LMFDB - Embedded newform 882.6.a.ba.1.1

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,-70,0,0,-128,0,280,-62] 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{79}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			$-8.88819$ 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)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -106.106 q^{5} -64.0000 q^{8} +424.422 q^{10} +93.4347 q^{11} +661.131 q^{13} +256.000 q^{16} -455.919 q^{17} +1106.28 q^{19} -1697.69 q^{20} -373.739 q^{22} -748.390 q^{23} +8133.39 q^{25} -2644.52 q^{26} -2804.78 q^{29} -359.299 q^{31} -1024.00 q^{32} +1823.68 q^{34} -6812.82 q^{37} -4425.12 q^{38} +6790.76 q^{40} -2319.39 q^{41} -19965.7 q^{43} +1494.96 q^{44} +2993.56 q^{46} +14209.5 q^{47} -32533.6 q^{50} +10578.1 q^{52} -26144.2 q^{53} -9913.94 q^{55} +11219.1 q^{58} -4904.35 q^{59} -13203.3 q^{61} +1437.19 q^{62} +4096.00 q^{64} -70149.6 q^{65} -59658.1 q^{67} -7294.71 q^{68} -8906.43 q^{71} -10480.6 q^{73} +27251.3 q^{74} +17700.5 q^{76} -7230.02 q^{79} -27163.0 q^{80} +9277.57 q^{82} +100461. q^{83} +48375.6 q^{85} +79862.9 q^{86} -5979.82 q^{88} -20071.8 q^{89} -11974.2 q^{92} -56838.2 q^{94} -117382. q^{95} -23320.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 70 q^{5} - 128 q^{8} + 280 q^{10} - 62 q^{11} + 1820 q^{13} + 512 q^{16} - 1694 q^{17} + 826 q^{19} - 1120 q^{20} + 248 q^{22} + 2734 q^{23} + 6312 q^{25} - 7280 q^{26} + 2852 q^{29}+ \cdots - 8316 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 70 * q^5 - 128 * q^8 + 280 * q^10 - 62 * q^11 + 1820 * q^13 + 512 * q^16 - 1694 * q^17 + 826 * q^19 - 1120 * q^20 + 248 * q^22 + 2734 * q^23 + 6312 * q^25 - 7280 * q^26 + 2852 * q^29 - 2674 * q^31 - 2048 * q^32 + 6776 * q^34 - 9146 * q^37 - 3304 * q^38 + 4480 * q^40 - 6132 * q^41 - 16040 * q^43 - 992 * q^44 - 10936 * q^46 + 25326 * q^47 - 25248 * q^50 + 29120 * q^52 - 14958 * q^53 - 15526 * q^55 - 11408 * q^58 + 1106 * q^59 - 28042 * q^61 + 10696 * q^62 + 8192 * q^64 - 28308 * q^65 - 102642 * q^67 - 27104 * q^68 + 11056 * q^71 + 35070 * q^73 + 36584 * q^74 + 13216 * q^76 + 101762 * q^79 - 17920 * q^80 + 24528 * q^82 + 44632 * q^83 + 3674 * q^85 + 64160 * q^86 + 3968 * q^88 + 75474 * q^89 + 43744 * q^92 - 101304 * q^94 - 127502 * q^95 - 8316 * 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
$5$	−106.106	−1.89807	−0.949037	−	0.315165i	$-0.897940\pi$
$5$	−106.106	−1.89807	−0.949037	+	0.315165i	$0.897940\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	424.422	1.34214
$11$	93.4347	0.232823	0.116412	−	0.993201i	$-0.462861\pi$
$11$	93.4347	0.232823	0.116412	+	0.993201i	$0.462861\pi$
$12$	0	0
$13$	661.131	1.08500	0.542499	−	0.840057i	$-0.317478\pi$
$13$	661.131	1.08500	0.542499	+	0.840057i	$0.317478\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−455.919	−0.382618	−0.191309	−	0.981530i	$-0.561273\pi$
$17$	−455.919	−0.382618	−0.191309	+	0.981530i	$0.561273\pi$
$18$	0	0
$19$	1106.28	0.703041	0.351521	−	0.936180i	$-0.385665\pi$
$19$	1106.28	0.703041	0.351521	+	0.936180i	$0.385665\pi$
$20$	−1697.69	−0.949037
$21$	0	0
$22$	−373.739	−0.164631
$23$	−748.390	−0.294991	−0.147495	−	0.989063i	$-0.547121\pi$
$23$	−748.390	−0.294991	−0.147495	+	0.989063i	$0.547121\pi$
$24$	0	0
$25$	8133.39	2.60268
$26$	−2644.52	−0.767209
$27$	0	0
$28$	0	0
$29$	−2804.78	−0.619304	−0.309652	−	0.950850i	$-0.600213\pi$
$29$	−2804.78	−0.619304	−0.309652	+	0.950850i	$0.600213\pi$
$30$	0	0
$31$	−359.299	−0.0671508	−0.0335754	−	0.999436i	$-0.510689\pi$
$31$	−359.299	−0.0671508	−0.0335754	+	0.999436i	$0.510689\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	1823.68	0.270552
$35$	0	0
$36$	0	0
$37$	−6812.82	−0.818131	−0.409066	−	0.912505i	$-0.634145\pi$
$37$	−6812.82	−0.818131	−0.409066	+	0.912505i	$0.634145\pi$
$38$	−4425.12	−0.497125
$39$	0	0
$40$	6790.76	0.671070
$41$	−2319.39	−0.215484	−0.107742	−	0.994179i	$-0.534362\pi$
$41$	−2319.39	−0.215484	−0.107742	+	0.994179i	$0.534362\pi$
