801\pi$
$11$	−179.516	−0.447323	−0.223662	+	0.974667i	$0.571801\pi$
$12$	0	0
$13$	−56.8747	−0.0933385	−0.0466693	−	0.998910i	$-0.514861\pi$
$13$	−56.8747	−0.0933385	−0.0466693	+	0.998910i	$0.514861\pi$
$14$	0	0
$15$	−867.516	−0.995518
$16$	0	0
$17$	1416.89	1.18909	0.594545	−	0.804063i	$-0.297332\pi$
$17$	1416.89	1.18909	0.594545	+	0.804063i	$0.297332\pi$
$18$	0	0
$19$	−1292.59	−0.821445	−0.410722	−	0.911760i	$-0.634723\pi$
$19$	−1292.59	−0.821445	−0.410722	+	0.911760i	$0.634723\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4668.61	−1.84021	−0.920107	−	0.391668i	$-0.871898\pi$
$23$	−4668.61	−1.84021	−0.920107	+	0.391668i	$0.871898\pi$
$24$	0	0
$25$	6166.16	1.97317
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5003.10	−1.10470	−0.552349	−	0.833613i	$-0.686269\pi$
$29$	−5003.10	−1.10470	−0.552349	+	0.833613i	$0.686269\pi$
$30$	0	0
$31$	−7444.35	−1.39131	−0.695653	−	0.718378i	$-0.744885\pi$
$31$	−7444.35	−1.39131	−0.695653	+	0.718378i	$0.744885\pi$
$32$	0	0
$33$	1615.64	0.258262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9080.90	−1.09050	−0.545249	−	0.838274i	$-0.683565\pi$
$37$	−9080.90	−1.09050	−0.545249	+	0.838274i	$0.683565\pi$
$38$	0	0
$39$	511.872	0.0538890
$40$	0	0
$41$	6628.36	0.615809	0.307905	−	0.951417i	$-0.400372\pi$
$41$	6628.36	0.615809	0.307905	+	0.951417i	$0.400372\pi$
$42$	0	0
$43$	−12924.5	−1.06597	−0.532983	−	0.846126i	$-0.678929\pi$
$43$	−12924.5	−1.06597	−0.532983	+	0.846126i	$0.678929\pi$
$44$	0	0
$45$	7807.64	0.574763
$46$	0	0
$47$	−5498.00	−0.363045	−0.181522	−	0.983387i	$-0.558103\pi$
$47$	−5498.00	−0.363045	−0.181522	+	0.983387i	$0.558103\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12752.0	−0.686521
$52$	0	0
$53$	29539.1	1.44447	0.722233	−	0.691649i	$-0.243116\pi$
$53$	29539.1	1.44447	0.722233	+	0.691649i	$0.243116\pi$
$54$	0	0
$55$	−17303.7	−0.771314
$56$	0	0
$57$	11633.4	0.474261
$58$	0	0
$59$	1058.51	0.0395880	0.0197940	−	0.999804i	$-0.493699\pi$
$59$	1058.51	0.0395880	0.0197940	+	0.999804i	$0.493699\pi$
$60$	0	0
$61$	9487.86	0.326470	0.163235	−	0.986587i	$-0.447807\pi$
$61$	9487.86	0.326470	0.163235	+	0.986587i	$0.447807\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5482.19	−0.160943
$66$	0	0
$67$	1331.03	0.0362244	0.0181122	−	0.999836i	$-0.494234\pi$
$67$	1331.03	0.0362244	0.0181122	+	0.999836i	$0.494234\pi$
$68$	0	0
$69$	42017.5	1.06245
$70$	0	0
$71$	49811.2	1.17268	0.586342	−	0.810064i	$-0.300567\pi$
$71$	49811.2	1.17268	0.586342	+	0.810064i	$0.300567\pi$
$72$	0	0
$73$	−75758.3	−1.66388	−0.831942	−	0.554863i	$-0.812771\pi$
$73$	−75758.3	−1.66388	−0.831942	+	0.554863i	$0.812771\pi$
$74$	0	0
$75$	−55495.4	−1.13921
$76$	0	0
$77$	0	0
$78$	0	0
$79$	62370.8	1.12438	0.562190	−	0.827008i	$-0.309959\pi$
$79$	62370.8	1.12438	0.562190	+	0.827008i	$0.309959\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	53221.5	0.847993	0.423996	−	0.905664i	$-0.360627\pi$
$83$	53221.5	0.847993	0.423996	+	0.905664i	$0.360627\pi$
$84$	0	0
$85$	136575.	2.05033
$86$	0	0
$87$	45027.9	0.637798
$88$	0	0
$89$	−118682.	−1.58822	−0.794108	−	0.607777i	$-0.792061\pi$
$89$	−118682.	−1.58822	−0.794108	+	0.607777i	$0.792061\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	66999.1	0.803271
$94$	0	0
$95$	−124594.	−1.41641
$96$	0	0
$97$	−136608.	−1.47416	−0.737081	−	0.675804i	$-0.763797\pi$
$97$	−136608.	−1.47416	−0.737081	+	0.675804i	$0.763797\pi$
$98$	0	0
$99$	−14540.8	−0.149108
$100$	0	0
$101$	12833.4	0.125181	0.0625903	−	0.998039i	$-0.480064\pi$
$101$	12833.4	0.125181	0.0625903	+	0.998039i	$0.480064\pi$
$102$	0	0
$103$	22369.4	0.207760	0.103880	−	0.994590i	$-0.466874\pi$
$103$	22369.4	0.207760	0.103880	+	0.994590i	$0.466874\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	73061.1	0.616917	0.308459	−	0.951238i	$-0.400187\pi$
$107$	73061.1	0.616917	0.308459	+	0.951238i	$0.400187\pi$
$108$	0	0
$109$	27451.2	0.221307	0.110653	−	0.993859i	$-0.464706\pi$
$109$	27451.2	0.221307	0.110653	+	0.993859i	$0.464706\pi$
$110$	0	0
$111$	81728.1	0.629599
$112$	0	0
$113$	−9821.49	−0.0723571	−0.0361786	−	0.999345i	$-0.511519\pi$
$113$	−9821.49	−0.0723571	−0.0361786	+	0.999345i	$0.511519\pi$
$114$	0	0
$115$	−450011.	−3.17306
$116$	0	0
$117$	−4606.85	−0.0311128
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−128825.	−0.799902
