LMFDB - Embedded newform 624.6.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,6,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("624.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.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:	$100.079503563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -61.4166 q^{5} +46.3165 q^{7} +81.0000 q^{9} -16.2992 q^{11} -169.000 q^{13} -552.749 q^{15} +659.231 q^{17} -17.7180 q^{19} +416.848 q^{21} +2141.20 q^{23} +646.996 q^{25} +729.000 q^{27} -3177.40 q^{29} -4033.87 q^{31} -146.693 q^{33} -2844.60 q^{35} +14286.6 q^{37} -1521.00 q^{39} -17898.8 q^{41} +3558.19 q^{43} -4974.74 q^{45} +25247.9 q^{47} -14661.8 q^{49} +5933.08 q^{51} -5306.59 q^{53} +1001.04 q^{55} -159.462 q^{57} +36537.0 q^{59} -6510.90 q^{61} +3751.63 q^{63} +10379.4 q^{65} -41120.1 q^{67} +19270.8 q^{69} -20914.4 q^{71} -71088.4 q^{73} +5822.96 q^{75} -754.922 q^{77} -49177.9 q^{79} +6561.00 q^{81} -99939.9 q^{83} -40487.7 q^{85} -28596.6 q^{87} +47291.4 q^{89} -7827.48 q^{91} -36304.8 q^{93} +1088.18 q^{95} -39729.5 q^{97} -1320.24 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} - 48 q^{5} - 72 q^{7} + 162 q^{9} + 596 q^{11} - 338 q^{13} - 432 q^{15} - 268 q^{17} - 1128 q^{19} - 648 q^{21} + 1768 q^{23} - 2298 q^{25} + 1458 q^{27} - 7612 q^{29} + 4160 q^{31} + 5364 q^{33}+ \cdots + 48276 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 - 48 * q^5 - 72 * q^7 + 162 * q^9 + 596 * q^11 - 338 * q^13 - 432 * q^15 - 268 * q^17 - 1128 * q^19 - 648 * q^21 + 1768 * q^23 - 2298 * q^25 + 1458 * q^27 - 7612 * q^29 + 4160 * q^31 + 5364 * q^33 - 4432 * q^35 + 17468 * q^37 - 3042 * q^39 - 28000 * q^41 + 24328 * q^43 - 3888 * q^45 + 18108 * q^47 - 17470 * q^49 - 2412 * q^51 + 1420 * q^53 + 9216 * q^55 - 10152 * q^57 - 6788 * q^59 - 37148 * q^61 - 5832 * q^63 + 8112 * q^65 - 106112 * q^67 + 15912 * q^69 + 30460 * q^71 - 37620 * q^73 - 20682 * q^75 - 73200 * q^77 + 2160 * q^79 + 13122 * q^81 - 207004 * q^83 - 52928 * q^85 - 68508 * q^87 - 74136 * q^89 + 12168 * q^91 + 37440 * q^93 - 13808 * q^95 - 121156 * q^97 + 48276 * 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
$5$	−61.4166	−1.09865	−0.549327	−	0.835608i	$-0.685116\pi$
$5$	−61.4166	−1.09865	−0.549327	+	0.835608i	$0.685116\pi$
$6$	0	0
$7$	46.3165	0.357265	0.178632	−	0.983916i	$-0.442833\pi$
$7$	46.3165	0.357265	0.178632	+	0.983916i	$0.442833\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−16.2992	−0.0406149	−0.0203074	−	0.999794i	$-0.506465\pi$
$11$	−16.2992	−0.0406149	−0.0203074	+	0.999794i	$0.506465\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	−552.749	−0.634308
$16$	0	0
$17$	659.231	0.553243	0.276621	−	0.960979i	$-0.410785\pi$
$17$	659.231	0.553243	0.276621	+	0.960979i	$0.410785\pi$
$18$	0	0
$19$	−17.7180	−0.0112598	−0.00562991	−	0.999984i	$-0.501792\pi$
$19$	−17.7180	−0.0112598	−0.00562991	+	0.999984i	$0.501792\pi$
$20$	0	0
$21$	416.848	0.206267
$22$	0	0
$23$	2141.20	0.843989	0.421995	−	0.906598i	$-0.361330\pi$
$23$	2141.20	0.843989	0.421995	+	0.906598i	$0.361330\pi$
$24$	0	0
$25$	646.996	0.207039
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−3177.40	−0.701580	−0.350790	−	0.936454i	$-0.614087\pi$
$29$	−3177.40	−0.701580	−0.350790	+	0.936454i	$0.614087\pi$
$30$	0	0
$31$	−4033.87	−0.753906	−0.376953	−	0.926232i	$-0.623028\pi$
$31$	−4033.87	−0.753906	−0.376953	+	0.926232i	$0.623028\pi$
$32$	0	0
$33$	−146.693	−0.0234490
$34$	0	0
$35$	−2844.60	−0.392510
$36$	0	0
$37$	14286.6	1.71564	0.857818	−	0.513954i	$-0.171820\pi$
$37$	14286.6	1.71564	0.857818	+	0.513954i	$0.171820\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	−17898.8	−1.66289	−0.831447	−	0.555604i	$-0.812487\pi$
$41$	−17898.8	−1.66289	−0.831447	+	0.555604i	$0.812487\pi$
$42$	0	0
$43$	3558.19	0.293466	0.146733	−	0.989176i	$-0.453124\pi$
$43$	3558.19	0.293466	0.146733	+	0.989176i	$0.453124\pi$
$44$	0	0
$45$	−4974.74	−0.366218
$46$	0	0
$47$	25247.9	1.66717	0.833586	−	0.552389i	$-0.186284\pi$
$47$	25247.9	1.66717	0.833586	+	0.552389i	$0.186284\pi$
$48$	0	0
$49$	−14661.8	−0.872362
$50$	0	0
$51$	5933.08	0.319415
$52$	0	0
$53$	−5306.59	−0.259493	−0.129746	−	0.991547i	$-0.541416\pi$
$53$	−5306.59	−0.259493	−0.129746	+	0.991547i	$0.541416\pi$
$54$	0	0
$55$	1001.04	0.0446217
$56$	0	0
$57$	−159.462	−0.00650086
$58$	0	0
$59$	36537.0	1.36648	0.683239	−	0.730195i	$-0.260571\pi$
$59$	36537.0	1.36648	0.683239	+	0.730195i	$0.260571\pi$
$60$	0	0
$61$	−6510.90	−0.224035	−0.112018	−	0.993706i	$-0.535731\pi$
$61$	−6510.90	−0.224035	−0.112018	+	0.993706i	$0.535731\pi$
$62$	0	0
$63$	3751.63	0.119088
$64$	0	0
$65$	10379.4	0.304712
$66$	0	0
$67$	−41120.1	−1.11910	−0.559548	−	0.828798i	$-0.689025\pi$
$67$	−41120.1	−1.11910	−0.559548	+	0.828798i	$0.689025\pi$
$68$	0	0
$69$	19270.8	0.487278
$70$	0	0