$42$	0	0
$43$	−19965.7	−1.64670	−0.823349	−	0.567535i	$-0.807897\pi$
$43$	−19965.7	−1.64670	−0.823349	+	0.567535i	$0.807897\pi$
$44$	1494.96	0.116412
$45$	0	0
$46$	2993.56	0.208590
$47$	14209.5	0.938287	0.469143	−	0.883122i	$-0.344563\pi$
$47$	14209.5	0.938287	0.469143	+	0.883122i	$0.344563\pi$
$48$	0	0
$49$	0	0
$50$	−32533.6	−1.84038
$51$	0	0
$52$	10578.1	0.542499
$53$	−26144.2	−1.27846	−0.639228	−	0.769017i	$-0.720746\pi$
$53$	−26144.2	−1.27846	−0.639228	+	0.769017i	$0.720746\pi$
$54$	0	0
$55$	−9913.94	−0.441916
$56$	0	0
$57$	0	0
$58$	11219.1	0.437914
$59$	−4904.35	−0.183422	−0.0917110	−	0.995786i	$-0.529234\pi$
$59$	−4904.35	−0.183422	−0.0917110	+	0.995786i	$0.529234\pi$
$60$	0	0
$61$	−13203.3	−0.454315	−0.227158	−	0.973858i	$-0.572943\pi$
$61$	−13203.3	−0.454315	−0.227158	+	0.973858i	$0.572943\pi$
$62$	1437.19	0.0474828
$63$	0	0
$64$	4096.00	0.125000
$65$	−70149.6	−2.05941
$66$	0	0
$67$	−59658.1	−1.62361	−0.811807	−	0.583926i	$-0.801516\pi$
$67$	−59658.1	−1.62361	−0.811807	+	0.583926i	$0.801516\pi$
$68$	−7294.71	−0.191309
$69$	0	0
$70$	0	0
$71$	−8906.43	−0.209680	−0.104840	−	0.994489i	$-0.533433\pi$
$71$	−8906.43	−0.209680	−0.104840	+	0.994489i	$0.533433\pi$
$72$	0	0
$73$	−10480.6	−0.230186	−0.115093	−	0.993355i	$-0.536717\pi$
$73$	−10480.6	−0.230186	−0.115093	+	0.993355i	$0.536717\pi$
$74$	27251.3	0.578506
$75$	0	0
$76$	17700.5	0.351521
$77$	0	0
$78$	0	0
$79$	−7230.02	−0.130338	−0.0651691	−	0.997874i	$-0.520759\pi$
$79$	−7230.02	−0.130338	−0.0651691	+	0.997874i	$0.520759\pi$
$80$	−27163.0	−0.474518
$81$	0	0
$82$	9277.57	0.152370
$83$	100461.	1.60067	0.800336	−	0.599552i	$-0.204655\pi$
$83$	100461.	1.60067	0.800336	+	0.599552i	$0.204655\pi$
$84$	0	0
$85$	48375.6	0.726238
$86$	79862.9	1.16439
$87$	0	0
$88$	−5979.82	−0.0823155
$89$	−20071.8	−0.268603	−0.134302	−	0.990940i	$-0.542879\pi$
$89$	−20071.8	−0.268603	−0.134302	+	0.990940i	$0.542879\pi$
$90$	0	0
$91$	0	0
$92$	−11974.2	−0.147495
$93$	0	0
$94$	−56838.2	−0.663469
$95$	−117382.	−1.33442
$96$	0	0
$97$	−23320.9	−0.251662	−0.125831	−	0.992052i	$-0.540160\pi$
$97$	−23320.9	−0.251662	−0.125831	+	0.992052i	$0.540160\pi$
$98$	0	0
$99$	0	0
$100$	130134.	1.30134
$101$	68974.1	0.672795	0.336398	−	0.941720i	$-0.390791\pi$
$101$	68974.1	0.672795	0.336398	+	0.941720i	$0.390791\pi$
$102$	0	0
$103$	113725.	1.05624	0.528121	−	0.849169i	$-0.322897\pi$
$103$	113725.	1.05624	0.528121	+	0.849169i	$0.322897\pi$
$104$	−42312.4	−0.383605
$105$	0	0
$106$	104577.	0.904005
$107$	121287.	1.02413	0.512066	−	0.858946i	$-0.328880\pi$
$107$	121287.	1.02413	0.512066	+	0.858946i	$0.328880\pi$
$108$	0	0
$109$	−61373.9	−0.494786	−0.247393	−	0.968915i	$-0.579574\pi$
$109$	−61373.9	−0.494786	−0.247393	+	0.968915i	$0.579574\pi$
$110$	39655.8	0.312482
$111$	0	0
$112$	0	0
$113$	242939.	1.78979	0.894893	−	0.446281i	$-0.147252\pi$
$113$	242939.	1.78979	0.894893	+	0.446281i	$0.147252\pi$
$114$	0	0
$115$	79408.4	0.559914
$116$	−44876.5	−0.309652
$117$	0	0
$118$	19617.4	0.129699
$119$	0	0
$120$	0	0
$121$	−152321.	−0.945793
$122$	52813.1	0.321249
$123$	0	0
$124$	−5748.78	−0.0335754
$125$	−531418.	−3.04201
$126$	0	0
$127$	201922.	1.11090	0.555448	−	0.831551i	$-0.312547\pi$
$127$	201922.	1.11090	0.555448	+	0.831551i	$0.312547\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	280598.	1.45622
$131$	−283521.	−1.44347	−0.721734	−	0.692171i	$-0.756655\pi$
$131$	−283521.	−1.44347	−0.721734	+	0.692171i	$0.756655\pi$
$132$	0	0
$133$	0	0
$134$	238633.	1.14807
$135$	0	0
$136$	29178.8	0.135276
$137$	−188471.	−0.857911	−0.428955	−	0.903326i	$-0.641118\pi$
$137$	−188471.	−0.857911	−0.428955	+	0.903326i	$0.641118\pi$
$138$	0	0
$139$	−211579.	−0.928830	−0.464415	−	0.885618i	$-0.653735\pi$
$139$	−211579.	−0.928830	−0.464415	+	0.885618i	$0.653735\pi$
$140$	0	0
$141$	0	0
$142$	35625.7	0.148266
$143$	61772.5	0.252613
$144$	0	0
$145$	297603.	1.17548
$146$	41922.4	0.162766
$147$	0	0
$148$	−109005.	−0.409066