$122$	0	0
$123$	−59655.2	−0.355538
$124$	0	0
$125$	293139.	1.67803
$126$	0	0
$127$	−216087.	−1.18883	−0.594414	−	0.804159i	$-0.702616\pi$
$127$	−216087.	−1.18883	−0.594414	+	0.804159i	$0.702616\pi$
$128$	0	0
$129$	116321.	0.615435
$130$	0	0
$131$	−295522.	−1.50457	−0.752284	−	0.658839i	$-0.771048\pi$
$131$	−295522.	−1.50457	−0.752284	+	0.658839i	$0.771048\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−70268.8	−0.331839
$136$	0	0
$137$	335551.	1.52742	0.763708	−	0.645562i	$-0.223377\pi$
$137$	335551.	1.52742	0.763708	+	0.645562i	$0.223377\pi$
$138$	0	0
$139$	34781.8	0.152692	0.0763459	−	0.997081i	$-0.475675\pi$
$139$	34781.8	0.152692	0.0763459	+	0.997081i	$0.475675\pi$
$140$	0	0
$141$	49482.0	0.209604
$142$	0	0
$143$	10209.9	0.0417525
$144$	0	0
$145$	−482252.	−1.90482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−166526.	−0.614492	−0.307246	−	0.951630i	$-0.599407\pi$
$149$	−166526.	−0.614492	−0.307246	+	0.951630i	$0.599407\pi$
$150$	0	0
$151$	−480481.	−1.71488	−0.857440	−	0.514585i	$-0.827946\pi$
$151$	−480481.	−1.71488	−0.857440	+	0.514585i	$0.827946\pi$
$152$	0	0
$153$	114768.	0.396363
$154$	0	0
$155$	−717566.	−2.39901
$156$	0	0
$157$	−598780.	−1.93873	−0.969367	−	0.245618i	$-0.921009\pi$
$157$	−598780.	−1.93873	−0.969367	+	0.245618i	$0.921009\pi$
$158$	0	0
$159$	−265852.	−0.833963
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13667.1	0.0402909	0.0201455	−	0.999797i	$-0.493587\pi$
$163$	13667.1	0.0402909	0.0201455	+	0.999797i	$0.493587\pi$
$164$	0	0
$165$	155733.	0.445318
$166$	0	0
$167$	3324.13	0.00922332	0.00461166	−	0.999989i	$-0.498532\pi$
$167$	3324.13	0.00922332	0.00461166	+	0.999989i	$0.498532\pi$
$168$	0	0
$169$	−368058.	−0.991288
$170$	0	0
$171$	−104700.	−0.273815
$172$	0	0
$173$	−287815.	−0.731136	−0.365568	−	0.930785i	$-0.619125\pi$
$173$	−287815.	−0.731136	−0.365568	+	0.930785i	$0.619125\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9526.55	−0.0228561
$178$	0	0
$179$	−397283.	−0.926760	−0.463380	−	0.886160i	$-0.653363\pi$
$179$	−397283.	−0.926760	−0.463380	+	0.886160i	$0.653363\pi$
$180$	0	0
$181$	267931.	0.607891	0.303946	−	0.952689i	$-0.401696\pi$
$181$	267931.	0.607891	0.303946	+	0.952689i	$0.401696\pi$
$182$	0	0
$183$	−85390.7	−0.188488
$184$	0	0
$185$	−875314.	−1.88033
$186$	0	0
$187$	−254355.	−0.531907
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−595611.	−1.18135	−0.590676	−	0.806909i	$-0.701139\pi$
$191$	−595611.	−1.18135	−0.590676	+	0.806909i	$0.701139\pi$
$192$	0	0
$193$	900426.	1.74002	0.870011	−	0.493032i	$-0.164111\pi$
$193$	900426.	1.74002	0.870011	+	0.493032i	$0.164111\pi$
$194$	0	0
$195$	49339.7	0.0929202
$196$	0	0
$197$	483809.	0.888195	0.444097	−	0.895978i	$-0.353524\pi$
$197$	483809.	0.888195	0.444097	+	0.895978i	$0.353524\pi$
$198$	0	0
$199$	647610.	1.15926	0.579630	−	0.814880i	$-0.303197\pi$
$199$	647610.	1.15926	0.579630	+	0.814880i	$0.303197\pi$
$200$	0	0
$201$	−11979.3	−0.0209142
$202$	0	0
$203$	0	0
$204$	0	0
$205$	638912.	1.06183
$206$	0	0
$207$	−378158.	−0.613404
$208$	0	0
$209$	232041.	0.367451
$210$	0	0
$211$	1.24416e6	1.92384	0.961919	−	0.273333i	$-0.0881262\pi$
$211$	1.24416e6	1.92384	0.961919	+	0.273333i	$0.0881262\pi$
$212$	0	0
$213$	−448301.	−0.677049
$214$	0	0
$215$	−1.24580e6	−1.83803
$216$	0	0
$217$	0	0
$218$	0	0
$219$	681825.	0.960644
$220$	0	0
$221$	−80585.3	−0.110988
$222$	0	0
$223$	565408.	0.761377	0.380689	−	0.924703i	$-0.375687\pi$
$223$	565408.	0.761377	0.380689	+	0.924703i	$0.375687\pi$
$224$	0	0
$225$	499459.	0.657724
$226$	0	0
$227$	−1.35812e6	−1.74933	−0.874666	−	0.484726i	$-0.838919\pi$
$227$	−1.35812e6	−1.74933	−0.874666	+	0.484726i	$0.838919\pi$
$228$	0	0
$229$	−1.06386e6	−1.34058	−0.670292	−	0.742098i	$-0.733831\pi$
$229$	−1.06386e6	−1.34058	−0.670292	+	0.742098i	$0.733831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	809030.	0.976281	0.488140	−	0.872765i	$-0.337675\pi$
$233$	809030.	0.976281	0.488140	+	0.872765i	$0.337675\pi$
$234$	0	0
$235$	−529956.	−0.625994
$236$	0	0
$237$	−561337.	−0.649161
$238$	0	0
$239$	290751.	0.329251	0.164625	−	0.986356i	$-0.447358\pi$
$239$	290751.	0.329251	0.164625	+	0.986356i	$0.447358\pi$
$240$	0	0
$241$	−159098.	−0.176450	−0.0882248	−	0.996101i	$-0.528119\pi$