$71$	−20914.4	−0.492379	−0.246190	−	0.969222i	$-0.579179\pi$
$71$	−20914.4	−0.492379	−0.246190	+	0.969222i	$0.579179\pi$
$72$	0	0
$73$	−71088.4	−1.56132	−0.780660	−	0.624956i	$-0.785117\pi$
$73$	−71088.4	−1.56132	−0.780660	+	0.624956i	$0.785117\pi$
$74$	0	0
$75$	5822.96	0.119534
$76$	0	0
$77$	−754.922	−0.0145103
$78$	0	0
$79$	−49177.9	−0.886549	−0.443274	−	0.896386i	$-0.646183\pi$
$79$	−49177.9	−0.886549	−0.443274	+	0.896386i	$0.646183\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−99939.9	−1.59237	−0.796185	−	0.605054i	$-0.793152\pi$
$83$	−99939.9	−1.59237	−0.796185	+	0.605054i	$0.793152\pi$
$84$	0	0
$85$	−40487.7	−0.607822
$86$	0	0
$87$	−28596.6	−0.405057
$88$	0	0
$89$	47291.4	0.632859	0.316430	−	0.948616i	$-0.397516\pi$
$89$	47291.4	0.632859	0.316430	+	0.948616i	$0.397516\pi$
$90$	0	0
$91$	−7827.48	−0.0990874
$92$	0	0
$93$	−36304.8	−0.435268
$94$	0	0
$95$	1088.18	0.0123706
$96$	0	0
$97$	−39729.5	−0.428730	−0.214365	−	0.976754i	$-0.568768\pi$
$97$	−39729.5	−0.428730	−0.214365	+	0.976754i	$0.568768\pi$
$98$	0	0
$99$	−1320.24	−0.0135383
$100$	0	0
$101$	−75862.3	−0.739984	−0.369992	−	0.929035i	$-0.620640\pi$
$101$	−75862.3	−0.739984	−0.369992	+	0.929035i	$0.620640\pi$
$102$	0	0
$103$	−133421.	−1.23917	−0.619586	−	0.784928i	$-0.712700\pi$
$103$	−133421.	−1.23917	−0.619586	+	0.784928i	$0.712700\pi$
$104$	0	0
$105$	−25601.4	−0.226616
$106$	0	0
$107$	2628.95	0.0221985	0.0110992	−	0.999938i	$-0.496467\pi$
$107$	2628.95	0.0221985	0.0110992	+	0.999938i	$0.496467\pi$
$108$	0	0
$109$	189286.	1.52599	0.762997	−	0.646401i	$-0.223727\pi$
$109$	189286.	1.52599	0.762997	+	0.646401i	$0.223727\pi$
$110$	0	0
$111$	128580.	0.990523
$112$	0	0
$113$	−151984.	−1.11970	−0.559848	−	0.828595i	$-0.689141\pi$
$113$	−151984.	−1.11970	−0.559848	+	0.828595i	$0.689141\pi$
$114$	0	0
$115$	−131505.	−0.927252
$116$	0	0
$117$	−13689.0	−0.0924500
$118$	0	0
$119$	30533.3	0.197654
$120$	0	0
$121$	−160785.	−0.998350
$122$	0	0
$123$	−161089.	−0.960072
$124$	0	0
$125$	152191.	0.871190
$126$	0	0
$127$	−254370.	−1.39945	−0.699724	−	0.714414i	$-0.746694\pi$
$127$	−254370.	−1.39945	−0.699724	+	0.714414i	$0.746694\pi$
$128$	0	0
$129$	32023.7	0.169433
$130$	0	0
$131$	254479.	1.29561	0.647804	−	0.761807i	$-0.275687\pi$
$131$	254479.	1.29561	0.647804	+	0.761807i	$0.275687\pi$
$132$	0	0
$133$	−820.636	−0.00402274
$134$	0	0
$135$	−44772.7	−0.211436
$136$	0	0
$137$	−248583.	−1.13154	−0.565769	−	0.824564i	$-0.691421\pi$
$137$	−248583.	−1.13154	−0.565769	+	0.824564i	$0.691421\pi$
$138$	0	0
$139$	−42062.4	−0.184653	−0.0923267	−	0.995729i	$-0.529430\pi$
$139$	−42062.4	−0.184653	−0.0923267	+	0.995729i	$0.529430\pi$
$140$	0	0
$141$	227231.	0.962542
$142$	0	0
$143$	2754.57	0.0112645
$144$	0	0
$145$	195145.	0.770793
$146$	0	0
$147$	−131956.	−0.503658
$148$	0	0
$149$	183520.	0.677200	0.338600	−	0.940930i	$-0.390047\pi$
$149$	183520.	0.677200	0.338600	+	0.940930i	$0.390047\pi$
$150$	0	0
$151$	−9544.14	−0.0340639	−0.0170319	−	0.999855i	$-0.505422\pi$
$151$	−9544.14	−0.0340639	−0.0170319	+	0.999855i	$0.505422\pi$
$152$	0	0
$153$	53397.7	0.184414
$154$	0	0
$155$	247746.	0.828282
$156$	0	0
$157$	−250307.	−0.810445	−0.405223	−	0.914218i	$-0.632806\pi$
$157$	−250307.	−0.810445	−0.405223	+	0.914218i	$0.632806\pi$
$158$	0	0
$159$	−47759.3	−0.149818
$160$	0	0
$161$	99172.7	0.301528
$162$	0	0
$163$	219671.	0.647595	0.323798	−	0.946126i	$-0.395040\pi$
$163$	219671.	0.647595	0.323798	+	0.946126i	$0.395040\pi$
$164$	0	0
$165$	9009.38	0.0257623
$166$	0	0
$167$	−92356.1	−0.256256	−0.128128	−	0.991758i	$-0.540897\pi$
$167$	−92356.1	−0.256256	−0.128128	+	0.991758i	$0.540897\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−1435.16	−0.00375327
$172$	0	0
$173$	−236712.	−0.601319	−0.300660	−	0.953731i	$-0.597207\pi$
$173$	−236712.	−0.601319	−0.300660	+	0.953731i	$0.597207\pi$
$174$	0	0
$175$	29966.5	0.0739676
$176$	0	0
$177$	328833.	0.788936
$178$	0	0
$179$	−525225.	−1.22522	−0.612608	−	0.790387i	$-0.709880\pi$
$179$	−525225.	−1.22522	−0.612608	+	0.790387i	$0.709880\pi$
$180$	0	0
$181$	−784923.	−1.78086	−0.890432	−	0.455116i	$-0.849598\pi$
$181$	−784923.	−1.78086	−0.890432	+	0.455116i	$0.849598\pi$
$182$	0	0
$183$	−58598.1	−0.129347
$184$	0	0
$185$	−877435.	−1.88489
$186$	0	0
$187$	−10745.0	−0.0224699
$188$	0	0
$189$	33764.7	0.0687557
$190$	0	0
$191$	571365.	1.13326	0.566630	−	0.823972i	$-0.308247\pi$
$191$	571365.	1.13326	0.566630	+	0.823972i	$0.308247\pi$
$192$	0	0
$193$	−726656.	−1.40422	−0.702111	−	0.712068i	$-0.747759\pi$
$193$	−726656.	−1.40422	−0.702111	+	0.712068i	$0.747759\pi$
$194$	0	0
$195$	93414.6	0.175925
$196$	0	0
$197$	48808.7	0.0896049	0.0448025	−	0.998996i	$-0.485734\pi$
$197$	48808.7	0.0896049	0.0448025	+	0.998996i	$0.485734\pi$
$198$	0	0
$199$	1.04968e6	1.87898	0.939491	−	0.342574i	$-0.111299\pi$
