LMFDB - Embedded newform 784.6.a.bd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,28,0,-42,0,0,0,124] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.55969 q^{3} +41.6418 q^{5} -230.329 q^{9} -110.754 q^{11} -179.135 q^{13} +148.232 q^{15} +355.567 q^{17} +1955.96 q^{19} -1546.73 q^{23} -1390.96 q^{25} -1684.90 q^{27} +6273.94 q^{29} +6004.18 q^{31} -394.249 q^{33} -9688.45 q^{37} -637.665 q^{39} -10577.3 q^{41} -6716.00 q^{43} -9591.31 q^{45} +27240.8 q^{47} +1265.71 q^{51} -32679.4 q^{53} -4611.98 q^{55} +6962.63 q^{57} -492.663 q^{59} -40552.2 q^{61} -7459.50 q^{65} +7688.15 q^{67} -5505.87 q^{69} -77879.3 q^{71} -73915.9 q^{73} -4951.38 q^{75} +43933.1 q^{79} +49972.1 q^{81} +41194.2 q^{83} +14806.5 q^{85} +22333.3 q^{87} -65724.6 q^{89} +21373.0 q^{93} +81449.9 q^{95} +68534.5 q^{97} +25509.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 28 q^{3} - 42 q^{5} + 124 q^{9} - 660 q^{11} + 644 q^{13} - 1896 q^{15} + 210 q^{17} + 3724 q^{19} - 24 q^{23} + 2480 q^{25} + 1036 q^{27} + 5532 q^{29} + 2800 q^{31} - 13818 q^{33} - 13238 q^{37}+ \cdots - 169104 q^{99}+O(q^{100}) $ 2 * q + 28 * q^3 - 42 * q^5 + 124 * q^9 - 660 * q^11 + 644 * q^13 - 1896 * q^15 + 210 * q^17 + 3724 * q^19 - 24 * q^23 + 2480 * q^25 + 1036 * q^27 + 5532 * q^29 + 2800 * q^31 - 13818 * q^33 - 13238 * q^37 + 19480 * q^39 - 4116 * q^41 - 13432 * q^43 - 39228 * q^45 + 8064 * q^47 - 2292 * q^51 - 53958 * q^53 + 41328 * q^55 + 50174 * q^57 + 36036 * q^59 - 83986 * q^61 - 76308 * q^65 + 2660 * q^67 + 31710 * q^69 - 71568 * q^71 - 31318 * q^73 + 89656 * q^75 - 51136 * q^79 + 30370 * q^81 + 6216 * q^83 + 26982 * q^85 + 4200 * q^87 - 166278 * q^89 - 56938 * q^93 - 66432 * q^95 - 8260 * q^97 - 169104 * 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
$3$	3.55969	0.228354	0.114177	−	0.993460i	$-0.463577\pi$
$3$	3.55969	0.228354	0.114177	+	0.993460i	$0.463577\pi$
$4$	0	0
$5$	41.6418	0.744912	0.372456	−	0.928050i	$-0.378516\pi$
$5$	41.6418	0.744912	0.372456	+	0.928050i	$0.378516\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−230.329	−0.947854
$10$	0	0
$11$	−110.754	−0.275979	−0.137989	−	0.990434i	$-0.544064\pi$
$11$	−110.754	−0.275979	−0.137989	+	0.990434i	$0.544064\pi$
$12$	0	0
$13$	−179.135	−0.293982	−0.146991	−	0.989138i	$-0.546959\pi$
$13$	−179.135	−0.293982	−0.146991	+	0.989138i	$0.546959\pi$
$14$	0	0
$15$	148.232	0.170104
$16$	0	0
$17$	355.567	0.298401	0.149200	−	0.988807i	$-0.452330\pi$
$17$	355.567	0.298401	0.149200	+	0.988807i	$0.452330\pi$
$18$	0	0
$19$	1955.96	1.24302	0.621508	−	0.783408i	$-0.286520\pi$
$19$	1955.96	1.24302	0.621508	+	0.783408i	$0.286520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1546.73	−0.609668	−0.304834	−	0.952405i	$-0.598601\pi$
$23$	−1546.73	−0.609668	−0.304834	+	0.952405i	$0.598601\pi$
$24$	0	0
$25$	−1390.96	−0.445106
$26$	0	0
$27$	−1684.90	−0.444801
$28$	0	0
$29$	6273.94	1.38531	0.692653	−	0.721271i	$-0.256442\pi$
$29$	6273.94	1.38531	0.692653	+	0.721271i	$0.256442\pi$
$30$	0	0
$31$	6004.18	1.12215	0.561073	−	0.827767i	$-0.310389\pi$
$31$	6004.18	1.12215	0.561073	+	0.827767i	$0.310389\pi$
$32$	0	0
$33$	−394.249	−0.0630210
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9688.45	−1.16346	−0.581728	−	0.813383i	$-0.697623\pi$
$37$	−9688.45	−1.16346	−0.581728	+	0.813383i	$0.697623\pi$
$38$	0	0
$39$	−637.665	−0.0671322
$40$	0	0
$41$	−10577.3	−0.982686	−0.491343	−	0.870966i	$-0.663494\pi$
$41$	−10577.3	−0.982686	−0.491343	+	0.870966i	$0.663494\pi$
$42$	0	0
$43$	−6716.00	−0.553910	−0.276955	−	0.960883i	$-0.589325\pi$
$43$	−6716.00	−0.553910	−0.276955	+	0.960883i	$0.589325\pi$
$44$	0	0
$45$	−9591.31	−0.706068
$46$	0	0
$47$	27240.8	1.79877	0.899384	−	0.437159i	$-0.144015\pi$
$47$	27240.8	1.79877	0.899384	+	0.437159i	$0.144015\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1265.71	0.0681411
$52$	0	0
$53$	−32679.4	−1.59803	−0.799014	−	0.601312i	$-0.794645\pi$
$53$	−32679.4	−1.59803	−0.799014	+	0.601312i	$0.794645\pi$
$54$	0	0
$55$	−4611.98	−0.205580
$56$	0	0
$57$	6962.63	0.283848
$58$	0	0
$59$	−492.663	−0.0184255	−0.00921277	−	0.999958i	$-0.502933\pi$
$59$	−492.663	−0.0184255	−0.00921277	+	0.999958i	$0.502933\pi$
$60$	0	0
$61$	−40552.2	−1.39537	−0.697686	−	0.716403i	$-0.745787\pi$
$61$	−40552.2	−1.39537	−0.697686	+	0.716403i	$0.745787\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7459.50	−0.218991
$66$	0	0
$67$	7688.15	0.209235	0.104618	−	0.994513i	$-0.466638\pi$
$67$	7688.15	0.209235	0.104618	+	0.994513i	$0.466638\pi$
$68$	0	0
$69$	−5505.87	−0.139220
$70$	0	0
$71$	−77879.3	−1.83348	−0.916740	−	0.399484i	$-0.869189\pi$
$71$	−77879.3	−1.83348	−0.916740	+	0.399484i	$0.869189\pi$
$72$	0	0
$73$	−73915.9	−1.62342	−0.811710	−	0.584061i	$-0.801463\pi$
$73$	−73915.9	−1.62342	−0.811710	+	0.584061i	$0.801463\pi$
