LMFDB - Embedded newform 882.6.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,6,Mod(1,882)] 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("882.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,8,0,32,-42,0,0,128,0,-168,294] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.458529075$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-11.4018$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -21.0000 q^{5} +64.0000 q^{8} -84.0000 q^{10} +625.874 q^{11} +206.821 q^{13} +256.000 q^{16} +1061.46 q^{17} +1883.98 q^{19} -336.000 q^{20} +2503.49 q^{22} -3717.37 q^{23} -2684.00 q^{25} +827.284 q^{26} +123.747 q^{29} +9109.26 q^{31} +1024.00 q^{32} +4245.85 q^{34} -6028.73 q^{37} +7535.92 q^{38} -1344.00 q^{40} -17201.9 q^{41} +5401.98 q^{43} +10014.0 q^{44} -14869.5 q^{46} +1875.24 q^{47} -10736.0 q^{50} +3309.14 q^{52} -18707.2 q^{53} -13143.3 q^{55} +494.989 q^{58} -2534.78 q^{59} +2094.71 q^{61} +36437.1 q^{62} +4096.00 q^{64} -4343.24 q^{65} +58620.8 q^{67} +16983.4 q^{68} +31279.5 q^{71} +7150.47 q^{73} -24114.9 q^{74} +30143.7 q^{76} +2979.81 q^{79} -5376.00 q^{80} -68807.4 q^{82} +45954.6 q^{83} -22290.7 q^{85} +21607.9 q^{86} +40055.9 q^{88} +99040.0 q^{89} -59477.9 q^{92} +7500.97 q^{94} -39563.6 q^{95} +115548. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 42 q^{5} + 128 q^{8} - 168 q^{10} + 294 q^{11} + 140 q^{13} + 512 q^{16} + 1302 q^{17} + 1442 q^{19} - 672 q^{20} + 1176 q^{22} - 2646 q^{23} - 5368 q^{25} + 560 q^{26} - 1668 q^{29}+ \cdots + 43652 q^{97}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 42 * q^5 + 128 * q^8 - 168 * q^10 + 294 * q^11 + 140 * q^13 + 512 * q^16 + 1302 * q^17 + 1442 * q^19 - 672 * q^20 + 1176 * q^22 - 2646 * q^23 - 5368 * q^25 + 560 * q^26 - 1668 * q^29 + 14798 * q^31 + 2048 * q^32 + 5208 * q^34 + 5182 * q^37 + 5768 * q^38 - 2688 * q^40 - 5124 * q^41 - 4520 * q^43 + 4704 * q^44 - 10584 * q^46 - 14994 * q^47 - 21472 * q^50 + 2240 * q^52 - 24006 * q^53 - 6174 * q^55 - 6672 * q^58 + 38850 * q^59 + 23618 * q^61 + 59192 * q^62 + 8192 * q^64 - 2940 * q^65 + 32002 * q^67 + 20832 * q^68 + 89376 * q^71 + 47138 * q^73 + 20728 * q^74 + 23072 * q^76 - 40970 * q^79 - 10752 * q^80 - 20496 * q^82 + 68376 * q^83 - 27342 * q^85 - 18080 * q^86 + 18816 * q^88 + 123102 * q^89 - 42336 * q^92 - 59976 * q^94 - 30282 * q^95 + 43652 * q^97

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−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
$8$	64.0000	0.353553
$9$	0	0
$10$	−84.0000	−0.265631
$11$	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	$-0.445717\pi$
$13$	206.821	0.339419	0.169710	+	0.985494i	$0.445717\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	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	$0.295710\pi$
$20$	−336.000	−0.187830
$21$	0	0
$22$	2503.49	1.10278
$23$	−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
$26$	827.284	0.240006
$27$	0	0
$28$	0	0
$29$	123.747	0.0273238	0.0136619	−	0.999907i	$-0.495651\pi$
$29$	123.747	0.0273238	0.0136619	+	0.999907i	$0.495651\pi$
$30$	0	0
$31$	9109.26	1.70247	0.851234	−	0.524786i	$-0.175855\pi$
$31$	9109.26	1.70247	0.851234	+	0.524786i	$0.175855\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	4245.85	0.629894
$35$	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	$0.617901\pi$
$38$	7535.92	0.846598
$39$	0	0
$40$	−1344.00	−0.132816
$41$	−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	$-0.428491\pi$
$43$	5401.98	0.445535	0.222767	+	0.974872i	$0.428491\pi$
$44$	10014.0	0.779785
$45$	0	0
$46$	−14869.5	−1.03610
$47$	1875.24	0.123826	0.0619131	−	0.998082i	$-0.480280\pi$
$47$	1875.24	0.123826	0.0619131	+	0.998082i	$0.480280\pi$
$48$	0	0
$49$	0	0
$50$	−10736.0	−0.607320
$51$	0	0
$52$	3309.14	0.169710
$53$	−18707.2	−0.914787	−0.457393	−	0.889264i	$-0.651217\pi$
$53$	−18707.2	−0.914787	−0.457393	+	0.889264i	$0.651217\pi$
$54$	0	0
$55$	−13143.3	−0.585867
$56$	0	0
$57$	0	0
$58$	494.989	0.0193208
$59$	−2534.78	−0.0948004	−0.0474002	−	0.998876i	$-0.515094\pi$
$59$	−2534.78	−0.0948004	−0.0474002	+	0.998876i	$0.515094\pi$
$60$	0	0
$61$	2094.71	0.0720773	0.0360386	−	0.999350i	$-0.488526\pi$
$61$	2094.71	0.0720773	0.0360386	+	0.999350i	$0.488526\pi$