$199$	1.04968e6	1.87898	0.939491	+	0.342574i	$0.111299\pi$
$200$	0	0
$201$	−370081.	−0.646110
$202$	0	0
$203$	−147166.	−0.250650
$204$	0	0
$205$	1.09928e6	1.82694
$206$	0	0
$207$	173437.	0.281330
$208$	0	0
$209$	288.790	0.000457316 0
$210$	0	0
$211$	−431486.	−0.667206	−0.333603	−	0.942714i	$-0.608265\pi$
$211$	−431486.	−0.667206	−0.333603	+	0.942714i	$0.608265\pi$
$212$	0	0
$213$	−188230.	−0.284275
$214$	0	0
$215$	−218532.	−0.322417
$216$	0	0
$217$	−186835.	−0.269344
$218$	0	0
$219$	−639796.	−0.901428
$220$	0	0
$221$	−111410.	−0.153442
$222$	0	0
$223$	−1.41878e6	−1.91052	−0.955261	−	0.295765i	$-0.904425\pi$
$223$	−1.41878e6	−1.91052	−0.955261	+	0.295765i	$0.904425\pi$
$224$	0	0
$225$	52406.6	0.0690129
$226$	0	0
$227$	−261098.	−0.336309	−0.168154	−	0.985761i	$-0.553781\pi$
$227$	−261098.	−0.336309	−0.168154	+	0.985761i	$0.553781\pi$
$228$	0	0
$229$	−76335.3	−0.0961915	−0.0480957	−	0.998843i	$-0.515315\pi$
$229$	−76335.3	−0.0961915	−0.0480957	+	0.998843i	$0.515315\pi$
$230$	0	0
$231$	−6794.30	−0.00837751
$232$	0	0
$233$	614891.	0.742008	0.371004	−	0.928631i	$-0.379014\pi$
$233$	614891.	0.742008	0.371004	+	0.928631i	$0.379014\pi$
$234$	0	0
$235$	−1.55064e6	−1.83164
$236$	0	0
$237$	−442601.	−0.511849
$238$	0	0
$239$	−241610.	−0.273602	−0.136801	−	0.990599i	$-0.543682\pi$
$239$	−241610.	−0.273602	−0.136801	+	0.990599i	$0.543682\pi$
$240$	0	0
$241$	−906551.	−1.00543	−0.502713	−	0.864454i	$-0.667665\pi$
$241$	−906551.	−1.00543	−0.502713	+	0.864454i	$0.667665\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	900477.	0.958423
$246$	0	0
$247$	2994.35	0.00312291
$248$	0	0
$249$	−899459.	−0.919355
$250$	0	0
$251$	−1.55237e6	−1.55529	−0.777643	−	0.628706i	$-0.783585\pi$
$251$	−1.55237e6	−1.55529	−0.777643	+	0.628706i	$0.783585\pi$
$252$	0	0
$253$	−34899.8	−0.0342785
$254$	0	0
$255$	−364390.	−0.350926
$256$	0	0
$257$	−1.37303e6	−1.29672	−0.648361	−	0.761333i	$-0.724545\pi$
$257$	−1.37303e6	−1.29672	−0.648361	+	0.761333i	$0.724545\pi$
$258$	0	0
$259$	661706.	0.612936
$260$	0	0
$261$	−257370.	−0.233860
$262$	0	0
$263$	457871.	0.408182	0.204091	−	0.978952i	$-0.434576\pi$
$263$	457871.	0.408182	0.204091	+	0.978952i	$0.434576\pi$
$264$	0	0
$265$	325912.	0.285093
$266$	0	0
$267$	425623.	0.365382
$268$	0	0
$269$	−1.91682e6	−1.61511	−0.807553	−	0.589795i	$-0.799209\pi$
$269$	−1.91682e6	−1.61511	−0.807553	+	0.589795i	$0.799209\pi$
$270$	0	0
$271$	1.81342e6	1.49995	0.749973	−	0.661469i	$-0.230067\pi$
$271$	1.81342e6	1.49995	0.749973	+	0.661469i	$0.230067\pi$
$272$	0	0
$273$	−70447.3	−0.0572082
$274$	0	0
$275$	−10545.5	−0.00840885
$276$	0	0
$277$	−792085.	−0.620258	−0.310129	−	0.950694i	$-0.600372\pi$
$277$	−792085.	−0.620258	−0.310129	+	0.950694i	$0.600372\pi$
$278$	0	0
$279$	−326743.	−0.251302
$280$	0	0
$281$	−90711.0	−0.0685321	−0.0342661	−	0.999413i	$-0.510909\pi$
$281$	−90711.0	−0.0685321	−0.0342661	+	0.999413i	$0.510909\pi$
$282$	0	0
$283$	809381.	0.600741	0.300370	−	0.953823i	$-0.402890\pi$
$283$	809381.	0.600741	0.300370	+	0.953823i	$0.402890\pi$
$284$	0	0
$285$	9793.62	0.00714219
$286$	0	0
$287$	−829009.	−0.594093
$288$	0	0
$289$	−985271.	−0.693923
$290$	0	0
$291$	−357565.	−0.247527
$292$	0	0
$293$	1.22134e6	0.831125	0.415562	−	0.909565i	$-0.363585\pi$
$293$	1.22134e6	0.831125	0.415562	+	0.909565i	$0.363585\pi$
$294$	0	0
$295$	−2.24398e6	−1.50128
$296$	0	0
$297$	−11882.1	−0.00781634
$298$	0	0
$299$	−361862.	−0.234081
$300$	0	0
$301$	164803.	0.104845
$302$	0	0
$303$	−682760.	−0.427230
$304$	0	0
$305$	399877.	0.246137
$306$	0	0
$307$	−385603.	−0.233504	−0.116752	−	0.993161i	$-0.537248\pi$
$307$	−385603.	−0.233504	−0.116752	+	0.993161i	$0.537248\pi$
$308$	0	0
$309$	−1.20079e6	−0.715437
$310$	0	0
$311$	1.14737e6	0.672670	0.336335	−	0.941742i	$-0.390813\pi$
$311$	1.14737e6	0.672670	0.336335	+	0.941742i	$0.390813\pi$
$312$	0	0
$313$	3.13172e6	1.80685	0.903425	−	0.428746i	$-0.141044\pi$
$313$	3.13172e6	1.80685	0.903425	+	0.428746i	$0.141044\pi$
$314$	0	0
$315$	−230412.	−0.130837
$316$	0	0
$317$	867095.	0.484639	0.242320	−	0.970196i	$-0.422092\pi$
$317$	867095.	0.484639	0.242320	+	0.970196i	$0.422092\pi$
$318$	0	0
$319$	51789.2	0.0284946
$320$	0	0
$321$	23660.6	0.0128163
$322$	0	0
$323$	−11680.3	−0.00622941
$324$	0	0
$325$	−109342.	−0.0574222
$326$	0	0
$327$	1.70358e6	0.881034
$328$	0	0
$329$	1.16939e6	0.595622
$330$	0	0
$331$	−454460.	−0.227995	−0.113998	−	0.993481i	$-0.536366\pi$
$331$	−454460.	−0.227995	−0.113998	+	0.993481i	$0.536366\pi$
$332$	0	0
$333$	1.15722e6	0.571879
$334$	0	0
$335$	2.52546e6	1.22950
$336$	0	0
$337$	3.55249e6	1.70396	0.851978	−	0.523577i	$-0.175403\pi$
$337$	3.55249e6	1.70396	0.851978	+	0.523577i	$0.175403\pi$
$338$	0	0
$339$	−1.36785e6	−0.646457
$340$	0	0
$341$	65748.9	0.0306198
$342$	0	0
$343$	−1.45752e6	−0.668929
$344$	0	0
$345$	−1.18354e6	−0.535349
$346$	0	0
