LMFDB - Embedded newform 224.6.a.i.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [224,6,Mod(1,224)] 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("224.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	224.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.9259756381$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 $ x^5 - 229x^3 - 272x^2 + 7973*x - 13998
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.97668$ of defining polynomial
Character	$\chi$	$=$	224.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.95336 q^{3} +11.6830 q^{5} +49.0000 q^{7} -207.557 q^{9} +12.4129 q^{11} -8.71531 q^{13} +69.5530 q^{15} +2127.81 q^{17} +2222.88 q^{19} +291.715 q^{21} -2947.98 q^{23} -2988.51 q^{25} -2682.33 q^{27} +2466.27 q^{29} +9139.71 q^{31} +73.8983 q^{33} +572.466 q^{35} +6680.55 q^{37} -51.8854 q^{39} +11344.4 q^{41} -11812.9 q^{43} -2424.89 q^{45} -4124.21 q^{47} +2401.00 q^{49} +12667.6 q^{51} +36812.7 q^{53} +145.019 q^{55} +13233.6 q^{57} -1642.24 q^{59} +41959.0 q^{61} -10170.3 q^{63} -101.821 q^{65} -50736.8 q^{67} -17550.4 q^{69} +23385.3 q^{71} +52990.4 q^{73} -17791.7 q^{75} +608.230 q^{77} +62433.7 q^{79} +34467.6 q^{81} +7745.46 q^{83} +24859.1 q^{85} +14682.6 q^{87} +5087.49 q^{89} -427.050 q^{91} +54412.0 q^{93} +25969.8 q^{95} +98615.3 q^{97} -2576.38 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) $ 5 * q - 10 * q^3 + 36 * q^5 + 245 * q^7 + 637 * q^9 + 116 * q^11 + 40 * q^13 + 16 * q^15 - 402 * q^17 - 3582 * q^19 - 490 * q^21 + 472 * q^23 + 9615 * q^25 + 356 * q^27 + 4754 * q^29 - 10500 * q^31 + 15864 * q^33 + 1764 * q^35 + 19642 * q^37 + 10872 * q^39 + 23398 * q^41 + 22044 * q^43 + 49476 * q^45 + 16004 * q^47 + 12005 * q^49 - 45676 * q^51 + 54246 * q^53 + 53456 * q^55 + 109556 * q^57 - 74366 * q^59 + 68316 * q^61 + 31213 * q^63 + 152568 * q^65 - 26560 * q^67 + 214720 * q^69 + 93072 * q^71 + 136098 * q^73 - 124510 * q^75 + 5684 * q^77 + 96080 * q^79 + 104801 * q^81 - 145894 * q^83 + 117352 * q^85 + 168876 * q^87 + 188554 * q^89 + 1960 * q^91 + 86296 * q^93 + 74736 * q^95 - 88146 * q^97 - 260236 * 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
$3$	5.95336	0.381908	0.190954	−	0.981599i	$-0.438842\pi$
$3$	5.95336	0.381908	0.190954	+	0.981599i	$0.438842\pi$
$4$	0	0
$5$	11.6830	0.208991	0.104496	−	0.994525i	$-0.466677\pi$
$5$	11.6830	0.208991	0.104496	+	0.994525i	$0.466677\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−207.557	−0.854146
$10$	0	0
$11$	12.4129	0.0309307	0.0154654	−	0.999880i	$-0.495077\pi$
$11$	12.4129	0.0309307	0.0154654	+	0.999880i	$0.495077\pi$
$12$	0	0
$13$	−8.71531	−0.0143029	−0.00715146	−	0.999974i	$-0.502276\pi$
$13$	−8.71531	−0.0143029	−0.00715146	+	0.999974i	$0.502276\pi$
$14$	0	0
$15$	69.5530	0.0798156
$16$	0	0
$17$	2127.81	1.78570	0.892852	−	0.450350i	$-0.148701\pi$
$17$	2127.81	1.78570	0.892852	+	0.450350i	$0.148701\pi$
$18$	0	0
$19$	2222.88	1.41264	0.706320	−	0.707893i	$-0.250354\pi$
$19$	2222.88	1.41264	0.706320	+	0.707893i	$0.250354\pi$
$20$	0	0
$21$	291.715	0.144348
$22$	0	0
$23$	−2947.98	−1.16200	−0.580999	−	0.813904i	$-0.697338\pi$
$23$	−2947.98	−1.16200	−0.580999	+	0.813904i	$0.697338\pi$
$24$	0	0
$25$	−2988.51	−0.956323
$26$	0	0
$27$	−2682.33	−0.708114
$28$	0	0
$29$	2466.27	0.544560	0.272280	−	0.962218i	$-0.412222\pi$
$29$	2466.27	0.544560	0.272280	+	0.962218i	$0.412222\pi$
$30$	0	0
$31$	9139.71	1.70816	0.854079	−	0.520143i	$-0.174121\pi$
$31$	9139.71	1.70816	0.854079	+	0.520143i	$0.174121\pi$
$32$	0	0
$33$	73.8983	0.0118127
$34$	0	0
$35$	572.466	0.0789913
$36$	0	0
$37$	6680.55	0.802247	0.401123	−	0.916024i	$-0.368620\pi$
$37$	6680.55	0.802247	0.401123	+	0.916024i	$0.368620\pi$
$38$	0	0
$39$	−51.8854	−0.00546240
$40$	0	0
$41$	11344.4	1.05396	0.526978	−	0.849879i	$-0.323325\pi$
$41$	11344.4	1.05396	0.526978	+	0.849879i	$0.323325\pi$
$42$	0	0
$43$	−11812.9	−0.974282	−0.487141	−	0.873323i	$-0.661960\pi$
$43$	−11812.9	−0.974282	−0.487141	+	0.873323i	$0.661960\pi$
$44$	0	0
$45$	−2424.89	−0.178509
$46$	0	0
$47$	−4124.21	−0.272331	−0.136165	−	0.990686i	$-0.543478\pi$
$47$	−4124.21	−0.272331	−0.136165	+	0.990686i	$0.543478\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	12667.6	0.681975
$52$	0	0
$53$	36812.7	1.80015	0.900073	−	0.435739i	$-0.143513\pi$
$53$	36812.7	1.80015	0.900073	+	0.435739i	$0.143513\pi$
$54$	0	0
$55$	145.019	0.00646426
$56$	0	0
$57$	13233.6	0.539499
$58$	0	0
$59$	−1642.24	−0.0614196	−0.0307098	−	0.999528i	$-0.509777\pi$
$59$	−1642.24	−0.0614196	−0.0307098	+	0.999528i	$0.509777\pi$
$60$	0	0
$61$	41959.0	1.44378	0.721889	−	0.692009i	$-0.243274\pi$
$61$	41959.0	1.44378	0.721889	+	0.692009i	$0.243274\pi$
$62$	0	0
$63$	−10170.3	−0.322837
$64$	0	0
$65$	−101.821	−0.00298919
$66$	0	0
$67$	−50736.8	−1.38082	−0.690408	−	0.723420i	$-0.742569\pi$