$241$	−159098.	−0.176450	−0.0882248	+	0.996101i	$0.528119\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	73515.9	0.0766724
$248$	0	0
$249$	−478994.	−0.489589
$250$	0	0
$251$	104144.	0.104340	0.0521699	−	0.998638i	$-0.483386\pi$
$251$	104144.	0.104340	0.0521699	+	0.998638i	$0.483386\pi$
$252$	0	0
$253$	838090.	0.823170
$254$	0	0
$255$	−1.22918e6	−1.18376
$256$	0	0
$257$	880928.	0.831970	0.415985	−	0.909372i	$-0.363437\pi$
$257$	880928.	0.831970	0.415985	+	0.909372i	$0.363437\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−405251.	−0.368233
$262$	0	0
$263$	1.80154e6	1.60603	0.803015	−	0.595959i	$-0.203228\pi$
$263$	1.80154e6	1.60603	0.803015	+	0.595959i	$0.203228\pi$
$264$	0	0
$265$	2.84729e6	2.49068
$266$	0	0
$267$	1.06814e6	0.916957
$268$	0	0
$269$	1.05044e6	0.885095	0.442548	−	0.896745i	$-0.354075\pi$
$269$	1.05044e6	0.885095	0.442548	+	0.896745i	$0.354075\pi$
$270$	0	0
$271$	−1.23127e6	−1.01842	−0.509212	−	0.860641i	$-0.670063\pi$
$271$	−1.23127e6	−1.01842	−0.509212	+	0.860641i	$0.670063\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.10692e6	−0.882645
$276$	0	0
$277$	−873312.	−0.683864	−0.341932	−	0.939725i	$-0.611081\pi$
$277$	−873312.	−0.683864	−0.341932	+	0.939725i	$0.611081\pi$
$278$	0	0
$279$	−602992.	−0.463768
$280$	0	0
$281$	−1.90392e6	−1.43841	−0.719204	−	0.694799i	$-0.755493\pi$
$281$	−1.90392e6	−1.43841	−0.719204	+	0.694799i	$0.755493\pi$
$282$	0	0
$283$	1.89580e6	1.40710	0.703551	−	0.710644i	$-0.251596\pi$
$283$	1.89580e6	1.40710	0.703551	+	0.710644i	$0.251596\pi$
$284$	0	0
$285$	1.12135e6	0.817763
$286$	0	0
$287$	0	0
$288$	0	0
$289$	587725.	0.413933
$290$	0	0
$291$	1.22947e6	0.851108
$292$	0	0
$293$	−1.85424e6	−1.26182	−0.630908	−	0.775858i	$-0.717317\pi$
$293$	−1.85424e6	−1.26182	−0.630908	+	0.775858i	$0.717317\pi$
$294$	0	0
$295$	102030.	0.0682611
$296$	0	0
$297$	130867.	0.0860874
$298$	0	0
$299$	265526.	0.171763
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−115500.	−0.0722730
$304$	0	0
$305$	914541.	0.562929
$306$	0	0
$307$	−1.66209e6	−1.00649	−0.503245	−	0.864144i	$-0.667861\pi$
$307$	−1.66209e6	−1.00649	−0.503245	+	0.864144i	$0.667861\pi$
$308$	0	0
$309$	−201325.	−0.119950
$310$	0	0
$311$	−1.90632e6	−1.11762	−0.558811	−	0.829295i	$-0.688742\pi$
$311$	−1.90632e6	−1.11762	−0.558811	+	0.829295i	$0.688742\pi$
$312$	0	0
$313$	−1.33752e6	−0.771683	−0.385841	−	0.922565i	$-0.626089\pi$
$313$	−1.33752e6	−0.771683	−0.385841	+	0.922565i	$0.626089\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.34837e6	−1.87148	−0.935741	−	0.352689i	$-0.885267\pi$
$317$	−3.34837e6	−1.87148	−0.935741	+	0.352689i	$0.885267\pi$
$318$	0	0
$319$	898135.	0.494157
$320$	0	0
$321$	−657550.	−0.356177
$322$	0	0
$323$	−1.83147e6	−0.976771
$324$	0	0
$325$	−350699.	−0.184173
$326$	0	0
$327$	−247061.	−0.127772
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.60437e6	−0.804889	−0.402444	−	0.915444i	$-0.631839\pi$
$331$	−1.60437e6	−0.804889	−0.402444	+	0.915444i	$0.631839\pi$
$332$	0	0
$333$	−735553.	−0.363499
$334$	0	0
$335$	128299.	0.0624614
$336$	0	0
$337$	−1.93575e6	−0.928483	−0.464242	−	0.885709i	$-0.653673\pi$
$337$	−1.93575e6	−0.928483	−0.464242	+	0.885709i	$0.653673\pi$
$338$	0	0
$339$	88393.4	0.0417754
$340$	0	0
$341$	1.33638e6	0.622363
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.05010e6	1.83197
$346$	0	0
$347$	42619.3	0.0190013	0.00950064	−	0.999955i	$-0.496976\pi$
$347$	42619.3	0.0190013	0.00950064	+	0.999955i	$0.496976\pi$
$348$	0	0
$349$	2.12211e6	0.932618	0.466309	−	0.884622i	$-0.345584\pi$
$349$	2.12211e6	0.932618	0.466309	+	0.884622i	$0.345584\pi$
$350$	0	0
$351$	41461.7	0.0179630
$352$	0	0
$353$	138714.	0.0592492	0.0296246	−	0.999561i	$-0.490569\pi$
$353$	138714.	0.0592492	0.0296246	+	0.999561i	$0.490569\pi$
$354$	0	0
$355$	4.80133e6	2.02205
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.27021e6	−1.33918	−0.669592	−	0.742729i	$-0.733531\pi$
$359$	−3.27021e6	−1.33918	−0.669592	+	0.742729i	$0.733531\pi$
$360$	0	0
$361$	−805298.	−0.325229
$362$	0	0
$363$	1.15943e6	0.461824
$364$	0	0
$365$	−7.30239e6	−2.86902
$366$	0	0
$367$	4.91951e6	1.90659	0.953294	−	0.302043i	$-0.0976686\pi$
$367$	4.91951e6	1.90659	0.953294	+	0.302043i	$0.0976686\pi$
$368$	0	0
$369$	536897.	0.205270
