LMFDB - Embedded newform 784.6.a.bl.1.3

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 = [5,0,5,0,81,0,0,0,390] 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:	$0$
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} - 200x^{3} - 99x^{2} + 5803x - 3615 $ x^5 - 200x^3 - 99x^2 + 5803*x - 3615
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.638887$ 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)

$f(q)$	$=$	$q+2.27777 q^{3} -50.5110 q^{5} -237.812 q^{9} -192.028 q^{11} +656.798 q^{13} -115.053 q^{15} -2189.37 q^{17} -279.330 q^{19} +3713.60 q^{23} -573.636 q^{25} -1095.18 q^{27} -5276.51 q^{29} -6485.80 q^{31} -437.396 q^{33} +2950.08 q^{37} +1496.04 q^{39} +2394.37 q^{41} -17261.5 q^{43} +12012.1 q^{45} +7720.77 q^{47} -4986.89 q^{51} -8497.26 q^{53} +9699.52 q^{55} -636.251 q^{57} -8935.01 q^{59} +22389.6 q^{61} -33175.6 q^{65} -52695.0 q^{67} +8458.73 q^{69} +25479.0 q^{71} +22908.1 q^{73} -1306.61 q^{75} +45919.1 q^{79} +55293.7 q^{81} -97180.7 q^{83} +110587. q^{85} -12018.7 q^{87} +59131.4 q^{89} -14773.2 q^{93} +14109.3 q^{95} +73491.5 q^{97} +45666.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 81 q^{5} + 390 q^{9} - 361 q^{11} + 342 q^{13} + 1049 q^{15} + 1809 q^{17} - 1277 q^{19} - 911 q^{23} + 3940 q^{25} + 4751 q^{27} + 5442 q^{29} - 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots - 249798 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 81 * q^5 + 390 * q^9 - 361 * q^11 + 342 * q^13 + 1049 * q^15 + 1809 * q^17 - 1277 * q^19 - 911 * q^23 + 3940 * q^25 + 4751 * q^27 + 5442 * q^29 - 2187 * q^31 - 5553 * q^33 - 8181 * q^37 + 3422 * q^39 + 16578 * q^41 - 6332 * q^43 + 41310 * q^45 - 16101 * q^47 + 67865 * q^51 + 16047 * q^53 - 45629 * q^55 + 22347 * q^57 - 71027 * q^59 + 31093 * q^61 - 64370 * q^65 - 47981 * q^67 + 137249 * q^69 - 22512 * q^71 + 123333 * q^73 + 45460 * q^75 + 212481 * q^79 + 52917 * q^81 - 87460 * q^83 + 222141 * q^85 - 318070 * q^87 + 129045 * q^89 - 252835 * q^93 - 300417 * q^95 + 328274 * q^97 - 249798 * 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
$3$	2.27777	0.146119	0.0730596	−	0.997328i	$-0.476724\pi$
$3$	2.27777	0.146119	0.0730596	+	0.997328i	$0.476724\pi$
$4$	0	0
$5$	−50.5110	−0.903569	−0.451784	−	0.892127i	$-0.649212\pi$
$5$	−50.5110	−0.903569	−0.451784	+	0.892127i	$0.649212\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−237.812	−0.978649
$10$	0	0
$11$	−192.028	−0.478500	−0.239250	−	0.970958i	$-0.576902\pi$
$11$	−192.028	−0.478500	−0.239250	+	0.970958i	$0.576902\pi$
$12$	0	0
$13$	656.798	1.07789	0.538944	−	0.842342i	$-0.318823\pi$
$13$	656.798	1.07789	0.538944	+	0.842342i	$0.318823\pi$
$14$	0	0
$15$	−115.053	−0.132029
$16$	0	0
$17$	−2189.37	−1.83737	−0.918685	−	0.394991i	$-0.870748\pi$
$17$	−2189.37	−1.83737	−0.918685	+	0.394991i	$0.870748\pi$
$18$	0	0
$19$	−279.330	−0.177515	−0.0887573	−	0.996053i	$-0.528290\pi$
$19$	−279.330	−0.177515	−0.0887573	+	0.996053i	$0.528290\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3713.60	1.46378	0.731889	−	0.681424i	$-0.238639\pi$
$23$	3713.60	1.46378	0.731889	+	0.681424i	$0.238639\pi$
$24$	0	0
$25$	−573.636	−0.183564
$26$	0	0
$27$	−1095.18	−0.289119
$28$	0	0
$29$	−5276.51	−1.16507	−0.582535	−	0.812806i	$-0.697939\pi$
$29$	−5276.51	−1.16507	−0.582535	+	0.812806i	$0.697939\pi$
$30$	0	0
$31$	−6485.80	−1.21216	−0.606079	−	0.795405i	$-0.707258\pi$
$31$	−6485.80	−1.21216	−0.606079	+	0.795405i	$0.707258\pi$
$32$	0	0
$33$	−437.396	−0.0699181
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2950.08	0.354266	0.177133	−	0.984187i	$-0.443318\pi$
$37$	2950.08	0.354266	0.177133	+	0.984187i	$0.443318\pi$
$38$	0	0
$39$	1496.04	0.157500
$40$	0	0
$41$	2394.37	0.222449	0.111225	−	0.993795i	$-0.464523\pi$
$41$	2394.37	0.222449	0.111225	+	0.993795i	$0.464523\pi$
$42$	0	0
$43$	−17261.5	−1.42366	−0.711832	−	0.702350i	$-0.752134\pi$
$43$	−17261.5	−1.42366	−0.711832	+	0.702350i	$0.752134\pi$
$44$	0	0
$45$	12012.1	0.884277
$46$	0	0
$47$	7720.77	0.509819	0.254909	−	0.966965i	$-0.417954\pi$
$47$	7720.77	0.509819	0.254909	+	0.966965i	$0.417954\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4986.89	−0.268475
$52$	0	0
$53$	−8497.26	−0.415517	−0.207759	−	0.978180i	$-0.566617\pi$
$53$	−8497.26	−0.415517	−0.207759	+	0.978180i	$0.566617\pi$
$54$	0	0
$55$	9699.52	0.432358
$56$	0	0
$57$	−636.251	−0.0259383
$58$	0	0
$59$	−8935.01	−0.334168	−0.167084	−	0.985943i	$-0.553435\pi$
$59$	−8935.01	−0.334168	−0.167084	+	0.985943i	$0.553435\pi$
$60$	0	0
$61$	22389.6	0.770410	0.385205	−	0.922831i	$-0.374131\pi$
$61$	22389.6	0.770410	0.385205	+	0.922831i	$0.374131\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−33175.6	−0.973946
$66$	0	0
$67$	−52695.0	−1.43411	−0.717055	−	0.697017i	$-0.754510\pi$
$67$	−52695.0	−1.43411	−0.717055	+	0.697017i	$0.754510\pi$