$74$	0	0
$75$	−4951.38	−0.101642
$76$	0	0
$77$	0	0
$78$	0	0
$79$	43933.1	0.791998	0.395999	−	0.918251i	$-0.370398\pi$
$79$	43933.1	0.791998	0.395999	+	0.918251i	$0.370398\pi$
$80$	0	0
$81$	49972.1	0.846282
$82$	0	0
$83$	41194.2	0.656359	0.328179	−	0.944615i	$-0.393565\pi$
$83$	41194.2	0.656359	0.328179	+	0.944615i	$0.393565\pi$
$84$	0	0
$85$	14806.5	0.222282
$86$	0	0
$87$	22333.3	0.316341
$88$	0	0
$89$	−65724.6	−0.879534	−0.439767	−	0.898112i	$-0.644939\pi$
$89$	−65724.6	−0.879534	−0.439767	+	0.898112i	$0.644939\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	21373.0	0.256247
$94$	0	0
$95$	81449.9	0.925937
$96$	0	0
$97$	68534.5	0.739571	0.369786	−	0.929117i	$-0.379431\pi$
$97$	68534.5	0.739571	0.369786	+	0.929117i	$0.379431\pi$
$98$	0	0
$99$	25509.7	0.261588
$100$	0	0
$101$	117358.	1.14475	0.572375	−	0.819992i	$-0.306022\pi$
$101$	117358.	1.14475	0.572375	+	0.819992i	$0.306022\pi$
$102$	0	0
$103$	−10819.0	−0.100484	−0.0502418	−	0.998737i	$-0.515999\pi$
$103$	−10819.0	−0.100484	−0.0502418	+	0.998737i	$0.515999\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−110400.	−0.932203	−0.466101	−	0.884731i	$-0.654342\pi$
$107$	−110400.	−0.932203	−0.466101	+	0.884731i	$0.654342\pi$
$108$	0	0
$109$	12261.2	0.0988476	0.0494238	−	0.998778i	$-0.484262\pi$
$109$	12261.2	0.0988476	0.0494238	+	0.998778i	$0.484262\pi$
$110$	0	0
$111$	−34487.9	−0.265680
$112$	0	0
$113$	−122314.	−0.901117	−0.450558	−	0.892747i	$-0.648775\pi$
$113$	−122314.	−0.901117	−0.450558	+	0.892747i	$0.648775\pi$
$114$	0	0
$115$	−64408.5	−0.454149
$116$	0	0
$117$	41259.8	0.278652
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−148785.	−0.923836
$122$	0	0
$123$	−37651.9	−0.224401
$124$	0	0
$125$	−188053.	−1.07648
$126$	0	0
$127$	169073.	0.930174	0.465087	−	0.885265i	$-0.346023\pi$
$127$	169073.	0.930174	0.465087	+	0.885265i	$0.346023\pi$
$128$	0	0
$129$	−23906.9	−0.126488
$130$	0	0
$131$	−377451.	−1.92168	−0.960842	−	0.277096i	$-0.910628\pi$
$131$	−377451.	−1.92168	−0.960842	+	0.277096i	$0.910628\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−70162.5	−0.331338
$136$	0	0
$137$	316630.	1.44129	0.720643	−	0.693306i	$-0.243847\pi$
$137$	316630.	1.44129	0.720643	+	0.693306i	$0.243847\pi$
$138$	0	0
$139$	−229447.	−1.00727	−0.503634	−	0.863917i	$-0.668004\pi$
$139$	−229447.	−1.00727	−0.503634	+	0.863917i	$0.668004\pi$
$140$	0	0
$141$	96968.9	0.410757
$142$	0	0
$143$	19839.8	0.0811330
$144$	0	0
$145$	261259.	1.03193
$146$	0	0
$147$	0	0
$148$	0	0
$149$	157502.	0.581191	0.290596	−	0.956846i	$-0.406147\pi$
$149$	157502.	0.581191	0.290596	+	0.956846i	$0.406147\pi$
$150$	0	0
$151$	146785.	0.523888	0.261944	−	0.965083i	$-0.415636\pi$
$151$	146785.	0.523888	0.261944	+	0.965083i	$0.415636\pi$
$152$	0	0
$153$	−81897.3	−0.282840
$154$	0	0
$155$	250025.	0.835899
$156$	0	0
$157$	327485.	1.06033	0.530166	−	0.847894i	$-0.322130\pi$
$157$	327485.	1.06033	0.530166	+	0.847894i	$0.322130\pi$
$158$	0	0
$159$	−116329.	−0.364917
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−348187.	−1.02646	−0.513231	−	0.858250i	$-0.671552\pi$
$163$	−348187.	−1.02646	−0.513231	+	0.858250i	$0.671552\pi$
$164$	0	0
$165$	−16417.2	−0.0469451
$166$	0	0
$167$	−547771.	−1.51987	−0.759937	−	0.649996i	$-0.774770\pi$
$167$	−547771.	−1.51987	−0.759937	+	0.649996i	$0.774770\pi$
$168$	0	0
$169$	−339204.	−0.913574
$170$	0	0
$171$	−450514.	−1.17820
$172$	0	0
$173$	−182541.	−0.463708	−0.231854	−	0.972751i	$-0.574479\pi$
$173$	−182541.	−0.463708	−0.231854	+	0.972751i	$0.574479\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1753.73	−0.00420755
$178$	0	0
$179$	−136123.	−0.317541	−0.158770	−	0.987316i	$-0.550753\pi$
$179$	−136123.	−0.317541	−0.158770	+	0.987316i	$0.550753\pi$
$180$	0	0
$181$	−591381.	−1.34175	−0.670874	−	0.741571i	$-0.734081\pi$
$181$	−591381.	−1.34175	−0.670874	+	0.741571i	$0.734081\pi$
$182$	0	0
$183$	−144354.	−0.318640
$184$	0	0
$185$	−403445.	−0.866672
$186$	0	0
$187$	−39380.4	−0.0823523
$188$	0	0
$189$	0	0
$190$	0	0
$191$	441191.	0.875071	0.437536	−	0.899201i	$-0.355851\pi$
$191$	441191.	0.875071	0.437536	+	0.899201i	$0.355851\pi$
$192$	0	0
$193$	448375.	0.866459	0.433230	−	0.901284i	$-0.357374\pi$
$193$	448375.	0.866459	0.433230	+	0.901284i	$0.357374\pi$
$194$	0	0
$195$	−26553.5	−0.0500076
$196$	0	0
$197$	−542019.	−0.995059	−0.497530	−	0.867447i	$-0.665759\pi$
$197$	−542019.	−0.995059	−0.497530	+	0.867447i	$0.665759\pi$
$198$	0	0
$199$	−197959.	−0.354358	−0.177179	−	0.984179i	$-0.556697\pi$
$199$	−197959.	−0.354358	−0.177179	+	0.984179i	$0.556697\pi$
$200$	0	0
$201$	27367.4	0.0477798
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−440458.	−0.732014
$206$	0	0
$207$	356255.	0.577877
$208$	0	0
$209$	−216630.	−0.343046
$210$	0	0
