LMFDB - Embedded newform 448.6.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
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:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.97668$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.95336 q^{3} -11.6830 q^{5} +49.0000 q^{7} -207.557 q^{9} +O(q^{10})$	$q-5.95336 q^{3} -11.6830 q^{5} +49.0000 q^{7} -207.557 q^{9} -12.4129 q^{11} +8.71531 q^{13} +69.5530 q^{15} +2127.81 q^{17} -2222.88 q^{19} -291.715 q^{21} -2947.98 q^{23} -2988.51 q^{25} +2682.33 q^{27} -2466.27 q^{29} +9139.71 q^{31} +73.8983 q^{33} -572.466 q^{35} -6680.55 q^{37} -51.8854 q^{39} +11344.4 q^{41} +11812.9 q^{43} +2424.89 q^{45} -4124.21 q^{47} +2401.00 q^{49} -12667.6 q^{51} -36812.7 q^{53} +145.019 q^{55} +13233.6 q^{57} +1642.24 q^{59} -41959.0 q^{61} -10170.3 q^{63} -101.821 q^{65} +50736.8 q^{67} +17550.4 q^{69} +23385.3 q^{71} +52990.4 q^{73} +17791.7 q^{75} -608.230 q^{77} +62433.7 q^{79} +34467.6 q^{81} -7745.46 q^{83} -24859.1 q^{85} +14682.6 q^{87} +5087.49 q^{89} +427.050 q^{91} -54412.0 q^{93} +25969.8 q^{95} +98615.3 q^{97} +2576.38 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9}+O(q^{10}) $ 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−5.95336	−0.381908	−0.190954	−	0.981599i	$-0.561158\pi$
$3$	−5.95336	−0.381908	−0.190954	+	0.981599i	$0.561158\pi$
$4$	0	0
$5$	−11.6830	−0.208991	−0.104496	−	0.994525i	$-0.533323\pi$
$5$	−11.6830	−0.208991	−0.104496	+	0.994525i	$0.533323\pi$
$6$	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	$-0.504923\pi$
$11$	−12.4129	−0.0309307	−0.0154654	+	0.999880i	$0.504923\pi$
$12$	0	0
$13$	8.71531	0.0143029	0.00715146	−	0.999974i	$-0.497724\pi$
$13$	8.71531	0.0143029	0.00715146	+	0.999974i	$0.497724\pi$
$14$	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	$-0.749646\pi$
$19$	−2222.88	−1.41264	−0.706320	+	0.707893i	$0.749646\pi$
$20$	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	$-0.587778\pi$
$29$	−2466.27	−0.544560	−0.272280	+	0.962218i	$0.587778\pi$
$30$	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	$-0.631380\pi$
$37$	−6680.55	−0.802247	−0.401123	+	0.916024i	$0.631380\pi$
$38$	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	$-0.338040\pi$
$43$	11812.9	0.974282	0.487141	+	0.873323i	$0.338040\pi$
$44$	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	$-0.856487\pi$
$53$	−36812.7	−1.80015	−0.900073	+	0.435739i	$0.856487\pi$
$54$	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	$-0.490223\pi$
$59$	1642.24	0.0614196	0.0307098	+	0.999528i	$0.490223\pi$
$60$	0	0
$61$	−41959.0	−1.44378	−0.721889	−	0.692009i	$-0.756726\pi$
$61$	−41959.0	−1.44378	−0.721889	+	0.692009i	$0.756726\pi$
$62$	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	$-0.257431\pi$
$67$	50736.8	1.38082	0.690408	+	0.723420i	$0.257431\pi$
$68$	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	$-0.519654\pi$
$83$	−7745.46	−0.123410	−0.0617052	+	0.998094i	$0.519654\pi$
$84$	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	$-0.602670\pi$
$101$	−64993.8	−0.633969	−0.316985	+	0.948431i	$0.602670\pi$
$102$	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	$-0.229974\pi$
$107$	177683.	1.50033	0.750165	+	0.661250i	$0.229974\pi$
$108$	0	0
$109$	153149.	1.23466	0.617329	−	0.786705i	$-0.288215\pi$
$109$	153149.	1.23466	0.617329	+	0.786705i	$0.288215\pi$
$110$	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	$-0.347065\pi$
$131$	181562.	0.924374	0.462187	+	0.886783i	$0.347065\pi$
$132$	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	$-0.287443\pi$
$139$	282112.	1.23847	0.619233	+	0.785207i	$0.287443\pi$
$140$	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	$-0.259263\pi$
$149$	371936.	1.37247	0.686234	+	0.727381i	$0.259263\pi$
$150$	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	$-0.405957\pi$
$157$	179854.	0.582333	0.291166	+	0.956672i	$0.405957\pi$
$158$	0	0
$159$	219159.	0.687491
$160$	0	0
$161$	−144451.	−0.439194
$162$	0	0
$163$	−201617.	−0.594371	−0.297185	−	0.954820i	$-0.596048\pi$
$163$	−201617.	−0.594371	−0.297185	+	0.954820i	$0.596048\pi$
$164$	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	$-0.455386\pi$
$173$	109988.	0.279403	0.139702	+	0.990194i	$0.455386\pi$
$174$	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	$-0.130927\pi$
$179$	785850.	1.83319	0.916594	+	0.399818i	$0.130927\pi$
$180$	0	0
$181$	117176.	0.265854	0.132927	−	0.991126i	$-0.457562\pi$
$181$	117176.	0.265854	0.132927	+	0.991126i	$0.457562\pi$
