LMFDB - Embedded newform 171.6.a.j.1.2

Error: no document with id 228956422 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [171,6,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	171.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,53,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$27.4256331880$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 90x^{2} + 118x + 1412 $ x^4 - x^3 - 90x^2 + 118x + 1412
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.82184$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.82184 q^{2} +1.89382 q^{4} -23.6174 q^{5} -250.612 q^{7} +175.273 q^{8} +137.497 q^{10} -414.956 q^{11} -920.892 q^{13} +1459.02 q^{14} -1081.02 q^{16} +238.401 q^{17} -361.000 q^{19} -44.7271 q^{20} +2415.81 q^{22} -1138.97 q^{23} -2567.22 q^{25} +5361.29 q^{26} -474.613 q^{28} -7031.41 q^{29} +495.673 q^{31} +684.752 q^{32} -1387.93 q^{34} +5918.80 q^{35} +2981.99 q^{37} +2101.68 q^{38} -4139.51 q^{40} -9339.27 q^{41} -11537.4 q^{43} -785.851 q^{44} +6630.91 q^{46} +19955.6 q^{47} +45999.2 q^{49} +14945.9 q^{50} -1744.00 q^{52} +20089.6 q^{53} +9800.20 q^{55} -43925.5 q^{56} +40935.7 q^{58} +48844.4 q^{59} +24559.4 q^{61} -2885.73 q^{62} +30606.0 q^{64} +21749.1 q^{65} -59482.1 q^{67} +451.487 q^{68} -34458.3 q^{70} +4635.76 q^{71} -26670.8 q^{73} -17360.6 q^{74} -683.668 q^{76} +103993. q^{77} -68240.6 q^{79} +25530.8 q^{80} +54371.8 q^{82} -2260.79 q^{83} -5630.41 q^{85} +67169.0 q^{86} -72730.8 q^{88} -749.444 q^{89} +230786. q^{91} -2157.00 q^{92} -116179. q^{94} +8525.89 q^{95} +88106.3 q^{97} -267800. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 53 q^{4} + 8 q^{5} - 142 q^{7} + 147 q^{8} - 496 q^{10} + 714 q^{11} - 74 q^{13} + 2790 q^{14} - 2839 q^{16} + 3690 q^{17} - 1444 q^{19} + 5158 q^{20} - 1582 q^{22} - 862 q^{23} + 3282 q^{25}+ \cdots - 178319 q^{98}+O(q^{100}) $ 4 * q - q^2 + 53 * q^4 + 8 * q^5 - 142 * q^7 + 147 * q^8 - 496 * q^10 + 714 * q^11 - 74 * q^13 + 2790 * q^14 - 2839 * q^16 + 3690 * q^17 - 1444 * q^19 + 5158 * q^20 - 1582 * q^22 - 862 * q^23 + 3282 * q^25 - 870 * q^26 + 5488 * q^28 - 992 * q^29 - 7382 * q^31 + 6103 * q^32 + 10288 * q^34 + 10242 * q^35 + 19114 * q^37 + 361 * q^38 + 19470 * q^40 + 5792 * q^41 - 18634 * q^43 + 5316 * q^44 + 62256 * q^46 + 35424 * q^47 + 15910 * q^49 - 14309 * q^50 + 48170 * q^52 + 20256 * q^53 + 23538 * q^55 - 27516 * q^56 + 37566 * q^58 + 76944 * q^59 + 11562 * q^61 + 104740 * q^62 - 15455 * q^64 + 157464 * q^65 - 110044 * q^67 + 70926 * q^68 + 24138 * q^70 + 71184 * q^71 - 138830 * q^73 - 91586 * q^74 - 19133 * q^76 + 76122 * q^77 - 59470 * q^79 - 81842 * q^80 + 17030 * q^82 + 191600 * q^83 + 23682 * q^85 - 56246 * q^86 - 134640 * q^88 - 158920 * q^89 + 267196 * q^91 - 34460 * q^92 + 35946 * q^94 - 2888 * q^95 + 47460 * q^97 - 178319 * q^98

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 $

$2$	−5.82184	−1.02917	−0.514583	−	0.857441i	$-0.672053\pi$
$2$	−5.82184	−1.02917	−0.514583	+	0.857441i	$0.672053\pi$
$3$	0	0
$3$	0	0
$4$	1.89382	0.0591818
$5$	−23.6174	−0.422482	−0.211241	−	0.977434i	$-0.567750\pi$
$5$	−23.6174	−0.422482	−0.211241	+	0.977434i	$0.567750\pi$
$6$	0	0
$7$	−250.612	−1.93311	−0.966554	−	0.256464i	$-0.917443\pi$
$7$	−250.612	−1.93311	−0.966554	+	0.256464i	$0.917443\pi$
$8$	175.273	0.968258
$9$	0	0
$10$	137.497	0.434803
$11$	−414.956	−1.03400	−0.517000	−	0.855985i	$-0.672951\pi$
$11$	−414.956	−1.03400	−0.517000	+	0.855985i	$0.672951\pi$
$12$	0	0
$13$	−920.892	−1.51130	−0.755650	−	0.654976i	$-0.772679\pi$
$13$	−920.892	−1.51130	−0.755650	+	0.654976i	$0.772679\pi$
$14$	1459.02	1.98949
$15$	0	0
$16$	−1081.02	−1.05568
$17$	238.401	0.200071	0.100036	−	0.994984i	$-0.468104\pi$
$17$	238.401	0.200071	0.100036	+	0.994984i	$0.468104\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	−44.7271	−0.0250032
$21$	0	0
$22$	2415.81	1.06416
$23$	−1138.97	−0.448945	−0.224472	−	0.974480i	$-0.572066\pi$
$23$	−1138.97	−0.448945	−0.224472	+	0.974480i	$0.572066\pi$
$24$	0	0
$25$	−2567.22	−0.821509
$26$	5361.29	1.55538
$27$	0	0
$28$	−474.613	−0.114405
$29$	−7031.41	−1.55256	−0.776278	−	0.630391i	$-0.782895\pi$
$29$	−7031.41	−1.55256	−0.776278	+	0.630391i	$0.782895\pi$
$30$	0	0
$31$	495.673	0.0926384	0.0463192	−	0.998927i	$-0.485251\pi$
$31$	495.673	0.0926384	0.0463192	+	0.998927i	$0.485251\pi$
$32$	684.752	0.118211
$33$	0	0
$34$	−1387.93	−0.205907
$35$	5918.80	0.816702
$36$	0	0
$37$	2981.99	0.358098	0.179049	−	0.983840i	$-0.442698\pi$
$37$	2981.99	0.358098	0.179049	+	0.983840i	$0.442698\pi$
$38$	2101.68	0.236107
$39$	0	0
$40$	−4139.51	−0.409071
$41$	−9339.27	−0.867668	−0.433834	−	0.900993i	$-0.642840\pi$
$41$	−9339.27	−0.867668	−0.433834	+	0.900993i	$0.642840\pi$
$42$	0	0
$43$	−11537.4	−0.951563	−0.475781	−	0.879564i	$-0.657835\pi$
$43$	−11537.4	−0.951563	−0.475781	+	0.879564i	$0.657835\pi$
$44$	−785.851	−0.0611940
$45$	0	0
$46$	6630.91	0.462039
$47$	19955.6	1.31771	0.658857	−	0.752268i	$-0.271040\pi$
$47$	19955.6	1.31771	0.658857	+	0.752268i	$0.271040\pi$
$48$	0	0
$49$	45999.2	2.73690
$50$	14945.9	0.845469
$51$	0	0
$52$	−1744.00	−0.0894414
$53$	20089.6	0.982385	0.491193	−	0.871051i	$-0.336561\pi$
$53$	20089.6	0.982385	0.491193	+	0.871051i	$0.336561\pi$
$54$	0	0
$55$	9800.20	0.436846
$56$	−43925.5	−1.87175
