LMFDB - Embedded newform 784.6.a.bc.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 = [2,0,14,0,-42,0,0,0,652] 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:	$2$
Coefficient field:	$\Q(\sqrt{130}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 130 $ x^2 - 130
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-11.4018$ 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)

$f(q)$	$=$	$q-15.8035 q^{3} -21.0000 q^{5} +6.75088 q^{9} +625.874 q^{11} -206.821 q^{13} +331.874 q^{15} +1061.46 q^{17} +1883.98 q^{19} -3717.37 q^{23} -2684.00 q^{25} +3733.56 q^{27} -123.747 q^{29} +9109.26 q^{31} -9891.00 q^{33} -6028.73 q^{37} +3268.50 q^{39} -17201.9 q^{41} -5401.98 q^{43} -141.769 q^{45} -1875.24 q^{47} -16774.8 q^{51} +18707.2 q^{53} -13143.3 q^{55} -29773.5 q^{57} +2534.78 q^{59} -2094.71 q^{61} +4343.24 q^{65} -58620.8 q^{67} +58747.5 q^{69} +31279.5 q^{71} -7150.47 q^{73} +42416.6 q^{75} -2979.81 q^{79} -60643.9 q^{81} -45954.6 q^{83} -22290.7 q^{85} +1955.64 q^{87} +99040.0 q^{89} -143958. q^{93} -39563.6 q^{95} -115548. q^{97} +4225.20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{3} - 42 q^{5} + 652 q^{9} + 294 q^{11} - 140 q^{13} - 294 q^{15} + 1302 q^{17} + 1442 q^{19} - 2646 q^{23} - 5368 q^{25} + 15722 q^{27} + 1668 q^{29} + 14798 q^{31} - 19782 q^{33} + 5182 q^{37}+ \cdots - 209916 q^{99}+O(q^{100}) $ 2 * q + 14 * q^3 - 42 * q^5 + 652 * q^9 + 294 * q^11 - 140 * q^13 - 294 * q^15 + 1302 * q^17 + 1442 * q^19 - 2646 * q^23 - 5368 * q^25 + 15722 * q^27 + 1668 * q^29 + 14798 * q^31 - 19782 * q^33 + 5182 * q^37 + 5260 * q^39 - 5124 * q^41 + 4520 * q^43 - 13692 * q^45 + 14994 * q^47 - 9606 * q^51 + 24006 * q^53 - 6174 * q^55 - 42946 * q^57 - 38850 * q^59 - 23618 * q^61 + 2940 * q^65 - 32002 * q^67 + 90678 * q^69 + 89376 * q^71 - 47138 * q^73 - 37576 * q^75 + 40970 * q^79 + 139858 * q^81 - 68376 * q^83 - 27342 * q^85 + 55356 * q^87 + 123102 * q^89 + 25586 * q^93 - 30282 * q^95 - 43652 * q^97 - 209916 * 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
$3$	−15.8035	−1.01380	−0.506898	−	0.862006i	$-0.669208\pi$
$3$	−15.8035	−1.01380	−0.506898	+	0.862006i	$0.669208\pi$
$4$	0	0
$5$	−21.0000	−0.375659	−0.187830	−	0.982202i	$-0.560145\pi$
$5$	−21.0000	−0.375659	−0.187830	+	0.982202i	$0.560145\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.75088	0.0277814
$10$	0	0
$11$	625.874	1.55957	0.779785	−	0.626047i	$-0.215328\pi$
$11$	625.874	1.55957	0.779785	+	0.626047i	$0.215328\pi$
$12$	0	0
$13$	−206.821	−0.339419	−0.169710	−	0.985494i	$-0.554283\pi$
$13$	−206.821	−0.339419	−0.169710	+	0.985494i	$0.554283\pi$
$14$	0	0
$15$	331.874	0.380842
$16$	0	0
$17$	1061.46	0.890805	0.445402	−	0.895330i	$-0.353061\pi$
$17$	1061.46	0.890805	0.445402	+	0.895330i	$0.353061\pi$
$18$	0	0
$19$	1883.98	1.19727	0.598635	−	0.801022i	$-0.295710\pi$
$19$	1883.98	1.19727	0.598635	+	0.801022i	$0.295710\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3717.37	−1.46526	−0.732632	−	0.680625i	$-0.761708\pi$
$23$	−3717.37	−1.46526	−0.732632	+	0.680625i	$0.761708\pi$
$24$	0	0
$25$	−2684.00	−0.858880
$26$	0	0
$27$	3733.56	0.985631
$28$	0	0
$29$	−123.747	−0.0273238	−0.0136619	−	0.999907i	$-0.504349\pi$
$29$	−123.747	−0.0273238	−0.0136619	+	0.999907i	$0.504349\pi$
$30$	0	0
$31$	9109.26	1.70247	0.851234	−	0.524786i	$-0.175855\pi$
$31$	9109.26	1.70247	0.851234	+	0.524786i	$0.175855\pi$
$32$	0	0
$33$	−9891.00	−1.58109
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6028.73	−0.723971	−0.361986	−	0.932184i	$-0.617901\pi$
$37$	−6028.73	−0.723971	−0.361986	+	0.932184i	$0.617901\pi$
$38$	0	0
$39$	3268.50	0.344102
$40$	0	0
$41$	−17201.9	−1.59814	−0.799071	−	0.601236i	$-0.794675\pi$
$41$	−17201.9	−1.59814	−0.799071	+	0.601236i	$0.794675\pi$
$42$	0	0
$43$	−5401.98	−0.445535	−0.222767	−	0.974872i	$-0.571509\pi$
$43$	−5401.98	−0.445535	−0.222767	+	0.974872i	$0.571509\pi$
$44$	0	0
$45$	−141.769	−0.0104363
$46$	0	0
$47$	−1875.24	−0.123826	−0.0619131	−	0.998082i	$-0.519720\pi$
$47$	−1875.24	−0.123826	−0.0619131	+	0.998082i	$0.519720\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−16774.8	−0.903094
$52$	0	0
$53$	18707.2	0.914787	0.457393	−	0.889264i	$-0.348783\pi$
$53$	18707.2	0.914787	0.457393	+	0.889264i	$0.348783\pi$
$54$	0	0
$55$	−13143.3	−0.585867
$56$	0	0
$57$	−29773.5	−1.21379
$58$	0	0
$59$	2534.78	0.0948004	0.0474002	−	0.998876i	$-0.484906\pi$
$59$	2534.78	0.0948004	0.0474002	+	0.998876i	$0.484906\pi$
$60$	0	0
$61$	−2094.71	−0.0720773	−0.0360386	−	0.999350i	$-0.511474\pi$
$61$	−2094.71	−0.0720773	−0.0360386	+	0.999350i	$0.511474\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4343.24	0.127506
$66$	0	0
$67$	−58620.8	−1.59538	−0.797691	−	0.603067i	$-0.793945\pi$
$67$	−58620.8	−1.59538	−0.797691	+	0.603067i	$0.793945\pi$
$68$	0	0
$69$	58747.5	1.48548