$62$	36437.1	1.20383
$63$	0	0
$64$	4096.00	0.125000
$65$	−4343.24	−0.127506
$66$	0	0
$67$	58620.8	1.59538	0.797691	−	0.603067i	$-0.206055\pi$
$67$	58620.8	1.59538	0.797691	+	0.603067i	$0.206055\pi$
$68$	16983.4	0.445402
$69$	0	0
$70$	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	$-0.474980\pi$
$73$	7150.47	0.157046	0.0785231	+	0.996912i	$0.474980\pi$
$74$	−24114.9	−0.511925
$75$	0	0
$76$	30143.7	0.598635
$77$	0	0
$78$	0	0
$79$	2979.81	0.0537181	0.0268591	−	0.999639i	$-0.491449\pi$
$79$	2979.81	0.0537181	0.0268591	+	0.999639i	$0.491449\pi$
$80$	−5376.00	−0.0939149
$81$	0	0
$82$	−68807.4	−1.13006
$83$	45954.6	0.732207	0.366103	−	0.930574i	$-0.380692\pi$
$83$	45954.6	0.732207	0.366103	+	0.930574i	$0.380692\pi$
$84$	0	0
$85$	−22290.7	−0.334639
$86$	21607.9	0.315041
$87$	0	0
$88$	40055.9	0.551391
$89$	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
$92$	−59477.9	−0.732632
$93$	0	0
$94$	7500.97	0.0875584
$95$	−39563.6	−0.449766
$96$	0	0
$97$	115548.	1.24691	0.623454	−	0.781860i	$-0.285729\pi$
$97$	115548.	1.24691	0.623454	+	0.781860i	$0.285729\pi$
$98$	0	0
$99$	0	0
$100$	−42944.0	−0.429440
$101$	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	$0.279113\pi$
$104$	13236.5	0.120003
$105$	0	0
$106$	−74828.9	−0.646852
$107$	−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	$0.442558\pi$
$110$	−52573.4	−0.414271
$111$	0	0
$112$	0	0
$113$	−90456.5	−0.666413	−0.333207	−	0.942854i	$-0.608131\pi$
$113$	−90456.5	−0.666413	−0.333207	+	0.942854i	$0.608131\pi$
$114$	0	0
$115$	78064.7	0.550440
$116$	1979.96	0.0136619
$117$	0	0
$118$	−10139.1	−0.0670340
$119$	0	0
$120$	0	0
$121$	230667.	1.43226
$122$	8378.82	0.0509663
$123$	0	0
$124$	145748.	0.851234
$125$	121989.	0.698306
$126$	0	0
$127$	187707.	1.03269	0.516346	−	0.856380i	$-0.327292\pi$
$127$	187707.	1.03269	0.516346	+	0.856380i	$0.327292\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−17373.0	−0.0901604
$131$	154412.	0.786147	0.393073	−	0.919507i	$-0.371412\pi$
$131$	154412.	0.786147	0.393073	+	0.919507i	$0.371412\pi$
$132$	0	0
$133$	0	0
$134$	234483.	1.12810
$135$	0	0
$136$	67933.6	0.314947
$137$	−109238.	−0.497246	−0.248623	−	0.968600i	$-0.579978\pi$
$137$	−109238.	−0.497246	−0.248623	+	0.968600i	$0.579978\pi$
$138$	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
$141$	0	0
$142$	125118.	0.520714
$143$	129444.	0.529348
$144$	0	0
$145$	−2598.69	−0.0102644
$146$	28601.9	0.111048
$147$	0	0
$148$	−96459.6	−0.361986
$149$	406308.	1.49930	0.749651	−	0.661833i	$-0.230221\pi$
$149$	406308.	1.49930	0.749651	+	0.661833i	$0.230221\pi$
$150$	0	0
$151$	−416838.	−1.48773	−0.743866	−	0.668328i	$-0.767010\pi$
$151$	−416838.	−1.48773	−0.743866	+	0.668328i	$0.767010\pi$
$152$	120575.	0.423299
$153$	0	0
$154$	0	0
$155$	−191295.	−0.639548
$156$	0	0
$157$	−119291.	−0.386240	−0.193120	−	0.981175i	$-0.561861\pi$
$157$	−119291.	−0.386240	−0.193120	+	0.981175i	$0.561861\pi$
$158$	11919.2	0.0379845
$159$	0	0
$160$	−21504.0	−0.0664078
$161$	0	0
$162$	0	0
$163$	47372.4	0.139655	0.0698275	−	0.997559i	$-0.477755\pi$
$163$	47372.4	0.139655	0.0698275	+	0.997559i	$0.477755\pi$
$164$	−275230.	−0.799071
$165$	0	0
$166$	183818.	0.517749
$167$	−231669.	−0.642802	−0.321401	−	0.946943i	$-0.604154\pi$
$167$	−231669.	−0.642802	−0.321401	+	0.946943i	$0.604154\pi$
$168$	0	0
$169$	−328518.	−0.884795
$170$	−89162.9	−0.236626
$171$	0	0
$172$	86431.7	0.222767
$173$	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
$176$	160224.	0.389893
$177$	0	0
$178$	396160.	0.937175
$179$	−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	$-0.889804\pi$
$181$	−829210.	−1.88134	−0.940672	+	0.339317i	$0.889804\pi$
$182$	0	0
$183$	0	0
$184$	−237912.	−0.518049
$185$	126603.	0.271967
$186$	0	0
$187$	664342.	1.38927
$188$	30003.9	0.0619131
$189$	0	0
$190$	−158254.	−0.318032
$191$	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	$0.268388\pi$
$194$	462194.	0.881698
$195$	0	0
$196$	0	0
$197$	−311915.	−0.572625	−0.286313	−	0.958136i	$-0.592430\pi$
$197$	−311915.	−0.572625	−0.286313	+	0.958136i	$0.592430\pi$
$198$	0	0
$199$	287212.	0.514126	0.257063	−	0.966395i	$-0.417245\pi$
$199$	287212.	0.514126	0.257063	+	0.966395i	$0.417245\pi$
$200$	−171776.	−0.303660