$57$	0	0
$58$	40935.7	1.59784
$59$	48844.4	1.82677	0.913387	−	0.407093i	$-0.133457\pi$
$59$	48844.4	1.82677	0.913387	+	0.407093i	$0.133457\pi$
$60$	0	0
$61$	24559.4	0.845069	0.422535	−	0.906347i	$-0.361140\pi$
$61$	24559.4	0.845069	0.422535	+	0.906347i	$0.361140\pi$
$62$	−2885.73	−0.0953402
$63$	0	0
$64$	30606.0	0.934021
$65$	21749.1	0.638496
$66$	0	0
$67$	−59482.1	−1.61882	−0.809411	−	0.587243i	$-0.800213\pi$
$67$	−59482.1	−1.61882	−0.809411	+	0.587243i	$0.800213\pi$
$68$	451.487	0.0118406
$69$	0	0
$70$	−34458.3	−0.840522
$71$	4635.76	0.109138	0.0545689	−	0.998510i	$-0.482622\pi$
$71$	4635.76	0.109138	0.0545689	+	0.998510i	$0.482622\pi$
$72$	0	0
$73$	−26670.8	−0.585772	−0.292886	−	0.956147i	$-0.594616\pi$
$73$	−26670.8	−0.585772	−0.292886	+	0.956147i	$0.594616\pi$
$74$	−17360.6	−0.368542
$75$	0	0
$76$	−683.668	−0.0135772
$77$	103993.	1.99883
$78$	0	0
$79$	−68240.6	−1.23020	−0.615100	−	0.788449i	$-0.710884\pi$
$79$	−68240.6	−1.23020	−0.615100	+	0.788449i	$0.710884\pi$
$80$	25530.8	0.446005
$81$	0	0
$82$	54371.8	0.892974
$83$	−2260.79	−0.0360217	−0.0180109	−	0.999838i	$-0.505733\pi$
$83$	−2260.79	−0.0360217	−0.0180109	+	0.999838i	$0.505733\pi$
$84$	0	0
$85$	−5630.41	−0.0845265
$86$	67169.0	0.979315
$87$	0	0
$88$	−72730.8	−1.00118
$89$	−749.444	−0.0100291	−0.00501457	−	0.999987i	$-0.501596\pi$
$89$	−749.444	−0.0100291	−0.00501457	+	0.999987i	$0.501596\pi$
$90$	0	0
$91$	230786.	2.92150
$92$	−2157.00	−0.0265694
$93$	0	0
$94$	−116179.	−1.35615
$95$	8525.89	0.0969239
$96$	0	0
$97$	88106.3	0.950775	0.475387	−	0.879777i	$-0.342308\pi$
$97$	88106.3	0.950775	0.475387	+	0.879777i	$0.342308\pi$
$98$	−267800.	−2.81673
$99$	0	0
$100$	−4861.84	−0.0486184
$101$	125404.	1.22323	0.611613	−	0.791157i	$-0.290521\pi$
$101$	125404.	1.22323	0.611613	+	0.791157i	$0.290521\pi$
$102$	0	0
$103$	−110218.	−1.02367	−0.511835	−	0.859084i	$-0.671034\pi$
$103$	−110218.	−1.02367	−0.511835	+	0.859084i	$0.671034\pi$
$104$	−161408.	−1.46333
$105$	0	0
$106$	−116958.	−1.01104
$107$	−160783.	−1.35763	−0.678813	−	0.734311i	$-0.737505\pi$
$107$	−160783.	−1.35763	−0.678813	+	0.734311i	$0.737505\pi$
$108$	0	0
$109$	−111776.	−0.901122	−0.450561	−	0.892746i	$-0.648776\pi$
$109$	−111776.	−0.901122	−0.450561	+	0.892746i	$0.648776\pi$
$110$	−57055.2	−0.449587
$111$	0	0
$112$	270915.	2.04074
$113$	−116128.	−0.855544	−0.427772	−	0.903887i	$-0.640701\pi$
$113$	−116128.	−0.855544	−0.427772	+	0.903887i	$0.640701\pi$
$114$	0	0
$115$	26899.6	0.189671
$116$	−13316.2	−0.0918830
$117$	0	0
$118$	−284364.	−1.88005
$119$	−59746.0	−0.386760
$120$	0	0
$121$	11137.7	0.0691561
$122$	−142981.	−0.869716
$123$	0	0
$124$	938.714	0.00548250
$125$	134436.	0.769554
$126$	0	0
$127$	−6062.47	−0.0333534	−0.0166767	−	0.999861i	$-0.505309\pi$
$127$	−6062.47	−0.0333534	−0.0166767	+	0.999861i	$0.505309\pi$
$128$	−200095.	−1.07947
$129$	0	0
$130$	−126620.	−0.657118
$131$	−115145.	−0.586228	−0.293114	−	0.956078i	$-0.594692\pi$
$131$	−115145.	−0.586228	−0.293114	+	0.956078i	$0.594692\pi$
$132$	0	0
$133$	90470.8	0.443485
$134$	346295.	1.66604
$135$	0	0
$136$	41785.3	0.193721
$137$	−326643.	−1.48687	−0.743434	−	0.668810i	$-0.766804\pi$
$137$	−326643.	−1.48687	−0.743434	+	0.668810i	$0.766804\pi$
$138$	0	0
$139$	−228617.	−1.00362	−0.501812	−	0.864977i	$-0.667333\pi$
$139$	−228617.	−1.00362	−0.501812	+	0.864977i	$0.667333\pi$
$140$	11209.1	0.0483339
$141$	0	0
$142$	−26988.6	−0.112321
$143$	382130.	1.56268
$144$	0	0
$145$	166064.	0.655926
$146$	155273.	0.602857
$147$	0	0
$148$	5647.34	0.0211929
$149$	−225286.	−0.831321	−0.415661	−	0.909520i	$-0.636450\pi$
$149$	−225286.	−0.831321	−0.415661	+	0.909520i	$0.636450\pi$
$150$	0	0
$151$	102524.	0.365918	0.182959	−	0.983121i	$-0.441432\pi$
$151$	102524.	0.365918	0.182959	+	0.983121i	$0.441432\pi$
$152$	−63273.7	−0.222134
$153$	0	0
$154$	−605430.	−2.05713
$155$	−11706.5	−0.0391380
$156$	0	0
$157$	441599.	1.42981	0.714906	−	0.699220i	$-0.246469\pi$
$157$	441599.	1.42981	0.714906	+	0.699220i	$0.246469\pi$
$158$	397286.	1.26608
$159$	0	0
$160$	−16172.1	−0.0499420
$161$	285439.	0.867859
$162$	0	0
$163$	−451041.	−1.32968	−0.664839	−	0.746986i	$-0.731500\pi$
$163$	−451041.	−1.32968	−0.664839	+	0.746986i	$0.731500\pi$
$164$	−17686.9	−0.0513501
$165$	0	0
$166$	13161.9	0.0370723
$167$	−447874.	−1.24270	−0.621348	−	0.783535i	$-0.713415\pi$
$167$	−447874.	−1.24270	−0.621348	+	0.783535i	$0.713415\pi$
$168$	0	0
$169$	476750.	1.28403
$170$	32779.4	0.0869918
$171$	0	0
$172$	−21849.8	−0.0563152
$173$	381949.	0.970266	0.485133	−	0.874440i	$-0.338771\pi$
$173$	381949.	0.970266	0.485133	+	0.874440i	$0.338771\pi$
$174$	0	0
$175$	643374.	1.58807
$176$	448574.	1.09157
$177$	0	0
$178$	4363.14	0.0103217
$179$	−193497.	−0.451379	−0.225690	−	0.974199i	$-0.572464\pi$
$179$	−193497.	−0.451379	−0.225690	+	0.974199i	$0.572464\pi$
$180$	0	0
$181$	52655.8	0.119468	0.0597338	−	0.998214i	$-0.480975\pi$
$181$	52655.8	0.119468	0.0597338	+	0.998214i	$0.480975\pi$
$182$	−1.34360e6	−3.00671
$183$	0	0
$184$	−199631.	−0.434694
$185$	−70426.9	−0.151290
$186$	0	0
$187$	−98925.8	−0.206874
$188$	37792.4	0.0779847
$189$	0	0
$190$	−49636.4	−0.0997507
$191$	303888.	0.602739	0.301370	−	0.953507i	$-0.402556\pi$
$191$	303888.	0.602739	0.301370	+	0.953507i	$0.402556\pi$
$192$	0	0