$70$	0	0
$71$	31279.5	0.736401	0.368201	−	0.929746i	$-0.379974\pi$
$71$	31279.5	0.736401	0.368201	+	0.929746i	$0.379974\pi$
$72$	0	0
$73$	−7150.47	−0.157046	−0.0785231	−	0.996912i	$-0.525020\pi$
$73$	−7150.47	−0.157046	−0.0785231	+	0.996912i	$0.525020\pi$
$74$	0	0
$75$	42416.6	0.870729
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2979.81	−0.0537181	−0.0268591	−	0.999639i	$-0.508551\pi$
$79$	−2979.81	−0.0537181	−0.0268591	+	0.999639i	$0.508551\pi$
$80$	0	0
$81$	−60643.9	−1.02701
$82$	0	0
$83$	−45954.6	−0.732207	−0.366103	−	0.930574i	$-0.619308\pi$
$83$	−45954.6	−0.732207	−0.366103	+	0.930574i	$0.619308\pi$
$84$	0	0
$85$	−22290.7	−0.334639
$86$	0	0
$87$	1955.64	0.0277007
$88$	0	0
$89$	99040.0	1.32536	0.662682	−	0.748900i	$-0.269418\pi$
$89$	99040.0	1.32536	0.662682	+	0.748900i	$0.269418\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−143958.	−1.72595
$94$	0	0
$95$	−39563.6	−0.449766
$96$	0	0
$97$	−115548.	−1.24691	−0.623454	−	0.781860i	$-0.714271\pi$
$97$	−115548.	−1.24691	−0.623454	+	0.781860i	$0.714271\pi$
$98$	0	0
$99$	4225.20	0.0433271
$100$	0	0
$101$	10951.5	0.106824	0.0534120	−	0.998573i	$-0.482990\pi$
$101$	10951.5	0.106824	0.0534120	+	0.998573i	$0.482990\pi$
$102$	0	0
$103$	137724.	1.27914	0.639570	−	0.768733i	$-0.279113\pi$
$103$	137724.	1.27914	0.639570	+	0.768733i	$0.279113\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−75573.0	−0.638127	−0.319064	−	0.947733i	$-0.603368\pi$
$107$	−75573.0	−0.638127	−0.319064	+	0.947733i	$0.603368\pi$
$108$	0	0
$109$	44526.3	0.358963	0.179482	−	0.983761i	$-0.442558\pi$
$109$	44526.3	0.358963	0.179482	+	0.983761i	$0.442558\pi$
$110$	0	0
$111$	95275.0	0.733959
$112$	0	0
$113$	90456.5	0.666413	0.333207	−	0.942854i	$-0.391869\pi$
$113$	90456.5	0.666413	0.333207	+	0.942854i	$0.391869\pi$
$114$	0	0
$115$	78064.7	0.550440
$116$	0	0
$117$	−1396.22	−0.00942954
$118$	0	0
$119$	0	0
$120$	0	0
$121$	230667.	1.43226
$122$	0	0
$123$	271850.	1.62019
$124$	0	0
$125$	121989.	0.698306
$126$	0	0
$127$	−187707.	−1.03269	−0.516346	−	0.856380i	$-0.672708\pi$
$127$	−187707.	−1.03269	−0.516346	+	0.856380i	$0.672708\pi$
$128$	0	0
$129$	85370.2	0.451681
$130$	0	0
$131$	−154412.	−0.786147	−0.393073	−	0.919507i	$-0.628588\pi$
$131$	−154412.	−0.786147	−0.393073	+	0.919507i	$0.628588\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−78404.9	−0.370262
$136$	0	0
$137$	109238.	0.497246	0.248623	−	0.968600i	$-0.420022\pi$
$137$	109238.	0.497246	0.248623	+	0.968600i	$0.420022\pi$
$138$	0	0
$139$	204695.	0.898609	0.449305	−	0.893379i	$-0.351672\pi$
$139$	204695.	0.898609	0.449305	+	0.893379i	$0.351672\pi$
$140$	0	0
$141$	29635.4	0.125534
$142$	0	0
$143$	−129444.	−0.529348
$144$	0	0
$145$	2598.69	0.0102644
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−406308.	−1.49930	−0.749651	−	0.661833i	$-0.769779\pi$
$149$	−406308.	−1.49930	−0.749651	+	0.661833i	$0.769779\pi$
$150$	0	0
$151$	416838.	1.48773	0.743866	−	0.668328i	$-0.232990\pi$
$151$	416838.	1.48773	0.743866	+	0.668328i	$0.232990\pi$
$152$	0	0
$153$	7165.81	0.0247478
$154$	0	0
$155$	−191295.	−0.639548
$156$	0	0
$157$	119291.	0.386240	0.193120	−	0.981175i	$-0.438139\pi$
$157$	119291.	0.386240	0.193120	+	0.981175i	$0.438139\pi$
$158$	0	0
$159$	−295640.	−0.927407
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−47372.4	−0.139655	−0.0698275	−	0.997559i	$-0.522245\pi$
$163$	−47372.4	−0.139655	−0.0698275	+	0.997559i	$0.522245\pi$
$164$	0	0
$165$	207711.	0.593950
$166$	0	0
$167$	231669.	0.642802	0.321401	−	0.946943i	$-0.395846\pi$
$167$	231669.	0.642802	0.321401	+	0.946943i	$0.395846\pi$
$168$	0	0
$169$	−328518.	−0.884795
$170$	0	0
$171$	12718.5	0.0332618
$172$	0	0
$173$	134339.	0.341262	0.170631	−	0.985335i	$-0.445419\pi$
$173$	134339.	0.341262	0.170631	+	0.985335i	$0.445419\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−40058.4	−0.0961082
$178$	0	0
$179$	−46584.4	−0.108669	−0.0543347	−	0.998523i	$-0.517304\pi$
$179$	−46584.4	−0.108669	−0.0543347	+	0.998523i	$0.517304\pi$
$180$	0	0
$181$	829210.	1.88134	0.940672	−	0.339317i	$-0.110196\pi$
$181$	829210.	1.88134	0.940672	+	0.339317i	$0.110196\pi$
$182$	0	0
$183$	33103.7	0.0730716
$184$	0	0
$185$	126603.	0.271967
$186$	0	0
$187$	664342.	1.38927
$188$	0	0
$189$	0	0
$190$	0	0
$191$	471917.	0.936013	0.468006	−	0.883725i	$-0.344972\pi$
$191$	471917.	0.936013	0.468006	+	0.883725i	$0.344972\pi$
$192$	0	0
$193$	688354.	1.33021	0.665103	−	0.746752i	$-0.268388\pi$
$193$	688354.	1.33021	0.665103	+	0.746752i	$0.268388\pi$
$194$	0	0
$195$	−68638.5	−0.129265
$196$	0	0
$197$	311915.	0.572625	0.286313	−	0.958136i	$-0.407570\pi$
$197$	311915.	0.572625	0.286313	+	0.958136i	$0.407570\pi$
$198$	0	0
$199$	287212.	0.514126	0.257063	−	0.966395i	$-0.417245\pi$
$199$	287212.	0.514126	0.257063	+	0.966395i	$0.417245\pi$
$200$	0	0