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.28061e6	0.476589	0.238295	−	0.971193i	$-0.423412\pi$
$373$	1.28061e6	0.476589	0.238295	+	0.971193i	$0.423412\pi$
$374$	0	0
$375$	−2.63825e6	−0.968810
$376$	0	0
$377$	284550.	0.103111
$378$	0	0
$379$	2.54359e6	0.909596	0.454798	−	0.890595i	$-0.349711\pi$
$379$	2.54359e6	0.909596	0.454798	+	0.890595i	$0.349711\pi$
$380$	0	0
$381$	1.94478e6	0.686370
$382$	0	0
$383$	−112734.	−0.0392697	−0.0196348	−	0.999807i	$-0.506250\pi$
$383$	−112734.	−0.0392697	−0.0196348	+	0.999807i	$0.506250\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.04689e6	−0.355322
$388$	0	0
$389$	−1.33653e6	−0.447820	−0.223910	−	0.974610i	$-0.571882\pi$
$389$	−1.33653e6	−0.447820	−0.223910	+	0.974610i	$0.571882\pi$
$390$	0	0
$391$	−6.61492e6	−2.18818
$392$	0	0
$393$	2.65970e6	0.868663
$394$	0	0
$395$	6.01196e6	1.93876
$396$	0	0
$397$	3.96291e6	1.26194	0.630968	−	0.775809i	$-0.282658\pi$
$397$	3.96291e6	1.26194	0.630968	+	0.775809i	$0.282658\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.05395e6	0.948419	0.474210	−	0.880412i	$-0.342734\pi$
$401$	3.05395e6	0.948419	0.474210	+	0.880412i	$0.342734\pi$
$402$	0	0
$403$	423395.	0.129862
$404$	0	0
$405$	632419.	0.191588
$406$	0	0
$407$	1.63017e6	0.487805
$408$	0	0
$409$	−4.42448e6	−1.30784	−0.653920	−	0.756564i	$-0.726877\pi$
$409$	−4.42448e6	−1.30784	−0.653920	+	0.756564i	$0.726877\pi$
$410$	0	0
$411$	−3.01996e6	−0.881854
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.13006e6	1.46218
$416$	0	0
$417$	−313037.	−0.0881566
$418$	0	0
$419$	−2.02795e6	−0.564315	−0.282158	−	0.959368i	$-0.591050\pi$
$419$	−2.02795e6	−0.564315	−0.282158	+	0.959368i	$0.591050\pi$
$420$	0	0
$421$	6.64038e6	1.82595	0.912973	−	0.408020i	$-0.133781\pi$
$421$	6.64038e6	1.82595	0.912973	+	0.408020i	$0.133781\pi$
$422$	0	0
$423$	−445338.	−0.121015
$424$	0	0
$425$	8.73678e6	2.34628
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−91889.3	−0.0241058
$430$	0	0
$431$	3.76085e6	0.975198	0.487599	−	0.873068i	$-0.337873\pi$
$431$	3.76085e6	0.975198	0.487599	+	0.873068i	$0.337873\pi$
$432$	0	0
$433$	−3.47031e6	−0.889505	−0.444753	−	0.895653i	$-0.646708\pi$
$433$	−3.47031e6	−0.889505	−0.444753	+	0.895653i	$0.646708\pi$
$434$	0	0
$435$	4.34027e6	1.09975
$436$	0	0
$437$	6.03462e6	1.51163
$438$	0	0
$439$	6.25398e6	1.54880	0.774399	−	0.632697i	$-0.218052\pi$
$439$	6.25398e6	1.54880	0.774399	+	0.632697i	$0.218052\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−410688.	−0.0994265	−0.0497133	−	0.998764i	$-0.515831\pi$
$443$	−410688.	−0.0994265	−0.0497133	+	0.998764i	$0.515831\pi$
$444$	0	0
$445$	−1.14398e7	−2.73854
$446$	0	0
$447$	1.49873e6	0.354777
$448$	0	0
$449$	−3.55144e6	−0.831359	−0.415680	−	0.909511i	$-0.636456\pi$
$449$	−3.55144e6	−0.831359	−0.415680	+	0.909511i	$0.636456\pi$
$450$	0	0
$451$	−1.18990e6	−0.275466
$452$	0	0
$453$	4.32433e6	0.990086
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.49553e6	1.45487	0.727435	−	0.686177i	$-0.240712\pi$
$457$	6.49553e6	1.45487	0.727435	+	0.686177i	$0.240712\pi$
$458$	0	0
$459$	−1.03291e6	−0.228840
$460$	0	0
$461$	7.57960e6	1.66109	0.830546	−	0.556950i	$-0.188028\pi$
$461$	7.57960e6	1.66109	0.830546	+	0.556950i	$0.188028\pi$
$462$	0	0
$463$	382670.	0.0829607	0.0414803	−	0.999139i	$-0.486793\pi$
$463$	382670.	0.0829607	0.0414803	+	0.999139i	$0.486793\pi$
$464$	0	0
$465$	6.45809e6	1.38507
$466$	0	0
$467$	−6.50195e6	−1.37959	−0.689797	−	0.724003i	$-0.742300\pi$
$467$	−6.50195e6	−1.37959	−0.689797	+	0.724003i	$0.742300\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.38902e6	1.11933
$472$	0	0
$473$	2.32016e6	0.476831
$474$	0	0
$475$	−7.97034e6	−1.62085
$476$	0	0
$477$	2.39267e6	0.481489
$478$	0	0
$479$	4.21928e6	0.840234	0.420117	−	0.907470i	$-0.361989\pi$
$479$	4.21928e6	0.840234	0.420117	+	0.907470i	$0.361989\pi$
$480$	0	0
$481$	516474.	0.101785
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.31677e7	−2.54188
$486$	0	0
$487$	2.54734e6	0.486704	0.243352	−	0.969938i	$-0.421753\pi$
$487$	2.54734e6	0.486704	0.243352	+	0.969938i	$0.421753\pi$
$488$	0	0
$489$	−123004.	−0.0232620
$490$	0	0
$491$	−6.76516e6	−1.26641	−0.633205	−	0.773984i	$-0.718261\pi$
$491$	−6.76516e6	−1.26641	−0.633205	+	0.773984i	$0.718261\pi$
$492$	0	0
$493$	−7.08885e6	−1.31359
$494$	0	0