$201$	0	0
$202$	43805.9	0.0755360
$203$	0	0
$204$	0	0
$205$	361239.	0.600357
$206$	550898.	0.904488
$207$	0	0
$208$	52946.2	0.0848548
$209$	1.17913e6	1.86723
$210$	0	0
$211$	−460493.	−0.712061	−0.356031	−	0.934474i	$-0.615870\pi$
$211$	−460493.	−0.712061	−0.356031	+	0.934474i	$0.615870\pi$
$212$	−299316.	−0.457393
$213$	0	0
$214$	−302292.	−0.451224
$215$	−113442.	−0.167369
$216$	0	0
$217$	0	0
$218$	178105.	0.253825
$219$	0	0
$220$	−210294.	−0.292934
$221$	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
$225$	0	0
$226$	−361826.	−0.471225
$227$	894561.	1.15225	0.576123	−	0.817363i	$-0.304565\pi$
$227$	894561.	1.15225	0.576123	+	0.817363i	$0.304565\pi$
$228$	0	0
$229$	259256.	0.326693	0.163347	−	0.986569i	$-0.447771\pi$
$229$	259256.	0.326693	0.163347	+	0.986569i	$0.447771\pi$
$230$	312259.	0.389220
$231$	0	0
$232$	7919.83	0.00966042
$233$	−211315.	−0.255000	−0.127500	−	0.991839i	$-0.540695\pi$
$233$	−211315.	−0.255000	−0.127500	+	0.991839i	$0.540695\pi$
$234$	0	0
$235$	−39380.1	−0.0465165
$236$	−40556.5	−0.0474002
$237$	0	0
$238$	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	$-0.550767\pi$
$241$	−286395.	−0.317631	−0.158815	+	0.987308i	$0.550767\pi$
$242$	922667.	1.01276
$243$	0	0
$244$	33515.3	0.0360386
$245$	0	0
$246$	0	0
$247$	389647.	0.406376
$248$	582993.	0.601913
$249$	0	0
$250$	487956.	0.493777
$251$	−1.37168e6	−1.37426	−0.687129	−	0.726535i	$-0.741130\pi$
$251$	−1.37168e6	−1.37426	−0.687129	+	0.726535i	$0.741130\pi$
$252$	0	0
$253$	−2.32660e6	−2.28518
$254$	750828.	0.730224
$255$	0	0
$256$	65536.0	0.0625000
$257$	−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
$260$	−69491.9	−0.0637530
$261$	0	0
$262$	617649.	0.555890
$263$	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
$267$	0	0
$268$	937932.	0.797691
$269$	−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	$0.177493\pi$
$272$	271735.	0.222701
$273$	0	0
$274$	−436951.	−0.351606
$275$	−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	$0.389267\pi$
$278$	818781.	0.635413
$279$	0	0
$280$	0	0
$281$	−2.35412e6	−1.77854	−0.889270	−	0.457383i	$-0.848787\pi$
$281$	−2.35412e6	−1.77854	−0.889270	+	0.457383i	$0.848787\pi$
$282$	0	0
$283$	2.19035e6	1.62573	0.812863	−	0.582454i	$-0.197908\pi$
$283$	2.19035e6	1.62573	0.812863	+	0.582454i	$0.197908\pi$
$284$	500473.	0.368201
$285$	0	0
$286$	517775.	0.374306
$287$	0	0
$288$	0	0
$289$	−293153.	−0.206467
$290$	−10394.8	−0.00725805
$291$	0	0
$292$	114408.	0.0785231
$293$	−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
$296$	−385838.	−0.255962
$297$	0	0
$298$	1.62523e6	1.06017
$299$	−768830.	−0.497339
$300$	0	0
$301$	0	0
$302$	−1.66735e6	−1.05199
$303$	0	0
$304$	482299.	0.299317
$305$	−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
$309$	0	0
$310$	−765178.	−0.452229
$311$	−291709.	−0.171021	−0.0855104	−	0.996337i	$-0.527252\pi$
$311$	−291709.	−0.171021	−0.0855104	+	0.996337i	$0.527252\pi$
$312$	0	0
$313$	1.99598e6	1.15158	0.575791	−	0.817597i	$-0.304694\pi$
$313$	1.99598e6	1.15158	0.575791	+	0.817597i	$0.304694\pi$
$314$	−477162.	−0.273113
$315$	0	0
$316$	47677.0	0.0268591
$317$	2.80957e6	1.57033	0.785167	−	0.619285i	$-0.212577\pi$
$317$	2.80957e6	1.57033	0.785167	+	0.619285i	$0.212577\pi$
$318$	0	0
$319$	77450.2	0.0426134
$320$	−86016.0	−0.0469574
$321$	0	0
$322$	0	0
$323$	1.99977e6	1.06653
$324$	0	0
$325$	−555108.	−0.291520
$326$	189490.	0.0987510
$327$	0	0
$328$	−1.10092e6	−0.565029
$329$	0	0
$330$	0	0
$331$	1.38050e6	0.692572	0.346286	−	0.938129i	$-0.387443\pi$
$331$	1.38050e6	0.692572	0.346286	+	0.938129i	$0.387443\pi$
$332$	735274.	0.366103
$333$	0	0
$334$	−926677.	−0.454530
$335$	−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	$0.456625\pi$
$338$	−1.31407e6	−0.625644
$339$	0	0
$340$	−356652.	−0.167320
$341$	5.70125e6	2.65512
$342$	0	0
$343$	0	0
$344$	345727.	0.157520
$345$	0	0
$346$	537357.	0.241308
$347$	−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	$-0.624737\pi$
$349$	−1.73807e6	−0.763841	−0.381921	+	0.924195i	$0.624737\pi$
$350$	0	0
$351$	0	0
$352$	640895.	0.275696
$353$	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
$356$	1.58464e6	0.662682
$357$	0	0
$358$	−186337.	−0.0768409