$347$	1.52993e6	0.682102	0.341051	−	0.940045i	$-0.389217\pi$
$347$	1.52993e6	0.682102	0.341051	+	0.940045i	$0.389217\pi$
$348$	0	0
$349$	−1.45386e6	−0.638939	−0.319469	−	0.947597i	$-0.603505\pi$
$349$	−1.45386e6	−0.638939	−0.319469	+	0.947597i	$0.603505\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	1.73292e6	0.740189	0.370095	−	0.928994i	$-0.379325\pi$
$353$	1.73292e6	0.740189	0.370095	+	0.928994i	$0.379325\pi$
$354$	0	0
$355$	1.28449e6	0.540954
$356$	0	0
$357$	274799.	0.114116
$358$	0	0
$359$	1.66707e6	0.682683	0.341341	−	0.939939i	$-0.389119\pi$
$359$	1.66707e6	0.682683	0.341341	+	0.939939i	$0.389119\pi$
$360$	0	0
$361$	−2.47579e6	−0.999873
$362$	0	0
$363$	−1.44707e6	−0.576398
$364$	0	0
$365$	4.36601e6	1.71535
$366$	0	0
$367$	−2.97554e6	−1.15319	−0.576595	−	0.817030i	$-0.695619\pi$
$367$	−2.97554e6	−1.15319	−0.576595	+	0.817030i	$0.695619\pi$
$368$	0	0
$369$	−1.44980e6	−0.554298
$370$	0	0
$371$	−245782.	−0.0927077
$372$	0	0
$373$	−51413.5	−0.0191340	−0.00956698	−	0.999954i	$-0.503045\pi$
$373$	−51413.5	−0.0191340	−0.00956698	+	0.999954i	$0.503045\pi$
$374$	0	0
$375$	1.36971e6	0.502981
$376$	0	0
$377$	536981.	0.194583
$378$	0	0
$379$	3.85498e6	1.37856	0.689278	−	0.724497i	$-0.257928\pi$
$379$	3.85498e6	1.37856	0.689278	+	0.724497i	$0.257928\pi$
$380$	0	0
$381$	−2.28933e6	−0.807971
$382$	0	0
$383$	−3.13730e6	−1.09285	−0.546423	−	0.837510i	$-0.684011\pi$
$383$	−3.13730e6	−1.09285	−0.546423	+	0.837510i	$0.684011\pi$
$384$	0	0
$385$	46364.7	0.0159417
$386$	0	0
$387$	288213.	0.0978220
$388$	0	0
$389$	1.85567e6	0.621766	0.310883	−	0.950448i	$-0.399375\pi$
$389$	1.85567e6	0.621766	0.310883	+	0.950448i	$0.399375\pi$
$390$	0	0
$391$	1.41154e6	0.466931
$392$	0	0
$393$	2.29031e6	0.748020
$394$	0	0
$395$	3.02034e6	0.974010
$396$	0	0
$397$	3.91561e6	1.24687	0.623437	−	0.781873i	$-0.285736\pi$
$397$	3.91561e6	1.24687	0.623437	+	0.781873i	$0.285736\pi$
$398$	0	0
$399$	−7385.72	−0.00232253
$400$	0	0
$401$	318965.	0.0990562	0.0495281	−	0.998773i	$-0.484228\pi$
$401$	318965.	0.0990562	0.0495281	+	0.998773i	$0.484228\pi$
$402$	0	0
$403$	681724.	0.209096
$404$	0	0
$405$	−402954.	−0.122073
$406$	0	0
$407$	−232861.	−0.0696803
$408$	0	0
$409$	−3.37547e6	−0.997760	−0.498880	−	0.866671i	$-0.666255\pi$
$409$	−3.37547e6	−0.997760	−0.498880	+	0.866671i	$0.666255\pi$
$410$	0	0
$411$	−2.23724e6	−0.653294
$412$	0	0
$413$	1.69226e6	0.488194
$414$	0	0
$415$	6.13797e6	1.74946
$416$	0	0
$417$	−378562.	−0.106610
$418$	0	0
$419$	1.18257e6	0.329072	0.164536	−	0.986371i	$-0.447387\pi$
$419$	1.18257e6	0.329072	0.164536	+	0.986371i	$0.447387\pi$
$420$	0	0
$421$	2.06389e6	0.567521	0.283761	−	0.958895i	$-0.408418\pi$
$421$	2.06389e6	0.567521	0.283761	+	0.958895i	$0.408418\pi$
$422$	0	0
$423$	2.04508e6	0.555724
$424$	0	0
$425$	426520.	0.114543
$426$	0	0
$427$	−301562.	−0.0800399
$428$	0	0
$429$	24791.1	0.00650358
$430$	0	0
$431$	5.94590e6	1.54179	0.770893	−	0.636965i	$-0.219810\pi$
$431$	5.94590e6	1.54179	0.770893	+	0.636965i	$0.219810\pi$
$432$	0	0
$433$	−6.20053e6	−1.58931	−0.794655	−	0.607061i	$-0.792348\pi$
$433$	−6.20053e6	−1.58931	−0.794655	+	0.607061i	$0.792348\pi$
$434$	0	0
$435$	1.75631e6	0.445017
$436$	0	0
$437$	−37937.8	−0.00950316
$438$	0	0
$439$	2.81407e6	0.696906	0.348453	−	0.937326i	$-0.386707\pi$
$439$	2.81407e6	0.696906	0.348453	+	0.937326i	$0.386707\pi$
$440$	0	0
$441$	−1.18760e6	−0.290787
$442$	0	0
$443$	−6.68727e6	−1.61897	−0.809486	−	0.587139i	$-0.800254\pi$
$443$	−6.68727e6	−1.61897	−0.809486	+	0.587139i	$0.800254\pi$
$444$	0	0
$445$	−2.90448e6	−0.695293
$446$	0	0
$447$	1.65168e6	0.390982
$448$	0	0
$449$	7.64990e6	1.79077	0.895385	−	0.445294i	$-0.146901\pi$
$449$	7.64990e6	1.79077	0.895385	+	0.445294i	$0.146901\pi$
$450$	0	0
$451$	291737.	0.0675382
$452$	0	0
$453$	−85897.2	−0.0196668
$454$	0	0
$455$	480737.	0.108863
$456$	0	0
$457$	−2.67357e6	−0.598826	−0.299413	−	0.954124i	$-0.596791\pi$
$457$	−2.67357e6	−0.598826	−0.299413	+	0.954124i	$0.596791\pi$
$458$	0	0
$459$	480580.	0.106472
$460$	0	0
$461$	1.32953e6	0.291370	0.145685	−	0.989331i	$-0.453461\pi$
$461$	1.32953e6	0.291370	0.145685	+	0.989331i	$0.453461\pi$
$462$	0	0
$463$	−5.26984e6	−1.14247	−0.571235	−	0.820786i	$-0.693536\pi$
$463$	−5.26984e6	−1.14247	−0.571235	+	0.820786i	$0.693536\pi$
$464$	0	0
$465$	2.22972e6	0.478209
$466$	0	0
$467$	7.33908e6	1.55722	0.778609	−	0.627510i	$-0.215926\pi$
$467$	7.33908e6	1.55722	0.778609	+	0.627510i	$0.215926\pi$
$468$	0	0
$469$	−1.90454e6	−0.399814
$470$	0	0
$471$	−2.25276e6	−0.467911
$472$	0	0
$473$	−57995.7	−0.0119191
$474$	0	0
$475$	−11463.5	−0.00233122
$476$	0	0
$477$	−429833.	−0.0864976
$478$	0	0
$479$	1.52236e6	0.303165	0.151582	−	0.988445i	$-0.451563\pi$
$479$	1.52236e6	0.303165	0.151582	+	0.988445i	$0.451563\pi$
$480$	0	0
$481$	−2.41444e6	−0.475832
$482$	0	0