$67$	−50736.8	−1.38082	−0.690408	+	0.723420i	$0.742569\pi$
$68$	0	0
$69$	−17550.4	−0.443777
$70$	0	0
$71$	23385.3	0.550551	0.275276	−	0.961365i	$-0.411231\pi$
$71$	23385.3	0.550551	0.275276	+	0.961365i	$0.411231\pi$
$72$	0	0
$73$	52990.4	1.16383	0.581915	−	0.813250i	$-0.302304\pi$
$73$	52990.4	1.16383	0.581915	+	0.813250i	$0.302304\pi$
$74$	0	0
$75$	−17791.7	−0.365228
$76$	0	0
$77$	608.230	0.0116907
$78$	0	0
$79$	62433.7	1.12551	0.562757	−	0.826622i	$-0.309741\pi$
$79$	62433.7	1.12551	0.562757	+	0.826622i	$0.309741\pi$
$80$	0	0
$81$	34467.6	0.583711
$82$	0	0
$83$	7745.46	0.123410	0.0617052	−	0.998094i	$-0.480346\pi$
$83$	7745.46	0.123410	0.0617052	+	0.998094i	$0.480346\pi$
$84$	0	0
$85$	24859.1	0.373197
$86$	0	0
$87$	14682.6	0.207972
$88$	0	0
$89$	5087.49	0.0680814	0.0340407	−	0.999420i	$-0.489162\pi$
$89$	5087.49	0.0680814	0.0340407	+	0.999420i	$0.489162\pi$
$90$	0	0
$91$	−427.050	−0.00540599
$92$	0	0
$93$	54412.0	0.652360
$94$	0	0
$95$	25969.8	0.295230
$96$	0	0
$97$	98615.3	1.06418	0.532090	−	0.846688i	$-0.321407\pi$
$97$	98615.3	1.06418	0.532090	+	0.846688i	$0.321407\pi$
$98$	0	0
$99$	−2576.38	−0.0264194
$100$	0	0
$101$	64993.8	0.633969	0.316985	−	0.948431i	$-0.397330\pi$
$101$	64993.8	0.633969	0.316985	+	0.948431i	$0.397330\pi$
$102$	0	0
$103$	86443.3	0.802857	0.401429	−	0.915890i	$-0.368514\pi$
$103$	86443.3	0.802857	0.401429	+	0.915890i	$0.368514\pi$
$104$	0	0
$105$	3408.10	0.0301675
$106$	0	0
$107$	−177683.	−1.50033	−0.750165	−	0.661250i	$-0.770026\pi$
$107$	−177683.	−1.50033	−0.750165	+	0.661250i	$0.770026\pi$
$108$	0	0
$109$	−153149.	−1.23466	−0.617329	−	0.786705i	$-0.711785\pi$
$109$	−153149.	−1.23466	−0.617329	+	0.786705i	$0.711785\pi$
$110$	0	0
$111$	39771.7	0.306385
$112$	0	0
$113$	−25761.5	−0.189791	−0.0948954	−	0.995487i	$-0.530252\pi$
$113$	−25761.5	−0.189791	−0.0948954	+	0.995487i	$0.530252\pi$
$114$	0	0
$115$	−34441.2	−0.242848
$116$	0	0
$117$	1808.93	0.0122168
$118$	0	0
$119$	104262.	0.674933
$120$	0	0
$121$	−160897.	−0.999043
$122$	0	0
$123$	67537.5	0.402515
$124$	0	0
$125$	−71424.0	−0.408855
$126$	0	0
$127$	72696.9	0.399951	0.199976	−	0.979801i	$-0.435914\pi$
$127$	72696.9	0.399951	0.199976	+	0.979801i	$0.435914\pi$
$128$	0	0
$129$	−70326.3	−0.372086
$130$	0	0
$131$	−181562.	−0.924374	−0.462187	−	0.886783i	$-0.652935\pi$
$131$	−181562.	−0.924374	−0.462187	+	0.886783i	$0.652935\pi$
$132$	0	0
$133$	108921.	0.533928
$134$	0	0
$135$	−31337.6	−0.147990
$136$	0	0
$137$	218519.	0.994690	0.497345	−	0.867553i	$-0.334308\pi$
$137$	218519.	0.994690	0.497345	+	0.867553i	$0.334308\pi$
$138$	0	0
$139$	−282112.	−1.23847	−0.619233	−	0.785207i	$-0.712557\pi$
$139$	−282112.	−1.23847	−0.619233	+	0.785207i	$0.712557\pi$
$140$	0	0
$141$	−24552.9	−0.104005
$142$	0	0
$143$	−108.182	−0.000442400 0
$144$	0	0
$145$	28813.4	0.113808
$146$	0	0
$147$	14294.0	0.0545583
$148$	0	0
$149$	−371936.	−1.37247	−0.686234	−	0.727381i	$-0.740737\pi$
$149$	−371936.	−1.37247	−0.686234	+	0.727381i	$0.740737\pi$
$150$	0	0
$151$	61561.3	0.219718	0.109859	−	0.993947i	$-0.464960\pi$
$151$	61561.3	0.219718	0.109859	+	0.993947i	$0.464960\pi$
$152$	0	0
$153$	−441642.	−1.52525
$154$	0	0
$155$	106779.	0.356990
$156$	0	0
$157$	−179854.	−0.582333	−0.291166	−	0.956672i	$-0.594043\pi$
$157$	−179854.	−0.582333	−0.291166	+	0.956672i	$0.594043\pi$
$158$	0	0
$159$	219159.	0.687491
$160$	0	0
$161$	−144451.	−0.439194
$162$	0	0
$163$	201617.	0.594371	0.297185	−	0.954820i	$-0.403952\pi$
$163$	201617.	0.594371	0.297185	+	0.954820i	$0.403952\pi$
$164$	0	0
$165$	863.352	0.00246876
$166$	0	0
$167$	165570.	0.459400	0.229700	−	0.973261i	$-0.426225\pi$
$167$	165570.	0.459400	0.229700	+	0.973261i	$0.426225\pi$
$168$	0	0
$169$	−371217.	−0.999795
$170$	0	0
$171$	−461375.	−1.20660
$172$	0	0
$173$	−109988.	−0.279403	−0.139702	−	0.990194i	$-0.544614\pi$
$173$	−109988.	−0.279403	−0.139702	+	0.990194i	$0.544614\pi$
$174$	0	0
$175$	−146437.	−0.361456
$176$	0	0
$177$	−9776.85	−0.0234566
$178$	0	0
$179$	−785850.	−1.83319	−0.916594	−	0.399818i	$-0.869073\pi$
$179$	−785850.	−1.83319	−0.916594	+	0.399818i	$0.869073\pi$
$180$	0	0
$181$	−117176.	−0.265854	−0.132927	−	0.991126i	$-0.542438\pi$
$181$	−117176.	−0.265854	−0.132927	+	0.991126i	$0.542438\pi$
$182$	0	0
$183$	249797.	0.551391
$184$	0	0
$185$	78048.7	0.167663
$186$	0	0
$187$	26412.2	0.0552332
$188$	0	0
$189$	−131434.	−0.267642
$190$	0	0
$191$	653452.	1.29608	0.648038	−	0.761608i	$-0.275590\pi$
$191$	653452.	1.29608	0.648038	+	0.761608i	$0.275590\pi$
$192$	0	0
$193$	−468276.	−0.904917	−0.452459	−	0.891785i	$-0.649453\pi$
$193$	−468276.	−0.904917	−0.452459	+	0.891785i	$0.649453\pi$
$194$	0	0
$195$	−606.176	−0.00114160
$196$	0	0
$197$	−372193.	−0.683286	−0.341643	−	0.939830i	$-0.610983\pi$