$211$	−1.13658e6	−1.75749	−0.878745	−	0.477292i	$-0.841618\pi$
$211$	−1.13658e6	−1.75749	−0.878745	+	0.477292i	$0.841618\pi$
$212$	0	0
$213$	−277226.	−0.418683
$214$	0	0
$215$	−279667.	−0.412614
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−263118.	−0.370715
$220$	0	0
$221$	−63694.5	−0.0877245
$222$	0	0
$223$	−497078.	−0.669364	−0.334682	−	0.942331i	$-0.608629\pi$
$223$	−497078.	−0.669364	−0.334682	+	0.942331i	$0.608629\pi$
$224$	0	0
$225$	320377.	0.421896
$226$	0	0
$227$	934539.	1.20374	0.601870	−	0.798594i	$-0.294422\pi$
$227$	934539.	1.20374	0.601870	+	0.798594i	$0.294422\pi$
$228$	0	0
$229$	−1.05206e6	−1.32572	−0.662860	−	0.748743i	$-0.730657\pi$
$229$	−1.05206e6	−1.32572	−0.662860	+	0.748743i	$0.730657\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	414582.	0.500289	0.250145	−	0.968209i	$-0.419522\pi$
$233$	414582.	0.500289	0.250145	+	0.968209i	$0.419522\pi$
$234$	0	0
$235$	1.13436e6	1.33992
$236$	0	0
$237$	156388.	0.180856
$238$	0	0
$239$	−481799.	−0.545596	−0.272798	−	0.962071i	$-0.587949\pi$
$239$	−481799.	−0.545596	−0.272798	+	0.962071i	$0.587949\pi$
$240$	0	0
$241$	−28961.0	−0.0321197	−0.0160598	−	0.999871i	$-0.505112\pi$
$241$	−28961.0	−0.0321197	−0.0160598	+	0.999871i	$0.505112\pi$
$242$	0	0
$243$	587317.	0.638053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−350381.	−0.365425
$248$	0	0
$249$	146639.	0.149882
$250$	0	0
$251$	553855.	0.554896	0.277448	−	0.960741i	$-0.410511\pi$
$251$	553855.	0.554896	0.277448	+	0.960741i	$0.410511\pi$
$252$	0	0
$253$	171305.	0.168256
$254$	0	0
$255$	52706.5	0.0507591
$256$	0	0
$257$	−1.87969e6	−1.77522	−0.887612	−	0.460592i	$-0.847637\pi$
$257$	−1.87969e6	−1.77522	−0.887612	+	0.460592i	$0.847637\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.44507e6	−1.31307
$262$	0	0
$263$	1.08337e6	0.965801	0.482900	−	0.875675i	$-0.339583\pi$
$263$	1.08337e6	0.965801	0.482900	+	0.875675i	$0.339583\pi$
$264$	0	0
$265$	−1.36083e6	−1.19039
$266$	0	0
$267$	−233959.	−0.200846
$268$	0	0
$269$	−1.10814e6	−0.933710	−0.466855	−	0.884334i	$-0.654613\pi$
$269$	−1.10814e6	−0.933710	−0.466855	+	0.884334i	$0.654613\pi$
$270$	0	0
$271$	−322007.	−0.266344	−0.133172	−	0.991093i	$-0.542516\pi$
$271$	−322007.	−0.266344	−0.133172	+	0.991093i	$0.542516\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	154053.	0.122840
$276$	0	0
$277$	1.45746e6	1.14130	0.570648	−	0.821195i	$-0.306692\pi$
$277$	1.45746e6	1.14130	0.570648	+	0.821195i	$0.306692\pi$
$278$	0	0
$279$	−1.38293e6	−1.06363
$280$	0	0
$281$	1.03123e6	0.779092	0.389546	−	0.921007i	$-0.372632\pi$
$281$	1.03123e6	0.779092	0.389546	+	0.921007i	$0.372632\pi$
$282$	0	0
$283$	79664.8	0.0591290	0.0295645	−	0.999563i	$-0.490588\pi$
$283$	79664.8	0.0591290	0.0295645	+	0.999563i	$0.490588\pi$
$284$	0	0
$285$	289937.	0.211442
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.29343e6	−0.910957
$290$	0	0
$291$	243962.	0.168884
$292$	0	0
$293$	−1.44146e6	−0.980920	−0.490460	−	0.871464i	$-0.663171\pi$
$293$	−1.44146e6	−0.980920	−0.490460	+	0.871464i	$0.663171\pi$
$294$	0	0
$295$	−20515.4	−0.0137254
$296$	0	0
$297$	186609.	0.122756
$298$	0	0
$299$	277072.	0.179232
$300$	0	0
$301$	0	0
$302$	0	0
$303$	417760.	0.261409
$304$	0	0
$305$	−1.68867e6	−1.03943
$306$	0	0
$307$	−1.71962e6	−1.04132	−0.520662	−	0.853763i	$-0.674315\pi$
$307$	−1.71962e6	−1.04132	−0.520662	+	0.853763i	$0.674315\pi$
$308$	0	0
$309$	−38512.4	−0.0229459
$310$	0	0
$311$	−159609.	−0.0935745	−0.0467873	−	0.998905i	$-0.514898\pi$
$311$	−159609.	−0.0935745	−0.0467873	+	0.998905i	$0.514898\pi$
$312$	0	0
$313$	−452617.	−0.261138	−0.130569	−	0.991439i	$-0.541680\pi$
$313$	−452617.	−0.261138	−0.130569	+	0.991439i	$0.541680\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.82631e6	1.02076	0.510382	−	0.859948i	$-0.329504\pi$
$317$	1.82631e6	1.02076	0.510382	+	0.859948i	$0.329504\pi$
$318$	0	0
$319$	−694862.	−0.382315
$320$	0	0
$321$	−392991.	−0.212873
$322$	0	0
$323$	695477.	0.370917
$324$	0	0
$325$	249169.	0.130853
$326$	0	0
$327$	43646.0	0.0225723
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.08212e6	−1.04457	−0.522283	−	0.852772i	$-0.674919\pi$
$331$	−2.08212e6	−1.04457	−0.522283	+	0.852772i	$0.674919\pi$
$332$	0	0
$333$	2.23153e6	1.10279
$334$	0	0
$335$	320149.	0.155862
$336$	0	0
$337$	329493.	0.158041	0.0790207	−	0.996873i	$-0.474821\pi$
$337$	329493.	0.158041	0.0790207	+	0.996873i	$0.474821\pi$
$338$	0	0
$339$	−435401.	−0.205774
$340$	0	0
$341$	−664984.	−0.309689
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−229274.	−0.103707
$346$	0	0
$347$	1.24318e6	0.554256	0.277128	−	0.960833i	$-0.410617\pi$
$347$	1.24318e6	0.554256	0.277128	+	0.960833i	$0.410617\pi$
$348$	0	0
$349$	1.07921e6	0.474287	0.237143	−	0.971475i	$-0.423789\pi$
$349$	1.07921e6	0.474287	0.237143	+	0.971475i	$0.423789\pi$