$359$	−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
$362$	−3.31684e6	−1.33031
$363$	0	0
$364$	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	$0.465773\pi$
$368$	−951646.	−0.366316
$369$	0	0
$370$	506413.	0.192309
$371$	0	0
$372$	0	0
$373$	−501478.	−0.186629	−0.0933146	−	0.995637i	$-0.529746\pi$
$373$	−501478.	−0.186629	−0.0933146	+	0.995637i	$0.529746\pi$
$374$	2.65737e6	0.982364
$375$	0	0
$376$	120015.	0.0437792
$377$	25593.6	0.00927422
$378$	0	0
$379$	999004.	0.357248	0.178624	−	0.983917i	$-0.442835\pi$
$379$	999004.	0.357248	0.178624	+	0.983917i	$0.442835\pi$
$380$	−633017.	−0.224883
$381$	0	0
$382$	1.88767e6	0.661861
$383$	652464.	0.227279	0.113640	−	0.993522i	$-0.463749\pi$
$383$	652464.	0.227279	0.113640	+	0.993522i	$0.463749\pi$
$384$	0	0
$385$	0	0
$386$	2.75342e6	0.940597
$387$	0	0
$388$	1.84877e6	0.623454
$389$	79253.0	0.0265547	0.0132774	−	0.999912i	$-0.495774\pi$
$389$	79253.0	0.0265547	0.0132774	+	0.999912i	$0.495774\pi$
$390$	0	0
$391$	−3.94585e6	−1.30526
$392$	0	0
$393$	0	0
$394$	−1.24766e6	−0.404907
$395$	−62576.0	−0.0201797
$396$	0	0
$397$	−4.00886e6	−1.27657	−0.638284	−	0.769801i	$-0.720356\pi$
$397$	−4.00886e6	−1.27657	−0.638284	+	0.769801i	$0.720356\pi$
$398$	1.14885e6	0.363542
$399$	0	0
$400$	−687104.	−0.214720
$401$	−674418.	−0.209444	−0.104722	−	0.994502i	$-0.533395\pi$
$401$	−674418.	−0.209444	−0.104722	+	0.994502i	$0.533395\pi$
$402$	0	0
$403$	1.88399e6	0.577850
$404$	175223.	0.0534120
$405$	0	0
$406$	0	0
$407$	−3.77322e6	−1.12908
$408$	0	0
$409$	2.85413e6	0.843656	0.421828	−	0.906676i	$-0.361389\pi$
$409$	2.85413e6	0.843656	0.421828	+	0.906676i	$0.361389\pi$
$410$	1.44496e6	0.424517
$411$	0	0
$412$	2.20359e6	0.639570
$413$	0	0
$414$	0	0
$415$	−965047.	−0.275060
$416$	211785.	0.0600014
$417$	0	0
$418$	4.71653e6	1.32033
$419$	−4.66553e6	−1.29827	−0.649136	−	0.760672i	$-0.724870\pi$
$419$	−4.66553e6	−1.29827	−0.649136	+	0.760672i	$0.724870\pi$
$420$	0	0
$421$	−3.73317e6	−1.02653	−0.513266	−	0.858229i	$-0.671565\pi$
$421$	−3.73317e6	−1.02653	−0.513266	+	0.858229i	$0.671565\pi$
$422$	−1.84197e6	−0.503503
$423$	0	0
$424$	−1.19726e6	−0.323426
$425$	−2.84897e6	−0.765095
$426$	0	0
$427$	0	0
$428$	−1.20917e6	−0.319064
$429$	0	0
$430$	−453766.	−0.118348
$431$	−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	$-0.208963\pi$
$433$	6.18096e6	1.58429	0.792147	+	0.610330i	$0.208963\pi$
$434$	0	0
$435$	0	0
$436$	712421.	0.179482
$437$	−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	$0.491520\pi$
$440$	−841174.	−0.207135
$441$	0	0
$442$	878132.	0.213798
$443$	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
$446$	4.79103e6	1.14049
$447$	0	0
$448$	0	0
$449$	−3.92153e6	−0.917994	−0.458997	−	0.888438i	$-0.651791\pi$
$449$	−3.92153e6	−0.917994	−0.458997	+	0.888438i	$0.651791\pi$
$450$	0	0
$451$	−1.07662e7	−2.49242
$452$	−1.44730e6	−0.333207
$453$	0	0
$454$	3.57824e6	0.814761
$455$	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	$0.452712\pi$
$458$	1.03702e6	0.231007
$459$	0	0
$460$	1.24904e6	0.275220
$461$	−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	$-0.615984\pi$
$463$	−3.28757e6	−0.712727	−0.356363	+	0.934347i	$0.615984\pi$
$464$	31679.3	0.00683095
$465$	0	0
$466$	−845259.	−0.180312
$467$	−643382.	−0.136514	−0.0682569	−	0.997668i	$-0.521744\pi$
$467$	−643382.	−0.136514	−0.0682569	+	0.997668i	$0.521744\pi$
$468$	0	0
$469$	0	0
$470$	−157520.	−0.0328921
$471$	0	0
$472$	−162226.	−0.0335170
$473$	3.38096e6	0.694843
$474$	0	0
$475$	−5.05660e6	−1.02831
$476$	0	0
$477$	0	0
$478$	−1852.07	−0.000370756 0
$479$	−7.42648e6	−1.47892	−0.739459	−	0.673202i	$-0.764919\pi$
$479$	−7.42648e6	−1.47892	−0.739459	+	0.673202i	$0.764919\pi$
$480$	0	0
$481$	−1.24687e6	−0.245730
$482$	−1.14558e6	−0.224599
$483$	0	0
$484$	3.69067e6	0.716130
$485$	−2.42652e6	−0.468413
$486$	0	0
$487$	−4.17085e6	−0.796896	−0.398448	−	0.917191i	$-0.630451\pi$
$487$	−4.17085e6	−0.796896	−0.398448	+	0.917191i	$0.630451\pi$
$488$	134061.	0.0254832
$489$	0	0
$490$	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
$494$	1.55859e6	0.287351
$495$	0	0
$496$	2.33197e6	0.425617
$497$	0	0
$498$	0	0
$499$	8.70862e6	1.56566	0.782831	−	0.622235i	$-0.213775\pi$