$182$	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	$-0.389017\pi$
$197$	372193.	0.683286	0.341643	+	0.939830i	$0.389017\pi$
$198$	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	$-0.445340\pi$
$211$	221013.	0.341753	0.170876	+	0.985292i	$0.445340\pi$
$212$	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	$-0.716864\pi$
$227$	−977910.	−1.25961	−0.629803	+	0.776755i	$0.716864\pi$
$228$	0	0
$229$	−760624.	−0.958476	−0.479238	−	0.877685i	$-0.659087\pi$
$229$	−760624.	−0.958476	−0.479238	+	0.877685i	$0.659087\pi$
$230$	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	$-0.341748\pi$
$251$	952077.	0.953867	0.476933	+	0.878939i	$0.341748\pi$
$252$	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	$-0.505321\pi$
$269$	−39673.8	−0.0334290	−0.0167145	+	0.999860i	$0.505321\pi$
$270$	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	$-0.719580\pi$
$277$	−1.62541e6	−1.27281	−0.636407	+	0.771354i	$0.719580\pi$
$278$	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	$-0.433418\pi$
$283$	559540.	0.415303	0.207652	+	0.978203i	$0.433418\pi$
$284$	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	$-0.560086\pi$
$293$	−551496.	−0.375295	−0.187648	+	0.982236i	$0.560086\pi$
$294$	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	$-0.472516\pi$
$307$	284816.	0.172472	0.0862358	+	0.996275i	$0.472516\pi$
$308$	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	$-0.625869\pi$
$317$	−1.37839e6	−0.770411	−0.385205	+	0.922831i	$0.625869\pi$
$318$	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	$-0.771613\pi$
$331$	−3.00370e6	−1.50690	−0.753452	+	0.657502i	$0.771613\pi$
$332$	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	$-0.249939\pi$
$347$	3.17264e6	1.41448	0.707241	+	0.706972i	$0.249939\pi$
$348$	0	0
$349$	−1.84732e6	−0.811856	−0.405928	−	0.913905i	$-0.633052\pi$
$349$	−1.84732e6	−0.811856	−0.405928	+	0.913905i	$0.633052\pi$
$350$	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	$-0.395701\pi$
$373$	1.72954e6	0.643664	0.321832	+	0.946797i	$0.395701\pi$
$374$	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	$-0.648618\pi$
$379$	−2.51742e6	−0.900238	−0.450119	+	0.892969i	$0.648618\pi$
$380$	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	$-0.162976\pi$
$389$	5.20359e6	1.74353	0.871764	+	0.489926i	$0.162976\pi$
$390$	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	$-0.481984\pi$
$397$	355287.	0.113136	0.0565682	+	0.998399i	$0.481984\pi$
$398$	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	$-0.274378\pi$
$419$	4.67844e6	1.30187	0.650933	+	0.759135i	$0.274378\pi$
$420$	0	0
$421$	−5.93307e6	−1.63145	−0.815727	−	0.578438i	$-0.803663\pi$
$421$	−5.93307e6	−1.63145	−0.815727	+	0.578438i	$0.803663\pi$
$422$	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	$-0.408384\pi$
$443$	2.34503e6	0.567726	0.283863	+	0.958865i	$0.408384\pi$
$444$	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	$-0.170885\pi$
$461$	7.84222e6	1.71865	0.859323	+	0.511433i	$0.170885\pi$
$462$	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	$-0.803693\pi$
$467$	−7.68946e6	−1.63156	−0.815782	+	0.578360i	$0.803693\pi$
$468$	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	$-0.329687\pi$
$491$	5.44764e6	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$492$	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	$-0.823658\pi$
$499$	−9.46063e6	−1.70086	−0.850430	+	0.526088i	$0.823658\pi$
$500$	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	$-0.801447\pi$
$509$	−9.48876e6	−1.62336	−0.811680	+	0.584102i	$0.801447\pi$
$510$	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	$-0.333331\pi$
$523$	6.25546e6	1.00001	0.500006	+	0.866022i	$0.333331\pi$
$524$	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	$-0.412564\pi$
$541$	3.69306e6	0.542492	0.271246	+	0.962510i	$0.412564\pi$
$542$	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	$-0.672353\pi$
$547$	−7.21330e6	−1.03078	−0.515390	+	0.856956i	$0.672353\pi$
$548$	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	$-0.491106\pi$
$557$	409138.	0.0558769	0.0279384	+	0.999610i	$0.491106\pi$
$558$	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	$-0.337617\pi$
$563$	7.34492e6	0.976599	0.488300	+	0.872676i	$0.337617\pi$
$564$	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	$-0.233553\pi$
$571$	1.15724e7	1.48537	0.742684	+	0.669643i	$0.233553\pi$
$572$	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	$-0.116940\pi$