$197$	−372193.	−0.683286	−0.341643	+	0.939830i	$0.610983\pi$
$198$	0	0
$199$	−581.174	−0.00104034	−0.000520168	−	1.00000i	$-0.500166\pi$
$199$	−581.174	−0.00104034	−0.000520168		1.00000i	$0.500166\pi$
$200$	0	0
$201$	−302054.	−0.527345
$202$	0	0
$203$	120847.	0.205824
$204$	0	0
$205$	132537.	0.220268
$206$	0	0
$207$	611876.	0.992516
$208$	0	0
$209$	27592.3	0.0436940
$210$	0	0
$211$	−221013.	−0.341753	−0.170876	−	0.985292i	$-0.554660\pi$
$211$	−221013.	−0.341753	−0.170876	+	0.985292i	$0.554660\pi$
$212$	0	0
$213$	139221.	0.210260
$214$	0	0
$215$	−138010.	−0.203617
$216$	0	0
$217$	447846.	0.645623
$218$	0	0
$219$	315471.	0.444477
$220$	0	0
$221$	−18544.5	−0.0255408
$222$	0	0
$223$	−1.36021e6	−1.83165	−0.915825	−	0.401578i	$-0.868462\pi$
$223$	−1.36021e6	−1.83165	−0.915825	+	0.401578i	$0.868462\pi$
$224$	0	0
$225$	620287.	0.816839
$226$	0	0
$227$	977910.	1.25961	0.629803	−	0.776755i	$-0.283136\pi$
$227$	977910.	1.25961	0.629803	+	0.776755i	$0.283136\pi$
$228$	0	0
$229$	760624.	0.958476	0.479238	−	0.877685i	$-0.340913\pi$
$229$	760624.	0.958476	0.479238	+	0.877685i	$0.340913\pi$
$230$	0	0
$231$	3621.02	0.00446478
$232$	0	0
$233$	−840644.	−1.01443	−0.507216	−	0.861819i	$-0.669325\pi$
$233$	−840644.	−1.01443	−0.507216	+	0.861819i	$0.669325\pi$
$234$	0	0
$235$	−48183.1	−0.0569148
$236$	0	0
$237$	371690.	0.429843
$238$	0	0
$239$	133752.	0.151463	0.0757314	−	0.997128i	$-0.475871\pi$
$239$	133752.	0.151463	0.0757314	+	0.997128i	$0.475871\pi$
$240$	0	0
$241$	436951.	0.484608	0.242304	−	0.970200i	$-0.422097\pi$
$241$	436951.	0.484608	0.242304	+	0.970200i	$0.422097\pi$
$242$	0	0
$243$	857005.	0.931038
$244$	0	0
$245$	28050.8	0.0298559
$246$	0	0
$247$	−19373.1	−0.0202049
$248$	0	0
$249$	46111.5	0.0471315
$250$	0	0
$251$	−952077.	−0.953867	−0.476933	−	0.878939i	$-0.658252\pi$
$251$	−952077.	−0.953867	−0.476933	+	0.878939i	$0.658252\pi$
$252$	0	0
$253$	−36592.9	−0.0359415
$254$	0	0
$255$	147995.	0.142527
$256$	0	0
$257$	−387585.	−0.366044	−0.183022	−	0.983109i	$-0.558588\pi$
$257$	−387585.	−0.366044	−0.183022	+	0.983109i	$0.558588\pi$
$258$	0	0
$259$	327347.	0.303221
$260$	0	0
$261$	−511893.	−0.465133
$262$	0	0
$263$	−1.58496e6	−1.41296	−0.706479	−	0.707734i	$-0.749717\pi$
$263$	−1.58496e6	−1.41296	−0.706479	+	0.707734i	$0.749717\pi$
$264$	0	0
$265$	430082.	0.376215
$266$	0	0
$267$	30287.7	0.0260008
$268$	0	0
$269$	39673.8	0.0334290	0.0167145	−	0.999860i	$-0.494679\pi$
$269$	39673.8	0.0334290	0.0167145	+	0.999860i	$0.494679\pi$
$270$	0	0
$271$	1.27837e6	1.05738	0.528691	−	0.848814i	$-0.322683\pi$
$271$	1.27837e6	1.05738	0.528691	+	0.848814i	$0.322683\pi$
$272$	0	0
$273$	−2542.39	−0.00206459
$274$	0	0
$275$	−37095.9	−0.0295798
$276$	0	0
$277$	1.62541e6	1.27281	0.636407	−	0.771354i	$-0.280420\pi$
$277$	1.62541e6	1.27281	0.636407	+	0.771354i	$0.280420\pi$
$278$	0	0
$279$	−1.89701e6	−1.45902
$280$	0	0
$281$	1.26168e6	0.953198	0.476599	−	0.879121i	$-0.341869\pi$
$281$	1.26168e6	0.953198	0.476599	+	0.879121i	$0.341869\pi$
$282$	0	0
$283$	−559540.	−0.415303	−0.207652	−	0.978203i	$-0.566582\pi$
$283$	−559540.	−0.415303	−0.207652	+	0.978203i	$0.566582\pi$
$284$	0	0
$285$	154608.	0.112751
$286$	0	0
$287$	555877.	0.398358
$288$	0	0
$289$	3.10770e6	2.18874
$290$	0	0
$291$	587092.	0.406419
$292$	0	0
$293$	551496.	0.375295	0.187648	−	0.982236i	$-0.439914\pi$
$293$	551496.	0.375295	0.187648	+	0.982236i	$0.439914\pi$
$294$	0	0
$295$	−19186.3	−0.0128362
$296$	0	0
$297$	−33295.4	−0.0219025
$298$	0	0
$299$	25692.6	0.0166200
$300$	0	0
$301$	−578831.	−0.368244
$302$	0	0
$303$	386931.	0.242118
$304$	0	0
$305$	490206.	0.301737
$306$	0	0
$307$	−284816.	−0.172472	−0.0862358	−	0.996275i	$-0.527484\pi$
$307$	−284816.	−0.172472	−0.0862358	+	0.996275i	$0.527484\pi$
$308$	0	0
$309$	514628.	0.306618
$310$	0	0
$311$	−2.68224e6	−1.57252	−0.786260	−	0.617895i	$-0.787986\pi$
$311$	−2.68224e6	−1.57252	−0.786260	+	0.617895i	$0.787986\pi$
$312$	0	0
$313$	1.96153e6	1.13171	0.565855	−	0.824505i	$-0.308546\pi$
$313$	1.96153e6	1.13171	0.565855	+	0.824505i	$0.308546\pi$
$314$	0	0
$315$	−118820.	−0.0674701
$316$	0	0
$317$	1.37839e6	0.770411	0.385205	−	0.922831i	$-0.374131\pi$
$317$	1.37839e6	0.770411	0.385205	+	0.922831i	$0.374131\pi$
$318$	0	0
$319$	30613.5	0.0168436
$320$	0	0
$321$	−1.05781e6	−0.572989
$322$	0	0
$323$	4.72985e6	2.52256
$324$	0	0
$325$	26045.8	0.0136782
$326$	0	0
$327$	−911749.	−0.471526
$328$	0	0
$329$	−202087.	−0.102931
$330$	0	0
$331$	3.00370e6	1.50690	0.753452	−	0.657502i	$-0.228387\pi$
$331$	3.00370e6	1.50690	0.753452	+	0.657502i	$0.228387\pi$
$332$	0	0
$333$	−1.38660e6	−0.685236
$334$	0	0
$335$	−592757.	−0.288579
$336$	0	0
$337$	−1.75936e6	−0.843877	−0.421938	−	0.906624i	$-0.638650\pi$
$337$	−1.75936e6	−0.843877	−0.421938	+	0.906624i	$0.638650\pi$
$338$	0	0
$339$	−153368.	−0.0724827
$340$	0	0