$201$	926414.	1.61739
$202$	0	0
$203$	0	0
$204$	0	0
$205$	361239.	0.600357
$206$	0	0
$207$	−25095.5	−0.0407071
$208$	0	0
$209$	1.17913e6	1.86723
$210$	0	0
$211$	460493.	0.712061	0.356031	−	0.934474i	$-0.384130\pi$
$211$	460493.	0.712061	0.356031	+	0.934474i	$0.384130\pi$
$212$	0	0
$213$	−494326.	−0.746560
$214$	0	0
$215$	113442.	0.167369
$216$	0	0
$217$	0	0
$218$	0	0
$219$	113003.	0.159213
$220$	0	0
$221$	−219533.	−0.302356
$222$	0	0
$223$	1.19776e6	1.61290	0.806449	−	0.591304i	$-0.201387\pi$
$223$	1.19776e6	1.61290	0.806449	+	0.591304i	$0.201387\pi$
$224$	0	0
$225$	−18119.4	−0.0238609
$226$	0	0
$227$	−894561.	−1.15225	−0.576123	−	0.817363i	$-0.695435\pi$
$227$	−894561.	−1.15225	−0.576123	+	0.817363i	$0.695435\pi$
$228$	0	0
$229$	−259256.	−0.326693	−0.163347	−	0.986569i	$-0.552229\pi$
$229$	−259256.	−0.326693	−0.163347	+	0.986569i	$0.552229\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	211315.	0.255000	0.127500	−	0.991839i	$-0.459305\pi$
$233$	211315.	0.255000	0.127500	+	0.991839i	$0.459305\pi$
$234$	0	0
$235$	39380.1	0.0465165
$236$	0	0
$237$	47091.5	0.0544592
$238$	0	0
$239$	−463.018	−0.000524328 0	−0.000262164	−	1.00000i	$-0.500083\pi$
$239$	−463.018	−0.000524328 0	−0.000262164		1.00000i	$0.500083\pi$
$240$	0	0
$241$	286395.	0.317631	0.158815	−	0.987308i	$-0.449233\pi$
$241$	286395.	0.317631	0.158815	+	0.987308i	$0.449233\pi$
$242$	0	0
$243$	51129.9	0.0555469
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−389647.	−0.406376
$248$	0	0
$249$	726244.	0.742308
$250$	0	0
$251$	1.37168e6	1.37426	0.687129	−	0.726535i	$-0.258870\pi$
$251$	1.37168e6	1.37426	0.687129	+	0.726535i	$0.258870\pi$
$252$	0	0
$253$	−2.32660e6	−2.28518
$254$	0	0
$255$	352272.	0.339256
$256$	0	0
$257$	−759223.	−0.717029	−0.358515	−	0.933524i	$-0.616717\pi$
$257$	−759223.	−0.717029	−0.358515	+	0.933524i	$0.616717\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−835.404	−0.000759093 0
$262$	0	0
$263$	1.07244e6	0.956061	0.478030	−	0.878343i	$-0.341351\pi$
$263$	1.07244e6	0.956061	0.478030	+	0.878343i	$0.341351\pi$
$264$	0	0
$265$	−392852.	−0.343648
$266$	0	0
$267$	−1.56518e6	−1.34365
$268$	0	0
$269$	−1.76611e6	−1.48812	−0.744059	−	0.668114i	$-0.767102\pi$
$269$	−1.76611e6	−1.48812	−0.744059	+	0.668114i	$0.767102\pi$
$270$	0	0
$271$	2.05171e6	1.69704	0.848522	−	0.529160i	$-0.177493\pi$
$271$	2.05171e6	1.69704	0.848522	+	0.529160i	$0.177493\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.67984e6	−1.33948
$276$	0	0
$277$	870682.	0.681805	0.340903	−	0.940099i	$-0.389267\pi$
$277$	870682.	0.681805	0.340903	+	0.940099i	$0.389267\pi$
$278$	0	0
$279$	61495.6	0.0472970
$280$	0	0
$281$	2.35412e6	1.77854	0.889270	−	0.457383i	$-0.151213\pi$
$281$	2.35412e6	1.77854	0.889270	+	0.457383i	$0.151213\pi$
$282$	0	0
$283$	2.19035e6	1.62573	0.812863	−	0.582454i	$-0.197908\pi$
$283$	2.19035e6	1.62573	0.812863	+	0.582454i	$0.197908\pi$
$284$	0	0
$285$	625243.	0.455970
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−293153.	−0.206467
$290$	0	0
$291$	1.82607e6	1.26411
$292$	0	0
$293$	−807700.	−0.549644	−0.274822	−	0.961495i	$-0.588619\pi$
$293$	−807700.	−0.549644	−0.274822	+	0.961495i	$0.588619\pi$
$294$	0	0
$295$	−53230.4	−0.0356127
$296$	0	0
$297$	2.33674e6	1.53716
$298$	0	0
$299$	768830.	0.497339
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−173072.	−0.108298
$304$	0	0
$305$	43988.8	0.0270765
$306$	0	0
$307$	−211516.	−0.128085	−0.0640424	−	0.997947i	$-0.520399\pi$
$307$	−211516.	−0.128085	−0.0640424	+	0.997947i	$0.520399\pi$
$308$	0	0
$309$	−2.17653e6	−1.29679
$310$	0	0
$311$	291709.	0.171021	0.0855104	−	0.996337i	$-0.472748\pi$
$311$	291709.	0.171021	0.0855104	+	0.996337i	$0.472748\pi$
$312$	0	0
$313$	−1.99598e6	−1.15158	−0.575791	−	0.817597i	$-0.695306\pi$
$313$	−1.99598e6	−1.15158	−0.575791	+	0.817597i	$0.695306\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.80957e6	−1.57033	−0.785167	−	0.619285i	$-0.787423\pi$
$317$	−2.80957e6	−1.57033	−0.785167	+	0.619285i	$0.787423\pi$
$318$	0	0
$319$	−77450.2	−0.0426134
$320$	0	0
$321$	1.19432e6	0.646930
$322$	0	0
$323$	1.99977e6	1.06653
$324$	0	0
$325$	555108.	0.291520
$326$	0	0
$327$	−703671.	−0.363916
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.38050e6	−0.692572	−0.346286	−	0.938129i	$-0.612557\pi$
$331$	−1.38050e6	−0.692572	−0.346286	+	0.938129i	$0.612557\pi$
$332$	0	0
$333$	−40699.2	−0.0201129
$334$	0	0
$335$	1.23104e6	0.599320
$336$	0	0
$337$	566429.	0.271688	0.135844	−	0.990730i	$-0.456625\pi$
$337$	566429.	0.271688	0.135844	+	0.990730i	$0.456625\pi$
$338$	0	0
$339$	−1.42953e6	−0.675607
$340$	0	0
$341$	5.70125e6	2.65512
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.23370e6	−0.558034
$346$	0	0
$347$	−540207.	−0.240845	−0.120422	−	0.992723i	$-0.538425\pi$
$347$	−540207.	−0.240845	−0.120422	+	0.992723i	$0.538425\pi$