$193$	134586.	0.260080	0.130040	−	0.991509i	$-0.458489\pi$
$193$	134586.	0.260080	0.130040	+	0.991509i	$0.458489\pi$
$194$	−512941.	−0.978505
$195$	0	0
$196$	87114.0	0.161975
$197$	651096.	1.19531	0.597654	−	0.801754i	$-0.296100\pi$
$197$	651096.	1.19531	0.597654	+	0.801754i	$0.296100\pi$
$198$	0	0
$199$	−291770.	−0.522286	−0.261143	−	0.965300i	$-0.584099\pi$
$199$	−291770.	−0.522286	−0.261143	+	0.965300i	$0.584099\pi$
$200$	−449965.	−0.795433
$201$	0	0
$202$	−730080.	−1.25890
$203$	1.76215e6	3.00126
$204$	0	0
$205$	220570.	0.366574
$206$	641673.	1.05353
$207$	0	0
$208$	995499.	1.59545
$209$	149799.	0.237216
$210$	0	0
$211$	1.05072e6	1.62472	0.812361	−	0.583154i	$-0.198182\pi$
$211$	1.05072e6	1.62472	0.812361	+	0.583154i	$0.198182\pi$
$212$	38046.1	0.0581393
$213$	0	0
$214$	936052.	1.39722
$215$	272484.	0.402018
$216$	0	0
$217$	−124221.	−0.179080
$218$	650744.	0.927404
$219$	0	0
$220$	18559.8	0.0258533
$221$	−219541.	−0.302368
$222$	0	0
$223$	−81993.8	−0.110413	−0.0552063	−	0.998475i	$-0.517582\pi$
$223$	−81993.8	−0.110413	−0.0552063	+	0.998475i	$0.517582\pi$
$224$	−171607.	−0.228515
$225$	0	0
$226$	676081.	0.880497
$227$	−911528.	−1.17410	−0.587051	−	0.809550i	$-0.699711\pi$
$227$	−911528.	−1.17410	−0.587051	+	0.809550i	$0.699711\pi$
$228$	0	0
$229$	−660815.	−0.832705	−0.416353	−	0.909203i	$-0.636692\pi$
$229$	−660815.	−0.832705	−0.416353	+	0.909203i	$0.636692\pi$
$230$	−156605.	−0.195203
$231$	0	0
$232$	−1.23242e6	−1.50327
$233$	−866250.	−1.04533	−0.522665	−	0.852538i	$-0.675062\pi$
$233$	−866250.	−1.04533	−0.522665	+	0.852538i	$0.675062\pi$
$234$	0	0
$235$	−471301.	−0.556710
$236$	92502.4	0.108112
$237$	0	0
$238$	347831.	0.398040
$239$	385761.	0.436841	0.218421	−	0.975855i	$-0.429910\pi$
$239$	385761.	0.436841	0.218421	+	0.975855i	$0.429910\pi$
$240$	0	0
$241$	−1.27179e6	−1.41050	−0.705252	−	0.708957i	$-0.749166\pi$
$241$	−1.27179e6	−1.41050	−0.705252	+	0.708957i	$0.749166\pi$
$242$	−64841.7	−0.0711731
$243$	0	0
$244$	46510.9	0.0500127
$245$	−1.08638e6	−1.15629
$246$	0	0
$247$	332442.	0.346716
$248$	86878.2	0.0896978
$249$	0	0
$250$	−782662.	−0.791999
$251$	1.13475e6	1.13689	0.568444	−	0.822722i	$-0.307546\pi$
$251$	1.13475e6	1.13689	0.568444	+	0.822722i	$0.307546\pi$
$252$	0	0
$253$	472623.	0.464209
$254$	35294.7	0.0343262
$255$	0	0
$256$	185531.	0.176936
$257$	129194.	0.122014	0.0610071	−	0.998137i	$-0.480569\pi$
$257$	129194.	0.122014	0.0610071	+	0.998137i	$0.480569\pi$
$258$	0	0
$259$	−747320.	−0.692241
$260$	41188.9	0.0377873
$261$	0	0
$262$	670355.	0.603326
$263$	−913758.	−0.814595	−0.407298	−	0.913295i	$-0.633529\pi$
$263$	−913758.	−0.814595	−0.407298	+	0.913295i	$0.633529\pi$
$264$	0	0
$265$	−474465.	−0.415040
$266$	−526706.	−0.456420
$267$	0	0
$268$	−112648.	−0.0958048
$269$	460736.	0.388214	0.194107	−	0.980980i	$-0.437819\pi$
$269$	460736.	0.388214	0.194107	+	0.980980i	$0.437819\pi$
$270$	0	0
$271$	−726289.	−0.600739	−0.300370	−	0.953823i	$-0.597110\pi$
$271$	−726289.	−0.600739	−0.300370	+	0.953823i	$0.597110\pi$
$272$	−257715.	−0.211211
$273$	0	0
$274$	1.90166e6	1.53023
$275$	1.06528e6	0.849441
$276$	0	0
$277$	−588940.	−0.461181	−0.230590	−	0.973051i	$-0.574066\pi$
$277$	−588940.	−0.461181	−0.230590	+	0.973051i	$0.574066\pi$
$278$	1.33097e6	1.03290
$279$	0	0
$280$	1.03741e6	0.790778
$281$	−111634.	−0.0843396	−0.0421698	−	0.999110i	$-0.513427\pi$
$281$	−111634.	−0.0843396	−0.0421698	+	0.999110i	$0.513427\pi$
$282$	0	0
$283$	−686287.	−0.509377	−0.254689	−	0.967023i	$-0.581973\pi$
$283$	−686287.	−0.509377	−0.254689	+	0.967023i	$0.581973\pi$
$284$	8779.28	0.00645897
$285$	0	0
$286$	−2.22470e6	−1.60826
$287$	2.34053e6	1.67730
$288$	0	0
$289$	−1.36302e6	−0.959971
$290$	−966797.	−0.675056
$291$	0	0
$292$	−50509.6	−0.0346670
$293$	1.87734e6	1.27754	0.638769	−	0.769399i	$-0.279444\pi$
$293$	1.87734e6	1.27754	0.638769	+	0.769399i	$0.279444\pi$
$294$	0	0
$295$	−1.15358e6	−0.771778
$296$	522663.	0.346731
$297$	0	0
$298$	1.31158e6	0.855567
$299$	1.04887e6	0.678490
$300$	0	0
$301$	2.89141e6	1.83947
$302$	−596879.	−0.376590
$303$	0	0
$304$	390247.	0.242189
$305$	−580029.	−0.357026
$306$	0	0
$307$	396709.	0.240229	0.120115	−	0.992760i	$-0.461674\pi$
$307$	396709.	0.240229	0.120115	+	0.992760i	$0.461674\pi$
$308$	196943.	0.118295
$309$	0	0
$310$	68153.5	0.0402795
$311$	1.18936e6	0.697290	0.348645	−	0.937255i	$-0.386642\pi$
$311$	1.18936e6	0.697290	0.348645	+	0.937255i	$0.386642\pi$
$312$	0	0
$313$	−778041.	−0.448892	−0.224446	−	0.974487i	$-0.572057\pi$
$313$	−778041.	−0.448892	−0.224446	+	0.974487i	$0.572057\pi$
$314$	−2.57092e6	−1.47151
$315$	0	0
$316$	−129235.	−0.0728054
$317$	−3.52584e6	−1.97067	−0.985335	−	0.170630i	$-0.945420\pi$
$317$	−3.52584e6	−1.97067	−0.985335	+	0.170630i	$0.945420\pi$
$318$	0	0
$319$	2.91773e6	1.60534
$320$	−722835.	−0.394606
$321$	0	0
$322$	−1.66178e6	−0.893171
$323$	−86062.7	−0.0458995
$324$	0	0
$325$	2.36413e6	1.24155
$326$	2.62589e6	1.36846
$327$	0	0
$328$	−1.63693e6	−0.840126
$329$	−5.00112e6	−2.54728
$330$	0	0
$331$	2.21135e6	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$331$	2.21135e6	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$332$	−4281.52	−0.00213183
$333$	0	0
$334$	2.60745e6	1.27894
$335$	1.40481e6	0.683922
$336$	0	0
$337$	3.52355e6	1.69008	0.845038	−	0.534706i	$-0.179578\pi$