$68$	0	0
$69$	8458.73	0.213886
$70$	0	0
$71$	25479.0	0.599842	0.299921	−	0.953964i	$-0.403040\pi$
$71$	25479.0	0.599842	0.299921	+	0.953964i	$0.403040\pi$
$72$	0	0
$73$	22908.1	0.503133	0.251566	−	0.967840i	$-0.419054\pi$
$73$	22908.1	0.503133	0.251566	+	0.967840i	$0.419054\pi$
$74$	0	0
$75$	−1306.61	−0.0268222
$76$	0	0
$77$	0	0
$78$	0	0
$79$	45919.1	0.827800	0.413900	−	0.910322i	$-0.364166\pi$
$79$	45919.1	0.827800	0.413900	+	0.910322i	$0.364166\pi$
$80$	0	0
$81$	55293.7	0.936403
$82$	0	0
$83$	−97180.7	−1.54841	−0.774203	−	0.632937i	$-0.781849\pi$
$83$	−97180.7	−1.54841	−0.774203	+	0.632937i	$0.781849\pi$
$84$	0	0
$85$	110587.	1.66019
$86$	0	0
$87$	−12018.7	−0.170239
$88$	0	0
$89$	59131.4	0.791304	0.395652	−	0.918401i	$-0.370519\pi$
$89$	59131.4	0.791304	0.395652	+	0.918401i	$0.370519\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14773.2	−0.177120
$94$	0	0
$95$	14109.3	0.160397
$96$	0	0
$97$	73491.5	0.793063	0.396531	−	0.918021i	$-0.370214\pi$
$97$	73491.5	0.793063	0.396531	+	0.918021i	$0.370214\pi$
$98$	0	0
$99$	45666.5	0.468284
$100$	0	0
$101$	178860.	1.74465	0.872326	−	0.488925i	$-0.162611\pi$
$101$	178860.	1.74465	0.872326	+	0.488925i	$0.162611\pi$
$102$	0	0
$103$	8711.96	0.0809138	0.0404569	−	0.999181i	$-0.487119\pi$
$103$	8711.96	0.0809138	0.0404569	+	0.999181i	$0.487119\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	172719.	1.45841	0.729206	−	0.684294i	$-0.239890\pi$
$107$	172719.	1.45841	0.729206	+	0.684294i	$0.239890\pi$
$108$	0	0
$109$	−166306.	−1.34073	−0.670367	−	0.742030i	$-0.733863\pi$
$109$	−166306.	−1.34073	−0.670367	+	0.742030i	$0.733863\pi$
$110$	0	0
$111$	6719.61	0.0517651
$112$	0	0
$113$	29604.7	0.218104	0.109052	−	0.994036i	$-0.465218\pi$
$113$	29604.7	0.218104	0.109052	+	0.994036i	$0.465218\pi$
$114$	0	0
$115$	−187578.	−1.32262
$116$	0	0
$117$	−156194.	−1.05487
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−124176.	−0.771037
$122$	0	0
$123$	5453.82	0.0325041
$124$	0	0
$125$	186822.	1.06943
$126$	0	0
$127$	−333136.	−1.83279	−0.916395	−	0.400275i	$-0.868915\pi$
$127$	−333136.	−1.83279	−0.916395	+	0.400275i	$0.868915\pi$
$128$	0	0
$129$	−39317.8	−0.208025
$130$	0	0
$131$	−58228.0	−0.296452	−0.148226	−	0.988954i	$-0.547356\pi$
$131$	−58228.0	−0.296452	−0.148226	+	0.988954i	$0.547356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	55318.7	0.261239
$136$	0	0
$137$	327685.	1.49161	0.745805	−	0.666164i	$-0.232065\pi$
$137$	327685.	1.49161	0.745805	+	0.666164i	$0.232065\pi$
$138$	0	0
$139$	165074.	0.724674	0.362337	−	0.932047i	$-0.381979\pi$
$139$	165074.	0.724674	0.362337	+	0.932047i	$0.381979\pi$
$140$	0	0
$141$	17586.2	0.0744943
$142$	0	0
$143$	−126124.	−0.515770
$144$	0	0
$145$	266522.	1.05272
$146$	0	0
$147$	0	0
$148$	0	0
$149$	117988.	0.435383	0.217692	−	0.976018i	$-0.430147\pi$
$149$	117988.	0.435383	0.217692	+	0.976018i	$0.430147\pi$
$150$	0	0
$151$	−415301.	−1.48225	−0.741123	−	0.671369i	$-0.765706\pi$
$151$	−415301.	−1.48225	−0.741123	+	0.671369i	$0.765706\pi$
$152$	0	0
$153$	520658.	1.79814
$154$	0	0
$155$	327604.	1.09527
$156$	0	0
$157$	453131.	1.46715	0.733575	−	0.679608i	$-0.237850\pi$
$157$	453131.	1.46715	0.733575	+	0.679608i	$0.237850\pi$
$158$	0	0
$159$	−19354.8	−0.0607151
$160$	0	0
$161$	0	0
$162$	0	0
$163$	432334.	1.27453	0.637265	−	0.770645i	$-0.280066\pi$
$163$	432334.	1.27453	0.637265	+	0.770645i	$0.280066\pi$
$164$	0	0
$165$	22093.3	0.0631758
$166$	0	0
$167$	343081.	0.951932	0.475966	−	0.879464i	$-0.342099\pi$
$167$	343081.	0.951932	0.475966	+	0.879464i	$0.342099\pi$
$168$	0	0
$169$	60091.2	0.161843
$170$	0	0
$171$	66428.0	0.173724
$172$	0	0
$173$	443505.	1.12663	0.563317	−	0.826241i	$-0.309525\pi$
$173$	443505.	1.12663	0.563317	+	0.826241i	$0.309525\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20351.9	−0.0488284
$178$	0	0
$179$	95299.6	0.222310	0.111155	−	0.993803i	$-0.464545\pi$
$179$	95299.6	0.222310	0.111155	+	0.993803i	$0.464545\pi$
$180$	0	0
$181$	78562.3	0.178245	0.0891226	−	0.996021i	$-0.471594\pi$
$181$	78562.3	0.178245	0.0891226	+	0.996021i	$0.471594\pi$
$182$	0	0
$183$	50998.4	0.112572
$184$	0	0
$185$	−149012.	−0.320104
$186$	0	0
$187$	420420.	0.879182
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−294761.	−0.584638	−0.292319	−	0.956321i	$-0.594427\pi$
$191$	−294761.	−0.584638	−0.292319	+	0.956321i	$0.594427\pi$
$192$	0	0
$193$	−874656.	−1.69022	−0.845111	−	0.534590i	$-0.820466\pi$
$193$	−874656.	−1.69022	−0.845111	+	0.534590i	$0.820466\pi$
$194$	0	0
$195$	−75566.4	−0.142312
$196$	0	0
$197$	338078.	0.620656	0.310328	−	0.950630i	$-0.399561\pi$
$197$	338078.	0.620656	0.310328	+	0.950630i	$0.399561\pi$
$198$	0	0
$199$	−353044.	−0.631970	−0.315985	−	0.948764i	$-0.602335\pi$
$199$	−353044.	−0.631970	−0.315985	+	0.948764i	$0.602335\pi$
$200$	0	0