$483$	892554.	0.174087
$484$	0	0
$485$	2.44005e6	0.471025
$486$	0	0
$487$	−8.83569e6	−1.68818	−0.844088	−	0.536204i	$-0.819858\pi$
$487$	−8.83569e6	−1.68818	−0.844088	+	0.536204i	$0.819858\pi$
$488$	0	0
$489$	1.97704e6	0.373889
$490$	0	0
$491$	−175648.	−0.0328805	−0.0164403	−	0.999865i	$-0.505233\pi$
$491$	−175648.	−0.0328805	−0.0164403	+	0.999865i	$0.505233\pi$
$492$	0	0
$493$	−2.09464e6	−0.388144
$494$	0	0
$495$	81084.4	0.0148739
$496$	0	0
$497$	−968682.	−0.175910
$498$	0	0
$499$	2.82489e6	0.507867	0.253934	−	0.967222i	$-0.418276\pi$
$499$	2.82489e6	0.507867	0.253934	+	0.967222i	$0.418276\pi$
$500$	0	0
$501$	−831205.	−0.147950
$502$	0	0
$503$	9.11142e6	1.60571	0.802854	−	0.596176i	$-0.203314\pi$
$503$	9.11142e6	1.60571	0.802854	+	0.596176i	$0.203314\pi$
$504$	0	0
$505$	4.65920e6	0.812986
$506$	0	0
$507$	257049.	0.0444116
$508$	0	0
$509$	4.05380e6	0.693534	0.346767	−	0.937951i	$-0.387280\pi$
$509$	4.05380e6	0.693534	0.346767	+	0.937951i	$0.387280\pi$
$510$	0	0
$511$	−3.29256e6	−0.557805
$512$	0	0
$513$	−12916.4	−0.00216695
$514$	0	0
$515$	8.19427e6	1.36142
$516$	0	0
$517$	−411521.	−0.0677120
$518$	0	0
$519$	−2.13041e6	−0.347172
$520$	0	0
$521$	−2.46630e6	−0.398062	−0.199031	−	0.979993i	$-0.563780\pi$
$521$	−2.46630e6	−0.398062	−0.199031	+	0.979993i	$0.563780\pi$
$522$	0	0
$523$	−3.47559e6	−0.555615	−0.277808	−	0.960637i	$-0.589608\pi$
$523$	−3.47559e6	−0.555615	−0.277808	+	0.960637i	$0.589608\pi$
$524$	0	0
$525$	269699.	0.0427052
$526$	0	0
$527$	−2.65925e6	−0.417093
$528$	0	0
$529$	−1.85162e6	−0.287682
$530$	0	0
$531$	2.95949e6	0.455492
$532$	0	0
$533$	3.02490e6	0.461204
$534$	0	0
$535$	−161461.	−0.0243884
$536$	0	0
$537$	−4.72702e6	−0.707379
$538$	0	0
$539$	238976.	0.0354309
$540$	0	0
$541$	7.20963e6	1.05906	0.529529	−	0.848292i	$-0.322369\pi$
$541$	7.20963e6	1.05906	0.529529	+	0.848292i	$0.322369\pi$
$542$	0	0
$543$	−7.06431e6	−1.02818
$544$	0	0
$545$	−1.16253e7	−1.67654
$546$	0	0
$547$	4.25767e6	0.608420	0.304210	−	0.952605i	$-0.401608\pi$
$547$	4.25767e6	0.608420	0.304210	+	0.952605i	$0.401608\pi$
$548$	0	0
$549$	−527383.	−0.0746784
$550$	0	0
$551$	56297.3	0.00789966
$552$	0	0
$553$	−2.27775e6	−0.316733
$554$	0	0
$555$	−7.89692e6	−1.08824
$556$	0	0
$557$	1.41670e7	1.93481	0.967405	−	0.253233i	$-0.0814938\pi$
$557$	1.41670e7	1.93481	0.967405	+	0.253233i	$0.0814938\pi$
$558$	0	0
$559$	−601334.	−0.0813928
$560$	0	0
$561$	−96704.6	−0.0129730
$562$	0	0
$563$	9.71182e6	1.29131	0.645654	−	0.763630i	$-0.276585\pi$
$563$	9.71182e6	1.29131	0.645654	+	0.763630i	$0.276585\pi$
$564$	0	0
$565$	9.33431e6	1.23016
$566$	0	0
$567$	303882.	0.0396961
$568$	0	0
$569$	1.11053e7	1.43796	0.718982	−	0.695029i	$-0.244608\pi$
$569$	1.11053e7	1.43796	0.718982	+	0.695029i	$0.244608\pi$
$570$	0	0
$571$	6.04167e6	0.775473	0.387737	−	0.921770i	$-0.373257\pi$
$571$	6.04167e6	0.775473	0.387737	+	0.921770i	$0.373257\pi$
$572$	0	0
$573$	5.14228e6	0.654288
$574$	0	0
$575$	1.38534e6	0.174738
$576$	0	0
$577$	1.19079e7	1.48900	0.744500	−	0.667623i	$-0.232688\pi$
$577$	1.19079e7	1.48900	0.744500	+	0.667623i	$0.232688\pi$
$578$	0	0
$579$	−6.53991e6	−0.810728
$580$	0	0
$581$	−4.62886e6	−0.568898
$582$	0	0
$583$	86493.2	0.0105393
$584$	0	0
$585$	840731.	0.101571
$586$	0	0
$587$	−6.16538e6	−0.738523	−0.369262	−	0.929325i	$-0.620389\pi$
$587$	−6.16538e6	−0.738523	−0.369262	+	0.929325i	$0.620389\pi$
$588$	0	0
$589$	71472.2	0.00848885
$590$	0	0
$591$	439278.	0.0517334
$592$	0	0
$593$	−2.25563e6	−0.263410	−0.131705	−	0.991289i	$-0.542045\pi$
$593$	−2.25563e6	−0.263410	−0.131705	+	0.991289i	$0.542045\pi$
$594$	0	0
$595$	−1.87525e6	−0.217153
$596$	0	0
$597$	9.44708e6	1.08483
$598$	0	0
$599$	4.27181e6	0.486458	0.243229	−	0.969969i	$-0.421793\pi$
$599$	4.27181e6	0.486458	0.243229	+	0.969969i	$0.421793\pi$
$600$	0	0
$601$	−4.97463e6	−0.561791	−0.280895	−	0.959738i	$-0.590631\pi$
$601$	−4.97463e6	−0.561791	−0.280895	+	0.959738i	$0.590631\pi$
$602$	0	0
$603$	−3.33073e6	−0.373032
$604$	0	0
$605$	9.87488e6	1.09684
$606$	0	0
$607$	−9.85476e6	−1.08561	−0.542806	−	0.839858i	$-0.682638\pi$
$607$	−9.85476e6	−1.08561	−0.542806	+	0.839858i	$0.682638\pi$
$608$	0	0
$609$	−1.32449e6	−0.144713
$610$	0	0
$611$	−4.26689e6	−0.462390
$612$	0	0
$613$	−9.84769e6	−1.05848	−0.529241	−	0.848472i	$-0.677523\pi$
$613$	−9.84769e6	−1.05848	−0.529241	+	0.848472i	$0.677523\pi$
$614$	0	0
$615$	9.89355e6	1.05479
$616$	0	0
$617$	−5.41851e6	−0.573016	−0.286508	−	0.958078i	$-0.592495\pi$
$617$	−5.41851e6	−0.573016	−0.286508	+	0.958078i	$0.592495\pi$
$618$	0	0
$619$	−6.15964e6	−0.646143	−0.323071	−	0.946375i	$-0.604715\pi$
$619$	−6.15964e6	−0.646143	−0.323071	+	0.946375i	$0.604715\pi$
$620$	0	0
$621$	1.56093e6	0.162426
$622$	0	0
$623$	2.19037e6	0.226098
$624$	0	0
$625$	−1.13689e7	−1.16417
$626$	0	0
$627$	2599.11	0.000264031 0
$628$	0	0