$337$	3.52355e6	1.69008	0.845038	+	0.534706i	$0.179578\pi$
$338$	−2.77556e6	−1.32147
$339$	0	0
$340$	−10663.0	−0.00500243
$341$	−205683.	−0.0957881
$342$	0	0
$343$	−7.31589e6	−3.35762
$344$	−2.02220e6	−0.921358
$345$	0	0
$346$	−2.22365e6	−0.998564
$347$	−3.53103e6	−1.57426	−0.787131	−	0.616785i	$-0.788435\pi$
$347$	−3.53103e6	−1.57426	−0.787131	+	0.616785i	$0.788435\pi$
$348$	0	0
$349$	−1.25322e6	−0.550760	−0.275380	−	0.961335i	$-0.588804\pi$
$349$	−1.25322e6	−0.550760	−0.275380	+	0.961335i	$0.588804\pi$
$350$	−3.74562e6	−1.63438
$351$	0	0
$352$	−284142.	−0.122230
$353$	−964634.	−0.412027	−0.206014	−	0.978549i	$-0.566049\pi$
$353$	−964634.	−0.412027	−0.206014	+	0.978549i	$0.566049\pi$
$354$	0	0
$355$	−109485.	−0.0461087
$356$	−1419.31	−0.000593543 0
$357$	0	0
$358$	1.12651e6	0.464544
$359$	1.26957e6	0.519900	0.259950	−	0.965622i	$-0.416294\pi$
$359$	1.26957e6	0.519900	0.259950	+	0.965622i	$0.416294\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	−306554.	−0.122952
$363$	0	0
$364$	437067.	0.172900
$365$	629896.	0.247478
$366$	0	0
$367$	−1.26919e6	−0.491881	−0.245941	−	0.969285i	$-0.579097\pi$
$367$	−1.26919e6	−0.491881	−0.245941	+	0.969285i	$0.579097\pi$
$368$	1.23125e6	0.473942
$369$	0	0
$370$	410014.	0.155702
$371$	−5.03469e6	−1.89906
$372$	0	0
$373$	3.47150e6	1.29195	0.645975	−	0.763359i	$-0.276451\pi$
$373$	3.47150e6	1.29195	0.645975	+	0.763359i	$0.276451\pi$
$374$	575930.	0.212908
$375$	0	0
$376$	3.49769e6	1.27589
$377$	6.47517e6	2.34638
$378$	0	0
$379$	−2.84273e6	−1.01657	−0.508286	−	0.861188i	$-0.669721\pi$
$379$	−2.84273e6	−1.01657	−0.508286	+	0.861188i	$0.669721\pi$
$380$	16146.5	0.00573613
$381$	0	0
$382$	−1.76918e6	−0.620319
$383$	1.59297e6	0.554894	0.277447	−	0.960741i	$-0.410512\pi$
$383$	1.59297e6	0.554894	0.277447	+	0.960741i	$0.410512\pi$
$384$	0	0
$385$	−2.45604e6	−0.844470
$386$	−783538.	−0.267665
$387$	0	0
$388$	166857.	0.0562686
$389$	5.24604e6	1.75775	0.878876	−	0.477050i	$-0.158294\pi$
$389$	5.24604e6	1.75775	0.878876	+	0.477050i	$0.158294\pi$
$390$	0	0
$391$	−271532.	−0.0898211
$392$	8.06243e6	2.65003
$393$	0	0
$394$	−3.79058e6	−1.23017
$395$	1.61167e6	0.519736
$396$	0	0
$397$	−361755.	−0.115196	−0.0575982	−	0.998340i	$-0.518344\pi$
$397$	−361755.	−0.115196	−0.0575982	+	0.998340i	$0.518344\pi$
$398$	1.69864e6	0.537518
$399$	0	0
$400$	2.77520e6	0.867250
$401$	3.07851e6	0.956049	0.478025	−	0.878346i	$-0.341353\pi$
$401$	3.07851e6	0.956049	0.478025	+	0.878346i	$0.341353\pi$
$402$	0	0
$403$	−456461.	−0.140004
$404$	237492.	0.0723927
$405$	0	0
$406$	−1.02590e7	−3.08879
$407$	−1.23739e6	−0.370273
$408$	0	0
$409$	4.13583e6	1.22252	0.611258	−	0.791432i	$-0.290664\pi$
$409$	4.13583e6	1.22252	0.611258	+	0.791432i	$0.290664\pi$
$410$	−1.28412e6	−0.377265
$411$	0	0
$412$	−208733.	−0.0605827
$413$	−1.22410e7	−3.53135
$414$	0	0
$415$	53394.0	0.0152185
$416$	−630583.	−0.178652
$417$	0	0
$418$	−872107.	−0.244134
$419$	−3.19560e6	−0.889238	−0.444619	−	0.895720i	$-0.646661\pi$
$419$	−3.19560e6	−0.889238	−0.444619	+	0.895720i	$0.646661\pi$
$420$	0	0
$421$	996574.	0.274034	0.137017	−	0.990569i	$-0.456249\pi$
$421$	996574.	0.274034	0.137017	+	0.990569i	$0.456249\pi$
$422$	−6.11710e6	−1.67211
$423$	0	0
$424$	3.52117e6	0.951202
$425$	−612026.	−0.164361
$426$	0	0
$427$	−6.15486e6	−1.63361
$428$	−304493.	−0.0803467
$429$	0	0
$430$	−1.58636e6	−0.413743
$431$	2.51955e6	0.653325	0.326662	−	0.945141i	$-0.394076\pi$
$431$	2.51955e6	0.653325	0.326662	+	0.945141i	$0.394076\pi$
$432$	0	0
$433$	−3.82101e6	−0.979395	−0.489698	−	0.871892i	$-0.662893\pi$
$433$	−3.82101e6	−0.979395	−0.489698	+	0.871892i	$0.662893\pi$
$434$	723197.	0.184303
$435$	0	0
$436$	−211684.	−0.0533300
$437$	411169.	0.102995
$438$	0	0
$439$	619538.	0.153429	0.0767144	−	0.997053i	$-0.475557\pi$
$439$	619538.	0.153429	0.0767144	+	0.997053i	$0.475557\pi$
$440$	1.71771e6	0.422979
$441$	0	0
$442$	1.27813e6	0.311187
$443$	1.08715e6	0.263198	0.131599	−	0.991303i	$-0.457989\pi$
$443$	1.08715e6	0.263198	0.131599	+	0.991303i	$0.457989\pi$
$444$	0	0
$445$	17699.9	0.00423713
$446$	477355.	0.113633
$447$	0	0
$448$	−7.67021e6	−1.80556
$449$	4.56352e6	1.06828	0.534139	−	0.845397i	$-0.320636\pi$
$449$	4.56352e6	1.06828	0.534139	+	0.845397i	$0.320636\pi$
$450$	0	0
$451$	3.87539e6	0.897169
$452$	−219926.	−0.0506327
$453$	0	0
$454$	5.30677e6	1.20834
$455$	−5.45058e6	−1.23428
$456$	0	0
$457$	605485.	0.135617	0.0678083	−	0.997698i	$-0.478399\pi$
$457$	605485.	0.135617	0.0678083	+	0.997698i	$0.478399\pi$
$458$	3.84716e6	0.856992
$459$	0	0
$460$	50942.9	0.0112251
$461$	7.26112e6	1.59130	0.795649	−	0.605759i	$-0.207130\pi$
$461$	7.26112e6	1.59130	0.795649	+	0.605759i	$0.207130\pi$
$462$	0	0
$463$	1.84692e6	0.400400	0.200200	−	0.979755i	$-0.435841\pi$
$463$	1.84692e6	0.400400	0.200200	+	0.979755i	$0.435841\pi$
$464$	7.60106e6	1.63900
$465$	0	0
$466$	5.04317e6	1.07582
$467$	−6.63166e6	−1.40712	−0.703559	−	0.710637i	$-0.748407\pi$
$467$	−6.63166e6	−1.40712	−0.703559	+	0.710637i	$0.748407\pi$
$468$	0	0
$469$	1.49069e7	3.12936
$470$	2.74384e6	0.572947
$471$	0	0
$472$	8.56112e6	1.76879
$473$	4.78752e6	0.983916
$474$	0	0
$475$	926765.	0.188467
$476$	−113148.	−0.0228891
$477$	0	0
$478$	−2.24584e6	−0.449582
$479$	2.39693e6	0.477328	0.238664	−	0.971102i	$-0.423291\pi$