$341$	113450.	0.0528346
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−205041.	−0.0927456
$346$	0	0
$347$	−3.17264e6	−1.41448	−0.707241	−	0.706972i	$-0.750061\pi$
$347$	−3.17264e6	−1.41448	−0.707241	+	0.706972i	$0.750061\pi$
$348$	0	0
$349$	1.84732e6	0.811856	0.405928	−	0.913905i	$-0.366948\pi$
$349$	1.84732e6	0.811856	0.405928	+	0.913905i	$0.366948\pi$
$350$	0	0
$351$	23377.4	0.0101281
$352$	0	0
$353$	721495.	0.308174	0.154087	−	0.988057i	$-0.450756\pi$
$353$	721495.	0.308174	0.154087	+	0.988057i	$0.450756\pi$
$354$	0	0
$355$	273210.	0.115061
$356$	0	0
$357$	620712.	0.257762
$358$	0	0
$359$	2.30499e6	0.943913	0.471957	−	0.881622i	$-0.343548\pi$
$359$	2.30499e6	0.943913	0.471957	+	0.881622i	$0.343548\pi$
$360$	0	0
$361$	2.46509e6	0.995552
$362$	0	0
$363$	−957878.	−0.381543
$364$	0	0
$365$	619085.	0.243231
$366$	0	0
$367$	−2.74774e6	−1.06490	−0.532452	−	0.846461i	$-0.678729\pi$
$367$	−2.74774e6	−1.06490	−0.532452	+	0.846461i	$0.678729\pi$
$368$	0	0
$369$	−2.35462e6	−0.900233
$370$	0	0
$371$	1.80382e6	0.680391
$372$	0	0
$373$	−1.72954e6	−0.643664	−0.321832	−	0.946797i	$-0.604299\pi$
$373$	−1.72954e6	−0.643664	−0.321832	+	0.946797i	$0.604299\pi$
$374$	0	0
$375$	−425213.	−0.156145
$376$	0	0
$377$	−21494.3	−0.00778879
$378$	0	0
$379$	2.51742e6	0.900238	0.450119	−	0.892969i	$-0.351382\pi$
$379$	2.51742e6	0.900238	0.450119	+	0.892969i	$0.351382\pi$
$380$	0	0
$381$	432791.	0.152745
$382$	0	0
$383$	1.02397e6	0.356689	0.178345	−	0.983968i	$-0.442926\pi$
$383$	1.02397e6	0.356689	0.178345	+	0.983968i	$0.442926\pi$
$384$	0	0
$385$	7105.94	0.00244326
$386$	0	0
$387$	2.45185e6	0.832179
$388$	0	0
$389$	−5.20359e6	−1.74353	−0.871764	−	0.489926i	$-0.837024\pi$
$389$	−5.20359e6	−1.74353	−0.871764	+	0.489926i	$0.837024\pi$
$390$	0	0
$391$	−6.27274e6	−2.07499
$392$	0	0
$393$	−1.08091e6	−0.353026
$394$	0	0
$395$	729411.	0.235223
$396$	0	0
$397$	−355287.	−0.113136	−0.0565682	−	0.998399i	$-0.518016\pi$
$397$	−355287.	−0.113136	−0.0565682	+	0.998399i	$0.518016\pi$
$398$	0	0
$399$	648446.	0.203912
$400$	0	0
$401$	−815077.	−0.253126	−0.126563	−	0.991959i	$-0.540395\pi$
$401$	−815077.	−0.253126	−0.126563	+	0.991959i	$0.540395\pi$
$402$	0	0
$403$	−79655.4	−0.0244316
$404$	0	0
$405$	402684.	0.121991
$406$	0	0
$407$	82924.8	0.0248141
$408$	0	0
$409$	−3.06950e6	−0.907317	−0.453659	−	0.891176i	$-0.649881\pi$
$409$	−3.06950e6	−0.907317	−0.453659	+	0.891176i	$0.649881\pi$
$410$	0	0
$411$	1.30092e6	0.379881
$412$	0	0
$413$	−80469.8	−0.0232144
$414$	0	0
$415$	90490.0	0.0257917
$416$	0	0
$417$	−1.67951e6	−0.472981
$418$	0	0
$419$	−4.67844e6	−1.30187	−0.650933	−	0.759135i	$-0.725622\pi$
$419$	−4.67844e6	−1.30187	−0.650933	+	0.759135i	$0.725622\pi$
$420$	0	0
$421$	5.93307e6	1.63145	0.815727	−	0.578438i	$-0.196337\pi$
$421$	5.93307e6	1.63145	0.815727	+	0.578438i	$0.196337\pi$
$422$	0	0
$423$	856012.	0.232610
$424$	0	0
$425$	−6.35896e6	−1.70771
$426$	0	0
$427$	2.05599e6	0.545697
$428$	0	0
$429$	−644.047	−0.000168956 0
$430$	0	0
$431$	−2.43836e6	−0.632273	−0.316136	−	0.948714i	$-0.602386\pi$
$431$	−2.43836e6	−0.632273	−0.316136	+	0.948714i	$0.602386\pi$
$432$	0	0
$433$	1.39843e6	0.358444	0.179222	−	0.983809i	$-0.442642\pi$
$433$	1.39843e6	0.358444	0.179222	+	0.983809i	$0.442642\pi$
$434$	0	0
$435$	171536.	0.0434643
$436$	0	0
$437$	−6.55301e6	−1.64149
$438$	0	0
$439$	2.68195e6	0.664186	0.332093	−	0.943247i	$-0.392245\pi$
$439$	2.68195e6	0.664186	0.332093	+	0.943247i	$0.392245\pi$
$440$	0	0
$441$	−498346.	−0.122021
$442$	0	0
$443$	−2.34503e6	−0.567726	−0.283863	−	0.958865i	$-0.591616\pi$
$443$	−2.34503e6	−0.567726	−0.283863	+	0.958865i	$0.591616\pi$
$444$	0	0
$445$	59437.0	0.0142284
$446$	0	0
$447$	−2.21427e6	−0.524157
$448$	0	0
$449$	−4.54290e6	−1.06345	−0.531726	−	0.846917i	$-0.678456\pi$
$449$	−4.54290e6	−1.06345	−0.531726	+	0.846917i	$0.678456\pi$
$450$	0	0
$451$	140817.	0.0325997
$452$	0	0
$453$	366497.	0.0839121
$454$	0	0
$455$	−4989.22	−0.00112981
$456$	0	0
$457$	3.49073e6	0.781854	0.390927	−	0.920422i	$-0.372154\pi$
$457$	3.49073e6	0.781854	0.390927	+	0.920422i	$0.372154\pi$
$458$	0	0
$459$	−5.70748e6	−1.26448
$460$	0	0
$461$	−7.84222e6	−1.71865	−0.859323	−	0.511433i	$-0.829115\pi$
$461$	−7.84222e6	−1.71865	−0.859323	+	0.511433i	$0.829115\pi$
$462$	0	0
$463$	8.89144e6	1.92761	0.963806	−	0.266606i	$-0.0859022\pi$
$463$	8.89144e6	1.92761	0.963806	+	0.266606i	$0.0859022\pi$
$464$	0	0
$465$	635694.	0.136338
$466$	0	0
$467$	7.68946e6	1.63156	0.815782	−	0.578360i	$-0.196307\pi$
$467$	7.68946e6	1.63156	0.815782	+	0.578360i	$0.196307\pi$
$468$	0	0
$469$	−2.48610e6	−0.521900
$470$	0	0
$471$	−1.07074e6	−0.222398
$472$	0	0
$473$	−146632.	−0.0301352
$474$	0	0
$475$	−6.64309e6	−1.35094
$476$	0	0
$477$	−7.64075e6	−1.53759
$478$	0	0
$479$	3.76195e6	0.749160	0.374580	−	0.927195i	$-0.377787\pi$