$499$	8.70862e6	1.56566	0.782831	+	0.622235i	$0.213775\pi$
$500$	1.95182e6	0.349153
$501$	0	0
$502$	−5.48672e6	−0.971748
$503$	3.28384e6	0.578711	0.289355	−	0.957222i	$-0.406559\pi$
$503$	3.28384e6	0.578711	0.289355	+	0.957222i	$0.406559\pi$
$504$	0	0
$505$	−229981.	−0.0401294
$506$	−9.30641e6	−1.61587
$507$	0	0
$508$	3.00331e6	0.516346
$509$	−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
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.03689e6	−0.507016
$515$	−2.89221e6	−0.480521
$516$	0	0
$517$	1.17366e6	0.193116
$518$	0	0
$519$	0	0
$520$	−277967.	−0.0450802
$521$	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	$0.516656\pi$
$524$	2.47060e6	0.393073
$525$	0	0
$526$	4.28978e6	0.676037
$527$	9.66915e6	1.51657
$528$	0	0
$529$	7.38248e6	1.14700
$530$	1.57141e6	0.242996
$531$	0	0
$532$	0	0
$533$	−3.55771e6	−0.542440
$534$	0	0
$535$	1.58703e6	0.239718
$536$	3.75173e6	0.564052
$537$	0	0
$538$	−7.06445e6	−1.05226
$539$	0	0
$540$	0	0
$541$	−863115.	−0.126787	−0.0633936	−	0.997989i	$-0.520192\pi$
$541$	−863115.	−0.126787	−0.0633936	+	0.997989i	$0.520192\pi$
$542$	8.20685e6	1.19999
$543$	0	0
$544$	1.08694e6	0.157474
$545$	−935052.	−0.134848
$546$	0	0
$547$	−5.45692e6	−0.779794	−0.389897	−	0.920859i	$-0.627489\pi$
$547$	−5.45692e6	−0.779794	−0.389897	+	0.920859i	$0.627489\pi$
$548$	−1.74781e6	−0.248623
$549$	0	0
$550$	−6.71938e6	−0.947158
$551$	233137.	0.0327139
$552$	0	0
$553$	0	0
$554$	3.48273e6	0.482109
$555$	0	0
$556$	3.27513e6	0.449305
$557$	−1.11776e7	−1.52655	−0.763275	−	0.646074i	$-0.776410\pi$
$557$	−1.11776e7	−1.52655	−0.763275	+	0.646074i	$0.776410\pi$
$558$	0	0
$559$	1.11724e6	0.151223
$560$	0	0
$561$	0	0
$562$	−9.41650e6	−1.25762
$563$	880265.	0.117042	0.0585211	−	0.998286i	$-0.481362\pi$
$563$	880265.	0.117042	0.0585211	+	0.998286i	$0.481362\pi$
$564$	0	0
$565$	1.89959e6	0.250344
$566$	8.76140e6	1.14956
$567$	0	0
$568$	2.00189e6	0.260357
$569$	1.54972e6	0.200665	0.100332	−	0.994954i	$-0.468009\pi$
$569$	1.54972e6	0.200665	0.100332	+	0.994954i	$0.468009\pi$
$570$	0	0
$571$	−1.09701e7	−1.40806	−0.704032	−	0.710168i	$-0.748619\pi$
$571$	−1.09701e7	−1.40806	−0.704032	+	0.710168i	$0.748619\pi$
$572$	2.07110e6	0.264674
$573$	0	0
$574$	0	0
$575$	9.97742e6	1.25849
$576$	0	0
$577$	1.25868e7	1.57390	0.786948	−	0.617019i	$-0.211660\pi$
$577$	1.25868e7	1.57390	0.786948	+	0.617019i	$0.211660\pi$
$578$	−1.17261e6	−0.145994
$579$	0	0
$580$	−41579.1	−0.00513222
$581$	0	0
$582$	0	0
$583$	−1.17084e7	−1.42667
$584$	457630.	0.0555242
$585$	0	0
$586$	−3.23080e6	−0.388657
$587$	1.19962e7	1.43697	0.718484	−	0.695544i	$-0.244836\pi$
$587$	1.19962e7	1.43697	0.718484	+	0.695544i	$0.244836\pi$
$588$	0	0
$589$	1.71617e7	2.03831
$590$	212921.	0.0251819
$591$	0	0
$592$	−1.54335e6	−0.180993
$593$	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
$596$	6.50092e6	0.749651
$597$	0	0
$598$	−3.07532e6	−0.351672
$599$	−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	$-0.580804\pi$
$601$	−4.44758e6	−0.502270	−0.251135	+	0.967952i	$0.580804\pi$
$602$	0	0
$603$	0	0
$604$	−6.66941e6	−0.743866
$605$	−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	$0.599317\pi$
$608$	1.92919e6	0.211649
$609$	0	0
$610$	−175955.	−0.0191460
$611$	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	$0.704024\pi$
$614$	−846064.	−0.0905696
$615$	0	0
$616$	0	0
$617$	1.07454e7	1.13634	0.568170	−	0.822911i	$-0.307651\pi$
$617$	1.07454e7	1.13634	0.568170	+	0.822911i	$0.307651\pi$
$618$	0	0
$619$	−1.35756e7	−1.42408	−0.712038	−	0.702141i	$-0.752228\pi$
$619$	−1.35756e7	−1.42408	−0.712038	+	0.702141i	$0.752228\pi$
$620$	−3.06071e6	−0.319774
$621$	0	0
$622$	−1.16684e6	−0.120930
$623$	0	0
$624$	0	0
$625$	5.82573e6	0.596555
$626$	7.98391e6	0.814291
$627$	0	0
$628$	−1.90865e6	−0.193120
$629$	−6.39927e6	−0.644917
$630$	0	0
$631$	1.27986e7	1.27964	0.639820	−	0.768525i	$-0.279009\pi$
$631$	1.27986e7	1.27964	0.639820	+	0.768525i	$0.279009\pi$
$632$	190708.	0.0189922
$633$	0	0
$634$	1.12383e7	1.11039
$635$	−3.94184e6	−0.387941
$636$	0	0
$637$	0	0
$638$	309801.	0.0301322
$639$	0	0
$640$	−344064.	−0.0332039
$641$	884833.	0.0850582	0.0425291	−	0.999095i	$-0.486458\pi$
$641$	884833.	0.0850582	0.0425291	+	0.999095i	$0.486458\pi$