$348$	0	0
$349$	1.73807e6	0.763841	0.381921	−	0.924195i	$-0.375263\pi$
$349$	1.73807e6	0.763841	0.381921	+	0.924195i	$0.375263\pi$
$350$	0	0
$351$	−772180.	−0.334542
$352$	0	0
$353$	1.07422e6	0.458834	0.229417	−	0.973328i	$-0.426318\pi$
$353$	1.07422e6	0.458834	0.229417	+	0.973328i	$0.426318\pi$
$354$	0	0
$355$	−656870.	−0.276636
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−121204.	−0.0496341	−0.0248170	−	0.999692i	$-0.507900\pi$
$359$	−121204.	−0.0496341	−0.0248170	+	0.999692i	$0.507900\pi$
$360$	0	0
$361$	1.07328e6	0.433455
$362$	0	0
$363$	−3.64535e6	−1.45202
$364$	0	0
$365$	150160.	0.0589959
$366$	0	0
$367$	553829.	0.214640	0.107320	−	0.994225i	$-0.465773\pi$
$367$	553829.	0.214640	0.107320	+	0.994225i	$0.465773\pi$
$368$	0	0
$369$	−116128.	−0.0443986
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−501478.	−0.186629	−0.0933146	−	0.995637i	$-0.529746\pi$
$373$	−501478.	−0.186629	−0.0933146	+	0.995637i	$0.529746\pi$
$374$	0	0
$375$	−1.92785e6	−0.707939
$376$	0	0
$377$	25593.6	0.00927422
$378$	0	0
$379$	−999004.	−0.357248	−0.178624	−	0.983917i	$-0.557165\pi$
$379$	−999004.	−0.357248	−0.178624	+	0.983917i	$0.557165\pi$
$380$	0	0
$381$	2.96643e6	1.04694
$382$	0	0
$383$	−652464.	−0.227279	−0.113640	−	0.993522i	$-0.536251\pi$
$383$	−652464.	−0.227279	−0.113640	+	0.993522i	$0.536251\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−36468.1	−0.0123776
$388$	0	0
$389$	−79253.0	−0.0265547	−0.0132774	−	0.999912i	$-0.504226\pi$
$389$	−79253.0	−0.0265547	−0.0132774	+	0.999912i	$0.504226\pi$
$390$	0	0
$391$	−3.94585e6	−1.30526
$392$	0	0
$393$	2.44026e6	0.796992
$394$	0	0
$395$	62576.0	0.0201797
$396$	0	0
$397$	4.00886e6	1.27657	0.638284	−	0.769801i	$-0.279644\pi$
$397$	4.00886e6	1.27657	0.638284	+	0.769801i	$0.279644\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	674418.	0.209444	0.104722	−	0.994502i	$-0.466605\pi$
$401$	674418.	0.209444	0.104722	+	0.994502i	$0.466605\pi$
$402$	0	0
$403$	−1.88399e6	−0.577850
$404$	0	0
$405$	1.27352e6	0.385806
$406$	0	0
$407$	−3.77322e6	−1.12908
$408$	0	0
$409$	−2.85413e6	−0.843656	−0.421828	−	0.906676i	$-0.638611\pi$
$409$	−2.85413e6	−0.843656	−0.421828	+	0.906676i	$0.638611\pi$
$410$	0	0
$411$	−1.72634e6	−0.504106
$412$	0	0
$413$	0	0
$414$	0	0
$415$	965047.	0.275060
$416$	0	0
$417$	−3.23490e6	−0.911006
$418$	0	0
$419$	4.66553e6	1.29827	0.649136	−	0.760672i	$-0.275130\pi$
$419$	4.66553e6	1.29827	0.649136	+	0.760672i	$0.275130\pi$
$420$	0	0
$421$	−3.73317e6	−1.02653	−0.513266	−	0.858229i	$-0.671565\pi$
$421$	−3.73317e6	−1.02653	−0.513266	+	0.858229i	$0.671565\pi$
$422$	0	0
$423$	−12659.5	−0.00344007
$424$	0	0
$425$	−2.84897e6	−0.765095
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.04567e6	0.536651
$430$	0	0
$431$	−964670.	−0.250141	−0.125071	−	0.992148i	$-0.539916\pi$
$431$	−964670.	−0.250141	−0.125071	+	0.992148i	$0.539916\pi$
$432$	0	0
$433$	−6.18096e6	−1.58429	−0.792147	−	0.610330i	$-0.791037\pi$
$433$	−6.18096e6	−1.58429	−0.792147	+	0.610330i	$0.791037\pi$
$434$	0	0
$435$	−41068.5	−0.0104060
$436$	0	0
$437$	−7.00344e6	−1.75432
$438$	0	0
$439$	215135.	0.0532782	0.0266391	−	0.999645i	$-0.491520\pi$
$439$	215135.	0.0532782	0.0266391	+	0.999645i	$0.491520\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.19867e6	1.74278	0.871391	−	0.490589i	$-0.163218\pi$
$443$	7.19867e6	1.74278	0.871391	+	0.490589i	$0.163218\pi$
$444$	0	0
$445$	−2.07984e6	−0.497886
$446$	0	0
$447$	6.42109e6	1.51999
$448$	0	0
$449$	3.92153e6	0.917994	0.458997	−	0.888438i	$-0.348209\pi$
$449$	3.92153e6	0.917994	0.458997	+	0.888438i	$0.348209\pi$
$450$	0	0
$451$	−1.07662e7	−2.49242
$452$	0	0
$453$	−6.58750e6	−1.50826
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.32168e6	0.296030	0.148015	−	0.988985i	$-0.452712\pi$
$457$	1.32168e6	0.296030	0.148015	+	0.988985i	$0.452712\pi$
$458$	0	0
$459$	3.96304e6	0.878005
$460$	0	0
$461$	−75459.1	−0.0165371	−0.00826855	−	0.999966i	$-0.502632\pi$
$461$	−75459.1	−0.0165371	−0.00826855	+	0.999966i	$0.502632\pi$
$462$	0	0
$463$	3.28757e6	0.712727	0.356363	−	0.934347i	$-0.384016\pi$
$463$	3.28757e6	0.712727	0.356363	+	0.934347i	$0.384016\pi$
$464$	0	0
$465$	3.02312e6	0.648371
$466$	0	0
$467$	643382.	0.136514	0.0682569	−	0.997668i	$-0.478256\pi$
$467$	643382.	0.136514	0.0682569	+	0.997668i	$0.478256\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.88521e6	−0.391568
$472$	0	0
$473$	−3.38096e6	−0.694843
$474$	0	0
$475$	−5.05660e6	−1.02831
$476$	0	0
$477$	126290.	0.0254141
$478$	0	0
$479$	7.42648e6	1.47892	0.739459	−	0.673202i	$-0.235081\pi$
$479$	7.42648e6	1.47892	0.739459	+	0.673202i	$0.235081\pi$
$480$	0	0
$481$	1.24687e6	0.245730
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.42652e6	0.468413
$486$	0	0
$487$	4.17085e6	0.796896	0.398448	−	0.917191i	$-0.369549\pi$