$479$	3.76195e6	0.749160	0.374580	+	0.927195i	$0.377787\pi$
$480$	0	0
$481$	−58223.1	−0.0114745
$482$	0	0
$483$	−859971.	−0.167732
$484$	0	0
$485$	1.15212e6	0.222404
$486$	0	0
$487$	−4.93372e6	−0.942653	−0.471326	−	0.881959i	$-0.656225\pi$
$487$	−4.93372e6	−0.942653	−0.471326	+	0.881959i	$0.656225\pi$
$488$	0	0
$489$	1.20030e6	0.226995
$490$	0	0
$491$	−5.44764e6	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$491$	−5.44764e6	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$492$	0	0
$493$	5.24774e6	0.972422
$494$	0	0
$495$	−30099.8	−0.00552142
$496$	0	0
$497$	1.14588e6	0.208089
$498$	0	0
$499$	9.46063e6	1.70086	0.850430	−	0.526088i	$-0.176342\pi$
$499$	9.46063e6	1.70086	0.850430	+	0.526088i	$0.176342\pi$
$500$	0	0
$501$	985700.	0.175449
$502$	0	0
$503$	−1.05175e7	−1.85350	−0.926750	−	0.375678i	$-0.877410\pi$
$503$	−1.05175e7	−1.85350	−0.926750	+	0.375678i	$0.877410\pi$
$504$	0	0
$505$	759321.	0.132494
$506$	0	0
$507$	−2.20999e6	−0.381830
$508$	0	0
$509$	9.48876e6	1.62336	0.811680	−	0.584102i	$-0.198553\pi$
$509$	9.48876e6	1.62336	0.811680	+	0.584102i	$0.198553\pi$
$510$	0	0
$511$	2.59653e6	0.439887
$512$	0	0
$513$	−5.96250e6	−1.00031
$514$	0	0
$515$	1.00991e6	0.167790
$516$	0	0
$517$	−51193.3	−0.00842339
$518$	0	0
$519$	−654800.	−0.106706
$520$	0	0
$521$	−461896.	−0.0745504	−0.0372752	−	0.999305i	$-0.511868\pi$
$521$	−461896.	−0.0745504	−0.0372752	+	0.999305i	$0.511868\pi$
$522$	0	0
$523$	−6.25546e6	−1.00001	−0.500006	−	0.866022i	$-0.666669\pi$
$523$	−6.25546e6	−1.00001	−0.500006	+	0.866022i	$0.666669\pi$
$524$	0	0
$525$	−871792.	−0.138043
$526$	0	0
$527$	1.94475e7	3.05027
$528$	0	0
$529$	2.25427e6	0.350241
$530$	0	0
$531$	340859.	0.0524613
$532$	0	0
$533$	−98870.2	−0.0150747
$534$	0	0
$535$	−2.07587e6	−0.313556
$536$	0	0
$537$	−4.67845e6	−0.700110
$538$	0	0
$539$	29803.3	0.00441868
$540$	0	0
$541$	−3.69306e6	−0.542492	−0.271246	−	0.962510i	$-0.587436\pi$
$541$	−3.69306e6	−0.542492	−0.271246	+	0.962510i	$0.587436\pi$
$542$	0	0
$543$	−697592.	−0.101532
$544$	0	0
$545$	−1.78923e6	−0.258033
$546$	0	0
$547$	7.21330e6	1.03078	0.515390	−	0.856956i	$-0.327647\pi$
$547$	7.21330e6	1.03078	0.515390	+	0.856956i	$0.327647\pi$
$548$	0	0
$549$	−8.70890e6	−1.23320
$550$	0	0
$551$	5.48221e6	0.769267
$552$	0	0
$553$	3.05925e6	0.425404
$554$	0	0
$555$	464652.	0.0640318
$556$	0	0
$557$	−409138.	−0.0558769	−0.0279384	−	0.999610i	$-0.508894\pi$
$557$	−409138.	−0.0558769	−0.0279384	+	0.999610i	$0.508894\pi$
$558$	0	0
$559$	102953.	0.0139351
$560$	0	0
$561$	157241.	0.0210940
$562$	0	0
$563$	−7.34492e6	−0.976599	−0.488300	−	0.872676i	$-0.662383\pi$
$563$	−7.34492e6	−0.976599	−0.488300	+	0.872676i	$0.662383\pi$
$564$	0	0
$565$	−300971.	−0.0396647
$566$	0	0
$567$	1.68891e6	0.220622
$568$	0	0
$569$	7.91777e6	1.02523	0.512616	−	0.858618i	$-0.328676\pi$
$569$	7.91777e6	1.02523	0.512616	+	0.858618i	$0.328676\pi$
$570$	0	0
$571$	−1.15724e7	−1.48537	−0.742684	−	0.669643i	$-0.766447\pi$
$571$	−1.15724e7	−1.48537	−0.742684	+	0.669643i	$0.766447\pi$
$572$	0	0
$573$	3.89024e6	0.494982
$574$	0	0
$575$	8.81008e6	1.11125
$576$	0	0
$577$	1.56068e7	1.95152	0.975762	−	0.218836i	$-0.0702258\pi$
$577$	1.56068e7	1.95152	0.975762	+	0.218836i	$0.0702258\pi$
$578$	0	0
$579$	−2.78782e6	−0.345596
$580$	0	0
$581$	379527.	0.0466448
$582$	0	0
$583$	456951.	0.0556798
$584$	0	0
$585$	21133.7	0.00255320
$586$	0	0
$587$	−1.55824e7	−1.86654	−0.933272	−	0.359169i	$-0.883060\pi$
$587$	−1.55824e7	−1.86654	−0.933272	+	0.359169i	$0.883060\pi$
$588$	0	0
$589$	2.03165e7	2.41301
$590$	0	0
$591$	−2.21580e6	−0.260953
$592$	0	0
$593$	−3.33313e6	−0.389238	−0.194619	−	0.980879i	$-0.562347\pi$
$593$	−3.33313e6	−0.389238	−0.194619	+	0.980879i	$0.562347\pi$
$594$	0	0
$595$	1.21810e6	0.141055
$596$	0	0
$597$	−3459.94	−0.000397313 0
$598$	0	0
$599$	7.91924e6	0.901813	0.450907	−	0.892571i	$-0.351101\pi$
$599$	7.91924e6	0.901813	0.450907	+	0.892571i	$0.351101\pi$
$600$	0	0
$601$	6.69762e6	0.756370	0.378185	−	0.925730i	$-0.376548\pi$
$601$	6.69762e6	0.756370	0.378185	+	0.925730i	$0.376548\pi$
$602$	0	0
$603$	1.05308e7	1.17942
$604$	0	0
$605$	−1.87976e6	−0.208792
$606$	0	0
$607$	9.79035e6	1.07852	0.539258	−	0.842141i	$-0.318705\pi$
$607$	9.79035e6	1.07852	0.539258	+	0.842141i	$0.318705\pi$
$608$	0	0
$609$	719447.	0.0786060
$610$	0	0
$611$	35943.8	0.00389512
$612$	0	0
$613$	−1.56969e6	−0.168719	−0.0843595	−	0.996435i	$-0.526884\pi$
$613$	−1.56969e6	−0.168719	−0.0843595	+	0.996435i	$0.526884\pi$
$614$	0	0
$615$	789039.	0.0841222
$616$	0	0
$617$	−1.92210e6	−0.203265	−0.101632	−	0.994822i	$-0.532407\pi$
$617$	−1.92210e6	−0.203265	−0.101632	+	0.994822i	$0.532407\pi$
$618$	0	0
$619$	−5.31663e6	−0.557712	−0.278856	−	0.960333i	$-0.589955\pi$
$619$	−5.31663e6	−0.557712	−0.278856	+	0.960333i	$0.589955\pi$