$642$	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
$645$	0	0
$646$	7.99910e6	0.754153
$647$	−1.69177e7	−1.58885	−0.794423	−	0.607365i	$-0.792227\pi$
$647$	−1.69177e7	−1.58885	−0.794423	+	0.607365i	$0.792227\pi$
$648$	0	0
$649$	−1.58645e6	−0.147848
$650$	−2.22043e6	−0.206136
$651$	0	0
$652$	757959.	0.0698275
$653$	−9.49307e6	−0.871212	−0.435606	−	0.900137i	$-0.643466\pi$
$653$	−9.49307e6	−0.871212	−0.435606	+	0.900137i	$0.643466\pi$
$654$	0	0
$655$	−3.24266e6	−0.295323
$656$	−4.40367e6	−0.399536
$657$	0	0
$658$	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	$-0.890624\pi$
$661$	−2.11531e7	−1.88309	−0.941543	+	0.336894i	$0.890624\pi$
$662$	5.52198e6	0.489722
$663$	0	0
$664$	2.94110e6	0.258874
$665$	0	0
$666$	0	0
$667$	−460015.	−0.0400366
$668$	−3.70671e6	−0.321401
$669$	0	0
$670$	−4.92414e6	−0.423783
$671$	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	$0.494344\pi$
$674$	2.26572e6	0.192112
$675$	0	0
$676$	−5.25629e6	−0.442397
$677$	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
$680$	−1.42661e6	−0.118313
$681$	0	0
$682$	2.28050e7	1.87745
$683$	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
$687$	0	0
$688$	1.38291e6	0.111384
$689$	−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	$0.556040\pi$
$692$	2.14943e6	0.170631
$693$	0	0
$694$	−2.16083e6	−0.170303
$695$	−4.29860e6	−0.337571
$696$	0	0
$697$	−1.82591e7	−1.42363
$698$	−6.95227e6	−0.540117
$699$	0	0
$700$	0	0
$701$	−6.51339e6	−0.500624	−0.250312	−	0.968165i	$-0.580533\pi$
$701$	−6.51339e6	−0.500624	−0.250312	+	0.968165i	$0.580533\pi$
$702$	0	0
$703$	−1.13580e7	−0.866789
$704$	2.56358e6	0.194946
$705$	0	0
$706$	4.29687e6	0.324445
$707$	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	$0.627846\pi$
$710$	−2.62748e6	−0.195611
$711$	0	0
$712$	6.33856e6	0.468587
$713$	−3.38625e7	−2.49457
$714$	0	0
$715$	−2.71832e6	−0.198855
$716$	−745350.	−0.0543347
$717$	0	0
$718$	−484815.	−0.0350966
$719$	−1.68149e7	−1.21303	−0.606516	−	0.795071i	$-0.707433\pi$
$719$	−1.68149e7	−1.21303	−0.606516	+	0.795071i	$0.707433\pi$
$720$	0	0
$721$	0	0
$722$	4.29311e6	0.306499
$723$	0	0
$724$	−1.32674e7	−0.940672
$725$	−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
$729$	0	0
$730$	−600640.	−0.0417164
$731$	5.73400e6	0.396885
$732$	0	0
$733$	−1.97365e7	−1.35679	−0.678393	−	0.734700i	$-0.737323\pi$
$733$	−1.97365e7	−1.35679	−0.678393	+	0.734700i	$0.737323\pi$
$734$	2.21531e6	0.151773
$735$	0	0
$736$	−3.80659e6	−0.259025
$737$	3.66892e7	2.48811
$738$	0	0
$739$	1.28719e7	0.867022	0.433511	−	0.901148i	$-0.357275\pi$
$739$	1.28719e7	0.867022	0.433511	+	0.901148i	$0.357275\pi$
$740$	2.02565e6	0.135983
$741$	0	0
$742$	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
$746$	−2.00591e6	−0.131967
$747$	0	0
$748$	1.06295e7	0.694636
$749$	0	0
$750$	0	0
$751$	−325965.	−0.0210898	−0.0105449	−	0.999944i	$-0.503357\pi$
$751$	−325965.	−0.0210898	−0.0105449	+	0.999944i	$0.503357\pi$
$752$	480062.	0.0309566
$753$	0	0
$754$	102374.	0.00655786
$755$	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	$0.300859\pi$
$758$	3.99602e6	0.252612
$759$	0	0
$760$	−2.53207e6	−0.159016
$761$	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
$764$	7.55066e6	0.468006
$765$	0	0
$766$	2.60985e6	0.160711
$767$	−524246.	−0.0321771
$768$	0	0
$769$	−1.63432e6	−0.0996602	−0.0498301	−	0.998758i	$-0.515868\pi$
$769$	−1.63432e6	−0.0996602	−0.0498301	+	0.998758i	$0.515868\pi$
$770$	0	0
$771$	0	0
$772$	1.10137e7	0.665103
$773$	−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
$776$	7.39510e6	0.440849
$777$	0	0
$778$	317012.	0.0187770
$779$	−3.24079e7	−1.91341
$780$	0	0
$781$	1.95770e7	1.14847
$782$	−1.57834e7	−0.922962
$783$	0	0
$784$	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	$0.427662\pi$
$788$	−4.99064e6	−0.286313
$789$	0	0
$790$	−250304.	−0.0142692
$791$	0	0
$792$	0	0
$793$	433229.	0.0244644
$794$	−1.60354e7	−0.902670
$795$	0	0
$796$	4.59539e6	0.257063
$797$	−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
$800$	−2.74842e6	−0.151830
$801$	0	0
$802$	−2.69767e6	−0.148099
$803$	4.47529e6	0.244925
$804$	0	0
$805$	0	0
$806$	7.53595e6	0.408602
$807$	0	0
$808$	700894.	0.0377680
$809$	−2.08181e7	−1.11833	−0.559165	−	0.829056i	$-0.688878\pi$