$149$	293324.	1.08238	0.541192	−	0.840899i	$-0.317973\pi$
$149$	293324.	1.08238	0.541192	+	0.840899i	$0.317973\pi$
$150$	0	0
$151$	−204476.	−0.729794	−0.364897	−	0.931048i	$-0.618896\pi$
$151$	−204476.	−0.729794	−0.364897	+	0.931048i	$0.618896\pi$
$152$	−70801.9	−0.248563
$153$	0	0
$154$	0	0
$155$	38123.6	0.127457
$156$	0	0
$157$	505915.	1.63805	0.819027	−	0.573755i	$-0.194514\pi$
$157$	505915.	1.63805	0.819027	+	0.573755i	$0.194514\pi$
$158$	28920.1	0.0921630
$159$	0	0
$160$	108652.	0.335535
$161$	0	0
$162$	0	0
$163$	−284200.	−0.837829	−0.418914	−	0.908026i	$-0.637589\pi$
$163$	−284200.	−0.837829	−0.418914	+	0.908026i	$0.637589\pi$
$164$	−37110.3	−0.107742
$165$	0	0
$166$	−401844.	−1.13185
$167$	13145.7	0.0364747	0.0182373	−	0.999834i	$-0.494195\pi$
$167$	13145.7	0.0364747	0.0182373	+	0.999834i	$0.494195\pi$
$168$	0	0
$169$	65800.6	0.177220
$170$	−193502.	−0.513528
$171$	0	0
$172$	−319452.	−0.823349
$173$	−51453.0	−0.130706	−0.0653530	−	0.997862i	$-0.520817\pi$
$173$	−51453.0	−0.130706	−0.0653530	+	0.997862i	$0.520817\pi$
$174$	0	0
$175$	0	0
$176$	23919.3	0.0582058
$177$	0	0
$178$	80287.3	0.189931
$179$	497735.	1.16109	0.580544	−	0.814229i	$-0.302840\pi$
$179$	497735.	1.16109	0.580544	+	0.814229i	$0.302840\pi$
$180$	0	0
$181$	−227120.	−0.515299	−0.257650	−	0.966238i	$-0.582948\pi$
$181$	−227120.	−0.515299	−0.257650	+	0.966238i	$0.582948\pi$
$182$	0	0
$183$	0	0
$184$	47897.0	0.104295
$185$	722879.	1.55287
$186$	0	0
$187$	−42598.7	−0.0890825
$188$	227353.	0.469143
$189$	0	0
$190$	469529.	0.943580
$191$	−502778.	−0.997225	−0.498612	−	0.866825i	$-0.666157\pi$
$191$	−502778.	−0.997225	−0.498612	+	0.866825i	$0.666157\pi$
$192$	0	0
$193$	140322.	0.271165	0.135582	−	0.990766i	$-0.456709\pi$
$193$	140322.	0.271165	0.135582	+	0.990766i	$0.456709\pi$
$194$	93283.8	0.177952
$195$	0	0
$196$	0	0
$197$	−378993.	−0.695769	−0.347885	−	0.937537i	$-0.613100\pi$
$197$	−378993.	−0.695769	−0.347885	+	0.937537i	$0.613100\pi$
$198$	0	0
$199$	−461022.	−0.825256	−0.412628	−	0.910900i	$-0.635389\pi$
$199$	−461022.	−0.825256	−0.412628	+	0.910900i	$0.635389\pi$
$200$	−520537.	−0.920188
$201$	0	0
$202$	−275897.	−0.475738
$203$	0	0
$204$	0	0
$205$	246100.	0.409004
$206$	−454901.	−0.746876
$207$	0	0
$208$	169249.	0.271249
$209$	103365.	0.163684
$210$	0	0
$211$	202139.	0.312567	0.156284	−	0.987712i	$-0.450049\pi$
$211$	202139.	0.312567	0.156284	+	0.987712i	$0.450049\pi$
$212$	−418307.	−0.639228
$213$	0	0
$214$	−485149.	−0.724170
$215$	2.11848e6	3.12556
$216$	0	0
$217$	0	0
$218$	245495.	0.349866
$219$	0	0
$220$	−158623.	−0.220958
$221$	−301422.	−0.415140
$222$	0	0
$223$	694340.	0.934997	0.467498	−	0.883994i	$-0.345155\pi$
$223$	694340.	0.934997	0.467498	+	0.883994i	$0.345155\pi$
$224$	0	0
$225$	0	0
$226$	−971756.	−1.26557
$227$	−1.26697e6	−1.63194	−0.815968	−	0.578096i	$-0.803796\pi$
$227$	−1.26697e6	−1.63194	−0.815968	+	0.578096i	$0.803796\pi$
$228$	0	0
$229$	856477.	1.07926	0.539631	−	0.841902i	$-0.318564\pi$
$229$	856477.	1.07926	0.539631	+	0.841902i	$0.318564\pi$
$230$	−317633.	−0.395919
$231$	0	0
$232$	179506.	0.218957
$233$	284358.	0.343143	0.171571	−	0.985172i	$-0.445116\pi$
$233$	284358.	0.343143	0.171571	+	0.985172i	$0.445116\pi$
$234$	0	0
$235$	−1.50771e6	−1.78094
$236$	−78469.6	−0.0917110
$237$	0	0
$238$	0	0
$239$	427941.	0.484606	0.242303	−	0.970201i	$-0.422097\pi$
$239$	427941.	0.484606	0.242303	+	0.970201i	$0.422097\pi$
$240$	0	0
$241$	809581.	0.897879	0.448940	−	0.893562i	$-0.351802\pi$
$241$	809581.	0.897879	0.448940	+	0.893562i	$0.351802\pi$
$242$	609284.	0.668777
$243$	0	0
$244$	−211253.	−0.227158
$245$	0	0
$246$	0	0
$247$	731395.	0.762798
$248$	22995.1	0.0237414
$249$	0	0
$250$	2.12567e6	2.15103
$251$	1.26305e6	1.26542	0.632711	−	0.774388i	$-0.281942\pi$
$251$	1.26305e6	1.26542	0.632711	+	0.774388i	$0.281942\pi$
$252$	0	0
$253$	−69925.6	−0.0686808
$254$	−807687.	−0.785523
$255$	0	0
$256$	65536.0	0.0625000
$257$	−56138.2	−0.0530183	−0.0265091	−	0.999649i	$-0.508439\pi$