$479$	2.39693e6	0.477328	0.238664	+	0.971102i	$0.423291\pi$
$480$	0	0
$481$	−2.74609e6	−0.541193
$482$	7.40418e6	1.45164
$483$	0	0
$484$	21092.7	0.00409278
$485$	−2.08084e6	−0.401685
$486$	0	0
$487$	−2.17359e6	−0.415294	−0.207647	−	0.978204i	$-0.566580\pi$
$487$	−2.17359e6	−0.415294	−0.207647	+	0.978204i	$0.566580\pi$
$488$	4.30460e6	0.818245
$489$	0	0
$490$	6.32474e6	1.19002
$491$	−9.09470e6	−1.70249	−0.851245	−	0.524768i	$-0.824152\pi$
$491$	−9.09470e6	−1.70249	−0.851245	+	0.524768i	$0.824152\pi$
$492$	0	0
$493$	−1.67629e6	−0.310622
$494$	−1.93542e6	−0.356828
$495$	0	0
$496$	−535830.	−0.0977964
$497$	−1.16177e6	−0.210975
$498$	0	0
$499$	8.69614e6	1.56342	0.781709	−	0.623644i	$-0.214348\pi$
$499$	8.69614e6	1.56342	0.781709	+	0.623644i	$0.214348\pi$
$500$	254596.	0.0455436
$501$	0	0
$502$	−6.60636e6	−1.17005
$503$	2.72206e6	0.479708	0.239854	−	0.970809i	$-0.422900\pi$
$503$	2.72206e6	0.479708	0.239854	+	0.970809i	$0.422900\pi$
$504$	0	0
$505$	−2.96171e6	−0.516790
$506$	−2.75154e6	−0.477748
$507$	0	0
$508$	−11481.2	−0.00197391
$509$	−7.17949e6	−1.22828	−0.614142	−	0.789195i	$-0.710498\pi$
$509$	−7.17949e6	−1.22828	−0.614142	+	0.789195i	$0.710498\pi$
$510$	0	0
$511$	6.68401e6	1.13236
$512$	5.32292e6	0.897377
$513$	0	0
$514$	−752149.	−0.125573
$515$	2.60307e6	0.432482
$516$	0	0
$517$	−8.28072e6	−1.36252
$518$	4.35078e6	0.712431
$519$	0	0
$520$	3.81204e6	0.618229
$521$	2.71363e6	0.437982	0.218991	−	0.975727i	$-0.429723\pi$
$521$	2.71363e6	0.437982	0.218991	+	0.975727i	$0.429723\pi$
$522$	0	0
$523$	−8.12813e6	−1.29938	−0.649690	−	0.760199i	$-0.725101\pi$
$523$	−8.12813e6	−1.29938	−0.649690	+	0.760199i	$0.725101\pi$
$524$	−218063.	−0.0346940
$525$	0	0
$526$	5.31975e6	0.838353
$527$	118169.	0.0185343
$528$	0	0
$529$	−5.13909e6	−0.798448
$530$	2.76226e6	0.427144
$531$	0	0
$532$	171335.	0.0262463
$533$	8.60047e6	1.31131
$534$	0	0
$535$	3.79728e6	0.573572
$536$	−1.04256e7	−1.56744
$537$	0	0
$538$	−2.68233e6	−0.399537
$539$	−1.90876e7	−2.82996
$540$	0	0
$541$	6.55456e6	0.962831	0.481416	−	0.876492i	$-0.340123\pi$
$541$	6.55456e6	0.962831	0.481416	+	0.876492i	$0.340123\pi$
$542$	4.22834e6	0.618260
$543$	0	0
$544$	163245.	0.0236507
$545$	2.63987e6	0.380707
$546$	0	0
$547$	4.13403e6	0.590752	0.295376	−	0.955381i	$-0.404555\pi$
$547$	4.13403e6	0.590752	0.295376	+	0.955381i	$0.404555\pi$
$548$	−618603.	−0.0879955
$549$	0	0
$550$	−6.20190e6	−0.874215
$551$	2.53834e6	0.356181
$552$	0	0
$553$	1.71019e7	2.37811
$554$	3.42871e6	0.474632
$555$	0	0
$556$	−432959.	−0.0593963
$557$	1.13647e7	1.55211	0.776053	−	0.630668i	$-0.217219\pi$
$557$	1.13647e7	1.55211	0.776053	+	0.630668i	$0.217219\pi$
$558$	0	0
$559$	1.06247e7	1.43810
$560$	−6.39832e6	−0.862176
$561$	0	0
$562$	649917.	0.0867994
$563$	−1.00795e7	−1.34019	−0.670095	−	0.742276i	$-0.733746\pi$
$563$	−1.00795e7	−1.34019	−0.670095	+	0.742276i	$0.733746\pi$
$564$	0	0
$565$	2.74266e6	0.361452
$566$	3.99545e6	0.524233
$567$	0	0
$568$	812525.	0.105673
$569$	−1.25833e7	−1.62934	−0.814671	−	0.579924i	$-0.803082\pi$
$569$	−1.25833e7	−1.62934	−0.814671	+	0.579924i	$0.803082\pi$
$570$	0	0
$571$	−1.32744e7	−1.70382	−0.851909	−	0.523690i	$-0.824555\pi$
$571$	−1.32744e7	−1.70382	−0.851909	+	0.523690i	$0.824555\pi$
$572$	723684.	0.0924824
$573$	0	0
$574$	−1.36262e7	−1.72621
$575$	2.92399e6	0.368813
$576$	0	0
$577$	−7.77405e6	−0.972093	−0.486046	−	0.873933i	$-0.661561\pi$
$577$	−7.77405e6	−0.972093	−0.486046	+	0.873933i	$0.661561\pi$
$578$	7.93530e6	0.987970
$579$	0	0
$580$	314494.	0.0388189
$581$	566580.	0.0696339
$582$	0	0
$583$	−8.33631e6	−1.01579
$584$	−4.67468e6	−0.567178
$585$	0	0
$586$	−1.09296e7	−1.31480
$587$	9.72112e6	1.16445	0.582225	−	0.813027i	$-0.302182\pi$
$587$	9.72112e6	1.16445	0.582225	+	0.813027i	$0.302182\pi$
$588$	0	0
$589$	−178938.	−0.0212527
$590$	6.71595e6	0.794287
$591$	0	0
$592$	−3.22357e6	−0.378036
$593$	9.51207e6	1.11081	0.555403	−	0.831581i	$-0.312564\pi$
$593$	9.51207e6	1.11081	0.555403	+	0.831581i	$0.312564\pi$
$594$	0	0
$595$	1.41105e6	0.163399
$596$	−426651.	−0.0491991
$597$	0	0
$598$	−6.10635e6	−0.698279
$599$	−2.79130e6	−0.317863	−0.158931	−	0.987290i	$-0.550805\pi$
$599$	−2.79130e6	−0.317863	−0.158931	+	0.987290i	$0.550805\pi$
$600$	0	0
$601$	−1.71913e6	−0.194143	−0.0970716	−	0.995277i	$-0.530948\pi$
$601$	−1.71913e6	−0.194143	−0.0970716	+	0.995277i	$0.530948\pi$
$602$	−1.68333e7	−1.89312
$603$	0	0
$604$	194162.	0.0216557
$605$	−263043.	−0.0292172
$606$	0	0
$607$	−1.11303e7	−1.22612	−0.613061	−	0.790035i	$-0.710062\pi$
$607$	−1.11303e7	−1.22612	−0.613061	+	0.790035i	$0.710062\pi$
$608$	−247195.	−0.0271195
$609$	0	0
$610$	3.37684e6	0.367439
$611$	−1.83770e7	−1.99146
$612$	0	0
$613$	998180.	0.107290	0.0536448	−	0.998560i	$-0.482916\pi$
$613$	998180.	0.107290	0.0536448	+	0.998560i	$0.482916\pi$
$614$	−2.30958e6	−0.247236
$615$	0	0
$616$	1.82272e7	1.93539
$617$	5.14322e6	0.543904	0.271952	−	0.962311i	$-0.412331\pi$
$617$	5.14322e6	0.543904	0.271952	+	0.962311i	$0.412331\pi$
$618$	0	0
$619$	−1.51122e7	−1.58526	−0.792632	−	0.609701i	$-0.791290\pi$
$619$	−1.51122e7	−1.58526	−0.792632	+	0.609701i	$0.791290\pi$
$620$	−22170.0	−0.00231626
$621$	0	0
$622$	−6.92428e6	−0.717626
$623$	187819.	0.0193874
$624$	0	0
$625$	4.84753e6	0.496387
$626$	4.52963e6	0.461984