$809$	−2.08181e7	−1.11833	−0.559165	+	0.829056i	$0.688878\pi$
$810$	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
$813$	0	0
$814$	−1.50929e7	−0.798383
$815$	−994821.	−0.0524627
$816$	0	0
$817$	1.01772e7	0.533426
$818$	1.14165e7	0.596555
$819$	0	0
$820$	5.77982e6	0.300179
$821$	−2.74387e7	−1.42071	−0.710355	−	0.703843i	$-0.751466\pi$
$821$	−2.74387e7	−1.42071	−0.710355	+	0.703843i	$0.751466\pi$
$822$	0	0
$823$	1.59126e7	0.818919	0.409459	−	0.912328i	$-0.365717\pi$
$823$	1.59126e7	0.818919	0.409459	+	0.912328i	$0.365717\pi$
$824$	8.81436e6	0.452244
$825$	0	0
$826$	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	$-0.686551\pi$
$829$	−2.18883e7	−1.10618	−0.553089	+	0.833122i	$0.686551\pi$
$830$	−3.86019e6	−0.194497
$831$	0	0
$832$	847139.	0.0424274
$833$	0	0
$834$	0	0
$835$	4.86505e6	0.241475
$836$	1.88661e7	0.933613
$837$	0	0
$838$	−1.86621e7	−0.918017
$839$	−1.98669e7	−0.974372	−0.487186	−	0.873298i	$-0.661977\pi$
$839$	−1.98669e7	−0.974372	−0.487186	+	0.873298i	$0.661977\pi$
$840$	0	0
$841$	−2.04958e7	−0.999253
$842$	−1.49327e7	−0.725868
$843$	0	0
$844$	−7.36790e6	−0.356031
$845$	6.89888e6	0.332381
$846$	0	0
$847$	0	0
$848$	−4.78905e6	−0.228697
$849$	0	0
$850$	−1.13959e7	−0.541004
$851$	2.24110e7	1.06081
$852$	0	0
$853$	−8.75258e6	−0.411873	−0.205937	−	0.978565i	$-0.566024\pi$
$853$	−8.75258e6	−0.411873	−0.205937	+	0.978565i	$0.566024\pi$
$854$	0	0
$855$	0	0
$856$	−4.83667e6	−0.225612
$857$	−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	$0.536764\pi$
$860$	−1.81506e6	−0.0836847
$861$	0	0
$862$	−3.85868e6	−0.176877
$863$	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
$866$	2.47238e7	1.12027
$867$	0	0
$868$	0	0
$869$	1.86498e6	0.0837772
$870$	0	0
$871$	1.21240e7	0.541503
$872$	2.84968e6	0.126913
$873$	0	0
$874$	−2.80138e7	−1.24049
$875$	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	$0.369207\pi$
$878$	860539.	0.0376733
$879$	0	0
$880$	−3.36470e6	−0.146467
$881$	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	$-0.488218\pi$
$883$	1.71479e6	0.0740131	0.0370065	+	0.999315i	$0.488218\pi$
$884$	3.51253e6	0.151178
$885$	0	0
$886$	2.87947e7	1.23233
$887$	−2.42436e7	−1.03464	−0.517318	−	0.855793i	$-0.673070\pi$
$887$	−2.42436e7	−1.03464	−0.517318	+	0.855793i	$0.673070\pi$
$888$	0	0
$889$	0	0
$890$	−8.31936e6	−0.352058
$891$	0	0
$892$	1.91641e7	0.806449
$893$	3.53292e6	0.148253
$894$	0	0
$895$	978272.	0.0408227
$896$	0	0
$897$	0	0
$898$	−1.56861e7	−0.649120
$899$	1.12725e6	0.0465179
$900$	0	0
$901$	−1.98570e7	−0.814897
$902$	−4.30647e7	−1.76240
$903$	0	0
$904$	−5.78921e6	−0.235613
$905$	1.74134e7	0.706745
$906$	0	0
$907$	−1.90432e7	−0.768639	−0.384319	−	0.923200i	$-0.625564\pi$
$907$	−1.90432e7	−0.768639	−0.384319	+	0.923200i	$0.625564\pi$
$908$	1.43130e7	0.576123
$909$	0	0
$910$	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
$914$	5.28671e6	0.209325
$915$	0	0
$916$	4.14810e6	0.163347
$917$	0	0
$918$	0	0
$919$	−2.94858e7	−1.15166	−0.575830	−	0.817569i	$-0.695321\pi$
$919$	−2.94858e7	−1.15166	−0.575830	+	0.817569i	$0.695321\pi$
$920$	4.99614e6	0.194610
$921$	0	0
$922$	−301836.	−0.0116935
$923$	6.46927e6	0.249949
$924$	0	0
$925$	1.61811e7	0.621804
$926$	−1.31503e7	−0.503974
$927$	0	0
$928$	126717.	0.00483021
$929$	−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
$932$	−3.38104e6	−0.127500
$933$	0	0
$934$	−2.57353e6	−0.0965299
$935$	−1.39512e7	−0.521893
$936$	0	0
$937$	−2.66734e7	−0.992498	−0.496249	−	0.868180i	$-0.665290\pi$
$937$	−2.66734e7	−0.992498	−0.496249	+	0.868180i	$0.665290\pi$
$938$	0	0
$939$	0	0
$940$	−630081.	−0.0232582
$941$	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
$944$	−648903.	−0.0237001
$945$	0	0
$946$	1.35238e7	0.491328
$947$	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
$950$	−2.02264e7	−0.727126
$951$	0	0
$952$	0	0
$953$	−2.97125e7	−1.05976	−0.529880	−	0.848073i	$-0.677763\pi$
$953$	−2.97125e7	−1.05976	−0.529880	+	0.848073i	$0.677763\pi$
$954$	0	0
$955$	−9.91025e6	−0.351622
$956$	−7408.29	−0.000262164 0
$957$	0	0
$958$	−2.97059e7	−1.04575
$959$	0	0
$960$	0	0
$961$	5.43495e7	1.89840
$962$	−4.98747e6	−0.173757
$963$	0	0
$964$	−4.58232e6	−0.158815
$965$	−1.44554e7	−0.499704