$350$	0	0
$351$	301825.	0.130764
$352$	0	0
$353$	526521.	0.224895	0.112447	−	0.993658i	$-0.464131\pi$
$353$	526521.	0.224895	0.112447	+	0.993658i	$0.464131\pi$
$354$	0	0
$355$	−3.24304e6	−1.36578
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.42758e6	−0.584606	−0.292303	−	0.956326i	$-0.594422\pi$
$359$	−1.42758e6	−0.584606	−0.292303	+	0.956326i	$0.594422\pi$
$360$	0	0
$361$	1.34969e6	0.545088
$362$	0	0
$363$	−529628.	−0.210962
$364$	0	0
$365$	−3.07799e6	−1.20930
$366$	0	0
$367$	−1.97129e6	−0.763986	−0.381993	−	0.924165i	$-0.624762\pi$
$367$	−1.97129e6	−0.763986	−0.381993	+	0.924165i	$0.624762\pi$
$368$	0	0
$369$	2.43625e6	0.931443
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.64517e6	−0.984422	−0.492211	−	0.870476i	$-0.663811\pi$
$373$	−2.64517e6	−0.984422	−0.492211	+	0.870476i	$0.663811\pi$
$374$	0	0
$375$	−669410.	−0.245818
$376$	0	0
$377$	−1.12388e6	−0.407255
$378$	0	0
$379$	1.17483e6	0.420123	0.210062	−	0.977688i	$-0.432634\pi$
$379$	1.17483e6	0.420123	0.210062	+	0.977688i	$0.432634\pi$
$380$	0	0
$381$	601847.	0.212409
$382$	0	0
$383$	−3.90876e6	−1.36158	−0.680788	−	0.732480i	$-0.738363\pi$
$383$	−3.90876e6	−1.36158	−0.680788	+	0.732480i	$0.738363\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.54689e6	0.525026
$388$	0	0
$389$	195479.	0.0654978	0.0327489	−	0.999464i	$-0.489574\pi$
$389$	195479.	0.0654978	0.0327489	+	0.999464i	$0.489574\pi$
$390$	0	0
$391$	−549965.	−0.181925
$392$	0	0
$393$	−1.34361e6	−0.438825
$394$	0	0
$395$	1.82946e6	0.589969
$396$	0	0
$397$	3.30425e6	1.05220	0.526099	−	0.850424i	$-0.323654\pi$
$397$	3.30425e6	1.05220	0.526099	+	0.850424i	$0.323654\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.38638e6	1.36221	0.681107	−	0.732184i	$-0.261499\pi$
$401$	4.38638e6	1.36221	0.681107	+	0.732184i	$0.261499\pi$
$402$	0	0
$403$	−1.07556e6	−0.329891
$404$	0	0
$405$	2.08093e6	0.630405
$406$	0	0
$407$	1.07303e6	0.321089
$408$	0	0
$409$	3.21445e6	0.950163	0.475081	−	0.879942i	$-0.342419\pi$
$409$	3.21445e6	0.950163	0.475081	+	0.879942i	$0.342419\pi$
$410$	0	0
$411$	1.12710e6	0.329124
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.71540e6	0.488929
$416$	0	0
$417$	−816761.	−0.230014
$418$	0	0
$419$	4.48815e6	1.24891	0.624457	−	0.781059i	$-0.285320\pi$
$419$	4.48815e6	1.24891	0.624457	+	0.781059i	$0.285320\pi$
$420$	0	0
$421$	−4.05150e6	−1.11406	−0.557032	−	0.830491i	$-0.688060\pi$
$421$	−4.05150e6	−1.11406	−0.557032	+	0.830491i	$0.688060\pi$
$422$	0	0
$423$	−6.27434e6	−1.70497
$424$	0	0
$425$	−494579.	−0.132820
$426$	0	0
$427$	0	0
$428$	0	0
$429$	70623.6	0.0185271
$430$	0	0
$431$	−2.83082e6	−0.734039	−0.367019	−	0.930213i	$-0.619622\pi$
$431$	−2.83082e6	−0.734039	−0.367019	+	0.930213i	$0.619622\pi$
$432$	0	0
$433$	3.39735e6	0.870804	0.435402	−	0.900236i	$-0.356606\pi$
$433$	3.39735e6	0.870804	0.435402	+	0.900236i	$0.356606\pi$
$434$	0	0
$435$	930000.	0.235646
$436$	0	0
$437$	−3.02534e6	−0.757827
$438$	0	0
$439$	3.39261e6	0.840181	0.420091	−	0.907482i	$-0.361998\pi$
$439$	3.39261e6	0.840181	0.420091	+	0.907482i	$0.361998\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.66957e6	−1.13049	−0.565246	−	0.824922i	$-0.691219\pi$
$443$	−4.66957e6	−1.13049	−0.565246	+	0.824922i	$0.691219\pi$
$444$	0	0
$445$	−2.73689e6	−0.655175
$446$	0	0
$447$	560657.	0.132718
$448$	0	0
$449$	−782382.	−0.183148	−0.0915742	−	0.995798i	$-0.529190\pi$
$449$	−782382.	−0.183148	−0.0915742	+	0.995798i	$0.529190\pi$
$450$	0	0
$451$	1.17147e6	0.271201
$452$	0	0
$453$	522508.	0.119632
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−531415.	−0.119026	−0.0595132	−	0.998228i	$-0.518955\pi$
$457$	−531415.	−0.119026	−0.0595132	+	0.998228i	$0.518955\pi$
$458$	0	0
$459$	−599097.	−0.132729
$460$	0	0
$461$	1.34721e6	0.295245	0.147623	−	0.989044i	$-0.452838\pi$
$461$	1.34721e6	0.295245	0.147623	+	0.989044i	$0.452838\pi$
$462$	0	0
$463$	−6.81207e6	−1.47682	−0.738409	−	0.674354i	$-0.764422\pi$
$463$	−6.81207e6	−1.47682	−0.738409	+	0.674354i	$0.764422\pi$
$464$	0	0
$465$	890012.	0.190881
$466$	0	0
$467$	−1.02154e6	−0.216752	−0.108376	−	0.994110i	$-0.534565\pi$
$467$	−1.02154e6	−0.216752	−0.108376	+	0.994110i	$0.534565\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.16575e6	0.242132
$472$	0	0
$473$	743821.	0.152868
$474$	0	0
$475$	−2.72066e6	−0.553274
$476$	0	0
$477$	7.52700e6	1.51470
$478$	0	0
$479$	8.31826e6	1.65651	0.828254	−	0.560353i	$-0.189334\pi$
$479$	8.31826e6	1.65651	0.828254	+	0.560353i	$0.189334\pi$
$480$	0	0
$481$	1.73554e6	0.342036
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.85390e6	0.550916
$486$	0	0
$487$	−602725.	−0.115159	−0.0575793	−	0.998341i	$-0.518338\pi$
$487$	−602725.	−0.115159	−0.0575793	+	0.998341i	$0.518338\pi$
$488$	0	0
$489$	−1.23944e6	−0.234397
$490$	0	0