$620$	0	0
$621$	7.90747e6	0.822827
$622$	0	0
$623$	249287.	0.0257323
$624$	0	0
$625$	8.50464e6	0.870875
$626$	0	0
$627$	164267.	0.0166871
$628$	0	0
$629$	1.42149e7	1.43258
$630$	0	0
$631$	−5.16106e6	−0.516019	−0.258009	−	0.966142i	$-0.583067\pi$
$631$	−5.16106e6	−0.516019	−0.258009	+	0.966142i	$0.583067\pi$
$632$	0	0
$633$	−1.31577e6	−0.130518
$634$	0	0
$635$	849317.	0.0835864
$636$	0	0
$637$	−20925.5	−0.00204327
$638$	0	0
$639$	−4.85380e6	−0.470251
$640$	0	0
$641$	1.15335e7	1.10871	0.554355	−	0.832281i	$-0.312965\pi$
$641$	1.15335e7	1.10871	0.554355	+	0.832281i	$0.312965\pi$
$642$	0	0
$643$	−1.36548e7	−1.30244	−0.651221	−	0.758888i	$-0.725743\pi$
$643$	−1.36548e7	−1.30244	−0.651221	+	0.758888i	$0.725743\pi$
$644$	0	0
$645$	−821621.	−0.0777629
$646$	0	0
$647$	−1.00104e7	−0.940132	−0.470066	−	0.882631i	$-0.655770\pi$
$647$	−1.00104e7	−0.940132	−0.470066	+	0.882631i	$0.655770\pi$
$648$	0	0
$649$	−20384.9	−0.00189975
$650$	0	0
$651$	2.66619e6	0.246569
$652$	0	0
$653$	−1.88670e7	−1.73149	−0.865745	−	0.500485i	$-0.833155\pi$
$653$	−1.88670e7	−1.73149	−0.865745	+	0.500485i	$0.833155\pi$
$654$	0	0
$655$	−2.12119e6	−0.193186
$656$	0	0
$657$	−1.09985e7	−0.994081
$658$	0	0
$659$	−5.13314e6	−0.460436	−0.230218	−	0.973139i	$-0.573944\pi$
$659$	−5.13314e6	−0.460436	−0.230218	+	0.973139i	$0.573944\pi$
$660$	0	0
$661$	1.72818e7	1.53846	0.769228	−	0.638974i	$-0.220641\pi$
$661$	1.72818e7	1.53846	0.769228	+	0.638974i	$0.220641\pi$
$662$	0	0
$663$	−110402.	−0.00975424
$664$	0	0
$665$	1.27252e6	0.111586
$666$	0	0
$667$	−7.27052e6	−0.632778
$668$	0	0
$669$	−8.09780e6	−0.699522
$670$	0	0
$671$	520831.	0.0446571
$672$	0	0
$673$	−5.02989e6	−0.428076	−0.214038	−	0.976825i	$-0.568662\pi$
$673$	−5.02989e6	−0.428076	−0.214038	+	0.976825i	$0.568662\pi$
$674$	0	0
$675$	8.01617e6	0.677185
$676$	0	0
$677$	1.32646e7	1.11230	0.556151	−	0.831081i	$-0.312278\pi$
$677$	1.32646e7	1.11230	0.556151	+	0.831081i	$0.312278\pi$
$678$	0	0
$679$	4.83215e6	0.402222
$680$	0	0
$681$	5.82186e6	0.481054
$682$	0	0
$683$	8.54838e6	0.701184	0.350592	−	0.936528i	$-0.385980\pi$
$683$	8.54838e6	0.701184	0.350592	+	0.936528i	$0.385980\pi$
$684$	0	0
$685$	2.55295e6	0.207882
$686$	0	0
$687$	4.52827e6	0.366050
$688$	0	0
$689$	−320834.	−0.0257473
$690$	0	0
$691$	1.44989e7	1.15515	0.577576	−	0.816337i	$-0.303999\pi$
$691$	1.44989e7	1.15515	0.577576	+	0.816337i	$0.303999\pi$
$692$	0	0
$693$	−126243.	−0.00998558
$694$	0	0
$695$	−3.29591e6	−0.258829
$696$	0	0
$697$	2.41387e7	1.88206
$698$	0	0
$699$	−5.00466e6	−0.387420
$700$	0	0
$701$	−1.46207e7	−1.12376	−0.561878	−	0.827220i	$-0.689921\pi$
$701$	−1.46207e7	−1.12376	−0.561878	+	0.827220i	$0.689921\pi$
$702$	0	0
$703$	1.48500e7	1.13329
$704$	0	0
$705$	−286851.	−0.0217362
$706$	0	0
$707$	3.18469e6	0.239618
$708$	0	0
$709$	1.16448e6	0.0869998	0.0434999	−	0.999053i	$-0.486149\pi$
$709$	1.16448e6	0.0869998	0.0434999	+	0.999053i	$0.486149\pi$
$710$	0	0
$711$	−1.29586e7	−0.961354
$712$	0	0
$713$	−2.69437e7	−1.98488
$714$	0	0
$715$	−1263.89	−9.24578e−5 0
$716$	0	0
$717$	796275.	0.0578449
$718$	0	0
$719$	−798699.	−0.0576184	−0.0288092	−	0.999585i	$-0.509172\pi$
$719$	−798699.	−0.0576184	−0.0288092	+	0.999585i	$0.509172\pi$
$720$	0	0
$721$	4.23572e6	0.303451
$722$	0	0
$723$	2.60133e6	0.185076
$724$	0	0
$725$	−7.37047e6	−0.520775
$726$	0	0
$727$	−5.01536e6	−0.351938	−0.175969	−	0.984396i	$-0.556306\pi$
$727$	−5.01536e6	−0.351938	−0.175969	+	0.984396i	$0.556306\pi$
$728$	0	0
$729$	−3.27356e6	−0.228140
$730$	0	0
$731$	−2.51355e7	−1.73978
$732$	0	0
$733$	1.99730e7	1.37304	0.686522	−	0.727109i	$-0.259137\pi$
$733$	1.99730e7	1.37304	0.686522	+	0.727109i	$0.259137\pi$
$734$	0	0
$735$	166997.	0.0114022
$736$	0	0
$737$	−629789.	−0.0427097
$738$	0	0
$739$	6.39496e6	0.430751	0.215376	−	0.976531i	$-0.430902\pi$
$739$	6.39496e6	0.430751	0.215376	+	0.976531i	$0.430902\pi$
$740$	0	0
$741$	−115335.	−0.00771641
$742$	0	0
$743$	1.85128e7	1.23027	0.615135	−	0.788422i	$-0.289101\pi$
$743$	1.85128e7	1.23027	0.615135	+	0.788422i	$0.289101\pi$
$744$	0	0
$745$	−4.34531e6	−0.286834
$746$	0	0
$747$	−1.60763e6	−0.105411
$748$	0	0
$749$	−8.70648e6	−0.567072
$750$	0	0
$751$	1.53034e7	0.990122	0.495061	−	0.868858i	$-0.335146\pi$
$751$	1.53034e7	0.990122	0.495061	+	0.868858i	$0.335146\pi$
$752$	0	0
$753$	−5.66806e6	−0.364290
$754$	0	0
$755$	719219.	0.0459191
$756$	0	0
$757$	1.09596e7	0.695112	0.347556	−	0.937659i	$-0.387012\pi$
$757$	1.09596e7	0.695112	0.347556	+	0.937659i	$0.387012\pi$
$758$	0	0
$759$	−217851.	−0.0137264
$760$	0	0
$761$	−2.00858e7	−1.25727	−0.628635	−	0.777701i	$-0.716386\pi$
$761$	−2.00858e7	−1.25727	−0.628635	+	0.777701i	$0.716386\pi$
$762$	0	0
$763$	−7.50428e6	−0.466657
$764$	0	0
$765$	−5.15969e6	−0.318765
$766$	0	0
$767$	14312.6	0.000878479 0