$257$	−56138.2	−0.0530183	−0.0265091	+	0.999649i	$0.508439\pi$
$258$	0	0
$259$	0	0
$260$	−1.12239e6	−1.02970
$261$	0	0
$262$	1.13408e6	1.02069
$263$	−1.54275e6	−1.37533	−0.687664	−	0.726029i	$-0.741364\pi$
$263$	−1.54275e6	−1.37533	−0.687664	+	0.726029i	$0.741364\pi$
$264$	0	0
$265$	2.77405e6	2.42660
$266$	0	0
$267$	0	0
$268$	−954530.	−0.811807
$269$	1.20535e6	1.01563	0.507813	−	0.861467i	$-0.330454\pi$
$269$	1.20535e6	1.01563	0.507813	+	0.861467i	$0.330454\pi$
$270$	0	0
$271$	1.76489e6	1.45980	0.729900	−	0.683554i	$-0.239567\pi$
$271$	1.76489e6	1.45980	0.729900	+	0.683554i	$0.239567\pi$
$272$	−116715.	−0.0956546
$273$	0	0
$274$	753882.	0.606634
$275$	759941.	0.605966
$276$	0	0
$277$	1.48319e6	1.16144	0.580719	−	0.814104i	$-0.302771\pi$
$277$	1.48319e6	1.16144	0.580719	+	0.814104i	$0.302771\pi$
$278$	846317.	0.656782
$279$	0	0
$280$	0	0
$281$	1.26812e6	0.958061	0.479031	−	0.877798i	$-0.340988\pi$
$281$	1.26812e6	0.958061	0.479031	+	0.877798i	$0.340988\pi$
$282$	0	0
$283$	−1.44657e6	−1.07368	−0.536838	−	0.843685i	$-0.680381\pi$
$283$	−1.44657e6	−1.07368	−0.536838	+	0.843685i	$0.680381\pi$
$284$	−142503.	−0.104840
$285$	0	0
$286$	−247090.	−0.178624
$287$	0	0
$288$	0	0
$289$	−1.21199e6	−0.853603
$290$	−1.19041e6	−0.831193
$291$	0	0
$292$	−167689.	−0.115093
$293$	−367016.	−0.249756	−0.124878	−	0.992172i	$-0.539854\pi$
$293$	−367016.	−0.249756	−0.124878	+	0.992172i	$0.539854\pi$
$294$	0	0
$295$	520379.	0.348149
$296$	436021.	0.289253
$297$	0	0
$298$	−1.17330e6	−0.765362
$299$	−494784.	−0.320064
$300$	0	0
$301$	0	0
$302$	817905.	0.516042
$303$	0	0
$304$	283207.	0.175760
$305$	1.40094e6	0.862324
$306$	0	0
$307$	−131281.	−0.0794979	−0.0397490	−	0.999210i	$-0.512656\pi$
$307$	−131281.	−0.0794979	−0.0397490	+	0.999210i	$0.512656\pi$
$308$	0	0
$309$	0	0
$310$	−152494.	−0.0901259
$311$	2.33471e6	1.36878	0.684389	−	0.729118i	$-0.260069\pi$
$311$	2.33471e6	1.36878	0.684389	+	0.729118i	$0.260069\pi$
$312$	0	0
$313$	−772801.	−0.445869	−0.222934	−	0.974833i	$-0.571564\pi$
$313$	−772801.	−0.445869	−0.222934	+	0.974833i	$0.571564\pi$
$314$	−2.02366e6	−1.15828
$315$	0	0
$316$	−115680.	−0.0651691
$317$	−524336.	−0.293063	−0.146532	−	0.989206i	$-0.546811\pi$
$317$	−524336.	−0.293063	−0.146532	+	0.989206i	$0.546811\pi$
$318$	0	0
$319$	−262064.	−0.144188
$320$	−434608.	−0.237259
$321$	0	0
$322$	0	0
$323$	−504374.	−0.268996
$324$	0	0
$325$	5.37723e6	2.82391
$326$	1.13680e6	0.592434
$327$	0	0
$328$	148441.	0.0761850
$329$	0	0
$330$	0	0
$331$	631635.	0.316881	0.158440	−	0.987369i	$-0.449353\pi$
$331$	631635.	0.316881	0.158440	+	0.987369i	$0.449353\pi$
$332$	1.60738e6	0.800336
$333$	0	0
$334$	−52582.7	−0.0257915
$335$	6.33006e6	3.08174
$336$	0	0
$337$	3.33479e6	1.59954	0.799768	−	0.600309i	$-0.204956\pi$
$337$	3.33479e6	1.59954	0.799768	+	0.600309i	$0.204956\pi$
$338$	−263202.	−0.125314
$339$	0	0
$340$	774009.	0.363119
$341$	−33571.0	−0.0156343
$342$	0	0
$343$	0	0
$344$	1.27781e6	0.582196
$345$	0	0
$346$	205812.	0.0924231
$347$	650351.	0.289951	0.144975	−	0.989435i	$-0.453690\pi$
$347$	650351.	0.289951	0.144975	+	0.989435i	$0.453690\pi$
$348$	0	0
$349$	−601041.	−0.264144	−0.132072	−	0.991240i	$-0.542163\pi$
$349$	−601041.	−0.264144	−0.132072	+	0.991240i	$0.542163\pi$
$350$	0	0
$351$	0	0
$352$	−95677.2	−0.0411577
$353$	2.88889e6	1.23394	0.616970	−	0.786987i	$-0.288360\pi$
$353$	2.88889e6	1.23394	0.616970	+	0.786987i	$0.288360\pi$
$354$	0	0
$355$	945021.	0.397989
$356$	−321149.	−0.134302
$357$	0	0
$358$	−1.99094e6	−0.821014
$359$	97830.5	0.0400625	0.0200313	−	0.999799i	$-0.493623\pi$
$359$	97830.5	0.0400625	0.0200313	+	0.999799i	$0.493623\pi$
$360$	0	0
$361$	−1.25225e6	−0.505733
$362$	908481.	0.364371
$363$	0	0
$364$	0	0
$365$	1.11205e6	0.436910
$366$	0	0
$367$	2.14775e6	0.832374	0.416187	−	0.909279i	$-0.363366\pi$
$367$	2.14775e6	0.832374	0.416187	+	0.909279i	$0.363366\pi$
$368$	−191588.	−0.0737477
$369$	0	0
$370$	−2.89151e6	−1.09805