$491$	5.27617e6	0.987678	0.493839	−	0.869553i	$-0.335593\pi$
$491$	5.27617e6	0.987678	0.493839	+	0.869553i	$0.335593\pi$
$492$	0	0
$493$	2.23081e6	0.413376
$494$	0	0
$495$	1.06227e6	0.194860
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.19577e6	1.47346	0.736730	−	0.676187i	$-0.236369\pi$
$499$	8.19577e6	1.47346	0.736730	+	0.676187i	$0.236369\pi$
$500$	0	0
$501$	−1.94990e6	−0.347070
$502$	0	0
$503$	3.81947e6	0.673106	0.336553	−	0.941665i	$-0.390739\pi$
$503$	3.81947e6	0.673106	0.336553	+	0.941665i	$0.390739\pi$
$504$	0	0
$505$	4.88702e6	0.852739
$506$	0	0
$507$	−1.20746e6	−0.208619
$508$	0	0
$509$	−2.00040e6	−0.342233	−0.171116	−	0.985251i	$-0.554737\pi$
$509$	−2.00040e6	−0.342233	−0.171116	+	0.985251i	$0.554737\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.29561e6	−0.552895
$514$	0	0
$515$	−450524.	−0.0748514
$516$	0	0
$517$	−3.01702e6	−0.496422
$518$	0	0
$519$	−649790.	−0.105890
$520$	0	0
$521$	−1.16663e7	−1.88295	−0.941477	−	0.337077i	$-0.890562\pi$
$521$	−1.16663e7	−1.88295	−0.941477	+	0.337077i	$0.890562\pi$
$522$	0	0
$523$	1.08222e7	1.73006	0.865029	−	0.501723i	$-0.167300\pi$
$523$	1.08222e7	1.73006	0.865029	+	0.501723i	$0.167300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.13489e6	0.334849
$528$	0	0
$529$	−4.04398e6	−0.628305
$530$	0	0
$531$	113474.	0.0174647
$532$	0	0
$533$	1.89476e6	0.288892
$534$	0	0
$535$	−4.59727e6	−0.694409
$536$	0	0
$537$	−484557.	−0.0725118
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.08714e6	0.159695	0.0798477	−	0.996807i	$-0.474557\pi$
$541$	1.08714e6	0.159695	0.0798477	+	0.996807i	$0.474557\pi$
$542$	0	0
$543$	−2.10513e6	−0.306394
$544$	0	0
$545$	510578.	0.0736327
$546$	0	0
$547$	−413333.	−0.0590652	−0.0295326	−	0.999564i	$-0.509402\pi$
$547$	−413333.	−0.0590652	−0.0295326	+	0.999564i	$0.509402\pi$
$548$	0	0
$549$	9.34034e6	1.32261
$550$	0	0
$551$	1.22716e7	1.72196
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.43614e6	−0.197908
$556$	0	0
$557$	−1.20182e7	−1.64135	−0.820673	−	0.571398i	$-0.806401\pi$
$557$	−1.20182e7	−1.64135	−0.820673	+	0.571398i	$0.806401\pi$
$558$	0	0
$559$	1.20307e6	0.162840
$560$	0	0
$561$	−140182.	−0.0188055
$562$	0	0
$563$	1.42532e7	1.89514	0.947569	−	0.319551i	$-0.103532\pi$
$563$	1.42532e7	1.89514	0.947569	+	0.319551i	$0.103532\pi$
$564$	0	0
$565$	−5.09339e6	−0.671253
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.73389e6	0.353998	0.176999	−	0.984211i	$-0.443361\pi$
$569$	2.73389e6	0.353998	0.176999	+	0.984211i	$0.443361\pi$
$570$	0	0
$571$	3.60896e6	0.463225	0.231612	−	0.972808i	$-0.425600\pi$
$571$	3.60896e6	0.463225	0.231612	+	0.972808i	$0.425600\pi$
$572$	0	0
$573$	1.57051e6	0.199826
$574$	0	0
$575$	2.15143e6	0.271367
$576$	0	0
$577$	1.62585e6	0.203302	0.101651	−	0.994820i	$-0.467587\pi$
$577$	1.62585e6	0.203302	0.101651	+	0.994820i	$0.467587\pi$
$578$	0	0
$579$	1.59608e6	0.197860
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.61936e6	0.441022
$584$	0	0
$585$	1.71814e6	0.207572
$586$	0	0
$587$	9.69220e6	1.16099	0.580493	−	0.814265i	$-0.302860\pi$
$587$	9.69220e6	1.16099	0.580493	+	0.814265i	$0.302860\pi$
$588$	0	0
$589$	1.17439e7	1.39484
$590$	0	0
$591$	−1.92942e6	−0.227226
$592$	0	0
$593$	3.64266e6	0.425384	0.212692	−	0.977119i	$-0.431777\pi$
$593$	3.64266e6	0.425384	0.212692	+	0.977119i	$0.431777\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−704672.	−0.0809191
$598$	0	0
$599$	−657286.	−0.0748492	−0.0374246	−	0.999299i	$-0.511915\pi$
$599$	−657286.	−0.0748492	−0.0374246	+	0.999299i	$0.511915\pi$
$600$	0	0
$601$	1.45544e7	1.64365	0.821824	−	0.569742i	$-0.192957\pi$
$601$	1.45544e7	1.64365	0.821824	+	0.569742i	$0.192957\pi$
$602$	0	0
$603$	−1.77080e6	−0.198324
$604$	0	0
$605$	−6.19567e6	−0.688176
$606$	0	0
$607$	−1.40843e7	−1.55154	−0.775770	−	0.631015i	$-0.782638\pi$
$607$	−1.40843e7	−1.55154	−0.775770	+	0.631015i	$0.782638\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.87977e6	−0.528806
$612$	0	0
$613$	3.44096e6	0.369853	0.184926	−	0.982752i	$-0.440795\pi$
$613$	3.44096e6	0.369853	0.184926	+	0.982752i	$0.440795\pi$
$614$	0	0
$615$	−1.56789e6	−0.167159
$616$	0	0
$617$	6.03183e6	0.637876	0.318938	−	0.947776i	$-0.396674\pi$
$617$	6.03183e6	0.637876	0.318938	+	0.947776i	$0.396674\pi$
$618$	0	0
$619$	1.14635e7	1.20252	0.601259	−	0.799054i	$-0.294666\pi$
$619$	1.14635e7	1.20252	0.601259	+	0.799054i	$0.294666\pi$
$620$	0	0
$621$	2.60608e6	0.271181
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.48412e6	−0.356774
$626$	0	0
$627$	−771136.	−0.0783361
$628$	0	0
$629$	−3.44490e6	−0.347176
$630$	0	0
$631$	−536107.	−0.0536016	−0.0268008	−	0.999641i	$-0.508532\pi$
$631$	−536107.	−0.0536016	−0.0268008	+	0.999641i	$0.508532\pi$
$632$	0	0
$633$	−4.04587e6	−0.401331
$634$	0	0
$635$	7.04050e6	0.692898