$201$	−120027.	−0.209551
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−120942.	−0.200998
$206$	0	0
$207$	−883137.	−1.43253
$208$	0	0
$209$	53639.2	0.0849408
$210$	0	0
$211$	−351876.	−0.544106	−0.272053	−	0.962282i	$-0.587703\pi$
$211$	−351876.	−0.544106	−0.272053	+	0.962282i	$0.587703\pi$
$212$	0	0
$213$	58035.4	0.0876484
$214$	0	0
$215$	871896.	1.28638
$216$	0	0
$217$	0	0
$218$	0	0
$219$	52179.5	0.0735174
$220$	0	0
$221$	−1.43797e6	−1.98048
$222$	0	0
$223$	434832.	0.585544	0.292772	−	0.956182i	$-0.405422\pi$
$223$	434832.	0.585544	0.292772	+	0.956182i	$0.405422\pi$
$224$	0	0
$225$	136417.	0.179644
$226$	0	0
$227$	1.07056e6	1.37894	0.689472	−	0.724312i	$-0.257843\pi$
$227$	1.07056e6	1.37894	0.689472	+	0.724312i	$0.257843\pi$
$228$	0	0
$229$	−131866.	−0.166167	−0.0830836	−	0.996543i	$-0.526477\pi$
$229$	−131866.	−0.166167	−0.0830836	+	0.996543i	$0.526477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	62663.0	0.0756174	0.0378087	−	0.999285i	$-0.487962\pi$
$233$	62663.0	0.0756174	0.0378087	+	0.999285i	$0.487962\pi$
$234$	0	0
$235$	−389984.	−0.460656
$236$	0	0
$237$	104593.	0.120957
$238$	0	0
$239$	1.29337e6	1.46462	0.732312	−	0.680969i	$-0.238441\pi$
$239$	1.29337e6	1.46462	0.732312	+	0.680969i	$0.238441\pi$
$240$	0	0
$241$	−116282.	−0.128965	−0.0644823	−	0.997919i	$-0.520540\pi$
$241$	−116282.	−0.128965	−0.0644823	+	0.997919i	$0.520540\pi$
$242$	0	0
$243$	392075.	0.425945
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−183464.	−0.191341
$248$	0	0
$249$	−221356.	−0.226252
$250$	0	0
$251$	360974.	0.361653	0.180827	−	0.983515i	$-0.442123\pi$
$251$	360974.	0.361653	0.180827	+	0.983515i	$0.442123\pi$
$252$	0	0
$253$	−713114.	−0.700419
$254$	0	0
$255$	251893.	0.242586
$256$	0	0
$257$	1.93335e6	1.82591	0.912953	−	0.408064i	$-0.133796\pi$
$257$	1.93335e6	1.82591	0.912953	+	0.408064i	$0.133796\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.25482e6	1.14020
$262$	0	0
$263$	−214788.	−0.191479	−0.0957396	−	0.995406i	$-0.530522\pi$
$263$	−214788.	−0.191479	−0.0957396	+	0.995406i	$0.530522\pi$
$264$	0	0
$265$	429205.	0.375449
$266$	0	0
$267$	134688.	0.115625
$268$	0	0
$269$	−1.90275e6	−1.60325	−0.801625	−	0.597827i	$-0.796031\pi$
$269$	−1.90275e6	−1.60325	−0.801625	+	0.597827i	$0.796031\pi$
$270$	0	0
$271$	−1.28492e6	−1.06281	−0.531403	−	0.847119i	$-0.678335\pi$
$271$	−1.28492e6	−1.06281	−0.531403	+	0.847119i	$0.678335\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	110154.	0.0878352
$276$	0	0
$277$	1.42670e6	1.11720	0.558602	−	0.829436i	$-0.311338\pi$
$277$	1.42670e6	1.11720	0.558602	+	0.829436i	$0.311338\pi$
$278$	0	0
$279$	1.54240e6	1.18628
$280$	0	0
$281$	492636.	0.372186	0.186093	−	0.982532i	$-0.440417\pi$
$281$	492636.	0.372186	0.186093	+	0.982532i	$0.440417\pi$
$282$	0	0
$283$	2.01296e6	1.49406	0.747031	−	0.664789i	$-0.231478\pi$
$283$	2.01296e6	1.49406	0.747031	+	0.664789i	$0.231478\pi$
$284$	0	0
$285$	32137.7	0.0234370
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.37348e6	2.37593
$290$	0	0
$291$	167397.	0.115882
$292$	0	0
$293$	58017.9	0.0394814	0.0197407	−	0.999805i	$-0.493716\pi$
$293$	58017.9	0.0394814	0.0197407	+	0.999805i	$0.493716\pi$
$294$	0	0
$295$	451317.	0.301944
$296$	0	0
$297$	210305.	0.138343
$298$	0	0
$299$	2.43909e6	1.57779
$300$	0	0
$301$	0	0
$302$	0	0
$303$	407402.	0.254927
$304$	0	0
$305$	−1.13092e6	−0.696118
$306$	0	0
$307$	3.08036e6	1.86533	0.932665	−	0.360744i	$-0.117477\pi$
$307$	3.08036e6	1.86533	0.932665	+	0.360744i	$0.117477\pi$
$308$	0	0
$309$	19843.9	0.0118231
$310$	0	0
$311$	−561853.	−0.329399	−0.164699	−	0.986344i	$-0.552665\pi$
$311$	−561853.	−0.329399	−0.164699	+	0.986344i	$0.552665\pi$
$312$	0	0
$313$	−2.97128e6	−1.71428	−0.857141	−	0.515082i	$-0.827761\pi$
$313$	−2.97128e6	−1.71428	−0.857141	+	0.515082i	$0.827761\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−750487.	−0.419465	−0.209732	−	0.977759i	$-0.567259\pi$
$317$	−750487.	−0.419465	−0.209732	+	0.977759i	$0.567259\pi$
$318$	0	0
$319$	1.01324e6	0.557487
$320$	0	0
$321$	393414.	0.213102
$322$	0	0
$323$	611557.	0.326160
$324$	0	0
$325$	−376763.	−0.197861
$326$	0	0
$327$	−378808.	−0.195907
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.68677e6	−1.34791	−0.673954	−	0.738773i	$-0.735405\pi$
$331$	−2.68677e6	−1.34791	−0.673954	+	0.738773i	$0.735405\pi$
$332$	0	0
$333$	−701564.	−0.346702
$334$	0	0
$335$	2.66168e6	1.29582
$336$	0	0
$337$	49365.1	0.0236780	0.0118390	−	0.999930i	$-0.496231\pi$
$337$	49365.1	0.0236780	0.0118390	+	0.999930i	$0.496231\pi$
$338$	0	0
$339$	67432.8	0.0318692
$340$	0	0
$341$	1.24545e6	0.580018
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−427259.	−0.193261
$346$	0	0
$347$	1.21898e6	0.543468	0.271734	−	0.962372i	$-0.412403\pi$
$347$	1.21898e6	0.543468	0.271734	+	0.962372i	$0.412403\pi$