$629$	9.41819e6	0.949163
$630$	0	0
$631$	−1.92146e6	−0.192114	−0.0960569	−	0.995376i	$-0.530623\pi$
$631$	−1.92146e6	−0.192114	−0.0960569	+	0.995376i	$0.530623\pi$
$632$	0	0
$633$	−3.88337e6	−0.385212
$634$	0	0
$635$	1.56225e7	1.53751
$636$	0	0
$637$	2.47784e6	0.241950
$638$	0	0
$639$	−1.69407e6	−0.164126
$640$	0	0
$641$	−1.24396e7	−1.19581	−0.597904	−	0.801568i	$-0.704000\pi$
$641$	−1.24396e7	−1.19581	−0.597904	+	0.801568i	$0.704000\pi$
$642$	0	0
$643$	2.05927e6	0.196420	0.0982102	−	0.995166i	$-0.468688\pi$
$643$	2.05927e6	0.196420	0.0982102	+	0.995166i	$0.468688\pi$
$644$	0	0
$645$	−1.96679e6	−0.186148
$646$	0	0
$647$	8.23244e6	0.773158	0.386579	−	0.922256i	$-0.373657\pi$
$647$	8.23244e6	0.773158	0.386579	+	0.922256i	$0.373657\pi$
$648$	0	0
$649$	−595524.	−0.0554993
$650$	0	0
$651$	−1.68151e6	−0.155506
$652$	0	0
$653$	−1.66572e7	−1.52869	−0.764345	−	0.644807i	$-0.776938\pi$
$653$	−1.66572e7	−1.52869	−0.764345	+	0.644807i	$0.776938\pi$
$654$	0	0
$655$	−1.56292e7	−1.42342
$656$	0	0
$657$	−5.75816e6	−0.520440
$658$	0	0
$659$	−7.94491e6	−0.712649	−0.356324	−	0.934362i	$-0.615970\pi$
$659$	−7.94491e6	−0.712649	−0.356324	+	0.934362i	$0.615970\pi$
$660$	0	0
$661$	1.76591e7	1.57205	0.786024	−	0.618196i	$-0.212136\pi$
$661$	1.76591e7	1.57205	0.786024	+	0.618196i	$0.212136\pi$
$662$	0	0
$663$	−1.00269e6	−0.0885897
$664$	0	0
$665$	50400.7	0.00441959
$666$	0	0
$667$	−6.80344e6	−0.592126
$668$	0	0
$669$	−1.27690e7	−1.10304
$670$	0	0
$671$	106123.	0.00909916
$672$	0	0
$673$	−6.75780e6	−0.575132	−0.287566	−	0.957761i	$-0.592846\pi$
$673$	−6.75780e6	−0.575132	−0.287566	+	0.957761i	$0.592846\pi$
$674$	0	0
$675$	471660.	0.0398446
$676$	0	0
$677$	6.59598e6	0.553105	0.276552	−	0.960999i	$-0.410808\pi$
$677$	6.59598e6	0.553105	0.276552	+	0.960999i	$0.410808\pi$
$678$	0	0
$679$	−1.84013e6	−0.153170
$680$	0	0
$681$	−2.34988e6	−0.194168
$682$	0	0
$683$	1.33104e7	1.09179	0.545897	−	0.837852i	$-0.316189\pi$
$683$	1.33104e7	1.09179	0.545897	+	0.837852i	$0.316189\pi$
$684$	0	0
$685$	1.52671e7	1.24317
$686$	0	0
$687$	−687018.	−0.0555362
$688$	0	0
$689$	896813.	0.0719704
$690$	0	0
$691$	1.55105e7	1.23575	0.617873	−	0.786278i	$-0.287994\pi$
$691$	1.55105e7	1.23575	0.617873	+	0.786278i	$0.287994\pi$
$692$	0	0
$693$	−61148.7	−0.00483676
$694$	0	0
$695$	2.58333e6	0.202870
$696$	0	0
$697$	−1.17995e7	−0.919983
$698$	0	0
$699$	5.53402e6	0.428399
$700$	0	0
$701$	1.02799e7	0.790124	0.395062	−	0.918654i	$-0.370723\pi$
$701$	1.02799e7	0.790124	0.395062	+	0.918654i	$0.370723\pi$
$702$	0	0
$703$	−253131.	−0.0193177
$704$	0	0
$705$	−1.39558e7	−1.05750
$706$	0	0
$707$	−3.51367e6	−0.264370
$708$	0	0
$709$	−1.47793e7	−1.10417	−0.552086	−	0.833787i	$-0.686168\pi$
$709$	−1.47793e7	−1.10417	−0.552086	+	0.833787i	$0.686168\pi$
$710$	0	0
$711$	−3.98341e6	−0.295516
$712$	0	0
$713$	−8.63731e6	−0.636289
$714$	0	0
$715$	−169176.	−0.0123758
$716$	0	0
$717$	−2.17449e6	−0.157964
$718$	0	0
$719$	−2.24114e7	−1.61677	−0.808383	−	0.588657i	$-0.799657\pi$
$719$	−2.24114e7	−1.61677	−0.808383	+	0.588657i	$0.799657\pi$
$720$	0	0
$721$	−6.17960e6	−0.442713
$722$	0	0
$723$	−8.15896e6	−0.580483
$724$	0	0
$725$	−2.05576e6	−0.145254
$726$	0	0
$727$	5.73926e6	0.402735	0.201368	−	0.979516i	$-0.435461\pi$
$727$	5.73926e6	0.402735	0.201368	+	0.979516i	$0.435461\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.34567e6	0.162358
$732$	0	0
$733$	1.21128e7	0.832694	0.416347	−	0.909206i	$-0.363310\pi$
$733$	1.21128e7	0.832694	0.416347	+	0.909206i	$0.363310\pi$
$734$	0	0
$735$	8.10429e6	0.553346
$736$	0	0
$737$	670226.	0.0454519
$738$	0	0
$739$	−1.12661e7	−0.758863	−0.379432	−	0.925220i	$-0.623880\pi$
$739$	−1.12661e7	−0.758863	−0.379432	+	0.925220i	$0.623880\pi$
$740$	0	0
$741$	26949.1	0.00180301
$742$	0	0
$743$	1.70415e7	1.13250	0.566248	−	0.824235i	$-0.308394\pi$
$743$	1.70415e7	1.13250	0.566248	+	0.824235i	$0.308394\pi$
$744$	0	0
$745$	−1.12712e7	−0.744008
$746$	0	0
$747$	−8.09514e6	−0.530790
$748$	0	0
$749$	121764.	0.00793074
$750$	0	0
$751$	−9.96885e6	−0.644979	−0.322489	−	0.946573i	$-0.604520\pi$
$751$	−9.96885e6	−0.644979	−0.322489	+	0.946573i	$0.604520\pi$
$752$	0	0
$753$	−1.39713e7	−0.897945
$754$	0	0
$755$	586168.	0.0374244
$756$	0	0
$757$	7.74370e6	0.491144	0.245572	−	0.969378i	$-0.421024\pi$
$757$	7.74370e6	0.491144	0.245572	+	0.969378i	$0.421024\pi$
$758$	0	0
$759$	−314099.	−0.0197907
$760$	0	0
$761$	−5.83133e6	−0.365011	−0.182505	−	0.983205i	$-0.558421\pi$
$761$	−5.83133e6	−0.365011	−0.182505	+	0.983205i	$0.558421\pi$
$762$	0	0
$763$	8.76708e6	0.545184
$764$	0	0
$765$	−3.27951e6	−0.202607
$766$	0	0
$767$	−6.17475e6	−0.378993
$768$	0	0
$769$	−2.48999e7	−1.51839	−0.759193	−	0.650866i	$-0.774406\pi$
$769$	−2.48999e7	−1.51839	−0.759193	+	0.650866i	$0.774406\pi$
$770$	0	0
$771$	−1.23573e7	−0.748663
$772$	0	0