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.79378e7	1.73787
$640$	0	0
$641$	2.91174e6	0.279903	0.139952	−	0.990158i	$-0.455305\pi$
$641$	2.91174e6	0.279903	0.139952	+	0.990158i	$0.455305\pi$
$642$	0	0
$643$	1.93974e7	1.85019	0.925096	−	0.379732i	$-0.123984\pi$
$643$	1.93974e7	1.85019	0.925096	+	0.379732i	$0.123984\pi$
$644$	0	0
$645$	−995527.	−0.0942223
$646$	0	0
$647$	−1.40085e7	−1.31562	−0.657811	−	0.753183i	$-0.728517\pi$
$647$	−1.40085e7	−1.31562	−0.657811	+	0.753183i	$0.728517\pi$
$648$	0	0
$649$	54564.2	0.00508506
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.11062e7	1.01926	0.509629	−	0.860394i	$-0.329783\pi$
$653$	1.11062e7	1.01926	0.509629	+	0.860394i	$0.329783\pi$
$654$	0	0
$655$	−1.57177e7	−1.43149
$656$	0	0
$657$	1.70249e7	1.53877
$658$	0	0
$659$	1.00615e7	0.902502	0.451251	−	0.892397i	$-0.350978\pi$
$659$	1.00615e7	0.902502	0.451251	+	0.892397i	$0.350978\pi$
$660$	0	0
$661$	1.42867e7	1.27183	0.635916	−	0.771758i	$-0.280623\pi$
$661$	1.42867e7	1.27183	0.635916	+	0.771758i	$0.280623\pi$
$662$	0	0
$663$	−226733.	−0.0200323
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.70406e6	−0.844577
$668$	0	0
$669$	−1.76944e6	−0.152852
$670$	0	0
$671$	4.49130e6	0.385094
$672$	0	0
$673$	−2.05348e7	−1.74764	−0.873822	−	0.486245i	$-0.838366\pi$
$673$	−2.05348e7	−1.74764	−0.873822	+	0.486245i	$0.838366\pi$
$674$	0	0
$675$	2.34363e6	0.197984
$676$	0	0
$677$	3.59295e6	0.301287	0.150643	−	0.988588i	$-0.451865\pi$
$677$	3.59295e6	0.301287	0.150643	+	0.988588i	$0.451865\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.32667e6	0.274879
$682$	0	0
$683$	1.19204e7	0.977773	0.488886	−	0.872347i	$-0.337403\pi$
$683$	1.19204e7	0.977773	0.488886	+	0.872347i	$0.337403\pi$
$684$	0	0
$685$	1.31850e7	1.07363
$686$	0	0
$687$	−3.74501e6	−0.302734
$688$	0	0
$689$	5.85402e6	0.469792
$690$	0	0
$691$	−4.28101e6	−0.341076	−0.170538	−	0.985351i	$-0.554551\pi$
$691$	−4.28101e6	−0.341076	−0.170538	+	0.985351i	$0.554551\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.55459e6	−0.750326
$696$	0	0
$697$	−3.76094e6	−0.293234
$698$	0	0
$699$	1.47579e6	0.114243
$700$	0	0
$701$	−8.79081e6	−0.675669	−0.337834	−	0.941206i	$-0.609694\pi$
$701$	−8.79081e6	−0.675669	−0.337834	+	0.941206i	$0.609694\pi$
$702$	0	0
$703$	−1.89502e7	−1.44619
$704$	0	0
$705$	4.03796e6	0.305978
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.97977e6	0.297332	0.148666	−	0.988887i	$-0.452502\pi$
$709$	3.97977e6	0.297332	0.148666	+	0.988887i	$0.452502\pi$
$710$	0	0
$711$	−1.01191e7	−0.750699
$712$	0	0
$713$	−9.28681e6	−0.684136
$714$	0	0
$715$	826166.	0.0604369
$716$	0	0
$717$	−1.71506e6	−0.124589
$718$	0	0
$719$	−2.88818e6	−0.208354	−0.104177	−	0.994559i	$-0.533221\pi$
$719$	−2.88818e6	−0.208354	−0.104177	+	0.994559i	$0.533221\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−103092.	−0.00733467
$724$	0	0
$725$	−8.72679e6	−0.616608
$726$	0	0
$727$	−2.12040e7	−1.48793	−0.743963	−	0.668221i	$-0.767056\pi$
$727$	−2.12040e7	−1.48793	−0.743963	+	0.668221i	$0.767056\pi$
$728$	0	0
$729$	−1.00526e7	−0.700580
$730$	0	0
$731$	−2.38799e6	−0.165287
$732$	0	0
$733$	1.95972e7	1.34720	0.673602	−	0.739094i	$-0.264746\pi$
$733$	1.95972e7	1.34720	0.673602	+	0.739094i	$0.264746\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−851490.	−0.0577445
$738$	0	0
$739$	−2.41132e7	−1.62422	−0.812108	−	0.583507i	$-0.801680\pi$
$739$	−2.41132e7	−1.62422	−0.812108	+	0.583507i	$0.801680\pi$
$740$	0	0
$741$	−1.24725e6	−0.0834464
$742$	0	0
$743$	7.79774e6	0.518199	0.259100	−	0.965851i	$-0.416574\pi$
$743$	7.79774e6	0.518199	0.259100	+	0.965851i	$0.416574\pi$
$744$	0	0
$745$	6.55865e6	0.432936
$746$	0	0
$747$	−9.48821e6	−0.622132
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.01687e7	1.30490	0.652450	−	0.757832i	$-0.273741\pi$
$751$	2.01687e7	1.30490	0.652450	+	0.757832i	$0.273741\pi$
$752$	0	0
$753$	1.97155e6	0.126713
$754$	0	0
$755$	6.11238e6	0.390250
$756$	0	0
$757$	−2.80945e6	−0.178190	−0.0890948	−	0.996023i	$-0.528397\pi$
$757$	−2.80945e6	−0.178190	−0.0890948	+	0.996023i	$0.528397\pi$
$758$	0	0
$759$	609794.	0.0384219
$760$	0	0
$761$	1.87055e7	1.17087	0.585433	−	0.810721i	$-0.300925\pi$
$761$	1.87055e7	1.17087	0.585433	+	0.810721i	$0.300925\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.41036e6	−0.210691
$766$	0	0
$767$	88253.1	0.00541679
$768$	0	0
$769$	−2.25184e7	−1.37316	−0.686580	−	0.727054i	$-0.740889\pi$
$769$	−2.25184e7	−1.37316	−0.686580	+	0.727054i	$0.740889\pi$
$770$	0	0
$771$	−6.69111e6	−0.405380
$772$	0	0
$773$	−2.24941e7	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$773$	−2.24941e7	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$774$	0	0
$775$	−8.35155e6	−0.499474
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.06888e7	−1.22149