$348$	0	0
$349$	3.25801e6	1.43182	0.715910	−	0.698193i	$-0.246012\pi$
$349$	3.25801e6	1.43182	0.715910	+	0.698193i	$0.246012\pi$
$350$	0	0
$351$	−719313.	−0.311638
$352$	0	0
$353$	635449.	0.271421	0.135711	−	0.990749i	$-0.456668\pi$
$353$	635449.	0.271421	0.135711	+	0.990749i	$0.456668\pi$
$354$	0	0
$355$	−1.28697e6	−0.541998
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−943982.	−0.386569	−0.193285	−	0.981143i	$-0.561914\pi$
$359$	−943982.	−0.386569	−0.193285	+	0.981143i	$0.561914\pi$
$360$	0	0
$361$	−2.39807e6	−0.968489
$362$	0	0
$363$	−282846.	−0.112663
$364$	0	0
$365$	−1.15711e6	−0.454615
$366$	0	0
$367$	4.01448e6	1.55584	0.777918	−	0.628366i	$-0.216276\pi$
$367$	4.01448e6	1.55584	0.777918	+	0.628366i	$0.216276\pi$
$368$	0	0
$369$	−569408.	−0.217700
$370$	0	0
$371$	0	0
$372$	0	0
$373$	167016.	0.0621566	0.0310783	−	0.999517i	$-0.490106\pi$
$373$	167016.	0.0621566	0.0310783	+	0.999517i	$0.490106\pi$
$374$	0	0
$375$	425538.	0.156264
$376$	0	0
$377$	−3.46561e6	−1.25582
$378$	0	0
$379$	2.51875e6	0.900716	0.450358	−	0.892848i	$-0.351296\pi$
$379$	2.51875e6	0.900716	0.450358	+	0.892848i	$0.351296\pi$
$380$	0	0
$381$	−758809.	−0.267806
$382$	0	0
$383$	1.14410e6	0.398535	0.199267	−	0.979945i	$-0.436144\pi$
$383$	1.14410e6	0.398535	0.199267	+	0.979945i	$0.436144\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.10499e6	1.39327
$388$	0	0
$389$	2.43472e6	0.815785	0.407892	−	0.913030i	$-0.366264\pi$
$389$	2.43472e6	0.815785	0.407892	+	0.913030i	$0.366264\pi$
$390$	0	0
$391$	−8.13044e6	−2.68950
$392$	0	0
$393$	−132630.	−0.0433173
$394$	0	0
$395$	−2.31942e6	−0.747974
$396$	0	0
$397$	2.75120e6	0.876084	0.438042	−	0.898954i	$-0.355672\pi$
$397$	2.75120e6	0.876084	0.438042	+	0.898954i	$0.355672\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−110452.	−0.0343016	−0.0171508	−	0.999853i	$-0.505460\pi$
$401$	−110452.	−0.0343016	−0.0171508	+	0.999853i	$0.505460\pi$
$402$	0	0
$403$	−4.25986e6	−1.30657
$404$	0	0
$405$	−2.79294e6	−0.846105
$406$	0	0
$407$	−566497.	−0.169516
$408$	0	0
$409$	170695.	0.0504560	0.0252280	−	0.999682i	$-0.491969\pi$
$409$	170695.	0.0504560	0.0252280	+	0.999682i	$0.491969\pi$
$410$	0	0
$411$	746393.	0.217953
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.90870e6	1.39909
$416$	0	0
$417$	376002.	0.105889
$418$	0	0
$419$	2852.47	0.000793754 0	0.000396877	−	1.00000i	$-0.499874\pi$
$419$	2852.47	0.000793754 0	0.000396877		1.00000i	$0.499874\pi$
$420$	0	0
$421$	−730555.	−0.200885	−0.100443	−	0.994943i	$-0.532026\pi$
$421$	−730555.	−0.200885	−0.100443	+	0.994943i	$0.532026\pi$
$422$	0	0
$423$	−1.83609e6	−0.498934
$424$	0	0
$425$	1.25590e6	0.337274
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−287281.	−0.0753639
$430$	0	0
$431$	−3.69410e6	−0.957889	−0.478945	−	0.877845i	$-0.658981\pi$
$431$	−3.69410e6	−0.957889	−0.478945	+	0.877845i	$0.658981\pi$
$432$	0	0
$433$	674656.	0.172927	0.0864635	−	0.996255i	$-0.472443\pi$
$433$	674656.	0.172927	0.0864635	+	0.996255i	$0.472443\pi$
$434$	0	0
$435$	607077.	0.153823
$436$	0	0
$437$	−1.03732e6	−0.259842
$438$	0	0
$439$	3.95626e6	0.979769	0.489884	−	0.871787i	$-0.337039\pi$
$439$	3.95626e6	0.979769	0.489884	+	0.871787i	$0.337039\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.29412e6	−1.76589	−0.882945	−	0.469477i	$-0.844443\pi$
$443$	−7.29412e6	−1.76589	−0.882945	+	0.469477i	$0.844443\pi$
$444$	0	0
$445$	−2.98679e6	−0.714997
$446$	0	0
$447$	268750.	0.0636179
$448$	0	0
$449$	5.49637e6	1.28665	0.643325	−	0.765593i	$-0.277554\pi$
$449$	5.49637e6	1.28665	0.643325	+	0.765593i	$0.277554\pi$
$450$	0	0
$451$	−459785.	−0.106442
$452$	0	0
$453$	−945960.	−0.216585
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.73755e6	−0.837137	−0.418568	−	0.908185i	$-0.637468\pi$
$457$	−3.73755e6	−0.837137	−0.418568	+	0.908185i	$0.637468\pi$
$458$	0	0
$459$	2.39775e6	0.531218
$460$	0	0
$461$	−414493.	−0.0908374	−0.0454187	−	0.998968i	$-0.514462\pi$
$461$	−414493.	−0.0908374	−0.0454187	+	0.998968i	$0.514462\pi$
$462$	0	0
$463$	−6.90895e6	−1.49782	−0.748910	−	0.662672i	$-0.769423\pi$
$463$	−6.90895e6	−1.49782	−0.748910	+	0.662672i	$0.769423\pi$
$464$	0	0
$465$	746208.	0.160040
$466$	0	0
$467$	−5.30705e6	−1.12606	−0.563029	−	0.826437i	$-0.690364\pi$
$467$	−5.30705e6	−1.12606	−0.563029	+	0.826437i	$0.690364\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.03213e6	0.214379
$472$	0	0
$473$	3.31469e6	0.681223
$474$	0	0
$475$	160234.	0.0325852
$476$	0	0
$477$	2.02075e6	0.406646
$478$	0	0
$479$	−9.03751e6	−1.79974	−0.899870	−	0.436157i	$-0.856339\pi$
$479$	−9.03751e6	−1.79974	−0.899870	+	0.436157i	$0.856339\pi$
$480$	0	0
$481$	1.93761e6	0.381859
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.71213e6	−0.716587
$486$	0	0
$487$	4.45282e6	0.850770	0.425385	−	0.905012i	$-0.360139\pi$