$495$	−1.40160e6	−0.257105
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.17337e6	0.750300	0.375150	−	0.926964i	$-0.377591\pi$
$499$	4.17337e6	0.750300	0.375150	+	0.926964i	$0.377591\pi$
$500$	0	0
$501$	−29917.2	−0.00532508
$502$	0	0
$503$	−1.07179e7	−1.88881	−0.944406	−	0.328781i	$-0.893362\pi$
$503$	−1.07179e7	−1.88881	−0.944406	+	0.328781i	$0.893362\pi$
$504$	0	0
$505$	1.23702e6	0.215847
$506$	0	0
$507$	3.31252e6	0.572320
$508$	0	0
$509$	6.03626e6	1.03270	0.516349	−	0.856378i	$-0.327291\pi$
$509$	6.03626e6	1.03270	0.516349	+	0.856378i	$0.327291\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	942301.	0.158087
$514$	0	0
$515$	2.15620e6	0.358238
$516$	0	0
$517$	986979.	0.162398
$518$	0	0
$519$	2.59034e6	0.422122
$520$	0	0
$521$	2.27593e6	0.367337	0.183669	−	0.982988i	$-0.441203\pi$
$521$	2.27593e6	0.367337	0.183669	+	0.982988i	$0.441203\pi$
$522$	0	0
$523$	6.74220e6	1.07782	0.538911	−	0.842362i	$-0.318836\pi$
$523$	6.74220e6	1.07782	0.538911	+	0.842362i	$0.318836\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.05478e7	−1.65439
$528$	0	0
$529$	1.53596e7	2.38638
$530$	0	0
$531$	85739.0	0.0131960
$532$	0	0
$533$	−376986.	−0.0574787
$534$	0	0
$535$	7.04241e6	1.06374
$536$	0	0
$537$	3.57555e6	0.535065
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.17845e7	−1.73108	−0.865541	−	0.500838i	$-0.833025\pi$
$541$	−1.17845e7	−1.73108	−0.865541	+	0.500838i	$0.833025\pi$
$542$	0	0
$543$	−2.41138e6	−0.350966
$544$	0	0
$545$	2.64604e6	0.381597
$546$	0	0
$547$	2.46712e6	0.352551	0.176275	−	0.984341i	$-0.443595\pi$
$547$	2.46712e6	0.352551	0.176275	+	0.984341i	$0.443595\pi$
$548$	0	0
$549$	768516.	0.108823
$550$	0	0
$551$	6.46697e6	0.907449
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.87783e6	1.08561
$556$	0	0
$557$	6.65458e6	0.908829	0.454415	−	0.890790i	$-0.349848\pi$
$557$	6.65458e6	0.908829	0.454415	+	0.890790i	$0.349848\pi$
$558$	0	0
$559$	735078.	0.0994956
$560$	0	0
$561$	2.28919e6	0.307097
$562$	0	0
$563$	3.12246e6	0.415170	0.207585	−	0.978217i	$-0.433440\pi$
$563$	3.12246e6	0.415170	0.207585	+	0.978217i	$0.433440\pi$
$564$	0	0
$565$	−946700.	−0.124765
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.20578e6	−0.544585	−0.272293	−	0.962214i	$-0.587782\pi$
$569$	−4.20578e6	−0.544585	−0.272293	+	0.962214i	$0.587782\pi$
$570$	0	0
$571$	−4.71461e6	−0.605140	−0.302570	−	0.953127i	$-0.597845\pi$
$571$	−4.71461e6	−0.605140	−0.302570	+	0.953127i	$0.597845\pi$
$572$	0	0
$573$	5.36050e6	0.682054
$574$	0	0
$575$	−2.87874e7	−3.63106
$576$	0	0
$577$	258113.	0.0322753	0.0161376	−	0.999870i	$-0.494863\pi$
$577$	258113.	0.0322753	0.0161376	+	0.999870i	$0.494863\pi$
$578$	0	0
$579$	−8.10384e6	−1.00460
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.30274e6	−0.646143
$584$	0	0
$585$	−444057.	−0.0536475
$586$	0	0
$587$	7.33876e6	0.879078	0.439539	−	0.898224i	$-0.355142\pi$
$587$	7.33876e6	0.879078	0.439539	+	0.898224i	$0.355142\pi$
$588$	0	0
$589$	9.62252e6	1.14288
$590$	0	0
$591$	−4.35428e6	−0.512800
$592$	0	0
$593$	162794.	0.0190108	0.00950542	−	0.999955i	$-0.496974\pi$
$593$	162794.	0.0190108	0.00950542	+	0.999955i	$0.496974\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.82849e6	−0.669299
$598$	0	0
$599$	9.51587e6	1.08363	0.541815	−	0.840498i	$-0.317737\pi$
$599$	9.51587e6	1.08363	0.541815	+	0.840498i	$0.317737\pi$
$600$	0	0
$601$	6.14487e6	0.693947	0.346974	−	0.937875i	$-0.387209\pi$
$601$	6.14487e6	0.693947	0.346974	+	0.937875i	$0.387209\pi$
$602$	0	0
$603$	107814.	0.0120748
$604$	0	0
$605$	−1.24175e7	−1.37926
$606$	0	0
$607$	−7.34089e6	−0.808681	−0.404341	−	0.914609i	$-0.632499\pi$
$607$	−7.34089e6	−0.808681	−0.404341	+	0.914609i	$0.632499\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	312697.	0.0338861
$612$	0	0
$613$	−3.51721e6	−0.378048	−0.189024	−	0.981973i	$-0.560532\pi$
$613$	−3.51721e6	−0.378048	−0.189024	+	0.981973i	$0.560532\pi$
$614$	0	0
$615$	−5.75020e6	−0.613049
$616$	0	0
$617$	−6.95703e6	−0.735717	−0.367859	−	0.929882i	$-0.619909\pi$
$617$	−6.95703e6	−0.735717	−0.367859	+	0.929882i	$0.619909\pi$
$618$	0	0
$619$	6.47022e6	0.678723	0.339361	−	0.940656i	$-0.389789\pi$
$619$	6.47022e6	0.678723	0.339361	+	0.940656i	$0.389789\pi$
$620$	0	0
$621$	3.40342e6	0.354149
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.98665e6	0.920233