$627$	0	0
$628$	836308.	0.0846189
$629$	710908.	0.0716451
$630$	0	0
$631$	−3.02455e6	−0.302404	−0.151202	−	0.988503i	$-0.548314\pi$
$631$	−3.02455e6	−0.302404	−0.151202	+	0.988503i	$0.548314\pi$
$632$	−1.19608e7	−1.19115
$633$	0	0
$634$	2.05269e7	2.02815
$635$	143180.	0.0140912
$636$	0	0
$637$	−4.23603e7	−4.13628
$638$	−1.69865e7	−1.65216
$639$	0	0
$640$	4.72574e6	0.456057
$641$	6.09294e6	0.585709	0.292855	−	0.956157i	$-0.405395\pi$
$641$	6.09294e6	0.585709	0.292855	+	0.956157i	$0.405395\pi$
$642$	0	0
$643$	7.43059e6	0.708755	0.354377	−	0.935103i	$-0.384693\pi$
$643$	7.43059e6	0.708755	0.354377	+	0.935103i	$0.384693\pi$
$644$	540570.	0.0513615
$645$	0	0
$646$	501043.	0.0472382
$647$	1.75431e6	0.164758	0.0823789	−	0.996601i	$-0.473748\pi$
$647$	1.75431e6	0.164758	0.0823789	+	0.996601i	$0.473748\pi$
$648$	0	0
$649$	−2.02683e7	−1.88888
$650$	−1.37636e7	−1.27776
$651$	0	0
$652$	−854189.	−0.0786928
$653$	6.02593e6	0.553021	0.276510	−	0.961011i	$-0.410822\pi$
$653$	6.02593e6	0.553021	0.276510	+	0.961011i	$0.410822\pi$
$654$	0	0
$655$	2.71943e6	0.247670
$656$	1.00959e7	0.915979
$657$	0	0
$658$	2.91157e7	2.62158
$659$	1.15725e7	1.03804	0.519018	−	0.854763i	$-0.326298\pi$
$659$	1.15725e7	1.03804	0.519018	+	0.854763i	$0.326298\pi$
$660$	0	0
$661$	−8.34628e6	−0.743001	−0.371500	−	0.928433i	$-0.621157\pi$
$661$	−8.34628e6	−0.743001	−0.371500	+	0.928433i	$0.621157\pi$
$662$	−1.28741e7	−1.14176
$663$	0	0
$664$	−396256.	−0.0348783
$665$	−2.13669e6	−0.187364
$666$	0	0
$667$	8.00857e6	0.697012
$668$	−848192.	−0.0735449
$669$	0	0
$670$	−8.17860e6	−0.703869
$671$	−1.01911e7	−0.873802
$672$	0	0
$673$	−5.33266e6	−0.453843	−0.226922	−	0.973913i	$-0.572866\pi$
$673$	−5.33266e6	−0.453843	−0.226922	+	0.973913i	$0.572866\pi$
$674$	−2.05136e7	−1.73937
$675$	0	0
$676$	902877.	0.0759909
$677$	−2.14247e7	−1.79656	−0.898282	−	0.439418i	$-0.855185\pi$
$677$	−2.14247e7	−1.79656	−0.898282	+	0.439418i	$0.855185\pi$
$678$	0	0
$679$	−2.20805e7	−1.83795
$680$	−986861.	−0.0818434
$681$	0	0
$682$	1.19745e6	0.0985818
$683$	−2.01614e7	−1.65375	−0.826874	−	0.562388i	$-0.809883\pi$
$683$	−2.01614e7	−1.65375	−0.826874	+	0.562388i	$0.809883\pi$
$684$	0	0
$685$	7.71448e6	0.628174
$686$	4.25920e7	3.45555
$687$	0	0
$688$	1.24721e7	1.00454
$689$	−1.85004e7	−1.48468
$690$	0	0
$691$	1.54453e7	1.23056	0.615279	−	0.788310i	$-0.289043\pi$
$691$	1.54453e7	1.23056	0.615279	+	0.788310i	$0.289043\pi$
$692$	723343.	0.0574221
$693$	0	0
$694$	2.05571e7	1.62018
$695$	5.39934e6	0.424013
$696$	0	0
$697$	−2.22649e6	−0.173596
$698$	7.29603e6	0.566823
$699$	0	0
$700$	1.21843e6	0.0939846
$701$	3.36415e6	0.258571	0.129285	−	0.991607i	$-0.458732\pi$
$701$	3.36415e6	0.258571	0.129285	+	0.991607i	$0.458732\pi$
$702$	0	0
$703$	−1.07650e6	−0.0821532
$704$	−1.27001e7	−0.965777
$705$	0	0
$706$	5.61595e6	0.424044
$707$	−3.14276e7	−2.36463
$708$	0	0
$709$	9.46957e6	0.707481	0.353740	−	0.935344i	$-0.384910\pi$
$709$	9.46957e6	0.707481	0.353740	+	0.935344i	$0.384910\pi$
$710$	637402.	0.0474535
$711$	0	0
$712$	−131358.	−0.00971080
$713$	−564557.	−0.0415895
$714$	0	0
$715$	−9.02493e6	−0.660205
$716$	−366448.	−0.0267135
$717$	0	0
$718$	−7.39122e6	−0.535063
$719$	1.17104e7	0.844790	0.422395	−	0.906412i	$-0.361189\pi$
$719$	1.17104e7	0.844790	0.422395	+	0.906412i	$0.361189\pi$
$720$	0	0
$721$	2.76219e7	1.97886
$722$	−758708.	−0.0541666
$723$	0	0
$724$	99720.6	0.00707031
$725$	1.80511e7	1.27544
$726$	0	0
$727$	1.55978e7	1.09453	0.547265	−	0.836959i	$-0.315669\pi$
$727$	1.55978e7	1.09453	0.547265	+	0.836959i	$0.315669\pi$
$728$	4.04507e7	2.82877
$729$	0	0
$730$	−3.66715e6	−0.254696
$731$	−2.75053e6	−0.190381
$732$	0	0
$733$	172411.	0.0118523	0.00592617	−	0.999982i	$-0.498114\pi$
$733$	172411.	0.0118523	0.00592617	+	0.999982i	$0.498114\pi$
$734$	7.38900e6	0.506227
$735$	0	0
$736$	−779913.	−0.0530703
$737$	2.46825e7	1.67386
$738$	0	0
$739$	−1.29022e6	−0.0869064	−0.0434532	−	0.999055i	$-0.513836\pi$
$739$	−1.29022e6	−0.0869064	−0.0434532	+	0.999055i	$0.513836\pi$
$740$	−133376.	−0.00895359
$741$	0	0
$742$	2.93111e7	1.95444
$743$	3.69707e6	0.245689	0.122844	−	0.992426i	$-0.460798\pi$
$743$	3.69707e6	0.245689	0.122844	+	0.992426i	$0.460798\pi$
$744$	0	0
$745$	5.32068e6	0.351218
$746$	−2.02105e7	−1.32963
$747$	0	0
$748$	−187348.	−0.0122432
$749$	4.02940e7	2.62444
$750$	0	0
$751$	−188742.	−0.0122115	−0.00610575	−	0.999981i	$-0.501944\pi$
$751$	−188742.	−0.0122115	−0.00610575	+	0.999981i	$0.501944\pi$
$752$	−2.15724e7	−1.39108
$753$	0	0
$754$	−3.76974e7	−2.41481
$755$	−2.42136e6	−0.154594
$756$	0	0
$757$	−2.13322e7	−1.35299	−0.676497	−	0.736446i	$-0.736503\pi$
$757$	−2.13322e7	−1.35299	−0.676497	+	0.736446i	$0.736503\pi$
$758$	1.65499e7	1.04622
$759$	0	0
$760$	1.49436e6	0.0938473
$761$	2.83469e7	1.77437	0.887185	−	0.461414i	$-0.152658\pi$
$761$	2.83469e7	1.77437	0.887185	+	0.461414i	$0.152658\pi$
$762$	0	0
$763$	2.80124e7	1.74197
$764$	575508.	0.0356712
$765$	0	0
$766$	−9.27400e6	−0.571078
$767$	−4.49804e7	−2.76080
$768$	0	0
$769$	2.37153e7	1.44615	0.723073	−	0.690772i	$-0.242729\pi$
$769$	2.37153e7	1.44615	0.723073	+	0.690772i	$0.242729\pi$
$770$	1.42987e7	0.869100
$771$	0	0
$772$	254881.	0.0153920
$773$	−92234.1	−0.00555192	−0.00277596	−	0.999996i	$-0.500884\pi$