$487$	4.17085e6	0.796896	0.398448	+	0.917191i	$0.369549\pi$
$488$	0	0
$489$	748651.	0.141582
$490$	0	0
$491$	3.73674e6	0.699502	0.349751	−	0.936843i	$-0.386266\pi$
$491$	3.73674e6	0.699502	0.349751	+	0.936843i	$0.386266\pi$
$492$	0	0
$493$	−131353.	−0.0243402
$494$	0	0
$495$	−88729.2	−0.0162762
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.70862e6	−1.56566	−0.782831	−	0.622235i	$-0.786225\pi$
$499$	−8.70862e6	−1.56566	−0.782831	+	0.622235i	$0.786225\pi$
$500$	0	0
$501$	−3.66119e6	−0.651670
$502$	0	0
$503$	−3.28384e6	−0.578711	−0.289355	−	0.957222i	$-0.593441\pi$
$503$	−3.28384e6	−0.578711	−0.289355	+	0.957222i	$0.593441\pi$
$504$	0	0
$505$	−229981.	−0.0401294
$506$	0	0
$507$	5.19174e6	0.897001
$508$	0	0
$509$	−9.43703e6	−1.61451	−0.807255	−	0.590203i	$-0.799048\pi$
$509$	−9.43703e6	−1.61451	−0.807255	+	0.590203i	$0.799048\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.03396e6	1.18007
$514$	0	0
$515$	−2.89221e6	−0.480521
$516$	0	0
$517$	−1.17366e6	−0.193116
$518$	0	0
$519$	−2.12303e6	−0.345970
$520$	0	0
$521$	5.90726e6	0.953437	0.476719	−	0.879056i	$-0.341826\pi$
$521$	5.90726e6	0.953437	0.476719	+	0.879056i	$0.341826\pi$
$522$	0	0
$523$	−654337.	−0.104604	−0.0523019	−	0.998631i	$-0.516656\pi$
$523$	−654337.	−0.104604	−0.0523019	+	0.998631i	$0.516656\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.66915e6	1.51657
$528$	0	0
$529$	7.38248e6	1.14700
$530$	0	0
$531$	17112.0	0.00263369
$532$	0	0
$533$	3.55771e6	0.542440
$534$	0	0
$535$	1.58703e6	0.239718
$536$	0	0
$537$	736196.	0.110169
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−863115.	−0.126787	−0.0633936	−	0.997989i	$-0.520192\pi$
$541$	−863115.	−0.126787	−0.0633936	+	0.997989i	$0.520192\pi$
$542$	0	0
$543$	−1.31044e7	−1.90730
$544$	0	0
$545$	−935052.	−0.134848
$546$	0	0
$547$	5.45692e6	0.779794	0.389897	−	0.920859i	$-0.372511\pi$
$547$	5.45692e6	0.779794	0.389897	+	0.920859i	$0.372511\pi$
$548$	0	0
$549$	−14141.1	−0.00200241
$550$	0	0
$551$	−233137.	−0.0327139
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.00078e6	−0.275719
$556$	0	0
$557$	1.11776e7	1.52655	0.763275	−	0.646074i	$-0.223590\pi$
$557$	1.11776e7	1.52655	0.763275	+	0.646074i	$0.223590\pi$
$558$	0	0
$559$	1.11724e6	0.151223
$560$	0	0
$561$	−1.04989e7	−1.40844
$562$	0	0
$563$	−880265.	−0.117042	−0.0585211	−	0.998286i	$-0.518638\pi$
$563$	−880265.	−0.117042	−0.0585211	+	0.998286i	$0.518638\pi$
$564$	0	0
$565$	−1.89959e6	−0.250344
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.54972e6	−0.200665	−0.100332	−	0.994954i	$-0.531991\pi$
$569$	−1.54972e6	−0.200665	−0.100332	+	0.994954i	$0.531991\pi$
$570$	0	0
$571$	1.09701e7	1.40806	0.704032	−	0.710168i	$-0.251381\pi$
$571$	1.09701e7	1.40806	0.704032	+	0.710168i	$0.251381\pi$
$572$	0	0
$573$	−7.45794e6	−0.948926
$574$	0	0
$575$	9.97742e6	1.25849
$576$	0	0
$577$	−1.25868e7	−1.57390	−0.786948	−	0.617019i	$-0.788340\pi$
$577$	−1.25868e7	−1.57390	−0.786948	+	0.617019i	$0.788340\pi$
$578$	0	0
$579$	−1.08784e7	−1.34856
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.17084e7	1.42667
$584$	0	0
$585$	29320.7	0.00354230
$586$	0	0
$587$	−1.19962e7	−1.43697	−0.718484	−	0.695544i	$-0.755164\pi$
$587$	−1.19962e7	−1.43697	−0.718484	+	0.695544i	$0.755164\pi$
$588$	0	0
$589$	1.71617e7	2.03831
$590$	0	0
$591$	−4.92935e6	−0.580525
$592$	0	0
$593$	5.32309e6	0.621623	0.310811	−	0.950472i	$-0.399399\pi$
$593$	5.32309e6	0.621623	0.310811	+	0.950472i	$0.399399\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.53895e6	−0.521218
$598$	0	0
$599$	−7.55513e6	−0.860350	−0.430175	−	0.902746i	$-0.641548\pi$
$599$	−7.55513e6	−0.860350	−0.430175	+	0.902746i	$0.641548\pi$
$600$	0	0
$601$	4.44758e6	0.502270	0.251135	−	0.967952i	$-0.419196\pi$
$601$	4.44758e6	0.502270	0.251135	+	0.967952i	$0.419196\pi$
$602$	0	0
$603$	−395742.	−0.0443219
$604$	0	0
$605$	−4.84400e6	−0.538042
$606$	0	0
$607$	−5.57320e6	−0.613950	−0.306975	−	0.951718i	$-0.599317\pi$
$607$	−5.57320e6	−0.613950	−0.306975	+	0.951718i	$0.599317\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	387840.	0.0420290
$612$	0	0
$613$	−1.11265e7	−1.19593	−0.597965	−	0.801522i	$-0.704024\pi$
$613$	−1.11265e7	−1.19593	−0.597965	+	0.801522i	$0.704024\pi$
$614$	0	0
$615$	−5.70884e6	−0.608640
$616$	0	0
$617$	−1.07454e7	−1.13634	−0.568170	−	0.822911i	$-0.692349\pi$
$617$	−1.07454e7	−1.13634	−0.568170	+	0.822911i	$0.692349\pi$
$618$	0	0
$619$	−1.35756e7	−1.42408	−0.712038	−	0.702141i	$-0.752228\pi$
$619$	−1.35756e7	−1.42408	−0.712038	+	0.702141i	$0.752228\pi$
$620$	0	0
$621$	−1.38790e7	−1.44421
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.82573e6	0.596555
$626$	0	0
$627$	−1.86344e7	−1.89299
$628$	0	0
$629$	−6.39927e6	−0.644917
$630$	0	0
$631$	−1.27986e7	−1.27964	−0.639820	−	0.768525i	$-0.720991\pi$