$768$	0	0
$769$	3.05867e6	0.186516	0.0932582	−	0.995642i	$-0.470272\pi$
$769$	3.05867e6	0.186516	0.0932582	+	0.995642i	$0.470272\pi$
$770$	0	0
$771$	−2.30743e6	−0.139795
$772$	0	0
$773$	−1.12284e7	−0.675879	−0.337939	−	0.941168i	$-0.609730\pi$
$773$	−1.12284e7	−0.675879	−0.337939	+	0.941168i	$0.609730\pi$
$774$	0	0
$775$	−2.73141e7	−1.63355
$776$	0	0
$777$	1.94882e6	0.115803
$778$	0	0
$779$	2.52173e7	1.48886
$780$	0	0
$781$	290279.	0.0170290
$782$	0	0
$783$	−6.61535e6	−0.385610
$784$	0	0
$785$	−2.10123e6	−0.121703
$786$	0	0
$787$	1.69505e7	0.975543	0.487772	−	0.872971i	$-0.337810\pi$
$787$	1.69505e7	0.975543	0.487772	+	0.872971i	$0.337810\pi$
$788$	0	0
$789$	−9.43584e6	−0.539620
$790$	0	0
$791$	−1.26231e6	−0.0717342
$792$	0	0
$793$	−365686.	−0.0206502
$794$	0	0
$795$	2.56043e6	0.143680
$796$	0	0
$797$	1.23975e7	0.691337	0.345669	−	0.938357i	$-0.387652\pi$
$797$	1.23975e7	0.691337	0.345669	+	0.938357i	$0.387652\pi$
$798$	0	0
$799$	−8.77552e6	−0.486302
$800$	0	0
$801$	−1.05595e6	−0.0581514
$802$	0	0
$803$	657762.	0.0359981
$804$	0	0
$805$	−1.68762e6	−0.0917878
$806$	0	0
$807$	236193.	0.0127668
$808$	0	0
$809$	−6.94674e6	−0.373173	−0.186586	−	0.982439i	$-0.559742\pi$
$809$	−6.94674e6	−0.373173	−0.186586	+	0.982439i	$0.559742\pi$
$810$	0	0
$811$	2.49740e7	1.33332	0.666662	−	0.745360i	$-0.267722\pi$
$811$	2.49740e7	1.33332	0.666662	+	0.745360i	$0.267722\pi$
$812$	0	0
$813$	7.61058e6	0.403823
$814$	0	0
$815$	2.35548e6	0.124218
$816$	0	0
$817$	−2.62586e7	−1.37631
$818$	0	0
$819$	88637.5	0.00461751
$820$	0	0
$821$	−2.51438e7	−1.30188	−0.650942	−	0.759127i	$-0.725626\pi$
$821$	−2.51438e7	−1.30188	−0.650942	+	0.759127i	$0.725626\pi$
$822$	0	0
$823$	−2.28787e7	−1.17742	−0.588711	−	0.808343i	$-0.700364\pi$
$823$	−2.28787e7	−1.17742	−0.588711	+	0.808343i	$0.700364\pi$
$824$	0	0
$825$	−220846.	−0.0112968
$826$	0	0
$827$	−2.11076e7	−1.07319	−0.536594	−	0.843841i	$-0.680289\pi$
$827$	−2.11076e7	−1.07319	−0.536594	+	0.843841i	$0.680289\pi$
$828$	0	0
$829$	−3.36314e7	−1.69965	−0.849824	−	0.527067i	$-0.823292\pi$
$829$	−3.36314e7	−1.69965	−0.849824	+	0.527067i	$0.823292\pi$
$830$	0	0
$831$	9.67668e6	0.486098
$832$	0	0
$833$	5.10886e6	0.255101
$834$	0	0
$835$	1.93435e6	0.0960107
$836$	0	0
$837$	−2.45157e7	−1.20957
$838$	0	0
$839$	5.71851e6	0.280464	0.140232	−	0.990119i	$-0.455215\pi$
$839$	5.71851e6	0.280464	0.140232	+	0.990119i	$0.455215\pi$
$840$	0	0
$841$	−1.44287e7	−0.703455
$842$	0	0
$843$	7.51123e6	0.364034
$844$	0	0
$845$	−4.33692e6	−0.208949
$846$	0	0
$847$	−7.88395e6	−0.377603
$848$	0	0
$849$	−3.33115e6	−0.158608
$850$	0	0
$851$	−1.96942e7	−0.932210
$852$	0	0
$853$	−2.04266e7	−0.961219	−0.480610	−	0.876935i	$-0.659585\pi$
$853$	−2.04266e7	−0.961219	−0.480610	+	0.876935i	$0.659585\pi$
$854$	0	0
$855$	−5.39023e6	−0.252169
$856$	0	0
$857$	3.13151e7	1.45647	0.728235	−	0.685328i	$-0.240341\pi$
$857$	3.13151e7	1.45647	0.728235	+	0.685328i	$0.240341\pi$
$858$	0	0
$859$	3.07708e6	0.142284	0.0711419	−	0.997466i	$-0.477336\pi$
$859$	3.07708e6	0.142284	0.0711419	+	0.997466i	$0.477336\pi$
$860$	0	0
$861$	3.30934e6	0.152136
$862$	0	0
$863$	−2.50508e7	−1.14497	−0.572485	−	0.819915i	$-0.694021\pi$
$863$	−2.50508e7	−1.14497	−0.572485	+	0.819915i	$0.694021\pi$
$864$	0	0
$865$	−1.28499e6	−0.0583928
$866$	0	0
$867$	1.85012e7	0.835898
$868$	0	0
$869$	774980.	0.0348130
$870$	0	0
$871$	442187.	0.0197497
$872$	0	0
$873$	−2.04683e7	−0.908964
$874$	0	0
$875$	−3.49977e6	−0.154533
$876$	0	0
$877$	1.20180e6	0.0527635	0.0263818	−	0.999652i	$-0.491601\pi$
$877$	1.20180e6	0.0527635	0.0263818	+	0.999652i	$0.491601\pi$
$878$	0	0
$879$	3.28325e6	0.143328
$880$	0	0
$881$	1.49531e7	0.649068	0.324534	−	0.945874i	$-0.394792\pi$
$881$	1.49531e7	0.649068	0.324534	+	0.945874i	$0.394792\pi$
$882$	0	0
$883$	1.73150e7	0.747344	0.373672	−	0.927561i	$-0.378099\pi$
$883$	1.73150e7	0.747344	0.373672	+	0.927561i	$0.378099\pi$
$884$	0	0
$885$	−114223.	−0.00490224
$886$	0	0
$887$	1.23187e7	0.525723	0.262862	−	0.964834i	$-0.415334\pi$
$887$	1.23187e7	0.525723	0.262862	+	0.964834i	$0.415334\pi$
$888$	0	0
$889$	3.56215e6	0.151167
$890$	0	0
$891$	427841.	0.0180546
$892$	0	0
$893$	−9.16762e6	−0.384705
$894$	0	0
$895$	−9.18107e6	−0.383121
$896$	0	0
$897$	152957.	0.00634730
$898$	0	0
$899$	2.25410e7	0.930194
$900$	0	0
$901$	7.83302e7	3.21453
$902$	0	0
$903$	−3.44599e6	−0.140635
$904$	0	0
$905$	−1.36897e6	−0.0555612
$906$	0	0
$907$	2.90810e7	1.17379	0.586896	−	0.809663i	$-0.300350\pi$
$907$	2.90810e7	1.17379	0.586896	+	0.809663i	$0.300350\pi$
$908$	0	0
$909$	−1.34899e7	−0.541502
$910$	0	0
$911$	−2.59902e7	−1.03756	−0.518780	−	0.854908i	$-0.673614\pi$
$911$	−2.59902e7	−1.03756	−0.518780	+	0.854908i	$0.673614\pi$
$912$	0	0
$913$	96143.3	0.00381718
$914$	0	0
$915$	2.91837e6	0.115236