$773$	−92234.1	−0.00555192	−0.00277596	+	0.999996i	$0.500884\pi$
$774$	0	0
$775$	−1.27250e6	−0.0761033
$776$	1.54427e7	0.920595
$777$	0	0
$778$	−3.05416e7	−1.80902
$779$	3.37148e6	0.199057
$780$	0	0
$781$	−1.92364e6	−0.112848
$782$	1.58081e6	0.0924408
$783$	0	0
$784$	−4.97258e7	−2.88929
$785$	−1.04294e7	−0.604070
$786$	0	0
$787$	350387.	0.0201656	0.0100828	−	0.999949i	$-0.496790\pi$
$787$	350387.	0.0201656	0.0100828	+	0.999949i	$0.496790\pi$
$788$	1.23306e6	0.0707404
$789$	0	0
$790$	−9.38288e6	−0.534895
$791$	2.91031e7	1.65386
$792$	0	0
$793$	−2.26165e7	−1.27715
$794$	2.10608e6	0.118556
$795$	0	0
$796$	−552559.	−0.0309098
$797$	−9.25461e6	−0.516074	−0.258037	−	0.966135i	$-0.583076\pi$
$797$	−9.25461e6	−0.516074	−0.258037	+	0.966135i	$0.583076\pi$
$798$	0	0
$799$	4.75744e6	0.263637
$800$	−1.75791e6	−0.0971115
$801$	0	0
$802$	−1.79226e7	−0.983933
$803$	1.10672e7	0.605688
$804$	0	0
$805$	−6.74134e6	−0.366654
$806$	2.65744e6	0.144088
$807$	0	0
$808$	2.19799e7	1.18440
$809$	2.94050e6	0.157961	0.0789805	−	0.996876i	$-0.474834\pi$
$809$	2.94050e6	0.157961	0.0789805	+	0.996876i	$0.474834\pi$
$810$	0	0
$811$	2.12110e7	1.13242	0.566211	−	0.824260i	$-0.308409\pi$
$811$	2.12110e7	1.13242	0.566211	+	0.824260i	$0.308409\pi$
$812$	3.33719e6	0.177620
$813$	0	0
$814$	7.20391e6	0.381072
$815$	1.06524e7	0.561765
$816$	0	0
$817$	4.16501e6	0.218303
$818$	−2.40781e7	−1.25817
$819$	0	0
$820$	417719.	0.0216945
$821$	28537.6	0.00147761	0.000738804	−	1.00000i	$-0.499765\pi$
$821$	28537.6	0.00147761	0.000738804		1.00000i	$0.499765\pi$
$822$	0	0
$823$	−1.60864e7	−0.827863	−0.413931	−	0.910308i	$-0.635845\pi$
$823$	−1.60864e7	−0.827863	−0.413931	+	0.910308i	$0.635845\pi$
$824$	−1.93183e7	−0.991177
$825$	0	0
$826$	7.12650e7	3.63434
$827$	1.43894e7	0.731607	0.365803	−	0.930692i	$-0.380794\pi$
$827$	1.43894e7	0.731607	0.365803	+	0.930692i	$0.380794\pi$
$828$	0	0
$829$	2.29442e7	1.15954	0.579771	−	0.814779i	$-0.303142\pi$
$829$	2.29442e7	1.15954	0.579771	+	0.814779i	$0.303142\pi$
$830$	−310851.	−0.0156624
$831$	0	0
$832$	−2.81848e7	−1.41158
$833$	1.09662e7	0.547577
$834$	0	0
$835$	1.05776e7	0.525016
$836$	283692.	0.0140389
$837$	0	0
$838$	1.86043e7	0.915173
$839$	3.34833e7	1.64219	0.821095	−	0.570792i	$-0.193364\pi$
$839$	3.34833e7	1.64219	0.821095	+	0.570792i	$0.193364\pi$
$840$	0	0
$841$	2.89295e7	1.41043
$842$	−5.80189e6	−0.282026
$843$	0	0
$844$	1.98986e6	0.0961540
$845$	−1.12596e7	−0.542477
$846$	0	0
$847$	−2.79123e6	−0.133686
$848$	−2.17172e7	−1.03708
$849$	0	0
$850$	3.56312e6	0.169154
$851$	−3.39640e6	−0.160766
$852$	0	0
$853$	−9.55702e6	−0.449728	−0.224864	−	0.974390i	$-0.572194\pi$
$853$	−9.55702e6	−0.449728	−0.224864	+	0.974390i	$0.572194\pi$
$854$	3.58326e7	1.68126
$855$	0	0
$856$	−2.81809e7	−1.31453
$857$	−1.69429e7	−0.788019	−0.394009	−	0.919106i	$-0.628912\pi$
$857$	−1.69429e7	−0.788019	−0.394009	+	0.919106i	$0.628912\pi$
$858$	0	0
$859$	−6.40368e6	−0.296106	−0.148053	−	0.988979i	$-0.547301\pi$
$859$	−6.40368e6	−0.296106	−0.148053	+	0.988979i	$0.547301\pi$
$860$	516035.	0.0237921
$861$	0	0
$862$	−1.46684e7	−0.672379
$863$	−3.12340e6	−0.142758	−0.0713791	−	0.997449i	$-0.522740\pi$
$863$	−3.12340e6	−0.142758	−0.0713791	+	0.997449i	$0.522740\pi$
$864$	0	0
$865$	−9.02067e6	−0.409919
$866$	2.22453e7	1.00796
$867$	0	0
$868$	−235253.	−0.0105983
$869$	2.83169e7	1.27203
$870$	0	0
$871$	5.47766e7	2.44652
$872$	−1.95914e7	−0.872518
$873$	0	0
$874$	−2.39376e6	−0.105999
$875$	−3.36911e7	−1.48763
$876$	0	0
$877$	3.31907e7	1.45719	0.728597	−	0.684943i	$-0.240173\pi$
$877$	3.31907e7	1.45719	0.728597	+	0.684943i	$0.240173\pi$
$878$	−3.60685e6	−0.157904
$879$	0	0
$880$	−1.05942e7	−0.461169
$881$	−3.73467e7	−1.62111	−0.810555	−	0.585662i	$-0.800835\pi$
$881$	−3.73467e7	−1.62111	−0.810555	+	0.585662i	$0.800835\pi$
$882$	0	0
$883$	−3.95144e7	−1.70551	−0.852753	−	0.522314i	$-0.825069\pi$
$883$	−3.95144e7	−1.70551	−0.852753	+	0.522314i	$0.825069\pi$
$884$	−415771.	−0.0178947
$885$	0	0
$886$	−6.32924e6	−0.270874
$887$	−3.39325e7	−1.44813	−0.724064	−	0.689732i	$-0.757728\pi$
$887$	−3.39325e7	−1.44813	−0.724064	+	0.689732i	$0.757728\pi$
$888$	0	0
$889$	1.51932e6	0.0644757
$890$	−103046.	−0.00436071
$891$	0	0
$892$	−155281.	−0.00653442
$893$	−7.20399e6	−0.302304
$894$	0	0
$895$	4.56990e6	0.190699
$896$	5.01462e7	2.08674
$897$	0	0
$898$	−2.65681e7	−1.09943
$899$	−3.48528e6	−0.143826
$900$	0	0
$901$	4.78938e6	0.196547
$902$	−2.25619e7	−0.923335
$903$	0	0
$904$	−2.03542e7	−0.828388
$905$	−1.24360e6	−0.0504729
$906$	0	0
$907$	−1.00269e7	−0.404716	−0.202358	−	0.979312i	$-0.564860\pi$
$907$	−1.00269e7	−0.404716	−0.202358	+	0.979312i	$0.564860\pi$
$908$	−1.72627e6	−0.0694854
$909$	0	0
$910$	3.17324e7	1.27028
$911$	6.57126e6	0.262333	0.131166	−	0.991360i	$-0.458128\pi$
$911$	6.57126e6	0.262333	0.131166	+	0.991360i	$0.458128\pi$
$912$	0	0
$913$	938128.	0.0372465
$914$	−3.52504e6	−0.139572
$915$	0	0
$916$	−1.25146e6	−0.0492810
$917$	2.88566e7	1.13324
$918$	0	0
$919$	−7.60638e6	−0.297091	−0.148545	−	0.988906i	$-0.547459\pi$
$919$	−7.60638e6	−0.297091	−0.148545	+	0.988906i	$0.547459\pi$
$920$	4.71478e6	0.183650
$921$	0	0
$922$	−4.22731e7	−1.63771
$923$	−4.26903e6	−0.164940
$924$	0	0
$925$	−7.65541e6	−0.294181
$926$	−1.07524e7	−0.412078
$927$	0	0
$928$	−4.81477e6	−0.183529