$487$	4.45282e6	0.850770	0.425385	+	0.905012i	$0.360139\pi$
$488$	0	0
$489$	984758.	0.186233
$490$	0	0
$491$	−2.27049e6	−0.425027	−0.212513	−	0.977158i	$-0.568165\pi$
$491$	−2.27049e6	−0.425027	−0.212513	+	0.977158i	$0.568165\pi$
$492$	0	0
$493$	1.15522e7	2.14067
$494$	0	0
$495$	−2.30666e6	−0.423127
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.47102e6	−1.34316	−0.671582	−	0.740931i	$-0.734385\pi$
$499$	−7.47102e6	−1.34316	−0.671582	+	0.740931i	$0.734385\pi$
$500$	0	0
$501$	781461.	0.139095
$502$	0	0
$503$	−6.64380e6	−1.17084	−0.585418	−	0.810731i	$-0.699070\pi$
$503$	−6.64380e6	−1.17084	−0.585418	+	0.810731i	$0.699070\pi$
$504$	0	0
$505$	−9.03438e6	−1.57641
$506$	0	0
$507$	136874.	0.0236484
$508$	0	0
$509$	−8.66213e6	−1.48194	−0.740970	−	0.671539i	$-0.765634\pi$
$509$	−8.66213e6	−1.48194	−0.740970	+	0.671539i	$0.765634\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	305917.	0.0513228
$514$	0	0
$515$	−440050.	−0.0731112
$516$	0	0
$517$	−1.48260e6	−0.243948
$518$	0	0
$519$	1.01020e6	0.164623
$520$	0	0
$521$	2.48870e6	0.401679	0.200839	−	0.979624i	$-0.435633\pi$
$521$	2.48870e6	0.401679	0.200839	+	0.979624i	$0.435633\pi$
$522$	0	0
$523$	−8.36398e6	−1.33708	−0.668542	−	0.743674i	$-0.733081\pi$
$523$	−8.36398e6	−1.33708	−0.668542	+	0.743674i	$0.733081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.41998e7	2.22718
$528$	0	0
$529$	7.35447e6	1.14265
$530$	0	0
$531$	2.12485e6	0.327033
$532$	0	0
$533$	1.57262e6	0.239775
$534$	0	0
$535$	−8.72421e6	−1.31778
$536$	0	0
$537$	217071.	0.0324837
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.12558e6	−0.752922	−0.376461	−	0.926433i	$-0.622859\pi$
$541$	−5.12558e6	−0.752922	−0.376461	+	0.926433i	$0.622859\pi$
$542$	0	0
$543$	178947.	0.0260450
$544$	0	0
$545$	8.40031e6	1.21145
$546$	0	0
$547$	−8.27719e6	−1.18281	−0.591404	−	0.806375i	$-0.701426\pi$
$547$	−8.27719e6	−1.18281	−0.591404	+	0.806375i	$0.701426\pi$
$548$	0	0
$549$	−5.32451e6	−0.753961
$550$	0	0
$551$	1.47389e6	0.206817
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−339415.	−0.0467733
$556$	0	0
$557$	1.14592e7	1.56501	0.782503	−	0.622646i	$-0.213942\pi$
$557$	1.14592e7	1.56501	0.782503	+	0.622646i	$0.213942\pi$
$558$	0	0
$559$	−1.13373e7	−1.53455
$560$	0	0
$561$	957621.	0.128465
$562$	0	0
$563$	9.23070e6	1.22734	0.613668	−	0.789564i	$-0.289693\pi$
$563$	9.23070e6	1.22734	0.613668	+	0.789564i	$0.289693\pi$
$564$	0	0
$565$	−1.49536e6	−0.197072
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.29747e6	−1.07440	−0.537199	−	0.843456i	$-0.680518\pi$
$569$	−8.29747e6	−1.07440	−0.537199	+	0.843456i	$0.680518\pi$
$570$	0	0
$571$	1.00735e7	1.29298	0.646490	−	0.762922i	$-0.276236\pi$
$571$	1.00735e7	1.29298	0.646490	+	0.762922i	$0.276236\pi$
$572$	0	0
$573$	−671399.	−0.0854268
$574$	0	0
$575$	−2.13025e6	−0.268696
$576$	0	0
$577$	−6.42515e6	−0.803422	−0.401711	−	0.915767i	$-0.631584\pi$
$577$	−6.42515e6	−0.803422	−0.401711	+	0.915767i	$0.631584\pi$
$578$	0	0
$579$	−1.99227e6	−0.246974
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.63171e6	0.198825
$584$	0	0
$585$	7.88954e6	0.953152
$586$	0	0
$587$	−3.97124e6	−0.475698	−0.237849	−	0.971302i	$-0.576442\pi$
$587$	−3.97124e6	−0.475698	−0.237849	+	0.971302i	$0.576442\pi$
$588$	0	0
$589$	1.81168e6	0.215176
$590$	0	0
$591$	770064.	0.0906897
$592$	0	0
$593$	−4.89322e6	−0.571423	−0.285712	−	0.958316i	$-0.592230\pi$
$593$	−4.89322e6	−0.571423	−0.285712	+	0.958316i	$0.592230\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−804155.	−0.0923430
$598$	0	0
$599$	3.97986e6	0.453212	0.226606	−	0.973987i	$-0.427237\pi$
$599$	3.97986e6	0.453212	0.226606	+	0.973987i	$0.427237\pi$
$600$	0	0
$601$	−6.77069e6	−0.764622	−0.382311	−	0.924034i	$-0.624872\pi$
$601$	−6.77069e6	−0.764622	−0.382311	+	0.924034i	$0.624872\pi$
$602$	0	0
$603$	1.25315e7	1.40349
$604$	0	0
$605$	6.27227e6	0.696685
$606$	0	0
$607$	−1.52249e7	−1.67719	−0.838596	−	0.544754i	$-0.816623\pi$
$607$	−1.52249e7	−1.67719	−0.838596	+	0.544754i	$0.816623\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.07099e6	0.549528
$612$	0	0
$613$	−1.16036e7	−1.24721	−0.623606	−	0.781739i	$-0.714333\pi$
$613$	−1.16036e7	−1.24721	−0.623606	+	0.781739i	$0.714333\pi$
$614$	0	0
$615$	−275478.	−0.0293697
$616$	0	0
$617$	−4.51891e6	−0.477882	−0.238941	−	0.971034i	$-0.576800\pi$
$617$	−4.51891e6	−0.477882	−0.238941	+	0.971034i	$0.576800\pi$
$618$	0	0
$619$	−6.87077e6	−0.720741	−0.360370	−	0.932809i	$-0.617350\pi$
$619$	−6.87077e6	−0.720741	−0.360370	+	0.932809i	$0.617350\pi$
$620$	0	0
$621$	−4.06706e6	−0.423206
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.64395e6	−0.782741
$626$	0	0
$627$	122178.	0.0124115
$628$	0	0
$629$	−6.45882e6	−0.650918
$630$	0	0
$631$	−3.17504e6	−0.317450	−0.158725	−	0.987323i	$-0.550738\pi$