$773$	697284.	0.0419721	0.0209861	−	0.999780i	$-0.493319\pi$
$773$	697284.	0.0419721	0.0209861	+	0.999780i	$0.493319\pi$
$774$	0	0
$775$	−2.60989e6	−0.156088
$776$	0	0
$777$	5.95535e6	0.353879
$778$	0	0
$779$	317131.	0.0187239
$780$	0	0
$781$	340889.	0.0199979
$782$	0	0
$783$	−2.31633e6	−0.135019
$784$	0	0
$785$	1.53730e7	0.890398
$786$	0	0
$787$	−1.21605e7	−0.699865	−0.349933	−	0.936775i	$-0.613796\pi$
$787$	−1.21605e7	−0.699865	−0.349933	+	0.936775i	$0.613796\pi$
$788$	0	0
$789$	4.12084e6	0.235664
$790$	0	0
$791$	−7.03934e6	−0.400028
$792$	0	0
$793$	1.10034e6	0.0621362
$794$	0	0
$795$	2.93321e6	0.164598
$796$	0	0
$797$	9.61296e6	0.536058	0.268029	−	0.963411i	$-0.413628\pi$
$797$	9.61296e6	0.536058	0.268029	+	0.963411i	$0.413628\pi$
$798$	0	0
$799$	1.66442e7	0.922351
$800$	0	0
$801$	3.83060e6	0.210953
$802$	0	0
$803$	1.15869e6	0.0634128
$804$	0	0
$805$	−6.09085e6	−0.331274
$806$	0	0
$807$	−1.72514e7	−0.932482
$808$	0	0
$809$	−2.32130e7	−1.24698	−0.623492	−	0.781830i	$-0.714287\pi$
$809$	−2.32130e7	−1.24698	−0.623492	+	0.781830i	$0.714287\pi$
$810$	0	0
$811$	4.90107e6	0.261661	0.130830	−	0.991405i	$-0.458236\pi$
$811$	4.90107e6	0.261661	0.130830	+	0.991405i	$0.458236\pi$
$812$	0	0
$813$	1.63208e7	0.865994
$814$	0	0
$815$	−1.34914e7	−0.711482
$816$	0	0
$817$	−63044.1	−0.00330437
$818$	0	0
$819$	−634026.	−0.0330291
$820$	0	0
$821$	1.38159e7	0.715354	0.357677	−	0.933845i	$-0.383569\pi$
$821$	1.38159e7	0.715354	0.357677	+	0.933845i	$0.383569\pi$
$822$	0	0
$823$	2.74254e7	1.41141	0.705706	−	0.708505i	$-0.250630\pi$
$823$	2.74254e7	1.41141	0.705706	+	0.708505i	$0.250630\pi$
$824$	0	0
$825$	−94909.7	−0.00485485
$826$	0	0
$827$	−2.28348e7	−1.16100	−0.580502	−	0.814259i	$-0.697143\pi$
$827$	−2.28348e7	−1.16100	−0.580502	+	0.814259i	$0.697143\pi$
$828$	0	0
$829$	1.83487e7	0.927297	0.463649	−	0.886019i	$-0.346540\pi$
$829$	1.83487e7	0.927297	0.463649	+	0.886019i	$0.346540\pi$
$830$	0	0
$831$	−7.12877e6	−0.358106
$832$	0	0
$833$	−9.66551e6	−0.482628
$834$	0	0
$835$	5.67219e6	0.281537
$836$	0	0
$837$	−2.94069e6	−0.145089
$838$	0	0
$839$	−6.63289e6	−0.325310	−0.162655	−	0.986683i	$-0.552006\pi$
$839$	−6.63289e6	−0.325310	−0.162655	+	0.986683i	$0.552006\pi$
$840$	0	0
$841$	−1.04153e7	−0.507786
$842$	0	0
$843$	−816399.	−0.0395670
$844$	0	0
$845$	−1.75412e6	−0.0845118
$846$	0	0
$847$	−7.44701e6	−0.356676
$848$	0	0
$849$	7.28443e6	0.346838
$850$	0	0
$851$	3.05905e7	1.44798
$852$	0	0
$853$	−3.02938e7	−1.42554	−0.712772	−	0.701395i	$-0.752561\pi$
$853$	−3.02938e7	−1.42554	−0.712772	+	0.701395i	$0.752561\pi$
$854$	0	0
$855$	88142.6	0.00412354
$856$	0	0
$857$	−1.37741e7	−0.640636	−0.320318	−	0.947310i	$-0.603790\pi$
$857$	−1.37741e7	−0.640636	−0.320318	+	0.947310i	$0.603790\pi$
$858$	0	0
$859$	−2.06680e7	−0.955687	−0.477843	−	0.878445i	$-0.658581\pi$
$859$	−2.06680e7	−0.955687	−0.477843	+	0.878445i	$0.658581\pi$
$860$	0	0
$861$	−7.46108e6	−0.343000
$862$	0	0
$863$	1.17441e7	0.536775	0.268387	−	0.963311i	$-0.413509\pi$
$863$	1.17441e7	0.536775	0.268387	+	0.963311i	$0.413509\pi$
$864$	0	0
$865$	1.45380e7	0.660641
$866$	0	0
$867$	−8.86744e6	−0.400636
$868$	0	0
$869$	801562.	0.0360071
$870$	0	0
$871$	6.94930e6	0.310381
$872$	0	0
$873$	−3.21809e6	−0.142910
$874$	0	0
$875$	7.04893e6	0.311245
$876$	0	0
$877$	7.54480e6	0.331245	0.165622	−	0.986189i	$-0.447037\pi$
$877$	7.54480e6	0.331245	0.165622	+	0.986189i	$0.447037\pi$
$878$	0	0
$879$	1.09920e7	0.479850
$880$	0	0
$881$	−7.88797e6	−0.342394	−0.171197	−	0.985237i	$-0.554763\pi$
$881$	−7.88797e6	−0.342394	−0.171197	+	0.985237i	$0.554763\pi$
$882$	0	0
$883$	−2.26784e7	−0.978836	−0.489418	−	0.872049i	$-0.662791\pi$
$883$	−2.26784e7	−0.978836	−0.489418	+	0.872049i	$0.662791\pi$
$884$	0	0
$885$	−2.01958e7	−0.866767
$886$	0	0
$887$	−3.42780e7	−1.46287	−0.731436	−	0.681911i	$-0.761149\pi$
$887$	−3.42780e7	−1.46287	−0.731436	+	0.681911i	$0.761149\pi$
$888$	0	0
$889$	−1.17815e7	−0.499973
$890$	0	0
$891$	−106939.	−0.00451276
$892$	0	0
$893$	−447343.	−0.0187721
$894$	0	0
$895$	3.22575e7	1.34609
$896$	0	0
$897$	−3.25676e6	−0.135146
$898$	0	0
$899$	1.28172e7	0.528926
$900$	0	0
$901$	−3.49827e6	−0.143563
$902$	0	0
$903$	1.48322e6	0.0605323
$904$	0	0
$905$	4.82073e7	1.95655
$906$	0	0
$907$	−3.53458e7	−1.42666	−0.713328	−	0.700830i	$-0.752813\pi$
$907$	−3.53458e7	−1.42666	−0.713328	+	0.700830i	$0.752813\pi$
$908$	0	0
$909$	−6.14484e6	−0.246661
$910$	0	0
$911$	−1.18935e7	−0.474804	−0.237402	−	0.971411i	$-0.576296\pi$
$911$	−1.18935e7	−0.474804	−0.237402	+	0.971411i	$0.576296\pi$
$912$	0	0
$913$	1.62894e6	0.0646739
$914$	0	0
$915$	3.59889e6	0.142107
$916$	0	0
$917$	1.17866e7	0.462876
$918$	0	0
$919$	2.49027e7	0.972653	0.486327	−	0.873777i	$-0.338337\pi$
$919$	2.49027e7	0.972653	0.486327	+	0.873777i	$0.338337\pi$
$920$	0	0
$921$	−3.47042e6	−0.134814
$922$	0	0