$371$	0	0
$372$	0	0
$373$	4.20458e6	1.56477	0.782386	−	0.622794i	$-0.214003\pi$
$373$	4.20458e6	1.56477	0.782386	+	0.622794i	$0.214003\pi$
$374$	170395.	0.0629908
$375$	0	0
$376$	−909411.	−0.331734
$377$	−1.85433e6	−0.671943
$378$	0	0
$379$	−535586.	−0.191527	−0.0957637	−	0.995404i	$-0.530529\pi$
$379$	−535586.	−0.191527	−0.0957637	+	0.995404i	$0.530529\pi$
$380$	−1.87812e6	−0.667212
$381$	0	0
$382$	2.01111e6	0.705144
$383$	3.84697e6	1.34005	0.670027	−	0.742337i	$-0.266283\pi$
$383$	3.84697e6	1.34005	0.670027	+	0.742337i	$0.266283\pi$
$384$	0	0
$385$	0	0
$386$	−561289.	−0.191742
$387$	0	0
$388$	−373135.	−0.125831
$389$	1.69833e6	0.569045	0.284523	−	0.958669i	$-0.408165\pi$
$389$	1.69833e6	0.569045	0.284523	+	0.958669i	$0.408165\pi$
$390$	0	0
$391$	341206.	0.112869
$392$	0	0
$393$	0	0
$394$	1.51597e6	0.491983
$395$	767145.	0.247391
$396$	0	0
$397$	3.62378e6	1.15394	0.576972	−	0.816764i	$-0.304234\pi$
$397$	3.62378e6	1.15394	0.576972	+	0.816764i	$0.304234\pi$
$398$	1.84409e6	0.583544
$399$	0	0
$400$	2.08215e6	0.650671
$401$	2.79801e6	0.868937	0.434469	−	0.900687i	$-0.356936\pi$
$401$	2.79801e6	0.868937	0.434469	+	0.900687i	$0.356936\pi$
$402$	0	0
$403$	−237543.	−0.0728585
$404$	1.10359e6	0.336398
$405$	0	0
$406$	0	0
$407$	−636554.	−0.190480
$408$	0	0
$409$	−666281.	−0.196947	−0.0984735	−	0.995140i	$-0.531396\pi$
$409$	−666281.	−0.196947	−0.0984735	+	0.995140i	$0.531396\pi$
$410$	−984401.	−0.289210
$411$	0	0
$412$	1.81960e6	0.528121
$413$	0	0
$414$	0	0
$415$	−1.06595e7	−3.03819
$416$	−676998.	−0.191802
$417$	0	0
$418$	−413460.	−0.115742
$419$	−2.89203e6	−0.804762	−0.402381	−	0.915472i	$-0.631817\pi$
$419$	−2.89203e6	−0.804762	−0.402381	+	0.915472i	$0.631817\pi$
$420$	0	0
$421$	6.72027e6	1.84791	0.923956	−	0.382498i	$-0.124936\pi$
$421$	6.72027e6	1.84791	0.923956	+	0.382498i	$0.124936\pi$
$422$	−808555.	−0.221018
$423$	0	0
$424$	1.67323e6	0.452002
$425$	−3.70817e6	−0.995835
$426$	0	0
$427$	0	0
$428$	1.94060e6	0.512066
$429$	0	0
$430$	−8.47390e6	−2.21010
$431$	−4.72298e6	−1.22468	−0.612341	−	0.790594i	$-0.709772\pi$
$431$	−4.72298e6	−1.22468	−0.612341	+	0.790594i	$0.709772\pi$
$432$	0	0
$433$	−5.78136e6	−1.48187	−0.740935	−	0.671577i	$-0.765617\pi$
$433$	−5.78136e6	−1.48187	−0.740935	+	0.671577i	$0.765617\pi$
$434$	0	0
$435$	0	0
$436$	−981982.	−0.247393
$437$	−827929.	−0.207391
$438$	0	0
$439$	5.36873e6	1.32957	0.664784	−	0.747036i	$-0.268524\pi$
$439$	5.36873e6	1.32957	0.664784	+	0.747036i	$0.268524\pi$
$440$	634492.	0.156241
$441$	0	0
$442$	1.20569e6	0.293548
$443$	−4.31511e6	−1.04468	−0.522339	−	0.852738i	$-0.674941\pi$
$443$	−4.31511e6	−1.04468	−0.522339	+	0.852738i	$0.674941\pi$
$444$	0	0
$445$	2.12973e6	0.509829
$446$	−2.77736e6	−0.661143
$447$	0	0
$448$	0	0
$449$	−4.75292e6	−1.11261	−0.556307	−	0.830977i	$-0.687782\pi$
$449$	−4.75292e6	−1.11261	−0.556307	+	0.830977i	$0.687782\pi$
$450$	0	0
$451$	−216712.	−0.0501696
$452$	3.88702e6	0.894893
$453$	0	0
$454$	5.06790e6	1.15395
$455$	0	0
$456$	0	0
$457$	2.13728e6	0.478707	0.239354	−	0.970932i	$-0.423064\pi$
$457$	2.13728e6	0.478707	0.239354	+	0.970932i	$0.423064\pi$
$458$	−3.42591e6	−0.763153
$459$	0	0
$460$	1.27053e6	0.279957
$461$	1.72920e6	0.378959	0.189479	−	0.981885i	$-0.439320\pi$
$461$	1.72920e6	0.378959	0.189479	+	0.981885i	$0.439320\pi$
$462$	0	0
$463$	−6.54367e6	−1.41863	−0.709315	−	0.704892i	$-0.750996\pi$
$463$	−6.54367e6	−1.41863	−0.709315	+	0.704892i	$0.750996\pi$
$464$	−718024.	−0.154826
$465$	0	0
$466$	−1.13743e6	−0.242639
$467$	1.79419e6	0.380694	0.190347	−	0.981717i	$-0.439039\pi$
$467$	1.79419e6	0.380694	0.190347	+	0.981717i	$0.439039\pi$
$468$	0	0
$469$	0	0
$470$	6.03085e6	1.25931
$471$	0	0
$472$	313878.	0.0648495
$473$	−1.86549e6	−0.383390
$474$	0	0
$475$	8.99780e6	1.82979
$476$	0	0
$477$	0	0
$478$	−1.71176e6	−0.342669
$479$	−4.09403e6	−0.815289	−0.407645	−	0.913141i	$-0.633650\pi$
$479$	−4.09403e6	−0.815289	−0.407645	+	0.913141i	$0.633650\pi$