$626$	0	0
$627$	−2.08837e6	−0.212148
$628$	0	0
$629$	−1.28667e7	−1.29670
$630$	0	0
$631$	591973.	0.0591873	0.0295936	−	0.999562i	$-0.490579\pi$
$631$	591973.	0.0591873	0.0295936	+	0.999562i	$0.490579\pi$
$632$	0	0
$633$	−1.11974e7	−1.11073
$634$	0	0
$635$	−2.08287e7	−2.04988
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.03471e6	0.390895
$640$	0	0
$641$	1.20679e7	1.16007	0.580037	−	0.814590i	$-0.303038\pi$
$641$	1.20679e7	1.16007	0.580037	+	0.814590i	$0.303038\pi$
$642$	0	0
$643$	−4.48967e6	−0.428240	−0.214120	−	0.976807i	$-0.568688\pi$
$643$	−4.48967e6	−0.428240	−0.214120	+	0.976807i	$0.568688\pi$
$644$	0	0
$645$	1.12122e7	1.06119
$646$	0	0
$647$	−1.15701e7	−1.08662	−0.543309	−	0.839533i	$-0.682829\pi$
$647$	−1.15701e7	−1.08662	−0.543309	+	0.839533i	$0.682829\pi$
$648$	0	0
$649$	−190019.	−0.0177086
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.33400e7	−1.22426	−0.612129	−	0.790758i	$-0.709687\pi$
$653$	−1.33400e7	−1.22426	−0.612129	+	0.790758i	$0.709687\pi$
$654$	0	0
$655$	−2.84856e7	−2.59431
$656$	0	0
$657$	−6.13642e6	−0.554628
$658$	0	0
$659$	3.85674e6	0.345945	0.172972	−	0.984927i	$-0.444663\pi$
$659$	3.85674e6	0.345945	0.172972	+	0.984927i	$0.444663\pi$
$660$	0	0
$661$	5.07023e6	0.451361	0.225680	−	0.974201i	$-0.427539\pi$
$661$	5.07023e6	0.451361	0.225680	+	0.974201i	$0.427539\pi$
$662$	0	0
$663$	725268.	0.0640788
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.33575e7	2.03288
$668$	0	0
$669$	−5.08867e6	−0.439581
$670$	0	0
$671$	−1.70322e6	−0.146038
$672$	0	0
$673$	8.50347e6	0.723700	0.361850	−	0.932236i	$-0.382145\pi$
$673$	8.50347e6	0.723700	0.361850	+	0.932236i	$0.382145\pi$
$674$	0	0
$675$	−4.49513e6	−0.379737
$676$	0	0
$677$	1.05704e7	0.886383	0.443192	−	0.896427i	$-0.353846\pi$
$677$	1.05704e7	0.886383	0.443192	+	0.896427i	$0.353846\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.22230e7	1.00998
$682$	0	0
$683$	−4.81908e6	−0.395287	−0.197643	−	0.980274i	$-0.563329\pi$
$683$	−4.81908e6	−0.395287	−0.197643	+	0.980274i	$0.563329\pi$
$684$	0	0
$685$	3.23440e7	2.63371
$686$	0	0
$687$	9.57470e6	0.773986
$688$	0	0
$689$	−1.68003e6	−0.134824
$690$	0	0
$691$	−1.75621e7	−1.39920	−0.699602	−	0.714533i	$-0.746639\pi$
$691$	−1.75621e7	−1.39920	−0.699602	+	0.714533i	$0.746639\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.35264e6	0.263285
$696$	0	0
$697$	9.39166e6	0.732252
$698$	0	0
$699$	−7.28127e6	−0.563656
$700$	0	0
$701$	9.94577e6	0.764440	0.382220	−	0.924071i	$-0.375160\pi$
$701$	9.94577e6	0.764440	0.382220	+	0.924071i	$0.375160\pi$
$702$	0	0
$703$	1.17379e7	0.895784
$704$	0	0
$705$	4.76961e6	0.361418
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.99017e6	0.671664	0.335832	−	0.941922i	$-0.390982\pi$
$709$	8.99017e6	0.671664	0.335832	+	0.941922i	$0.390982\pi$
$710$	0	0
$711$	5.05203e6	0.374793
$712$	0	0
$713$	3.47548e7	2.56030
$714$	0	0
$715$	984141.	0.0719933
$716$	0	0
$717$	−2.61676e6	−0.190093
$718$	0	0
$719$	−2.13270e7	−1.53854	−0.769270	−	0.638924i	$-0.779380\pi$
$719$	−2.13270e7	−1.53854	−0.769270	+	0.638924i	$0.779380\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.43188e6	0.101873
$724$	0	0
$725$	−3.08499e7	−2.17976
$726$	0	0
$727$	−1.47016e6	−0.103164	−0.0515819	−	0.998669i	$-0.516426\pi$
$727$	−1.47016e6	−0.103164	−0.0515819	+	0.998669i	$0.516426\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.83126e7	−1.26753
$732$	0	0
$733$	−1.96893e7	−1.35354	−0.676770	−	0.736194i	$-0.736621\pi$
$733$	−1.96893e7	−1.35354	−0.676770	+	0.736194i	$0.736621\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−238941.	−0.0162040
$738$	0	0
$739$	3.29227e6	0.221761	0.110880	−	0.993834i	$-0.464633\pi$
$739$	3.29227e6	0.221761	0.110880	+	0.993834i	$0.464633\pi$
$740$	0	0
$741$	−661644.	−0.0442668
$742$	0	0
$743$	−2.05711e6	−0.136705	−0.0683527	−	0.997661i	$-0.521774\pi$
$743$	−2.05711e6	−0.136705	−0.0683527	+	0.997661i	$0.521774\pi$
$744$	0	0
$745$	−1.60515e7	−1.05956
$746$	0	0
$747$	4.31094e6	0.282664
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.39547e6	−0.154985	−0.0774927	−	0.996993i	$-0.524691\pi$
$751$	−2.39547e6	−0.154985	−0.0774927	+	0.996993i	$0.524691\pi$
$752$	0	0
$753$	−937296.	−0.0602406
$754$	0	0
$755$	−4.63138e7	−2.95695
$756$	0	0