$631$	−1.27986e7	−1.27964	−0.639820	+	0.768525i	$0.720991\pi$
$632$	0	0
$633$	−7.27741e6	−0.721885
$634$	0	0
$635$	3.94184e6	0.387941
$636$	0	0
$637$	0	0
$638$	0	0
$639$	211164.	0.0204583
$640$	0	0
$641$	−884833.	−0.0850582	−0.0425291	−	0.999095i	$-0.513542\pi$
$641$	−884833.	−0.0850582	−0.0425291	+	0.999095i	$0.513542\pi$
$642$	0	0
$643$	−6.66271e6	−0.635511	−0.317756	−	0.948173i	$-0.602929\pi$
$643$	−6.66271e6	−0.635511	−0.317756	+	0.948173i	$0.602929\pi$
$644$	0	0
$645$	−1.79277e6	−0.169678
$646$	0	0
$647$	1.69177e7	1.58885	0.794423	−	0.607365i	$-0.207773\pi$
$647$	1.69177e7	1.58885	0.794423	+	0.607365i	$0.207773\pi$
$648$	0	0
$649$	1.58645e6	0.147848
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.49307e6	0.871212	0.435606	−	0.900137i	$-0.356534\pi$
$653$	9.49307e6	0.871212	0.435606	+	0.900137i	$0.356534\pi$
$654$	0	0
$655$	3.24266e6	0.295323
$656$	0	0
$657$	−48272.0	−0.00436297
$658$	0	0
$659$	4.97563e6	0.446308	0.223154	−	0.974783i	$-0.428365\pi$
$659$	4.97563e6	0.446308	0.223154	+	0.974783i	$0.428365\pi$
$660$	0	0
$661$	2.11531e7	1.88309	0.941543	−	0.336894i	$-0.109376\pi$
$661$	2.11531e7	1.88309	0.941543	+	0.336894i	$0.109376\pi$
$662$	0	0
$663$	3.46939e6	0.306527
$664$	0	0
$665$	0	0
$666$	0	0
$667$	460015.	0.0400366
$668$	0	0
$669$	−1.89288e7	−1.63515
$670$	0	0
$671$	−1.31102e6	−0.112410
$672$	0	0
$673$	417573.	0.0355382	0.0177691	−	0.999842i	$-0.494344\pi$
$673$	417573.	0.0355382	0.0177691	+	0.999842i	$0.494344\pi$
$674$	0	0
$675$	−1.00209e7	−0.846539
$676$	0	0
$677$	2.62468e6	0.220092	0.110046	−	0.993926i	$-0.464900\pi$
$677$	2.62468e6	0.220092	0.110046	+	0.993926i	$0.464900\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.41372e7	1.16814
$682$	0	0
$683$	8.74455e6	0.717275	0.358637	−	0.933477i	$-0.383241\pi$
$683$	8.74455e6	0.717275	0.358637	+	0.933477i	$0.383241\pi$
$684$	0	0
$685$	−2.29399e6	−0.186795
$686$	0	0
$687$	4.09716e6	0.331200
$688$	0	0
$689$	−3.86905e6	−0.310496
$690$	0	0
$691$	−4.39667e6	−0.350291	−0.175146	−	0.984543i	$-0.556040\pi$
$691$	−4.39667e6	−0.350291	−0.175146	+	0.984543i	$0.556040\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.29860e6	−0.337571
$696$	0	0
$697$	−1.82591e7	−1.42363
$698$	0	0
$699$	−3.33951e6	−0.258518
$700$	0	0
$701$	6.51339e6	0.500624	0.250312	−	0.968165i	$-0.419467\pi$
$701$	6.51339e6	0.500624	0.250312	+	0.968165i	$0.419467\pi$
$702$	0	0
$703$	−1.13580e7	−0.866789
$704$	0	0
$705$	−622343.	−0.0471582
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.04651e7	−0.781859	−0.390929	−	0.920421i	$-0.627846\pi$
$709$	−1.04651e7	−0.781859	−0.390929	+	0.920421i	$0.627846\pi$
$710$	0	0
$711$	−20116.3	−0.00149237
$712$	0	0
$713$	−3.38625e7	−2.49457
$714$	0	0
$715$	2.71832e6	0.198855
$716$	0	0
$717$	7317.31	0.000531562 0
$718$	0	0
$719$	1.68149e7	1.21303	0.606516	−	0.795071i	$-0.292567\pi$
$719$	1.68149e7	1.21303	0.606516	+	0.795071i	$0.292567\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.52605e6	−0.322013
$724$	0	0
$725$	332138.	0.0234679
$726$	0	0
$727$	1.71928e7	1.20646	0.603228	−	0.797569i	$-0.293881\pi$
$727$	1.71928e7	1.20646	0.603228	+	0.797569i	$0.293881\pi$
$728$	0	0
$729$	1.39284e7	0.970696
$730$	0	0
$731$	−5.73400e6	−0.396885
$732$	0	0
$733$	1.97365e7	1.35679	0.678393	−	0.734700i	$-0.262677\pi$
$733$	1.97365e7	1.35679	0.678393	+	0.734700i	$0.262677\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.66892e7	−2.48811
$738$	0	0
$739$	−1.28719e7	−0.867022	−0.433511	−	0.901148i	$-0.642725\pi$
$739$	−1.28719e7	−0.867022	−0.433511	+	0.901148i	$0.642725\pi$
$740$	0	0
$741$	6.15778e6	0.411983
$742$	0	0
$743$	−2.45606e7	−1.63218	−0.816089	−	0.577927i	$-0.803862\pi$
$743$	−2.45606e7	−1.63218	−0.816089	+	0.577927i	$0.803862\pi$
$744$	0	0
$745$	8.53246e6	0.563227
$746$	0	0
$747$	−310234.	−0.0203417
$748$	0	0
$749$	0	0
$750$	0	0
$751$	325965.	0.0210898	0.0105449	−	0.999944i	$-0.496643\pi$
$751$	325965.	0.0210898	0.0105449	+	0.999944i	$0.496643\pi$
$752$	0	0
$753$	−2.16774e7	−1.39322
$754$	0	0
$755$	−8.75360e6	−0.558881
$756$	0	0
$757$	1.84659e7	1.17120	0.585599	−	0.810601i	$-0.300859\pi$
$757$	1.84659e7	1.17120	0.585599	+	0.810601i	$0.300859\pi$
$758$	0	0
$759$	3.67685e7	2.31671
$760$	0	0
$761$	5.12626e6	0.320877	0.160439	−	0.987046i	$-0.448709\pi$
$761$	5.12626e6	0.320877	0.160439	+	0.987046i	$0.448709\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−150482.	−0.00929675
$766$	0	0
$767$	−524246.	−0.0321771
$768$	0	0
$769$	1.63432e6	0.0996602	0.0498301	−	0.998758i	$-0.484132\pi$
$769$	1.63432e6	0.0996602	0.0498301	+	0.998758i	$0.484132\pi$
$770$	0	0
$771$	1.19984e7	0.726921
$772$	0	0
$773$	−4.28257e6	−0.257784	−0.128892	−	0.991659i	$-0.541142\pi$
$773$	−4.28257e6	−0.257784	−0.128892	+	0.991659i	$0.541142\pi$
$774$	0	0
$775$	−2.44493e7	−1.46222
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.24079e7	−1.91341