$929$	3.40362e7	1.29390	0.646951	−	0.762532i	$-0.276044\pi$
$929$	3.40362e7	1.29390	0.646951	+	0.762532i	$0.276044\pi$
$930$	0	0
$931$	−1.66057e7	−0.627889
$932$	−1.64052e6	−0.0618645
$933$	0	0
$934$	3.86085e7	1.44816
$935$	2.33637e6	0.0874004
$936$	0	0
$937$	1.77553e6	0.0660663	0.0330332	−	0.999454i	$-0.489483\pi$
$937$	1.77553e6	0.0660663	0.0330332	+	0.999454i	$0.489483\pi$
$938$	−8.67855e7	−3.22063
$939$	0	0
$940$	−892558.	−0.0329471
$941$	−1.16181e7	−0.427721	−0.213860	−	0.976864i	$-0.568604\pi$
$941$	−1.16181e7	−0.427721	−0.213860	+	0.976864i	$0.568604\pi$
$942$	0	0
$943$	1.06372e7	0.389535
$944$	−5.28016e7	−1.92849
$945$	0	0
$946$	−2.78722e7	−1.01261
$947$	−3.38853e7	−1.22782	−0.613912	−	0.789374i	$-0.710405\pi$
$947$	−3.38853e7	−1.22782	−0.613912	+	0.789374i	$0.710405\pi$
$948$	0	0
$949$	2.45609e7	0.885277
$950$	−5.39548e6	−0.193964
$951$	0	0
$952$	−1.04719e7	−0.374483
$953$	2.44407e7	0.871729	0.435864	−	0.900012i	$-0.356443\pi$
$953$	2.44407e7	0.871729	0.435864	+	0.900012i	$0.356443\pi$
$954$	0	0
$955$	−7.17705e6	−0.254646
$956$	730561.	0.0258530
$957$	0	0
$958$	−1.39546e7	−0.491250
$959$	8.18606e7	2.87427
$960$	0	0
$961$	−2.83835e7	−0.991418
$962$	1.59873e7	0.556977
$963$	0	0
$964$	−2.40855e6	−0.0834762
$965$	−3.17858e6	−0.109879
$966$	0	0
$967$	−3.67340e7	−1.26329	−0.631644	−	0.775258i	$-0.717620\pi$
$967$	−3.67340e7	−1.26329	−0.631644	+	0.775258i	$0.717620\pi$
$968$	1.95214e6	0.0669609
$969$	0	0
$970$	1.21143e7	0.413400
$971$	3.43205e7	1.16817	0.584085	−	0.811693i	$-0.301454\pi$
$971$	3.43205e7	1.16817	0.584085	+	0.811693i	$0.301454\pi$
$972$	0	0
$973$	5.72940e7	1.94011
$974$	1.26543e7	0.427406
$975$	0	0
$976$	−2.65490e7	−0.892122
$977$	8.40409e6	0.281679	0.140839	−	0.990032i	$-0.455020\pi$
$977$	8.40409e6	0.281679	0.140839	+	0.990032i	$0.455020\pi$
$978$	0	0
$979$	310986.	0.0103701
$980$	−2.05741e6	−0.0684314
$981$	0	0
$982$	5.29479e7	1.75215
$983$	−1.87755e7	−0.619738	−0.309869	−	0.950779i	$-0.600285\pi$
$983$	−1.87755e7	−0.619738	−0.309869	+	0.950779i	$0.600285\pi$
$984$	0	0
$985$	−1.53772e7	−0.504995
$986$	9.75910e6	0.319682
$987$	0	0
$988$	629585.	0.0205193
$989$	1.31408e7	0.427199
$990$	0	0
$991$	−1.98114e6	−0.0640814	−0.0320407	−	0.999487i	$-0.510201\pi$
$991$	−1.98114e6	−0.0640814	−0.0320407	+	0.999487i	$0.510201\pi$
$992$	339413.	0.0109509
$993$	0	0
$994$	6.76366e6	0.217128
$995$	6.89086e6	0.220656
$996$	0	0
$997$	1.55752e6	0.0496244	0.0248122	−	0.999692i	$-0.492101\pi$
$997$	1.55752e6	0.0496244	0.0248122	+	0.999692i	$0.492101\pi$
$998$	−5.06275e7	−1.60902
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.6.a.j.1.2		4
3.2	odd	2		57.6.a.f.1.3	✓	4
12.11	even	2		912.6.a.r.1.3		4
57.56	even	2		1083.6.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.6.a.f.1.3	✓	4	3.2	odd	2
171.6.a.j.1.2		4	1.1	even	1	trivial
912.6.a.r.1.3		4	12.11	even	2
1083.6.a.g.1.2		4	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q-5.82184 q^{2} +1.89382 q^{4} -23.6174 q^{5} -250.612 q^{7} +175.273 q^{8} +137.497 q^{10} -414.956 q^{11} -920.892 q^{13} +1459.02 q^{14} -1081.02 q^{16} +238.401 q^{17} -361.000 q^{19} -44.7271 q^{20} +2415.81 q^{22} -1138.97 q^{23} -2567.22 q^{25} +5361.29 q^{26} -474.613 q^{28} -7031.41 q^{29} +495.673 q^{31} +684.752 q^{32} -1387.93 q^{34} +5918.80 q^{35} +2981.99 q^{37} +2101.68 q^{38} -4139.51 q^{40} -9339.27 q^{41} -11537.4 q^{43} -785.851 q^{44} +6630.91 q^{46} +19955.6 q^{47} +45999.2 q^{49} +14945.9 q^{50} -1744.00 q^{52} +20089.6 q^{53} +9800.20 q^{55} -43925.5 q^{56} +40935.7 q^{58} +48844.4 q^{59} +24559.4 q^{61} -2885.73 q^{62} +30606.0 q^{64} +21749.1 q^{65} -59482.1 q^{67} +451.487 q^{68} -34458.3 q^{70} +4635.76 q^{71} -26670.8 q^{73} -17360.6 q^{74} -683.668 q^{76} +103993. q^{77} -68240.6 q^{79} +25530.8 q^{80} +54371.8 q^{82} -2260.79 q^{83} -5630.41 q^{85} +67169.0 q^{86} -72730.8 q^{88} -749.444 q^{89} +230786. q^{91} -2157.00 q^{92} -116179. q^{94} +8525.89 q^{95} +88106.3 q^{97} -267800. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 53 q^{4} + 8 q^{5} - 142 q^{7} + 147 q^{8} - 496 q^{10} + 714 q^{11} - 74 q^{13} + 2790 q^{14} - 2839 q^{16} + 3690 q^{17} - 1444 q^{19} + 5158 q^{20} - 1582 q^{22} - 862 q^{23} + 3282 q^{25}+ \cdots - 178319 q^{98}+O(q^{100}) \) 4 * q - q^2 + 53 * q^4 + 8 * q^5 - 142 * q^7 + 147 * q^8 - 496 * q^10 + 714 * q^11 - 74 * q^13 + 2790 * q^14 - 2839 * q^16 + 3690 * q^17 - 1444 * q^19 + 5158 * q^20 - 1582 * q^22 - 862 * q^23 + 3282 * q^25 - 870 * q^26 + 5488 * q^28 - 992 * q^29 - 7382 * q^31 + 6103 * q^32 + 10288 * q^34 + 10242 * q^35 + 19114 * q^37 + 361 * q^38 + 19470 * q^40 + 5792 * q^41 - 18634 * q^43 + 5316 * q^44 + 62256 * q^46 + 35424 * q^47 + 15910 * q^49 - 14309 * q^50 + 48170 * q^52 + 20256 * q^53 + 23538 * q^55 - 27516 * q^56 + 37566 * q^58 + 76944 * q^59 + 11562 * q^61 + 104740 * q^62 - 15455 * q^64 + 157464 * q^65 - 110044 * q^67 + 70926 * q^68 + 24138 * q^70 + 71184 * q^71 - 138830 * q^73 - 91586 * q^74 - 19133 * q^76 + 76122 * q^77 - 59470 * q^79 - 81842 * q^80 + 17030 * q^82 + 191600 * q^83 + 23682 * q^85 - 56246 * q^86 - 134640 * q^88 - 158920 * q^89 + 267196 * q^91 - 34460 * q^92 + 35946 * q^94 - 2888 * q^95 + 47460 * q^97 - 178319 * q^98

Introduction

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