$780$	0	0
$781$	8.62541e6	0.506002
$782$	0	0
$783$	−1.05710e7	−0.616186
$784$	0	0
$785$	1.36371e7	0.789854
$786$	0	0
$787$	9.80027e6	0.564029	0.282014	−	0.959410i	$-0.408997\pi$
$787$	9.80027e6	0.564029	0.282014	+	0.959410i	$0.408997\pi$
$788$	0	0
$789$	3.85647e6	0.220545
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.26431e6	0.410215
$794$	0	0
$795$	−4.84414e6	−0.271831
$796$	0	0
$797$	1.21486e7	0.677456	0.338728	−	0.940884i	$-0.390003\pi$
$797$	1.21486e7	0.677456	0.338728	+	0.940884i	$0.390003\pi$
$798$	0	0
$799$	9.68594e6	0.536753
$800$	0	0
$801$	1.51382e7	0.833670
$802$	0	0
$803$	8.18645e6	0.448030
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.94462e6	−0.213217
$808$	0	0
$809$	−5.73864e6	−0.308275	−0.154137	−	0.988049i	$-0.549260\pi$
$809$	−5.73864e6	−0.308275	−0.154137	+	0.988049i	$0.549260\pi$
$810$	0	0
$811$	1.14111e7	0.609220	0.304610	−	0.952477i	$-0.401474\pi$
$811$	1.14111e7	0.609220	0.304610	+	0.952477i	$0.401474\pi$
$812$	0	0
$813$	−1.14625e6	−0.0608208
$814$	0	0
$815$	−1.44991e7	−0.764624
$816$	0	0
$817$	−1.31362e7	−0.688519
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.53935e7	−0.797038	−0.398519	−	0.917160i	$-0.630476\pi$
$821$	−1.53935e7	−0.797038	−0.398519	+	0.917160i	$0.630476\pi$
$822$	0	0
$823$	1.69192e7	0.870722	0.435361	−	0.900256i	$-0.356621\pi$
$823$	1.69192e7	0.870722	0.435361	+	0.900256i	$0.356621\pi$
$824$	0	0
$825$	548383.	0.0280511
$826$	0	0
$827$	2.64225e7	1.34341	0.671707	−	0.740817i	$-0.265561\pi$
$827$	2.64225e7	1.34341	0.671707	+	0.740817i	$0.265561\pi$
$828$	0	0
$829$	1.97821e6	0.0999737	0.0499869	−	0.998750i	$-0.484082\pi$
$829$	1.97821e6	0.0999737	0.0499869	+	0.998750i	$0.484082\pi$
$830$	0	0
$831$	5.18812e6	0.260620
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.28102e7	−1.13217
$836$	0	0
$837$	−1.01165e7	−0.499132
$838$	0	0
$839$	−2.20079e7	−1.07938	−0.539689	−	0.841865i	$-0.681458\pi$
$839$	−2.20079e7	−1.07938	−0.539689	+	0.841865i	$0.681458\pi$
$840$	0	0
$841$	1.88512e7	0.919071
$842$	0	0
$843$	3.67086e6	0.177909
$844$	0	0
$845$	−1.41251e7	−0.680532
$846$	0	0
$847$	0	0
$848$	0	0
$849$	283582.	0.0135024
$850$	0	0
$851$	1.49854e7	0.709322
$852$	0	0
$853$	2.53601e6	0.119338	0.0596689	−	0.998218i	$-0.480996\pi$
$853$	2.53601e6	0.119338	0.0596689	+	0.998218i	$0.480996\pi$
$854$	0	0
$855$	−1.87602e7	−0.877653
$856$	0	0
$857$	1.14631e6	0.0533151	0.0266576	−	0.999645i	$-0.491514\pi$
$857$	1.14631e6	0.0533151	0.0266576	+	0.999645i	$0.491514\pi$
$858$	0	0
$859$	1.67759e7	0.775717	0.387858	−	0.921719i	$-0.373215\pi$
$859$	1.67759e7	0.775717	0.387858	+	0.921719i	$0.373215\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.84806e7	0.844672	0.422336	−	0.906439i	$-0.361210\pi$
$863$	1.84806e7	0.844672	0.422336	+	0.906439i	$0.361210\pi$
$864$	0	0
$865$	−7.60134e6	−0.345422
$866$	0	0
$867$	−4.60421e6	−0.208021
$868$	0	0
$869$	−4.86575e6	−0.218575
$870$	0	0
$871$	−1.37721e6	−0.0615115
$872$	0	0
$873$	−1.57855e7	−0.701006
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.61410e6	−0.290383	−0.145192	−	0.989404i	$-0.546380\pi$
$877$	−6.61410e6	−0.290383	−0.145192	+	0.989404i	$0.546380\pi$
$878$	0	0
$879$	−5.13115e6	−0.223997
$880$	0	0
$881$	−3.43282e7	−1.49009	−0.745043	−	0.667016i	$-0.767571\pi$
$881$	−3.43282e7	−1.49009	−0.745043	+	0.667016i	$0.767571\pi$
$882$	0	0
$883$	−5.51963e6	−0.238236	−0.119118	−	0.992880i	$-0.538007\pi$
$883$	−5.51963e6	−0.238236	−0.119118	+	0.992880i	$0.538007\pi$
$884$	0	0
$885$	−73028.6	−0.00313426
$886$	0	0
$887$	−7.55469e6	−0.322410	−0.161205	−	0.986921i	$-0.551538\pi$
$887$	−7.55469e6	−0.322410	−0.161205	+	0.986921i	$0.551538\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.53459e6	−0.233556
$892$	0	0
$893$	5.32820e7	2.23590
$894$	0	0
$895$	−5.66842e6	−0.236540
$896$	0	0
$897$	986292.	0.0409284
$898$	0	0
$899$	3.76699e7	1.55451
$900$	0	0
$901$	−1.16197e7	−0.476853
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.46262e7	−0.999484
$906$	0	0
$907$	3.38270e7	1.36536	0.682678	−	0.730720i	$-0.260815\pi$
$907$	3.38270e7	1.36536	0.682678	+	0.730720i	$0.260815\pi$
$908$	0	0
$909$	−2.70310e7	−1.08506
$910$	0	0
$911$	−3.27859e7	−1.30885	−0.654426	−	0.756126i	$-0.727090\pi$
$911$	−3.27859e7	−1.30885	−0.654426	+	0.756126i	$0.727090\pi$
$912$	0	0
$913$	−4.56241e6	−0.181141
$914$	0	0
$915$	−6.01115e6	−0.237358
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.69872e7	1.83523	0.917615	−	0.397470i	$-0.130112\pi$
$919$	4.69872e7	1.83523	0.917615	+	0.397470i	$0.130112\pi$
$920$	0	0
$921$	−6.12131e6	−0.237791
$922$	0	0
$923$	1.39509e7	0.539011
$924$	0	0
$925$	1.34762e7	0.517862
$926$	0	0
$927$	2.49193e6	0.0952438
$928$	0	0
$929$	−3.95603e7	−1.50390	−0.751952	−	0.659218i	$-0.770888\pi$