$631$	−3.17504e6	−0.317450	−0.158725	+	0.987323i	$0.550738\pi$
$632$	0	0
$633$	−801494.	−0.0795044
$634$	0	0
$635$	1.68271e7	1.65605
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.05921e6	−0.587034
$640$	0	0
$641$	3.71197e6	0.356829	0.178414	−	0.983955i	$-0.442903\pi$
$641$	3.71197e6	0.356829	0.178414	+	0.983955i	$0.442903\pi$
$642$	0	0
$643$	−2.23112e6	−0.212811	−0.106406	−	0.994323i	$-0.533934\pi$
$643$	−2.23112e6	−0.212811	−0.106406	+	0.994323i	$0.533934\pi$
$644$	0	0
$645$	1.98598e6	0.187964
$646$	0	0
$647$	−7.06649e6	−0.663656	−0.331828	−	0.943340i	$-0.607665\pi$
$647$	−7.06649e6	−0.663656	−0.331828	+	0.943340i	$0.607665\pi$
$648$	0	0
$649$	1.71577e6	0.159900
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.83924e7	1.68793	0.843966	−	0.536397i	$-0.180215\pi$
$653$	1.83924e7	1.68793	0.843966	+	0.536397i	$0.180215\pi$
$654$	0	0
$655$	2.94116e6	0.267864
$656$	0	0
$657$	−5.44782e6	−0.492390
$658$	0	0
$659$	−4.02375e6	−0.360926	−0.180463	−	0.983582i	$-0.557760\pi$
$659$	−4.02375e6	−0.360926	−0.180463	+	0.983582i	$0.557760\pi$
$660$	0	0
$661$	2.52922e6	0.225156	0.112578	−	0.993643i	$-0.464089\pi$
$661$	2.52922e6	0.225156	0.112578	+	0.993643i	$0.464089\pi$
$662$	0	0
$663$	−3.27538e6	−0.289386
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.95949e7	−1.70540
$668$	0	0
$669$	990449.	0.0855592
$670$	0	0
$671$	−4.29942e6	−0.368641
$672$	0	0
$673$	1.31377e7	1.11811	0.559053	−	0.829132i	$-0.311165\pi$
$673$	1.31377e7	1.11811	0.559053	+	0.829132i	$0.311165\pi$
$674$	0	0
$675$	628235.	0.0530716
$676$	0	0
$677$	−2.00815e6	−0.168393	−0.0841965	−	0.996449i	$-0.526832\pi$
$677$	−2.00815e6	−0.168393	−0.0841965	+	0.996449i	$0.526832\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.43849e6	0.201490
$682$	0	0
$683$	−3.35825e6	−0.275462	−0.137731	−	0.990470i	$-0.543981\pi$
$683$	−3.35825e6	−0.275462	−0.137731	+	0.990470i	$0.543981\pi$
$684$	0	0
$685$	−1.65517e7	−1.34777
$686$	0	0
$687$	−300362.	−0.0242802
$688$	0	0
$689$	−5.58099e6	−0.447881
$690$	0	0
$691$	−1.79752e6	−0.143211	−0.0716057	−	0.997433i	$-0.522812\pi$
$691$	−1.79752e6	−0.143211	−0.0716057	+	0.997433i	$0.522812\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.33808e6	−0.654793
$696$	0	0
$697$	−5.24215e6	−0.408722
$698$	0	0
$699$	142732.	0.0110492
$700$	0	0
$701$	−1.74200e7	−1.33891	−0.669457	−	0.742851i	$-0.733473\pi$
$701$	−1.74200e7	−1.33891	−0.669457	+	0.742851i	$0.733473\pi$
$702$	0	0
$703$	−824047.	−0.0628874
$704$	0	0
$705$	−888295.	−0.0673107
$706$	0	0
$707$	0	0
$708$	0	0
$709$	173533.	0.0129648	0.00648240	−	0.999979i	$-0.497937\pi$
$709$	173533.	0.0129648	0.00648240	+	0.999979i	$0.497937\pi$
$710$	0	0
$711$	−1.09201e7	−0.810125
$712$	0	0
$713$	−2.40856e7	−1.77433
$714$	0	0
$715$	6.37063e6	0.466034
$716$	0	0
$717$	2.94599e6	0.214010
$718$	0	0
$719$	−1.44188e7	−1.04018	−0.520089	−	0.854112i	$-0.674101\pi$
$719$	−1.44188e7	−1.04018	−0.520089	+	0.854112i	$0.674101\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−264865.	−0.0188442
$724$	0	0
$725$	3.02680e6	0.213864
$726$	0	0
$727$	−1.47752e7	−1.03681	−0.518403	−	0.855136i	$-0.673473\pi$
$727$	−1.47752e7	−1.03681	−0.518403	+	0.855136i	$0.673473\pi$
$728$	0	0
$729$	−1.25433e7	−0.874165
$730$	0	0
$731$	3.77918e7	2.61580
$732$	0	0
$733$	8.73446e6	0.600449	0.300225	−	0.953869i	$-0.402938\pi$
$733$	8.73446e6	0.600449	0.300225	+	0.953869i	$0.402938\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.01189e7	0.686222
$738$	0	0
$739$	2.56930e7	1.73063	0.865314	−	0.501229i	$-0.167119\pi$
$739$	2.56930e7	1.73063	0.865314	+	0.501229i	$0.167119\pi$
$740$	0	0
$741$	−417889.	−0.0279586
$742$	0	0
$743$	3.71107e6	0.246619	0.123310	−	0.992368i	$-0.460649\pi$
$743$	3.71107e6	0.246619	0.123310	+	0.992368i	$0.460649\pi$
$744$	0	0
$745$	−5.95969e6	−0.393399
$746$	0	0
$747$	2.31107e7	1.51535
$748$	0	0
$749$	0	0
$750$	0	0
$751$	876028.	0.0566785	0.0283392	−	0.999598i	$-0.490978\pi$
$751$	876028.	0.0566785	0.0283392	+	0.999598i	$0.490978\pi$
$752$	0	0
$753$	822218.	0.0528445
$754$	0	0
$755$	2.09773e7	1.33931
$756$	0	0
$757$	1.30192e7	0.825743	0.412871	−	0.910789i	$-0.364526\pi$
$757$	1.30192e7	0.825743	0.412871	+	0.910789i	$0.364526\pi$
$758$	0	0
$759$	−1.62431e6	−0.102345
$760$	0	0
$761$	−7.80270e6	−0.488409	−0.244204	−	0.969724i	$-0.578527\pi$
$761$	−7.80270e6	−0.488409	−0.244204	+	0.969724i	$0.578527\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.62990e7	−1.62474
$766$	0	0
$767$	−5.86850e6	−0.360196
$768$	0	0
$769$	1.99560e7	1.21691	0.608454	−	0.793589i	$-0.291790\pi$
$769$	1.99560e7	1.21691	0.608454	+	0.793589i	$0.291790\pi$
$770$	0	0
$771$	4.40374e6	0.266800
$772$	0	0
$773$	−2.74909e7	−1.65478	−0.827390	−	0.561628i	$-0.810175\pi$
$773$	−2.74909e7	−1.65478	−0.827390	+	0.561628i	$0.810175\pi$
$774$	0	0
$775$	3.72049e6	0.222508