$780$	0	0
$781$	1.95770e7	1.14847
$782$	0	0
$783$	−462019.	−0.0269312
$784$	0	0
$785$	−2.50510e6	−0.145095
$786$	0	0
$787$	7.82963e6	0.450614	0.225307	−	0.974288i	$-0.427662\pi$
$787$	7.82963e6	0.450614	0.225307	+	0.974288i	$0.427662\pi$
$788$	0	0
$789$	−1.69484e7	−0.969250
$790$	0	0
$791$	0	0
$792$	0	0
$793$	433229.	0.0244644
$794$	0	0
$795$	6.20844e6	0.348389
$796$	0	0
$797$	−3.68238e6	−0.205344	−0.102672	−	0.994715i	$-0.532739\pi$
$797$	−3.68238e6	−0.205344	−0.102672	+	0.994715i	$0.532739\pi$
$798$	0	0
$799$	−1.99050e6	−0.110305
$800$	0	0
$801$	668607.	0.0368205
$802$	0	0
$803$	−4.47529e6	−0.244925
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.79108e7	1.50865
$808$	0	0
$809$	2.08181e7	1.11833	0.559165	−	0.829056i	$-0.311122\pi$
$809$	2.08181e7	1.11833	0.559165	+	0.829056i	$0.311122\pi$
$810$	0	0
$811$	−3.03542e7	−1.62057	−0.810283	−	0.586039i	$-0.800687\pi$
$811$	−3.03542e7	−1.62057	−0.810283	+	0.586039i	$0.800687\pi$
$812$	0	0
$813$	−3.24243e7	−1.72046
$814$	0	0
$815$	994821.	0.0524627
$816$	0	0
$817$	−1.01772e7	−0.533426
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.74387e7	1.42071	0.710355	−	0.703843i	$-0.248534\pi$
$821$	2.74387e7	1.42071	0.710355	+	0.703843i	$0.248534\pi$
$822$	0	0
$823$	−1.59126e7	−0.818919	−0.409459	−	0.912328i	$-0.634283\pi$
$823$	−1.59126e7	−0.818919	−0.409459	+	0.912328i	$0.634283\pi$
$824$	0	0
$825$	2.65474e7	1.35796
$826$	0	0
$827$	−1.40824e7	−0.716001	−0.358001	−	0.933721i	$-0.616541\pi$
$827$	−1.40824e7	−0.716001	−0.358001	+	0.933721i	$0.616541\pi$
$828$	0	0
$829$	2.18883e7	1.10618	0.553089	−	0.833122i	$-0.313449\pi$
$829$	2.18883e7	1.10618	0.553089	+	0.833122i	$0.313449\pi$
$830$	0	0
$831$	−1.37598e7	−0.691211
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.86505e6	−0.241475
$836$	0	0
$837$	3.40100e7	1.67801
$838$	0	0
$839$	1.98669e7	0.974372	0.487186	−	0.873298i	$-0.338023\pi$
$839$	1.98669e7	0.974372	0.487186	+	0.873298i	$0.338023\pi$
$840$	0	0
$841$	−2.04958e7	−0.999253
$842$	0	0
$843$	−3.72034e7	−1.80308
$844$	0	0
$845$	6.89888e6	0.332381
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.46152e7	−1.64815
$850$	0	0
$851$	2.24110e7	1.06081
$852$	0	0
$853$	8.75258e6	0.411873	0.205937	−	0.978565i	$-0.433976\pi$
$853$	8.75258e6	0.411873	0.205937	+	0.978565i	$0.433976\pi$
$854$	0	0
$855$	−267089.	−0.0124951
$856$	0	0
$857$	−1.95477e6	−0.0909166	−0.0454583	−	0.998966i	$-0.514475\pi$
$857$	−1.95477e6	−0.0909166	−0.0454583	+	0.998966i	$0.514475\pi$
$858$	0	0
$859$	−4.98450e6	−0.230483	−0.115241	−	0.993338i	$-0.536764\pi$
$859$	−4.98450e6	−0.230483	−0.115241	+	0.993338i	$0.536764\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.69931e7	0.776685	0.388343	−	0.921515i	$-0.373048\pi$
$863$	1.69931e7	0.776685	0.388343	+	0.921515i	$0.373048\pi$
$864$	0	0
$865$	−2.82112e6	−0.128198
$866$	0	0
$867$	4.63285e6	0.209315
$868$	0	0
$869$	−1.86498e6	−0.0837772
$870$	0	0
$871$	1.21240e7	0.541503
$872$	0	0
$873$	−780054.	−0.0346409
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.81959e7	0.798869	0.399434	−	0.916762i	$-0.369207\pi$
$877$	1.81959e7	0.798869	0.399434	+	0.916762i	$0.369207\pi$
$878$	0	0
$879$	1.27645e7	0.557226
$880$	0	0
$881$	1.77637e7	0.771068	0.385534	−	0.922694i	$-0.374017\pi$
$881$	1.77637e7	0.771068	0.385534	+	0.922694i	$0.374017\pi$
$882$	0	0
$883$	−1.71479e6	−0.0740131	−0.0370065	−	0.999315i	$-0.511782\pi$
$883$	−1.71479e6	−0.0740131	−0.0370065	+	0.999315i	$0.511782\pi$
$884$	0	0
$885$	841226.	0.0361039
$886$	0	0
$887$	2.42436e7	1.03464	0.517318	−	0.855793i	$-0.326930\pi$
$887$	2.42436e7	1.03464	0.517318	+	0.855793i	$0.326930\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.79554e7	−1.60169
$892$	0	0
$893$	−3.53292e6	−0.148253
$894$	0	0
$895$	978272.	0.0408227
$896$	0	0
$897$	−1.21502e7	−0.504200
$898$	0	0
$899$	−1.12725e6	−0.0465179
$900$	0	0
$901$	1.98570e7	0.814897
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.74134e7	−0.706745
$906$	0	0
$907$	1.90432e7	0.768639	0.384319	−	0.923200i	$-0.374436\pi$
$907$	1.90432e7	0.768639	0.384319	+	0.923200i	$0.374436\pi$
$908$	0	0
$909$	73932.0	0.00296772
$910$	0	0
$911$	3.18731e7	1.27242	0.636208	−	0.771518i	$-0.280502\pi$
$911$	3.18731e7	1.27242	0.636208	+	0.771518i	$0.280502\pi$
$912$	0	0
$913$	−2.87618e7	−1.14193
$914$	0	0
$915$	−695178.	−0.0274500
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.94858e7	1.15166	0.575830	−	0.817569i	$-0.304679\pi$
$919$	2.94858e7	1.15166	0.575830	+	0.817569i	$0.304679\pi$
$920$	0	0
$921$	3.34270e6	0.129852
$922$	0	0
$923$	−6.46927e6	−0.249949
$924$	0	0
$925$	1.61811e7	0.621804
$926$	0	0
$927$	929761.	0.0355363
$928$	0	0
$929$	−9.73871e6	−0.370222	−0.185111	−	0.982718i	$-0.559264\pi$
$929$	−9.73871e6	−0.370222	−0.185111	+	0.982718i	$0.559264\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4.61003e6	−0.173380