$480$	0	0
$481$	−4.50417e6	−0.887670
$482$	−3.23833e6	−0.634897
$483$	0	0
$484$	−2.43714e6	−0.472897
$485$	2.47448e6	0.477672
$486$	0	0
$487$	−4.45210e6	−0.850634	−0.425317	−	0.905045i	$-0.639837\pi$
$487$	−4.45210e6	−0.850634	−0.425317	+	0.905045i	$0.639837\pi$
$488$	845010.	0.160625
$489$	0	0
$490$	0	0
$491$	−7.09674e6	−1.32848	−0.664240	−	0.747519i	$-0.731245\pi$
$491$	−7.09674e6	−1.32848	−0.664240	+	0.747519i	$0.731245\pi$
$492$	0	0
$493$	1.27875e6	0.236957
$494$	−2.92558e6	−0.539380
$495$	0	0
$496$	−91980.4	−0.0167877
$497$	0	0
$498$	0	0
$499$	9.41903e6	1.69338	0.846691	−	0.532085i	$-0.178591\pi$
$499$	9.41903e6	1.69338	0.846691	+	0.532085i	$0.178591\pi$
$500$	−8.50269e6	−1.52101
$501$	0	0
$502$	−5.05219e6	−0.894789
$503$	−957258.	−0.168698	−0.0843489	−	0.996436i	$-0.526881\pi$
$503$	−957258.	−0.168698	−0.0843489	+	0.996436i	$0.526881\pi$
$504$	0	0
$505$	−7.31854e6	−1.27702
$506$	279703.	0.0485646
$507$	0	0
$508$	3.23075e6	0.555448
$509$	4.01453e6	0.686816	0.343408	−	0.939186i	$-0.388419\pi$
$509$	4.01453e6	0.686816	0.343408	+	0.939186i	$0.388419\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	224553.	0.0374896
$515$	−1.20669e7	−2.00483
$516$	0	0
$517$	1.32766e6	0.218455
$518$	0	0
$519$	0	0
$520$	4.48958e6	0.728110
$521$	4.31820e6	0.696960	0.348480	−	0.937316i	$-0.386698\pi$
$521$	4.31820e6	0.696960	0.348480	+	0.937316i	$0.386698\pi$
$522$	0	0
$523$	1.34501e6	0.215017	0.107508	−	0.994204i	$-0.465713\pi$
$523$	1.34501e6	0.215017	0.107508	+	0.994204i	$0.465713\pi$
$524$	−4.53634e6	−0.721734
$525$	0	0
$526$	6.17100e6	0.972504
$527$	163811.	0.0256931
$528$	0	0
$529$	−5.87626e6	−0.912980
$530$	−1.10962e7	−1.71587
$531$	0	0
$532$	0	0
$533$	−1.53342e6	−0.233799
$534$	0	0
$535$	−1.28692e7	−1.94388
$536$	3.81812e6	0.574034
$537$	0	0
$538$	−4.82141e6	−0.718156
$539$	0	0
$540$	0	0
$541$	−7.06006e6	−1.03709	−0.518543	−	0.855051i	$-0.673526\pi$
$541$	−7.06006e6	−1.03709	−0.518543	+	0.855051i	$0.673526\pi$
$542$	−7.05954e6	−1.03223
$543$	0	0
$544$	466862.	0.0676380
$545$	6.51211e6	0.939140
$546$	0	0
$547$	−1.23520e7	−1.76510	−0.882549	−	0.470221i	$-0.844174\pi$
$547$	−1.23520e7	−1.76510	−0.882549	+	0.470221i	$0.844174\pi$
$548$	−3.01553e6	−0.428955
$549$	0	0
$550$	−3.03976e6	−0.428483
$551$	−3.10287e6	−0.435396
$552$	0	0
$553$	0	0
$554$	−5.93275e6	−0.821261
$555$	0	0
$556$	−3.38527e6	−0.464415
$557$	1.22251e7	1.66961	0.834803	−	0.550549i	$-0.185582\pi$
$557$	1.22251e7	1.66961	0.834803	+	0.550549i	$0.185582\pi$
$558$	0	0
$559$	−1.32000e7	−1.78666
$560$	0	0
$561$	0	0
$562$	−5.07247e6	−0.677452
$563$	812527.	0.108036	0.0540178	−	0.998540i	$-0.482797\pi$
$563$	812527.	0.108036	0.0540178	+	0.998540i	$0.482797\pi$
$564$	0	0
$565$	−2.57772e7	−3.39715
$566$	5.78628e6	0.759204
$567$	0	0
$568$	570011.	0.0741332
$569$	8.43881e6	1.09270	0.546350	−	0.837557i	$-0.316017\pi$
$569$	8.43881e6	1.09270	0.546350	+	0.837557i	$0.316017\pi$
$570$	0	0
$571$	3.71511e6	0.476849	0.238425	−	0.971161i	$-0.423369\pi$
$571$	3.71511e6	0.476849	0.238425	+	0.971161i	$0.423369\pi$
$572$	988361.	0.126306
$573$	0	0
$574$	0	0
$575$	−6.08695e6	−0.767768
$576$	0	0
$577$	1.30375e7	1.63025	0.815126	−	0.579284i	$-0.196668\pi$
$577$	1.30375e7	1.63025	0.815126	+	0.579284i	$0.196668\pi$
$578$	4.84798e6	0.603589
$579$	0	0
$580$	4.76164e6	0.587742
$581$	0	0
$582$	0	0
$583$	−2.44278e6	−0.297654
$584$	670758.	0.0813830
$585$	0	0
$586$	1.46806e6	0.176604
$587$	−1.15227e7	−1.38026	−0.690128	−	0.723687i	$-0.742446\pi$
$587$	−1.15227e7	−1.38026	−0.690128	+	0.723687i	$0.742446\pi$
$588$	0	0
$589$	−397485.	−0.0472098
$590$	−2.08152e6	−0.246178
$591$	0	0
$592$	−1.74408e6	−0.204533
$593$	1.20103e7	1.40254	0.701271	−	0.712895i	$-0.252616\pi$
$593$	1.20103e7	1.40254	0.701271	+	0.712895i	$0.252616\pi$
$594$	0	0
$595$	0	0
$596$	4.69318e6	0.541192
$597$	0	0
$598$	1.97913e6	0.226320
$599$	1.17460e6	0.133759	0.0668794	−	0.997761i	$-0.478696\pi$
$599$	1.17460e6	0.133759	0.0668794	+	0.997761i	$0.478696\pi$