$916$	0	0
$917$	−8.89656e6	−0.349380
$918$	0	0
$919$	−2.96126e7	−1.15661	−0.578307	−	0.815819i	$-0.696286\pi$
$919$	−2.96126e7	−1.15661	−0.578307	+	0.815819i	$0.696286\pi$
$920$	0	0
$921$	−1.69561e6	−0.0658684
$922$	0	0
$923$	−203811.	−0.00787449
$924$	0	0
$925$	−1.99649e7	−0.767207
$926$	0	0
$927$	−1.79420e7	−0.685757
$928$	0	0
$929$	−4.31754e7	−1.64134	−0.820668	−	0.571406i	$-0.806398\pi$
$929$	−4.31754e7	−1.64134	−0.820668	+	0.571406i	$0.806398\pi$
$930$	0	0
$931$	5.33713e6	0.201806
$932$	0	0
$933$	−1.59683e7	−0.600559
$934$	0	0
$935$	308573.	0.0115433
$936$	0	0
$937$	−8.42379e6	−0.313443	−0.156721	−	0.987643i	$-0.550092\pi$
$937$	−8.42379e6	−0.313443	−0.156721	+	0.987643i	$0.550092\pi$
$938$	0	0
$939$	1.16777e7	0.432209
$940$	0	0
$941$	−1.67707e7	−0.617414	−0.308707	−	0.951157i	$-0.599896\pi$
$941$	−1.67707e7	−0.617414	−0.308707	+	0.951157i	$0.599896\pi$
$942$	0	0
$943$	−3.34432e7	−1.22470
$944$	0	0
$945$	−1.53554e6	−0.0559349
$946$	0	0
$947$	−4.38904e7	−1.59036	−0.795178	−	0.606377i	$-0.792622\pi$
$947$	−4.38904e7	−1.59036	−0.795178	+	0.606377i	$0.792622\pi$
$948$	0	0
$949$	−461827.	−0.0166462
$950$	0	0
$951$	8.20603e6	0.294226
$952$	0	0
$953$	−2.68610e7	−0.958053	−0.479026	−	0.877800i	$-0.659010\pi$
$953$	−2.68610e7	−0.958053	−0.479026	+	0.877800i	$0.659010\pi$
$954$	0	0
$955$	7.63427e6	0.270869
$956$	0	0
$957$	182253.	0.00643272
$958$	0	0
$959$	1.07074e7	0.375958
$960$	0	0
$961$	5.49051e7	1.91780
$962$	0	0
$963$	3.68795e7	1.28150
$964$	0	0
$965$	−5.47086e6	−0.189120
$966$	0	0
$967$	−3.10989e7	−1.06950	−0.534748	−	0.845011i	$-0.679594\pi$
$967$	−3.10989e7	−1.06950	−0.534748	+	0.845011i	$0.679594\pi$
$968$	0	0
$969$	2.81585e7	0.963386
$970$	0	0
$971$	−8.08990e6	−0.275356	−0.137678	−	0.990477i	$-0.543964\pi$
$971$	−8.08990e6	−0.275356	−0.137678	+	0.990477i	$0.543964\pi$
$972$	0	0
$973$	−1.38235e7	−0.468096
$974$	0	0
$975$	155060.	0.00522382
$976$	0	0
$977$	4.18409e7	1.40238	0.701189	−	0.712976i	$-0.252653\pi$
$977$	4.18409e7	1.40238	0.701189	+	0.712976i	$0.252653\pi$
$978$	0	0
$979$	63150.3	0.00210581
$980$	0	0
$981$	3.17871e7	1.05458
$982$	0	0
$983$	4.55446e7	1.50333	0.751663	−	0.659547i	$-0.229252\pi$
$983$	4.55446e7	1.50333	0.751663	+	0.659547i	$0.229252\pi$
$984$	0	0
$985$	−4.34832e6	−0.142801
$986$	0	0
$987$	−1.20309e6	−0.0393103
$988$	0	0
$989$	3.48242e7	1.13211
$990$	0	0
$991$	2.27184e7	0.734841	0.367421	−	0.930055i	$-0.380241\pi$
$991$	2.27184e7	0.734841	0.367421	+	0.930055i	$0.380241\pi$
$992$	0	0
$993$	1.78821e7	0.575500
$994$	0	0
$995$	−6789.85	−0.000217421 0
$996$	0	0
$997$	−1.58021e7	−0.503472	−0.251736	−	0.967796i	$-0.581002\pi$
$997$	−1.58021e7	−0.503472	−0.251736	+	0.967796i	$0.581002\pi$
$998$	0	0
$999$	−1.79195e7	−0.568082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.6.a.i.1.4	✓	5
4.3	odd	2		224.6.a.j.1.2	yes	5
8.3	odd	2		448.6.a.be.1.4		5
8.5	even	2		448.6.a.bf.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.i.1.4	✓	5	1.1	even	1	trivial
224.6.a.j.1.2	yes	5	4.3	odd	2
448.6.a.be.1.4		5	8.3	odd	2
448.6.a.bf.1.2		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

\(f(q)\)	\(=\)	\(q+5.95336 q^{3} +11.6830 q^{5} +49.0000 q^{7} -207.557 q^{9} +12.4129 q^{11} -8.71531 q^{13} +69.5530 q^{15} +2127.81 q^{17} +2222.88 q^{19} +291.715 q^{21} -2947.98 q^{23} -2988.51 q^{25} -2682.33 q^{27} +2466.27 q^{29} +9139.71 q^{31} +73.8983 q^{33} +572.466 q^{35} +6680.55 q^{37} -51.8854 q^{39} +11344.4 q^{41} -11812.9 q^{43} -2424.89 q^{45} -4124.21 q^{47} +2401.00 q^{49} +12667.6 q^{51} +36812.7 q^{53} +145.019 q^{55} +13233.6 q^{57} -1642.24 q^{59} +41959.0 q^{61} -10170.3 q^{63} -101.821 q^{65} -50736.8 q^{67} -17550.4 q^{69} +23385.3 q^{71} +52990.4 q^{73} -17791.7 q^{75} +608.230 q^{77} +62433.7 q^{79} +34467.6 q^{81} +7745.46 q^{83} +24859.1 q^{85} +14682.6 q^{87} +5087.49 q^{89} -427.050 q^{91} +54412.0 q^{93} +25969.8 q^{95} +98615.3 q^{97} -2576.38 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) \) 5 * q - 10 * q^3 + 36 * q^5 + 245 * q^7 + 637 * q^9 + 116 * q^11 + 40 * q^13 + 16 * q^15 - 402 * q^17 - 3582 * q^19 - 490 * q^21 + 472 * q^23 + 9615 * q^25 + 356 * q^27 + 4754 * q^29 - 10500 * q^31 + 15864 * q^33 + 1764 * q^35 + 19642 * q^37 + 10872 * q^39 + 23398 * q^41 + 22044 * q^43 + 49476 * q^45 + 16004 * q^47 + 12005 * q^49 - 45676 * q^51 + 54246 * q^53 + 53456 * q^55 + 109556 * q^57 - 74366 * q^59 + 68316 * q^61 + 31213 * q^63 + 152568 * q^65 - 26560 * q^67 + 214720 * q^69 + 93072 * q^71 + 136098 * q^73 - 124510 * q^75 + 5684 * q^77 + 96080 * q^79 + 104801 * q^81 - 145894 * q^83 + 117352 * q^85 + 168876 * q^87 + 188554 * q^89 + 1960 * q^91 + 86296 * q^93 + 74736 * q^95 - 88146 * q^97 - 260236 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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