$587$	1.55824e7	1.86654	0.933272	+	0.359169i	$0.116940\pi$
$588$	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	$-0.473116\pi$
$613$	1.56969e6	0.168719	0.0843595	+	0.996435i	$0.473116\pi$
$614$	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	$-0.410045\pi$
$619$	5.31663e6	0.557712	0.278856	+	0.960333i	$0.410045\pi$
$620$	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	$-0.274257\pi$
$643$	1.36548e7	1.30244	0.651221	+	0.758888i	$0.274257\pi$
$644$	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	$-0.166845\pi$
$653$	1.88670e7	1.73149	0.865745	+	0.500485i	$0.166845\pi$
$654$	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	$-0.426056\pi$
$659$	5.13314e6	0.460436	0.230218	+	0.973139i	$0.426056\pi$
$660$	0	0
$661$	−1.72818e7	−1.53846	−0.769228	−	0.638974i	$-0.779359\pi$
$661$	−1.72818e7	−1.53846	−0.769228	+	0.638974i	$0.779359\pi$
$662$	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	$-0.687722\pi$
$677$	−1.32646e7	−1.11230	−0.556151	+	0.831081i	$0.687722\pi$
$678$	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	$-0.614020\pi$
$683$	−8.54838e6	−0.701184	−0.350592	+	0.936528i	$0.614020\pi$
$684$	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	$-0.696001\pi$
$691$	−1.44989e7	−1.15515	−0.577576	+	0.816337i	$0.696001\pi$
$692$	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	$-0.310079\pi$
$701$	1.46207e7	1.12376	0.561878	+	0.827220i	$0.310079\pi$
$702$	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	$-0.513851\pi$
$709$	−1.16448e6	−0.0869998	−0.0434999	+	0.999053i	$0.513851\pi$
$710$	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	$-0.740863\pi$
$733$	−1.99730e7	−1.37304	−0.686522	+	0.727109i	$0.740863\pi$
$734$	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	$-0.569098\pi$
$739$	−6.39496e6	−0.430751	−0.215376	+	0.976531i	$0.569098\pi$
$740$	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	$-0.612988\pi$
$757$	−1.09596e7	−0.695112	−0.347556	+	0.937659i	$0.612988\pi$
$758$	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	$-0.390270\pi$
$773$	1.12284e7	0.675879	0.337939	+	0.941168i	$0.390270\pi$
$774$	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	$-0.662190\pi$
$787$	−1.69505e7	−0.975543	−0.487772	+	0.872971i	$0.662190\pi$
$788$	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	$-0.612348\pi$
$797$	−1.23975e7	−0.691337	−0.345669	+	0.938357i	$0.612348\pi$
$798$	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	$-0.732278\pi$
$811$	−2.49740e7	−1.33332	−0.666662	+	0.745360i	$0.732278\pi$
$812$	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	$-0.274374\pi$
$821$	2.51438e7	1.30188	0.650942	+	0.759127i	$0.274374\pi$
$822$	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	$-0.319711\pi$
$827$	2.11076e7	1.07319	0.536594	+	0.843841i	$0.319711\pi$
$828$	0	0
$829$	3.36314e7	1.69965	0.849824	−	0.527067i	$-0.176708\pi$
$829$	3.36314e7	1.69965	0.849824	+	0.527067i	$0.176708\pi$
$830$	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	$-0.340415\pi$
$853$	2.04266e7	0.961219	0.480610	+	0.876935i	$0.340415\pi$
$854$	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	$-0.522664\pi$
$859$	−3.07708e6	−0.142284	−0.0711419	+	0.997466i	$0.522664\pi$
$860$	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	$-0.508399\pi$
$877$	−1.20180e6	−0.0527635	−0.0263818	+	0.999652i	$0.508399\pi$
$878$	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	$-0.621901\pi$
$883$	−1.73150e7	−0.747344	−0.373672	+	0.927561i	$0.621901\pi$
$884$	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	$-0.699650\pi$
$907$	−2.90810e7	−1.17379	−0.586896	+	0.809663i	$0.699650\pi$
$908$	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	$-0.400104\pi$
$941$	1.67707e7	0.617414	0.308707	+	0.951157i	$0.400104\pi$
$942$	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	$-0.207378\pi$
$947$	4.38904e7	1.59036	0.795178	+	0.606377i	$0.207378\pi$
$948$	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	$-0.456036\pi$
$971$	8.08990e6	0.275356	0.137678	+	0.990477i	$0.456036\pi$
$972$	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	$-0.418998\pi$
$997$	1.58021e7	0.503472	0.251736	+	0.967796i	$0.418998\pi$
$998$	0	0
$999$	−1.79195e7	−0.568082

Twists

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

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

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

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