$757$	7.29250e6	0.462527	0.231263	−	0.972891i	$-0.425714\pi$
$757$	7.29250e6	0.462527	0.231263	+	0.972891i	$0.425714\pi$
$758$	0	0
$759$	−7.54281e6	−0.475257
$760$	0	0
$761$	233537.	0.0146182	0.00730910	−	0.999973i	$-0.497673\pi$
$761$	233537.	0.0146182	0.00730910	+	0.999973i	$0.497673\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.10626e7	0.683444
$766$	0	0
$767$	−60202.2	−0.00369508
$768$	0	0
$769$	2.95723e7	1.80331	0.901654	−	0.432458i	$-0.142354\pi$
$769$	2.95723e7	1.80331	0.901654	+	0.432458i	$0.142354\pi$
$770$	0	0
$771$	−7.92835e6	−0.480338
$772$	0	0
$773$	−1.33291e7	−0.802327	−0.401164	−	0.916006i	$-0.631394\pi$
$773$	−1.33291e7	−0.802327	−0.401164	+	0.916006i	$0.631394\pi$
$774$	0	0
$775$	−4.59030e7	−2.74528
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.56778e6	−0.505853
$780$	0	0
$781$	−8.94190e6	−0.524569
$782$	0	0
$783$	3.64726e6	0.212599
$784$	0	0
$785$	−5.77168e7	−3.34294
$786$	0	0
$787$	−9.63918e6	−0.554758	−0.277379	−	0.960761i	$-0.589466\pi$
$787$	−9.63918e6	−0.554758	−0.277379	+	0.960761i	$0.589466\pi$
$788$	0	0
$789$	−1.62138e7	−0.927241
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−539619.	−0.0304722
$794$	0	0
$795$	−2.56256e7	−1.43799
$796$	0	0
$797$	1.06108e7	0.591699	0.295850	−	0.955235i	$-0.404397\pi$
$797$	1.06108e7	0.591699	0.295850	+	0.955235i	$0.404397\pi$
$798$	0	0
$799$	−7.79008e6	−0.431693
$800$	0	0
$801$	−9.61323e6	−0.529405
$802$	0	0
$803$	1.35998e7	0.744294
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.45395e6	−0.511010
$808$	0	0
$809$	1.70209e7	0.914350	0.457175	−	0.889377i	$-0.348861\pi$
$809$	1.70209e7	0.914350	0.457175	+	0.889377i	$0.348861\pi$
$810$	0	0
$811$	4.79571e6	0.256036	0.128018	−	0.991772i	$-0.459139\pi$
$811$	4.79571e6	0.256036	0.128018	+	0.991772i	$0.459139\pi$
$812$	0	0
$813$	1.10814e7	0.587987
$814$	0	0
$815$	1.31738e6	0.0694732
$816$	0	0
$817$	1.67062e7	0.875631
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.24534e7	−0.644809	−0.322404	−	0.946602i	$-0.604491\pi$
$821$	−1.24534e7	−0.644809	−0.322404	+	0.946602i	$0.604491\pi$
$822$	0	0
$823$	3.59326e7	1.84922	0.924610	−	0.380914i	$-0.124391\pi$
$823$	3.59326e7	1.84922	0.924610	+	0.380914i	$0.124391\pi$
$824$	0	0
$825$	9.96232e6	0.509595
$826$	0	0
$827$	1.39273e7	0.708112	0.354056	−	0.935224i	$-0.384802\pi$
$827$	1.39273e7	0.708112	0.354056	+	0.935224i	$0.384802\pi$
$828$	0	0
$829$	−8.83660e6	−0.446580	−0.223290	−	0.974752i	$-0.571680\pi$
$829$	−8.83660e6	−0.446580	−0.223290	+	0.974752i	$0.571680\pi$
$830$	0	0
$831$	7.85981e6	0.394829
$832$	0	0
$833$	0	0
$834$	0	0
$835$	320415.	0.0159037
$836$	0	0
$837$	5.42693e6	0.267757
$838$	0	0
$839$	3.34486e7	1.64049	0.820244	−	0.572014i	$-0.193838\pi$
$839$	3.34486e7	1.64049	0.820244	+	0.572014i	$0.193838\pi$
$840$	0	0
$841$	4.51982e6	0.220359
$842$	0	0
$843$	1.71352e7	0.830465
$844$	0	0
$845$	−3.54774e7	−1.70927
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.70622e7	−0.812391
$850$	0	0
$851$	4.23952e7	2.00675
$852$	0	0
$853$	−4.28936e6	−0.201846	−0.100923	−	0.994894i	$-0.532180\pi$
$853$	−4.28936e6	−0.201846	−0.100923	+	0.994894i	$0.532180\pi$
$854$	0	0
$855$	−1.00921e7	−0.472136
$856$	0	0
$857$	−3.88960e7	−1.80906	−0.904529	−	0.426411i	$-0.859778\pi$
$857$	−3.88960e7	−1.80906	−0.904529	+	0.426411i	$0.859778\pi$
$858$	0	0
$859$	1.58855e6	0.0734546	0.0367273	−	0.999325i	$-0.488307\pi$
$859$	1.58855e6	0.0734546	0.0367273	+	0.999325i	$0.488307\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.41298e7	−0.645815	−0.322907	−	0.946431i	$-0.604660\pi$
$863$	−1.41298e7	−0.645815	−0.322907	+	0.946431i	$0.604660\pi$
$864$	0	0
$865$	−2.77427e7	−1.26069
$866$	0	0
$867$	−5.28953e6	−0.238984
$868$	0	0
$869$	−1.11965e7	−0.502961
$870$	0	0
$871$	−75702.1	−0.00338113
$872$	0	0
$873$	−1.10652e7	−0.491388
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.66989e7	−1.61122	−0.805609	−	0.592448i	$-0.798162\pi$
$877$	−3.66989e7	−1.61122	−0.805609	+	0.592448i	$0.798162\pi$
$878$	0	0
$879$	1.66881e7	0.728510
$880$	0	0
$881$	−1.58408e7	−0.687602	−0.343801	−	0.939043i	$-0.611715\pi$
$881$	−1.58408e7	−0.687602	−0.343801	+	0.939043i	$0.611715\pi$
$882$	0	0
$883$	−2.18500e6	−0.0943083	−0.0471542	−	0.998888i	$-0.515015\pi$
$883$	−2.18500e6	−0.0943083	−0.0471542	+	0.998888i	$0.515015\pi$