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−668819.	−0.0394880
$780$	0	0
$781$	−4.89268e6	−0.287024
$782$	0	0
$783$	5.77873e6	0.336844
$784$	0	0
$785$	−2.28881e7	−1.32567
$786$	0	0
$787$	2.84767e7	1.63890	0.819451	−	0.573149i	$-0.194279\pi$
$787$	2.84767e7	1.63890	0.819451	+	0.573149i	$0.194279\pi$
$788$	0	0
$789$	−489239.	−0.0279788
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.47055e7	0.830415
$794$	0	0
$795$	977632.	0.0548603
$796$	0	0
$797$	1.32245e7	0.737450	0.368725	−	0.929538i	$-0.379794\pi$
$797$	1.32245e7	0.737450	0.368725	+	0.929538i	$0.379794\pi$
$798$	0	0
$799$	−1.69036e7	−0.936726
$800$	0	0
$801$	−1.40621e7	−0.774409
$802$	0	0
$803$	−4.39900e6	−0.240749
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.33403e6	−0.234266
$808$	0	0
$809$	−1.13419e7	−0.609276	−0.304638	−	0.952468i	$-0.598536\pi$
$809$	−1.13419e7	−0.609276	−0.304638	+	0.952468i	$0.598536\pi$
$810$	0	0
$811$	6.47233e6	0.345548	0.172774	−	0.984961i	$-0.444727\pi$
$811$	6.47233e6	0.345548	0.172774	+	0.984961i	$0.444727\pi$
$812$	0	0
$813$	−2.92677e6	−0.155296
$814$	0	0
$815$	−2.18376e7	−1.15163
$816$	0	0
$817$	4.82166e6	0.252721
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.83476e7	0.949996	0.474998	−	0.879987i	$-0.342449\pi$
$821$	1.83476e7	0.949996	0.474998	+	0.879987i	$0.342449\pi$
$822$	0	0
$823$	−1.00423e7	−0.516814	−0.258407	−	0.966036i	$-0.583197\pi$
$823$	−1.00423e7	−0.516814	−0.258407	+	0.966036i	$0.583197\pi$
$824$	0	0
$825$	250906.	0.0128344
$826$	0	0
$827$	334853.	0.0170251	0.00851256	−	0.999964i	$-0.497290\pi$
$827$	334853.	0.0170251	0.00851256	+	0.999964i	$0.497290\pi$
$828$	0	0
$829$	−3.12894e7	−1.58129	−0.790643	−	0.612277i	$-0.790254\pi$
$829$	−3.12894e7	−1.58129	−0.790643	+	0.612277i	$0.790254\pi$
$830$	0	0
$831$	3.24969e6	0.163245
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.73294e7	−0.860136
$836$	0	0
$837$	7.10312e6	0.350457
$838$	0	0
$839$	2.42772e7	1.19068	0.595339	−	0.803475i	$-0.297018\pi$
$839$	2.42772e7	1.19068	0.595339	+	0.803475i	$0.297018\pi$
$840$	0	0
$841$	7.33046e6	0.357389
$842$	0	0
$843$	1.12211e6	0.0543836
$844$	0	0
$845$	−3.03527e6	−0.146236
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.58506e6	0.218311
$850$	0	0
$851$	1.09554e7	0.518567
$852$	0	0
$853$	3.17995e7	1.49640	0.748200	−	0.663474i	$-0.230919\pi$
$853$	3.17995e7	1.49640	0.748200	+	0.663474i	$0.230919\pi$
$854$	0	0
$855$	−3.35535e6	−0.156972
$856$	0	0
$857$	3.42848e7	1.59459	0.797296	−	0.603588i	$-0.206263\pi$
$857$	3.42848e7	1.59459	0.797296	+	0.603588i	$0.206263\pi$
$858$	0	0
$859$	−7.63560e6	−0.353070	−0.176535	−	0.984294i	$-0.556489\pi$
$859$	−7.63560e6	−0.353070	−0.176535	+	0.984294i	$0.556489\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.60447e7	1.64746	0.823729	−	0.566983i	$-0.191890\pi$
$863$	3.60447e7	1.64746	0.823729	+	0.566983i	$0.191890\pi$
$864$	0	0
$865$	−2.24019e7	−1.01799
$866$	0	0
$867$	7.68402e6	0.347169
$868$	0	0
$869$	−8.81773e6	−0.396102
$870$	0	0
$871$	−3.46100e7	−1.54581
$872$	0	0
$873$	−1.74771e7	−0.776130
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.25755e7	1.43018	0.715092	−	0.699030i	$-0.246385\pi$
$877$	3.25755e7	1.43018	0.715092	+	0.699030i	$0.246385\pi$
$878$	0	0
$879$	132152.	0.00576900
$880$	0	0
$881$	−3.83192e7	−1.66332	−0.831661	−	0.555284i	$-0.812610\pi$
$881$	−3.83192e7	−1.66332	−0.831661	+	0.555284i	$0.812610\pi$
$882$	0	0
$883$	3.50607e7	1.51328	0.756640	−	0.653832i	$-0.226840\pi$
$883$	3.50607e7	1.51328	0.756640	+	0.653832i	$0.226840\pi$
$884$	0	0
$885$	1.02800e6	0.0441198
$886$	0	0
$887$	1.96135e7	0.837041	0.418521	−	0.908207i	$-0.362549\pi$
$887$	1.96135e7	0.837041	0.418521	+	0.908207i	$0.362549\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.06179e7	−0.448069
$892$	0	0
$893$	−2.15664e6	−0.0905002
$894$	0	0
$895$	−4.81368e6	−0.200872
$896$	0	0
$897$	5.55568e6	0.230545
$898$	0	0
$899$	3.42224e7	1.41225
$900$	0	0
$901$	1.86036e7	0.763459
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.96826e6	−0.161057
$906$	0	0
$907$	2.41242e7	0.973721	0.486860	−	0.873480i	$-0.338142\pi$
$907$	2.41242e7	0.973721	0.486860	+	0.873480i	$0.338142\pi$
$908$	0	0
$909$	−4.25349e7	−1.70740
$910$	0	0
$911$	346019.	0.0138135	0.00690676	−	0.999976i	$-0.497801\pi$
$911$	346019.	0.0138135	0.00690676	+	0.999976i	$0.497801\pi$
$912$	0	0
$913$	1.86614e7	0.740913
$914$	0	0
$915$	−2.57598e6	−0.101716
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.54640e6	0.255690	0.127845	−	0.991794i	$-0.459194\pi$
$919$	6.54640e6	0.255690	0.127845	+	0.991794i	$0.459194\pi$
$920$	0	0
$921$	7.01636e6	0.272560
$922$	0	0
$923$	1.67346e7	0.646562
$924$	0	0
$925$	−1.69227e6	−0.0650303
$926$	0	0
$927$	−2.07181e6	−0.0791863
$928$	0	0
$929$	1.84856e7	0.702740	0.351370	−	0.936237i	$-0.385716\pi$