$923$	3.53454e6	0.136561
$924$	0	0
$925$	9.24338e6	0.355203
$926$	0	0
$927$	−1.08071e7	−0.413058
$928$	0	0
$929$	−1.51280e7	−0.575097	−0.287549	−	0.957766i	$-0.592840\pi$
$929$	−1.51280e7	−0.575097	−0.287549	+	0.957766i	$0.592840\pi$
$930$	0	0
$931$	259778.	0.00982263
$932$	0	0
$933$	1.03263e7	0.388366
$934$	0	0
$935$	659918.	0.0246866
$936$	0	0
$937$	−4.75041e6	−0.176759	−0.0883797	−	0.996087i	$-0.528169\pi$
$937$	−4.75041e6	−0.176759	−0.0883797	+	0.996087i	$0.528169\pi$
$938$	0	0
$939$	2.81855e7	1.04319
$940$	0	0
$941$	−2.34813e7	−0.864466	−0.432233	−	0.901762i	$-0.642274\pi$
$941$	−2.34813e7	−0.864466	−0.432233	+	0.901762i	$0.642274\pi$
$942$	0	0
$943$	−3.83249e7	−1.40346
$944$	0	0
$945$	−2.07371e6	−0.0755386
$946$	0	0
$947$	7.01784e6	0.254289	0.127145	−	0.991884i	$-0.459419\pi$
$947$	7.01784e6	0.254289	0.127145	+	0.991884i	$0.459419\pi$
$948$	0	0
$949$	1.20139e7	0.433032
$950$	0	0
$951$	7.80385e6	0.279807
$952$	0	0
$953$	3.35891e7	1.19803	0.599013	−	0.800739i	$-0.295560\pi$
$953$	3.35891e7	1.19803	0.599013	+	0.800739i	$0.295560\pi$
$954$	0	0
$955$	−3.50913e7	−1.24506
$956$	0	0
$957$	466103.	0.0164514
$958$	0	0
$959$	−1.15135e7	−0.404259
$960$	0	0
$961$	−1.23571e7	−0.431625
$962$	0	0
$963$	212945.	0.00739950
$964$	0	0
$965$	4.46287e7	1.54275
$966$	0	0
$967$	6.65688e6	0.228931	0.114466	−	0.993427i	$-0.463484\pi$
$967$	6.65688e6	0.228931	0.114466	+	0.993427i	$0.463484\pi$
$968$	0	0
$969$	−105122.	−0.00359655
$970$	0	0
$971$	1.05043e6	0.0357537	0.0178769	−	0.999840i	$-0.494309\pi$
$971$	1.05043e6	0.0357537	0.0178769	+	0.999840i	$0.494309\pi$
$972$	0	0
$973$	−1.94818e6	−0.0659702
$974$	0	0
$975$	−984080.	−0.0331527
$976$	0	0
$977$	−7.59396e6	−0.254526	−0.127263	−	0.991869i	$-0.540619\pi$
$977$	−7.59396e6	−0.254526	−0.127263	+	0.991869i	$0.540619\pi$
$978$	0	0
$979$	−770813.	−0.0257035
$980$	0	0
$981$	1.53322e7	0.508665
$982$	0	0
$983$	−3.33190e7	−1.09978	−0.549892	−	0.835235i	$-0.685331\pi$
$983$	−3.33190e7	−1.09978	−0.549892	+	0.835235i	$0.685331\pi$
$984$	0	0
$985$	−2.99766e6	−0.0984447
$986$	0	0
$987$	1.05245e7	0.343883
$988$	0	0
$989$	7.61878e6	0.247682
$990$	0	0
$991$	−5.11503e7	−1.65449	−0.827245	−	0.561841i	$-0.810093\pi$
$991$	−5.11503e7	−1.65449	−0.827245	+	0.561841i	$0.810093\pi$
$992$	0	0
$993$	−4.09014e6	−0.131633
$994$	0	0
$995$	−6.44675e7	−2.06435
$996$	0	0
$997$	−5.11313e7	−1.62911	−0.814553	−	0.580089i	$-0.803018\pi$
$997$	−5.11313e7	−1.62911	−0.814553	+	0.580089i	$0.803018\pi$
$998$	0	0
$999$	1.04149e7	0.330174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.k.1.1		2
4.3	odd	2		39.6.a.b.1.2	✓	2
12.11	even	2		117.6.a.b.1.1		2
20.19	odd	2		975.6.a.c.1.1		2
52.51	odd	2		507.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.6.a.b.1.2	✓	2	4.3	odd	2
117.6.a.b.1.1		2	12.11	even	2
507.6.a.c.1.1		2	52.51	odd	2
624.6.a.k.1.1		2	1.1	even	1	trivial
975.6.a.c.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -61.4166 q^{5} +46.3165 q^{7} +81.0000 q^{9} -16.2992 q^{11} -169.000 q^{13} -552.749 q^{15} +659.231 q^{17} -17.7180 q^{19} +416.848 q^{21} +2141.20 q^{23} +646.996 q^{25} +729.000 q^{27} -3177.40 q^{29} -4033.87 q^{31} -146.693 q^{33} -2844.60 q^{35} +14286.6 q^{37} -1521.00 q^{39} -17898.8 q^{41} +3558.19 q^{43} -4974.74 q^{45} +25247.9 q^{47} -14661.8 q^{49} +5933.08 q^{51} -5306.59 q^{53} +1001.04 q^{55} -159.462 q^{57} +36537.0 q^{59} -6510.90 q^{61} +3751.63 q^{63} +10379.4 q^{65} -41120.1 q^{67} +19270.8 q^{69} -20914.4 q^{71} -71088.4 q^{73} +5822.96 q^{75} -754.922 q^{77} -49177.9 q^{79} +6561.00 q^{81} -99939.9 q^{83} -40487.7 q^{85} -28596.6 q^{87} +47291.4 q^{89} -7827.48 q^{91} -36304.8 q^{93} +1088.18 q^{95} -39729.5 q^{97} -1320.24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} - 48 q^{5} - 72 q^{7} + 162 q^{9} + 596 q^{11} - 338 q^{13} - 432 q^{15} - 268 q^{17} - 1128 q^{19} - 648 q^{21} + 1768 q^{23} - 2298 q^{25} + 1458 q^{27} - 7612 q^{29} + 4160 q^{31} + 5364 q^{33}+ \cdots + 48276 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 - 48 * q^5 - 72 * q^7 + 162 * q^9 + 596 * q^11 - 338 * q^13 - 432 * q^15 - 268 * q^17 - 1128 * q^19 - 648 * q^21 + 1768 * q^23 - 2298 * q^25 + 1458 * q^27 - 7612 * q^29 + 4160 * q^31 + 5364 * q^33 - 4432 * q^35 + 17468 * q^37 - 3042 * q^39 - 28000 * q^41 + 24328 * q^43 - 3888 * q^45 + 18108 * q^47 - 17470 * q^49 - 2412 * q^51 + 1420 * q^53 + 9216 * q^55 - 10152 * q^57 - 6788 * q^59 - 37148 * q^61 - 5832 * q^63 + 8112 * q^65 - 106112 * q^67 + 15912 * q^69 + 30460 * q^71 - 37620 * q^73 - 20682 * q^75 - 73200 * q^77 + 2160 * q^79 + 13122 * q^81 - 207004 * q^83 - 52928 * q^85 - 68508 * q^87 - 74136 * q^89 + 12168 * q^91 + 37440 * q^93 - 13808 * q^95 - 121156 * q^97 + 48276 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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