$600$	0	0
$601$	−1.62934e7	−1.84003	−0.920014	−	0.391886i	$-0.871823\pi$
$601$	−1.62934e7	−1.84003	−0.920014	+	0.391886i	$0.871823\pi$
$602$	0	0
$603$	0	0
$604$	−3.27162e6	−0.364897
$605$	1.61621e7	1.79519
$606$	0	0
$607$	−1.82629e6	−0.201186	−0.100593	−	0.994928i	$-0.532074\pi$
$607$	−1.82629e6	−0.201186	−0.100593	+	0.994928i	$0.532074\pi$
$608$	−1.13283e6	−0.124281
$609$	0	0
$610$	−5.60377e6	−0.609755
$611$	9.39436e6	1.01804
$612$	0	0
$613$	−5.10271e6	−0.548466	−0.274233	−	0.961663i	$-0.588424\pi$
$613$	−5.10271e6	−0.548466	−0.274233	+	0.961663i	$0.588424\pi$
$614$	525124.	0.0562135
$615$	0	0
$616$	0	0
$617$	8.03920e6	0.850158	0.425079	−	0.905156i	$-0.360246\pi$
$617$	8.03920e6	0.850158	0.425079	+	0.905156i	$0.360246\pi$
$618$	0	0
$619$	8.60132e6	0.902274	0.451137	−	0.892455i	$-0.351019\pi$
$619$	8.60132e6	0.902274	0.451137	+	0.892455i	$0.351019\pi$
$620$	609977.	0.0637286
$621$	0	0
$622$	−9.33886e6	−0.967872
$623$	0	0
$624$	0	0
$625$	3.09695e7	3.17128
$626$	3.09120e6	0.315277
$627$	0	0
$628$	8.09464e6	0.819027
$629$	3.10610e6	0.313032
$630$	0	0
$631$	6.21151e6	0.621046	0.310523	−	0.950566i	$-0.399496\pi$
$631$	6.21151e6	0.621046	0.310523	+	0.950566i	$0.399496\pi$
$632$	462721.	0.0460815
$633$	0	0
$634$	2.09734e6	0.207227
$635$	−2.14250e7	−2.10856
$636$	0	0
$637$	0	0
$638$	1.04826e6	0.101957
$639$	0	0
$640$	1.73843e6	0.167768
$641$	−4.45143e6	−0.427912	−0.213956	−	0.976843i	$-0.568635\pi$
$641$	−4.45143e6	−0.427912	−0.213956	+	0.976843i	$0.568635\pi$
$642$	0	0
$643$	1.58708e7	1.51381	0.756907	−	0.653523i	$-0.226710\pi$
$643$	1.58708e7	1.51381	0.756907	+	0.653523i	$0.226710\pi$
$644$	0	0
$645$	0	0
$646$	2.01750e6	0.190209
$647$	3.65619e6	0.343375	0.171687	−	0.985151i	$-0.445078\pi$
$647$	3.65619e6	0.343375	0.171687	+	0.985151i	$0.445078\pi$
$648$	0	0
$649$	−458237.	−0.0427049
$650$	−2.15089e7	−1.99680
$651$	0	0
$652$	−4.54720e6	−0.418914
$653$	4.15039e6	0.380896	0.190448	−	0.981697i	$-0.439006\pi$
$653$	4.15039e6	0.380896	0.190448	+	0.981697i	$0.439006\pi$
$654$	0	0
$655$	3.00832e7	2.73981
$656$	−593764.	−0.0538709
$657$	0	0
$658$	0	0
$659$	1.33739e7	1.19962	0.599810	−	0.800143i	$-0.295243\pi$
$659$	1.33739e7	1.19962	0.599810	+	0.800143i	$0.295243\pi$
$660$	0	0
$661$	−9.82641e6	−0.874764	−0.437382	−	0.899276i	$-0.644094\pi$
$661$	−9.82641e6	−0.874764	−0.437382	+	0.899276i	$0.644094\pi$
$662$	−2.52654e6	−0.224069
$663$	0	0
$664$	−6.42950e6	−0.565923
$665$	0	0
$666$	0	0
$667$	2.09907e6	0.182689
$668$	210331.	0.0182373
$669$	0	0
$670$	−2.53202e7	−2.17912
$671$	−1.23365e6	−0.105775
$672$	0	0
$673$	401148.	0.0341403	0.0170702	−	0.999854i	$-0.494566\pi$
$673$	401148.	0.0341403	0.0170702	+	0.999854i	$0.494566\pi$
$674$	−1.33392e7	−1.13104
$675$	0	0
$676$	1.05281e6	0.0886101
$677$	−2.07302e7	−1.73833	−0.869164	−	0.494524i	$-0.835343\pi$
$677$	−2.07302e7	−1.73833	−0.869164	+	0.494524i	$0.835343\pi$
$678$	0	0
$679$	0	0
$680$	−3.09604e6	−0.256764
$681$	0	0
$682$	134284.	0.0110551
$683$	1.91996e7	1.57485	0.787427	−	0.616409i	$-0.211413\pi$
$683$	1.91996e7	1.57485	0.787427	+	0.616409i	$0.211413\pi$
$684$	0	0
$685$	1.99978e7	1.62838
$686$	0	0
$687$	0	0
$688$	−5.11123e6	−0.411675
$689$	−1.72847e7	−1.38712
$690$	0	0
$691$	−1.13093e7	−0.901033	−0.450517	−	0.892768i	$-0.648760\pi$
$691$	−1.13093e7	−0.901033	−0.450517	+	0.892768i	$0.648760\pi$
$692$	−823248.	−0.0653530
$693$	0	0
$694$	−2.60140e6	−0.205026
$695$	2.24497e7	1.76299
$696$	0	0
$697$	1.05746e6	0.0824480
$698$	2.40417e6	0.186778
$699$	0	0
$700$	0	0
$701$	1.92159e7	1.47695	0.738473	−	0.674283i	$-0.235547\pi$
$701$	1.92159e7	1.47695	0.738473	+	0.674283i	$0.235547\pi$
$702$	0	0
$703$	−7.53689e6	−0.575180
$704$	382709.	0.0291029
$705$	0	0
$706$	−1.15555e7	−0.872527
$707$	0	0
$708$	0	0
$709$	5.64480e6	0.421729	0.210864	−	0.977515i	$-0.432372\pi$
$709$	5.64480e6	0.421729	0.210864	+	0.977515i	$0.432372\pi$
$710$	−3.78009e6	−0.281421
$711$	0	0
$712$	1.28460e6	0.0949657