$884$	0	0
$885$	−918271.	−0.0394105
$886$	0	0
$887$	−3.25170e7	−1.38772	−0.693860	−	0.720110i	$-0.744092\pi$
$887$	−3.25170e7	−1.38772	−0.693860	+	0.720110i	$0.744092\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.17780e6	−0.0497026
$892$	0	0
$893$	7.10669e6	0.298221
$894$	0	0
$895$	−3.82944e7	−1.59800
$896$	0	0
$897$	−2.38973e6	−0.0991673
$898$	0	0
$899$	3.72448e7	1.53697
$900$	0	0
$901$	4.18537e7	1.71760
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.58260e7	1.04818
$906$	0	0
$907$	−3.42544e6	−0.138261	−0.0691303	−	0.997608i	$-0.522022\pi$
$907$	−3.42544e6	−0.138261	−0.0691303	+	0.997608i	$0.522022\pi$
$908$	0	0
$909$	1.03950e6	0.0417269
$910$	0	0
$911$	6.40379e6	0.255647	0.127824	−	0.991797i	$-0.459201\pi$
$911$	6.40379e6	0.255647	0.127824	+	0.991797i	$0.459201\pi$
$912$	0	0
$913$	−9.55411e6	−0.379327
$914$	0	0
$915$	−8.23087e6	−0.325007
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−6.40524e6	−0.250176	−0.125088	−	0.992146i	$-0.539921\pi$
$919$	−6.40524e6	−0.250176	−0.125088	+	0.992146i	$0.539921\pi$
$920$	0	0
$921$	1.49588e7	0.581098
$922$	0	0
$923$	−2.83300e6	−0.109457
$924$	0	0
$925$	−5.59943e7	−2.15174
$926$	0	0
$927$	1.81192e6	0.0692532
$928$	0	0
$929$	1.93314e7	0.734892	0.367446	−	0.930045i	$-0.380232\pi$
$929$	1.93314e7	0.734892	0.367446	+	0.930045i	$0.380232\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.71569e7	0.645259
$934$	0	0
$935$	−2.45174e7	−0.917161
$936$	0	0
$937$	−4.35617e7	−1.62090	−0.810449	−	0.585809i	$-0.800777\pi$
$937$	−4.35617e7	−1.62090	−0.810449	+	0.585809i	$0.800777\pi$
$938$	0	0
$939$	1.20377e7	0.445531
$940$	0	0
$941$	1.43051e7	0.526644	0.263322	−	0.964708i	$-0.415182\pi$
$941$	1.43051e7	0.526644	0.263322	+	0.964708i	$0.415182\pi$
$942$	0	0
$943$	−3.09452e7	−1.13322
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.14953e7	−1.50357	−0.751785	−	0.659408i	$-0.770807\pi$
$947$	−4.14953e7	−1.50357	−0.751785	+	0.659408i	$0.770807\pi$
$948$	0	0
$949$	4.30873e6	0.155304
$950$	0	0
$951$	3.01353e7	1.08050
$952$	0	0
$953$	3.74252e7	1.33485	0.667424	−	0.744678i	$-0.267397\pi$
$953$	3.74252e7	1.33485	0.667424	+	0.744678i	$0.267397\pi$
$954$	0	0
$955$	−5.74114e7	−2.03699
$956$	0	0
$957$	−8.08322e6	−0.285302
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.67892e7	0.935731
$962$	0	0
$963$	5.91795e6	0.205639
$964$	0	0
$965$	8.67927e7	3.00030
$966$	0	0
$967$	4.22935e7	1.45448	0.727240	−	0.686383i	$-0.240803\pi$
$967$	4.22935e7	1.45448	0.727240	+	0.686383i	$0.240803\pi$
$968$	0	0
$969$	1.64832e7	0.563939
$970$	0	0
$971$	−2.51595e7	−0.856355	−0.428178	−	0.903695i	$-0.640844\pi$
$971$	−2.51595e7	−0.856355	−0.428178	+	0.903695i	$0.640844\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.15629e6	0.106332
$976$	0	0
$977$	−3.40511e7	−1.14129	−0.570643	−	0.821198i	$-0.693306\pi$
$977$	−3.40511e7	−1.14129	−0.570643	+	0.821198i	$0.693306\pi$
$978$	0	0
$979$	2.13053e7	0.710446
$980$	0	0
$981$	2.22354e6	0.0737689
$982$	0	0
$983$	2.51350e7	0.829649	0.414824	−	0.909902i	$-0.363843\pi$
$983$	2.51350e7	0.829649	0.414824	+	0.909902i	$0.363843\pi$
$984$	0	0
$985$	4.66347e7	1.53150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.03395e7	1.96160
$990$	0	0
$991$	2.38236e7	0.770589	0.385294	−	0.922794i	$-0.374100\pi$
$991$	2.38236e7	0.770589	0.385294	+	0.922794i	$0.374100\pi$
$992$	0	0
$993$	1.44394e7	0.464703
$994$	0	0
$995$	6.24235e7	1.99890
$996$	0	0
$997$	9.07764e6	0.289224	0.144612	−	0.989488i	$-0.453807\pi$
$997$	9.07764e6	0.289224	0.144612	+	0.989488i	$0.453807\pi$
$998$	0	0
$999$	6.61998e6	0.209866

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.a.i.1.2	✓	2
7.2	even	3		588.6.i.k.361.1		4
7.3	odd	6		588.6.i.j.373.2		4
7.4	even	3		588.6.i.k.373.1		4
7.5	odd	6		588.6.i.j.361.2		4
7.6	odd	2		588.6.a.j.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.6.a.i.1.2	✓	2	1.1	even	1	trivial
588.6.a.j.1.1	yes	2	7.6	odd	2
588.6.i.j.361.2		4	7.5	odd	6
588.6.i.j.373.2		4	7.3	odd	6
588.6.i.k.361.1		4	7.2	even	3
588.6.i.k.373.1		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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	588.6.a.i.1.2

Level	$588$
Weight	$6$
Character	588.1
Self dual	yes
Analytic conductor	$94.306$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$