$929$	−3.95603e7	−1.50390	−0.751952	+	0.659218i	$0.770888\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−568161.	−0.0213682
$934$	0	0
$935$	−1.63987e6	−0.0613452
$936$	0	0
$937$	2.90483e7	1.08086	0.540432	−	0.841387i	$-0.318261\pi$
$937$	2.90483e7	1.08086	0.540432	+	0.841387i	$0.318261\pi$
$938$	0	0
$939$	−1.61118e6	−0.0596320
$940$	0	0
$941$	−3.31199e7	−1.21931	−0.609656	−	0.792666i	$-0.708693\pi$
$941$	−3.31199e7	−1.21931	−0.609656	+	0.792666i	$0.708693\pi$
$942$	0	0
$943$	1.63602e7	0.599112
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.99131e7	−1.44624	−0.723121	−	0.690721i	$-0.757293\pi$
$947$	−3.99131e7	−1.44624	−0.723121	+	0.690721i	$0.757293\pi$
$948$	0	0
$949$	1.32409e7	0.477257
$950$	0	0
$951$	6.50109e6	0.233096
$952$	0	0
$953$	1.23250e7	0.439596	0.219798	−	0.975545i	$-0.429460\pi$
$953$	1.23250e7	0.439596	0.219798	+	0.975545i	$0.429460\pi$
$954$	0	0
$955$	1.83720e7	0.651851
$956$	0	0
$957$	−2.47349e6	−0.0873034
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.42097e6	0.259210
$962$	0	0
$963$	2.54283e7	0.883592
$964$	0	0
$965$	1.86712e7	0.645436
$966$	0	0
$967$	−4.13540e7	−1.42217	−0.711084	−	0.703107i	$-0.751796\pi$
$967$	−4.13540e7	−1.42217	−0.711084	+	0.703107i	$0.751796\pi$
$968$	0	0
$969$	2.47568e6	0.0847004
$970$	0	0
$971$	−2.88317e7	−0.981345	−0.490672	−	0.871344i	$-0.663249\pi$
$971$	−2.88317e7	−0.981345	−0.490672	+	0.871344i	$0.663249\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	886964.	0.0298810
$976$	0	0
$977$	−8.85143e6	−0.296672	−0.148336	−	0.988937i	$-0.547392\pi$
$977$	−8.85143e6	−0.296672	−0.148336	+	0.988937i	$0.547392\pi$
$978$	0	0
$979$	7.27923e6	0.242733
$980$	0	0
$981$	−2.82410e6	−0.0936931
$982$	0	0
$983$	−2.77705e7	−0.916643	−0.458322	−	0.888786i	$-0.651549\pi$
$983$	−2.77705e7	−0.916643	−0.458322	+	0.888786i	$0.651549\pi$
$984$	0	0
$985$	−2.25707e7	−0.741232
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.03878e7	0.337702
$990$	0	0
$991$	5.45725e7	1.76518	0.882591	−	0.470141i	$-0.155797\pi$
$991$	5.45725e7	1.76518	0.882591	+	0.470141i	$0.155797\pi$
$992$	0	0
$993$	−7.41171e6	−0.238531
$994$	0	0
$995$	−8.24336e6	−0.263965
$996$	0	0
$997$	−1.63991e7	−0.522494	−0.261247	−	0.965272i	$-0.584134\pi$
$997$	−1.63991e7	−0.522494	−0.261247	+	0.965272i	$0.584134\pi$
$998$	0	0
$999$	1.63241e7	0.517507

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bd.1.1		2
4.3	odd	2		196.6.a.h.1.2		2
7.3	odd	6		112.6.i.e.65.1		4
7.5	odd	6		112.6.i.e.81.1		4
7.6	odd	2		784.6.a.o.1.2		2
28.3	even	6		28.6.e.b.9.2	✓	4
28.11	odd	6		196.6.e.k.177.1		4
28.19	even	6		28.6.e.b.25.2	yes	4
28.23	odd	6		196.6.e.k.165.1		4
28.27	even	2		196.6.a.j.1.1		2
84.47	odd	6		252.6.k.d.109.1		4
84.59	odd	6		252.6.k.d.37.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.e.b.9.2	✓	4	28.3	even	6
28.6.e.b.25.2	yes	4	28.19	even	6
112.6.i.e.65.1		4	7.3	odd	6
112.6.i.e.81.1		4	7.5	odd	6
196.6.a.h.1.2		2	4.3	odd	2
196.6.a.j.1.1		2	28.27	even	2
196.6.e.k.165.1		4	28.23	odd	6
196.6.e.k.177.1		4	28.11	odd	6
252.6.k.d.37.1		4	84.59	odd	6
252.6.k.d.109.1		4	84.47	odd	6
784.6.a.o.1.2		2	7.6	odd	2
784.6.a.bd.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+3.55969 q^{3} +41.6418 q^{5} -230.329 q^{9} -110.754 q^{11} -179.135 q^{13} +148.232 q^{15} +355.567 q^{17} +1955.96 q^{19} -1546.73 q^{23} -1390.96 q^{25} -1684.90 q^{27} +6273.94 q^{29} +6004.18 q^{31} -394.249 q^{33} -9688.45 q^{37} -637.665 q^{39} -10577.3 q^{41} -6716.00 q^{43} -9591.31 q^{45} +27240.8 q^{47} +1265.71 q^{51} -32679.4 q^{53} -4611.98 q^{55} +6962.63 q^{57} -492.663 q^{59} -40552.2 q^{61} -7459.50 q^{65} +7688.15 q^{67} -5505.87 q^{69} -77879.3 q^{71} -73915.9 q^{73} -4951.38 q^{75} +43933.1 q^{79} +49972.1 q^{81} +41194.2 q^{83} +14806.5 q^{85} +22333.3 q^{87} -65724.6 q^{89} +21373.0 q^{93} +81449.9 q^{95} +68534.5 q^{97} +25509.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 28 q^{3} - 42 q^{5} + 124 q^{9} - 660 q^{11} + 644 q^{13} - 1896 q^{15} + 210 q^{17} + 3724 q^{19} - 24 q^{23} + 2480 q^{25} + 1036 q^{27} + 5532 q^{29} + 2800 q^{31} - 13818 q^{33} - 13238 q^{37}+ \cdots - 169104 q^{99}+O(q^{100}) \) 2 * q + 28 * q^3 - 42 * q^5 + 124 * q^9 - 660 * q^11 + 644 * q^13 - 1896 * q^15 + 210 * q^17 + 3724 * q^19 - 24 * q^23 + 2480 * q^25 + 1036 * q^27 + 5532 * q^29 + 2800 * q^31 - 13818 * q^33 - 13238 * q^37 + 19480 * q^39 - 4116 * q^41 - 13432 * q^43 - 39228 * q^45 + 8064 * q^47 - 2292 * q^51 - 53958 * q^53 + 41328 * q^55 + 50174 * q^57 + 36036 * q^59 - 83986 * q^61 - 76308 * q^65 + 2660 * q^67 + 31710 * q^69 - 71568 * q^71 - 31318 * q^73 + 89656 * q^75 - 51136 * q^79 + 30370 * q^81 + 6216 * q^83 + 26982 * q^85 + 4200 * q^87 - 166278 * q^89 - 56938 * q^93 - 66432 * q^95 - 8260 * q^97 - 169104 * q^99

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