$929$	1.84856e7	0.702740	0.351370	+	0.936237i	$0.385716\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.27977e6	−0.0481315
$934$	0	0
$935$	−2.12358e7	−0.794402
$936$	0	0
$937$	4.28083e7	1.59286	0.796432	−	0.604729i	$-0.206718\pi$
$937$	4.28083e7	1.59286	0.796432	+	0.604729i	$0.206718\pi$
$938$	0	0
$939$	−6.76789e6	−0.250490
$940$	0	0
$941$	1.84060e7	0.677619	0.338810	−	0.940855i	$-0.389976\pi$
$941$	1.84060e7	0.677619	0.338810	+	0.940855i	$0.389976\pi$
$942$	0	0
$943$	8.89171e6	0.325616
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.71884e7	−0.622818	−0.311409	−	0.950276i	$-0.600801\pi$
$947$	−1.71884e7	−0.622818	−0.311409	+	0.950276i	$0.600801\pi$
$948$	0	0
$949$	1.50460e7	0.542321
$950$	0	0
$951$	−1.70944e6	−0.0612918
$952$	0	0
$953$	3.40220e7	1.21347	0.606733	−	0.794905i	$-0.292480\pi$
$953$	3.40220e7	1.21347	0.606733	+	0.794905i	$0.292480\pi$
$954$	0	0
$955$	1.48887e7	0.528260
$956$	0	0
$957$	2.30792e6	0.0814595
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.34364e7	0.469327
$962$	0	0
$963$	−4.10746e7	−1.42727
$964$	0	0
$965$	4.41798e7	1.52723
$966$	0	0
$967$	1.22029e6	0.0419658	0.0209829	−	0.999780i	$-0.493320\pi$
$967$	1.22029e6	0.0419658	0.0209829	+	0.999780i	$0.493320\pi$
$968$	0	0
$969$	1.39299e6	0.0476582
$970$	0	0
$971$	−1.86907e7	−0.636176	−0.318088	−	0.948061i	$-0.603041\pi$
$971$	−1.86907e7	−0.636176	−0.318088	+	0.948061i	$0.603041\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−858181.	−0.0289113
$976$	0	0
$977$	−2.74507e7	−0.920062	−0.460031	−	0.887903i	$-0.652162\pi$
$977$	−2.74507e7	−0.920062	−0.460031	+	0.887903i	$0.652162\pi$
$978$	0	0
$979$	−1.13549e7	−0.378639
$980$	0	0
$981$	3.95496e7	1.31211
$982$	0	0
$983$	1.11626e7	0.368453	0.184227	−	0.982884i	$-0.441022\pi$
$983$	1.11626e7	0.368453	0.184227	+	0.982884i	$0.441022\pi$
$984$	0	0
$985$	−1.70766e7	−0.560805
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.41023e7	−2.08393
$990$	0	0
$991$	−4.48808e7	−1.45170	−0.725850	−	0.687853i	$-0.758553\pi$
$991$	−4.48808e7	−1.45170	−0.725850	+	0.687853i	$0.758553\pi$
$992$	0	0
$993$	−6.11985e6	−0.196955
$994$	0	0
$995$	1.78326e7	0.571028
$996$	0	0
$997$	2.06308e6	0.0657321	0.0328661	−	0.999460i	$-0.489537\pi$
$997$	2.06308e6	0.0657321	0.0328661	+	0.999460i	$0.489537\pi$
$998$	0	0
$999$	−3.23087e6	−0.102425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bl.1.3		5
4.3	odd	2		392.6.a.j.1.3		5
7.3	odd	6		112.6.i.f.65.3		10
7.5	odd	6		112.6.i.f.81.3		10
7.6	odd	2		784.6.a.bk.1.3		5
28.3	even	6		56.6.i.b.9.3	✓	10
28.11	odd	6		392.6.i.o.177.3		10
28.19	even	6		56.6.i.b.25.3	yes	10
28.23	odd	6		392.6.i.o.361.3		10
28.27	even	2		392.6.a.k.1.3		5
84.47	odd	6		504.6.s.b.361.5		10
84.59	odd	6		504.6.s.b.289.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.b.9.3	✓	10	28.3	even	6
56.6.i.b.25.3	yes	10	28.19	even	6
112.6.i.f.65.3		10	7.3	odd	6
112.6.i.f.81.3		10	7.5	odd	6
392.6.a.j.1.3		5	4.3	odd	2
392.6.a.k.1.3		5	28.27	even	2
392.6.i.o.177.3		10	28.11	odd	6
392.6.i.o.361.3		10	28.23	odd	6
504.6.s.b.289.5		10	84.59	odd	6
504.6.s.b.361.5		10	84.47	odd	6
784.6.a.bk.1.3		5	7.6	odd	2
784.6.a.bl.1.3		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.27777 q^{3} -50.5110 q^{5} -237.812 q^{9} -192.028 q^{11} +656.798 q^{13} -115.053 q^{15} -2189.37 q^{17} -279.330 q^{19} +3713.60 q^{23} -573.636 q^{25} -1095.18 q^{27} -5276.51 q^{29} -6485.80 q^{31} -437.396 q^{33} +2950.08 q^{37} +1496.04 q^{39} +2394.37 q^{41} -17261.5 q^{43} +12012.1 q^{45} +7720.77 q^{47} -4986.89 q^{51} -8497.26 q^{53} +9699.52 q^{55} -636.251 q^{57} -8935.01 q^{59} +22389.6 q^{61} -33175.6 q^{65} -52695.0 q^{67} +8458.73 q^{69} +25479.0 q^{71} +22908.1 q^{73} -1306.61 q^{75} +45919.1 q^{79} +55293.7 q^{81} -97180.7 q^{83} +110587. q^{85} -12018.7 q^{87} +59131.4 q^{89} -14773.2 q^{93} +14109.3 q^{95} +73491.5 q^{97} +45666.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 81 q^{5} + 390 q^{9} - 361 q^{11} + 342 q^{13} + 1049 q^{15} + 1809 q^{17} - 1277 q^{19} - 911 q^{23} + 3940 q^{25} + 4751 q^{27} + 5442 q^{29} - 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots - 249798 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 81 * q^5 + 390 * q^9 - 361 * q^11 + 342 * q^13 + 1049 * q^15 + 1809 * q^17 - 1277 * q^19 - 911 * q^23 + 3940 * q^25 + 4751 * q^27 + 5442 * q^29 - 2187 * q^31 - 5553 * q^33 - 8181 * q^37 + 3422 * q^39 + 16578 * q^41 - 6332 * q^43 + 41310 * q^45 - 16101 * q^47 + 67865 * q^51 + 16047 * q^53 - 45629 * q^55 + 22347 * q^57 - 71027 * q^59 + 31093 * q^61 - 64370 * q^65 - 47981 * q^67 + 137249 * q^69 - 22512 * q^71 + 123333 * q^73 + 45460 * q^75 + 212481 * q^79 + 52917 * q^81 - 87460 * q^83 + 222141 * q^85 - 318070 * q^87 + 129045 * q^89 - 252835 * q^93 - 300417 * q^95 + 328274 * q^97 - 249798 * q^99

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