$934$	0	0
$935$	−1.39512e7	−0.521893
$936$	0	0
$937$	2.66734e7	0.992498	0.496249	−	0.868180i	$-0.334710\pi$
$937$	2.66734e7	0.992498	0.496249	+	0.868180i	$0.334710\pi$
$938$	0	0
$939$	3.15434e7	1.16747
$940$	0	0
$941$	3.22865e6	0.118863	0.0594315	−	0.998232i	$-0.481071\pi$
$941$	3.22865e6	0.118863	0.0594315	+	0.998232i	$0.481071\pi$
$942$	0	0
$943$	6.39456e7	2.34170
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.59244e6	0.166406	0.0832029	−	0.996533i	$-0.473485\pi$
$947$	4.59244e6	0.166406	0.0832029	+	0.996533i	$0.473485\pi$
$948$	0	0
$949$	1.47887e6	0.0533045
$950$	0	0
$951$	4.44011e7	1.59200
$952$	0	0
$953$	2.97125e7	1.05976	0.529880	−	0.848073i	$-0.322237\pi$
$953$	2.97125e7	1.05976	0.529880	+	0.848073i	$0.322237\pi$
$954$	0	0
$955$	−9.91025e6	−0.351622
$956$	0	0
$957$	1.22399e6	0.0432012
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.43495e7	1.89840
$962$	0	0
$963$	−510184.	−0.0177281
$964$	0	0
$965$	−1.44554e7	−0.499704
$966$	0	0
$967$	−7.64435e6	−0.262890	−0.131445	−	0.991323i	$-0.541962\pi$
$967$	−7.64435e6	−0.262890	−0.131445	+	0.991323i	$0.541962\pi$
$968$	0	0
$969$	−3.16034e7	−1.08125
$970$	0	0
$971$	4.99621e7	1.70056	0.850281	−	0.526329i	$-0.176432\pi$
$971$	4.99621e7	1.70056	0.850281	+	0.526329i	$0.176432\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.77265e6	−0.295542
$976$	0	0
$977$	4.05394e7	1.35876	0.679378	−	0.733789i	$-0.262250\pi$
$977$	4.05394e7	1.35876	0.679378	+	0.733789i	$0.262250\pi$
$978$	0	0
$979$	6.19865e7	2.06700
$980$	0	0
$981$	300592.	0.00997251
$982$	0	0
$983$	4.13056e7	1.36341	0.681703	−	0.731629i	$-0.261239\pi$
$983$	4.13056e7	1.36341	0.681703	+	0.731629i	$0.261239\pi$
$984$	0	0
$985$	−6.55021e6	−0.215112
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.00811e7	0.652826
$990$	0	0
$991$	1.84084e7	0.595431	0.297715	−	0.954655i	$-0.403775\pi$
$991$	1.84084e7	0.595431	0.297715	+	0.954655i	$0.403775\pi$
$992$	0	0
$993$	2.18167e7	0.702126
$994$	0	0
$995$	−6.03144e6	−0.193136
$996$	0	0
$997$	−3.39708e7	−1.08235	−0.541175	−	0.840910i	$-0.682020\pi$
$997$	−3.39708e7	−1.08235	−0.541175	+	0.840910i	$0.682020\pi$
$998$	0	0
$999$	−2.25086e7	−0.713568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bc.1.1		2
4.3	odd	2		98.6.a.c.1.2		2
7.2	even	3		112.6.i.b.81.2		4
7.4	even	3		112.6.i.b.65.2		4
7.6	odd	2		784.6.a.r.1.2		2
12.11	even	2		882.6.a.bt.1.2		2
28.3	even	6		98.6.c.f.79.2		4
28.11	odd	6		14.6.c.b.9.1	✓	4
28.19	even	6		98.6.c.f.67.2		4
28.23	odd	6		14.6.c.b.11.1	yes	4
28.27	even	2		98.6.a.f.1.1		2
84.11	even	6		126.6.g.e.37.2		4
84.23	even	6		126.6.g.e.109.2		4
84.83	odd	2		882.6.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.1	✓	4	28.11	odd	6
14.6.c.b.11.1	yes	4	28.23	odd	6
98.6.a.c.1.2		2	4.3	odd	2
98.6.a.f.1.1		2	28.27	even	2
98.6.c.f.67.2		4	28.19	even	6
98.6.c.f.79.2		4	28.3	even	6
112.6.i.b.65.2		4	7.4	even	3
112.6.i.b.81.2		4	7.2	even	3
126.6.g.e.37.2		4	84.11	even	6
126.6.g.e.109.2		4	84.23	even	6
784.6.a.r.1.2		2	7.6	odd	2
784.6.a.bc.1.1		2	1.1	even	1	trivial
882.6.a.bl.1.2		2	84.83	odd	2
882.6.a.bt.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-15.8035 q^{3} -21.0000 q^{5} +6.75088 q^{9} +625.874 q^{11} -206.821 q^{13} +331.874 q^{15} +1061.46 q^{17} +1883.98 q^{19} -3717.37 q^{23} -2684.00 q^{25} +3733.56 q^{27} -123.747 q^{29} +9109.26 q^{31} -9891.00 q^{33} -6028.73 q^{37} +3268.50 q^{39} -17201.9 q^{41} -5401.98 q^{43} -141.769 q^{45} -1875.24 q^{47} -16774.8 q^{51} +18707.2 q^{53} -13143.3 q^{55} -29773.5 q^{57} +2534.78 q^{59} -2094.71 q^{61} +4343.24 q^{65} -58620.8 q^{67} +58747.5 q^{69} +31279.5 q^{71} -7150.47 q^{73} +42416.6 q^{75} -2979.81 q^{79} -60643.9 q^{81} -45954.6 q^{83} -22290.7 q^{85} +1955.64 q^{87} +99040.0 q^{89} -143958. q^{93} -39563.6 q^{95} -115548. q^{97} +4225.20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{3} - 42 q^{5} + 652 q^{9} + 294 q^{11} - 140 q^{13} - 294 q^{15} + 1302 q^{17} + 1442 q^{19} - 2646 q^{23} - 5368 q^{25} + 15722 q^{27} + 1668 q^{29} + 14798 q^{31} - 19782 q^{33} + 5182 q^{37}+ \cdots - 209916 q^{99}+O(q^{100}) \) 2 * q + 14 * q^3 - 42 * q^5 + 652 * q^9 + 294 * q^11 - 140 * q^13 - 294 * q^15 + 1302 * q^17 + 1442 * q^19 - 2646 * q^23 - 5368 * q^25 + 15722 * q^27 + 1668 * q^29 + 14798 * q^31 - 19782 * q^33 + 5182 * q^37 + 5260 * q^39 - 5124 * q^41 + 4520 * q^43 - 13692 * q^45 + 14994 * q^47 - 9606 * q^51 + 24006 * q^53 - 6174 * q^55 - 42946 * q^57 - 38850 * q^59 - 23618 * q^61 + 2940 * q^65 - 32002 * q^67 + 90678 * q^69 + 89376 * q^71 - 47138 * q^73 - 37576 * q^75 + 40970 * q^79 + 139858 * q^81 - 68376 * q^83 - 27342 * q^85 + 55356 * q^87 + 123102 * q^89 + 25586 * q^93 - 30282 * q^95 - 43652 * q^97 - 209916 * q^99

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