$966$	0	0
$967$	7.64435e6	0.262890	0.131445	−	0.991323i	$-0.458038\pi$
$967$	7.64435e6	0.262890	0.131445	+	0.991323i	$0.458038\pi$
$968$	1.47627e7	0.506380
$969$	0	0
$970$	−9.70607e6	−0.331218
$971$	−4.99621e7	−1.70056	−0.850281	−	0.526329i	$-0.823568\pi$
$971$	−4.99621e7	−1.70056	−0.850281	+	0.526329i	$0.823568\pi$
$972$	0	0
$973$	0	0
$974$	−1.66834e7	−0.563491
$975$	0	0
$976$	536245.	0.0180193
$977$	−4.05394e7	−1.35876	−0.679378	−	0.733789i	$-0.737750\pi$
$977$	−4.05394e7	−1.35876	−0.679378	+	0.733789i	$0.737750\pi$
$978$	0	0
$979$	6.19865e7	2.06700
$980$	0	0
$981$	0	0
$982$	1.49470e7	0.494623
$983$	−4.13056e7	−1.36341	−0.681703	−	0.731629i	$-0.738761\pi$
$983$	−4.13056e7	−1.36341	−0.681703	+	0.731629i	$0.738761\pi$
$984$	0	0
$985$	6.55021e6	0.215112
$986$	525413.	0.0172111
$987$	0	0
$988$	6.23434e6	0.203188
$989$	−2.00811e7	−0.652826
$990$	0	0
$991$	−1.84084e7	−0.595431	−0.297715	−	0.954655i	$-0.596225\pi$
$991$	−1.84084e7	−0.595431	−0.297715	+	0.954655i	$0.596225\pi$
$992$	9.32789e6	0.300957
$993$	0	0
$994$	0	0
$995$	−6.03144e6	−0.193136
$996$	0	0
$997$	3.39708e7	1.08235	0.541175	−	0.840910i	$-0.317980\pi$
$997$	3.39708e7	1.08235	0.541175	+	0.840910i	$0.317980\pi$
$998$	3.48345e7	1.10709
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bl.1.2		2
3.2	odd	2		98.6.a.f.1.1		2
7.3	odd	6		126.6.g.e.37.2		4
7.5	odd	6		126.6.g.e.109.2		4
7.6	odd	2		882.6.a.bt.1.2		2
12.11	even	2		784.6.a.r.1.2		2
21.2	odd	6		98.6.c.f.67.2		4
21.5	even	6		14.6.c.b.11.1	yes	4
21.11	odd	6		98.6.c.f.79.2		4
21.17	even	6		14.6.c.b.9.1	✓	4
21.20	even	2		98.6.a.c.1.2		2
84.47	odd	6		112.6.i.b.81.2		4
84.59	odd	6		112.6.i.b.65.2		4
84.83	odd	2		784.6.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.1	✓	4	21.17	even	6
14.6.c.b.11.1	yes	4	21.5	even	6
98.6.a.c.1.2		2	21.20	even	2
98.6.a.f.1.1		2	3.2	odd	2
98.6.c.f.67.2		4	21.2	odd	6
98.6.c.f.79.2		4	21.11	odd	6
112.6.i.b.65.2		4	84.59	odd	6
112.6.i.b.81.2		4	84.47	odd	6
126.6.g.e.37.2		4	7.3	odd	6
126.6.g.e.109.2		4	7.5	odd	6
784.6.a.r.1.2		2	12.11	even	2
784.6.a.bc.1.1		2	84.83	odd	2
882.6.a.bl.1.2		2	1.1	even	1	trivial
882.6.a.bt.1.2		2	7.6	odd	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -21.0000 q^{5} +64.0000 q^{8} -84.0000 q^{10} +625.874 q^{11} +206.821 q^{13} +256.000 q^{16} +1061.46 q^{17} +1883.98 q^{19} -336.000 q^{20} +2503.49 q^{22} -3717.37 q^{23} -2684.00 q^{25} +827.284 q^{26} +123.747 q^{29} +9109.26 q^{31} +1024.00 q^{32} +4245.85 q^{34} -6028.73 q^{37} +7535.92 q^{38} -1344.00 q^{40} -17201.9 q^{41} +5401.98 q^{43} +10014.0 q^{44} -14869.5 q^{46} +1875.24 q^{47} -10736.0 q^{50} +3309.14 q^{52} -18707.2 q^{53} -13143.3 q^{55} +494.989 q^{58} -2534.78 q^{59} +2094.71 q^{61} +36437.1 q^{62} +4096.00 q^{64} -4343.24 q^{65} +58620.8 q^{67} +16983.4 q^{68} +31279.5 q^{71} +7150.47 q^{73} -24114.9 q^{74} +30143.7 q^{76} +2979.81 q^{79} -5376.00 q^{80} -68807.4 q^{82} +45954.6 q^{83} -22290.7 q^{85} +21607.9 q^{86} +40055.9 q^{88} +99040.0 q^{89} -59477.9 q^{92} +7500.97 q^{94} -39563.6 q^{95} +115548. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 42 q^{5} + 128 q^{8} - 168 q^{10} + 294 q^{11} + 140 q^{13} + 512 q^{16} + 1302 q^{17} + 1442 q^{19} - 672 q^{20} + 1176 q^{22} - 2646 q^{23} - 5368 q^{25} + 560 q^{26} - 1668 q^{29}+ \cdots + 43652 q^{97}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 42 * q^5 + 128 * q^8 - 168 * q^10 + 294 * q^11 + 140 * q^13 + 512 * q^16 + 1302 * q^17 + 1442 * q^19 - 672 * q^20 + 1176 * q^22 - 2646 * q^23 - 5368 * q^25 + 560 * q^26 - 1668 * q^29 + 14798 * q^31 + 2048 * q^32 + 5208 * q^34 + 5182 * q^37 + 5768 * q^38 - 2688 * q^40 - 5124 * q^41 - 4520 * q^43 + 4704 * q^44 - 10584 * q^46 - 14994 * q^47 - 21472 * q^50 + 2240 * q^52 - 24006 * q^53 - 6174 * q^55 - 6672 * q^58 + 38850 * q^59 + 23618 * q^61 + 59192 * q^62 + 8192 * q^64 - 2940 * q^65 + 32002 * q^67 + 20832 * q^68 + 89376 * q^71 + 47138 * q^73 + 20728 * q^74 + 23072 * q^76 - 40970 * q^79 - 10752 * q^80 - 20496 * q^82 + 68376 * q^83 - 27342 * q^85 - 18080 * q^86 + 18816 * q^88 + 123102 * q^89 - 42336 * q^92 - 59976 * q^94 - 30282 * q^95 + 43652 * q^97

Introduction

L-functions

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