$713$	268896.	0.0198089
$714$	0	0
$715$	−6.55441e6	−0.479478
$716$	7.96376e6	0.580544
$717$	0	0
$718$	−391322.	−0.0283285
$719$	−4.42077e6	−0.318916	−0.159458	−	0.987205i	$-0.550975\pi$
$719$	−4.42077e6	−0.318916	−0.159458	+	0.987205i	$0.550975\pi$
$720$	0	0
$721$	0	0
$722$	5.00898e6	0.357607
$723$	0	0
$724$	−3.63392e6	−0.257650
$725$	−2.28124e7	−1.61185
$726$	0	0
$727$	1.92503e7	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$727$	1.92503e7	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$728$	0	0
$729$	0	0
$730$	−4.44819e6	−0.308942
$731$	9.10277e6	0.630057
$732$	0	0
$733$	1.30187e7	0.894966	0.447483	−	0.894292i	$-0.352320\pi$
$733$	1.30187e7	0.894966	0.447483	+	0.894292i	$0.352320\pi$
$734$	−8.59100e6	−0.588577
$735$	0	0
$736$	766352.	0.0521475
$737$	−5.57414e6	−0.378015
$738$	0	0
$739$	−1.70955e7	−1.15152	−0.575758	−	0.817620i	$-0.695293\pi$
$739$	−1.70955e7	−1.15152	−0.575758	+	0.817620i	$0.695293\pi$
$740$	1.15661e7	0.776437
$741$	0	0
$742$	0	0
$743$	−2.36546e7	−1.57197	−0.785984	−	0.618247i	$-0.787843\pi$
$743$	−2.36546e7	−1.57197	−0.785984	+	0.618247i	$0.787843\pi$
$744$	0	0
$745$	−3.11233e7	−2.05445
$746$	−1.68183e7	−1.10646
$747$	0	0
$748$	−681579.	−0.0445413
$749$	0	0
$750$	0	0
$751$	−166564.	−0.0107766	−0.00538828	−	0.999985i	$-0.501715\pi$
$751$	−166564.	−0.0107766	−0.00538828	+	0.999985i	$0.501715\pi$
$752$	3.63764e6	0.234572
$753$	0	0
$754$	7.41730e6	0.475136
$755$	2.16961e7	1.38520
$756$	0	0
$757$	−4.63842e6	−0.294192	−0.147096	−	0.989122i	$-0.546993\pi$
$757$	−4.63842e6	−0.294192	−0.147096	+	0.989122i	$0.546993\pi$
$758$	2.14234e6	0.135430
$759$	0	0
$760$	7.51247e6	0.471790
$761$	6.96999e6	0.436285	0.218143	−	0.975917i	$-0.430000\pi$
$761$	6.96999e6	0.436285	0.218143	+	0.975917i	$0.430000\pi$
$762$	0	0
$763$	0	0
$764$	−8.04445e6	−0.498612
$765$	0	0
$766$	−1.53879e7	−0.947561
$767$	−3.24242e6	−0.199013
$768$	0	0
$769$	3.02590e6	0.184518	0.0922590	−	0.995735i	$-0.470591\pi$
$769$	3.02590e6	0.184518	0.0922590	+	0.995735i	$0.470591\pi$
$770$	0	0
$771$	0	0
$772$	2.24516e6	0.135582
$773$	−2.78539e7	−1.67663	−0.838314	−	0.545188i	$-0.816458\pi$
$773$	−2.78539e7	−1.67663	−0.838314	+	0.545188i	$0.816458\pi$
$774$	0	0
$775$	−2.92232e6	−0.174772
$776$	1.49254e6	0.0889758
$777$	0	0
$778$	−6.79330e6	−0.402376
$779$	−2.56589e6	−0.151494
$780$	0	0
$781$	−832170.	−0.0488185
$782$	−1.36482e6	−0.0798104
$783$	0	0
$784$	0	0
$785$	−5.36804e7	−3.10915
$786$	0	0
$787$	2.87913e7	1.65701	0.828503	−	0.559985i	$-0.189193\pi$
$787$	2.87913e7	1.65701	0.828503	+	0.559985i	$0.189193\pi$
$788$	−6.06388e6	−0.347885
$789$	0	0
$790$	−3.06858e6	−0.174932
$791$	0	0
$792$	0	0
$793$	−8.72910e6	−0.492931
$794$	−1.44951e7	−0.815962
$795$	0	0
$796$	−7.37635e6	−0.412628
$797$	3.67489e6	0.204927	0.102463	−	0.994737i	$-0.467328\pi$
$797$	3.67489e6	0.204927	0.102463	+	0.994737i	$0.467328\pi$
$798$	0	0
$799$	−6.47841e6	−0.359006
$800$	−8.32859e6	−0.460094
$801$	0	0
$802$	−1.11920e7	−0.614431
$803$	−979251.	−0.0535926
$804$	0	0
$805$	0	0
$806$	950173.	0.0515187
$807$	0	0
$808$	−4.41435e6	−0.237869
$809$	1.04012e7	0.558744	0.279372	−	0.960183i	$-0.409874\pi$
$809$	1.04012e7	0.558744	0.279372	+	0.960183i	$0.409874\pi$
$810$	0	0
$811$	−1.53749e7	−0.820844	−0.410422	−	0.911896i	$-0.634619\pi$
$811$	−1.53749e7	−0.820844	−0.410422	+	0.911896i	$0.634619\pi$
$812$	0	0
$813$	0	0
$814$	2.54622e6	0.134690
$815$	3.01552e7	1.59026
$816$	0	0
$817$	−2.20877e7	−1.15770
$818$	2.66513e6	0.139263
$819$	0	0
$820$	3.93761e6	0.204502
$821$	−1.91243e7	−0.990209	−0.495104	−	0.868834i	$-0.664870\pi$
$821$	−1.91243e7	−0.990209	−0.495104	+	0.868834i	$0.664870\pi$
$822$	0	0
$823$	2.88895e6	0.148676	0.0743381	−	0.997233i	$-0.476316\pi$
$823$	2.88895e6	0.148676	0.0743381	+	0.997233i	$0.476316\pi$
$824$	−7.27841e6	−0.373438
$825$	0	0
$826$	0	0
$827$	1.99201e7	1.01281	0.506406	−	0.862295i	$-0.330974\pi$
$827$	1.99201e7	1.01281	0.506406	+	0.862295i	$0.330974\pi$
$828$	0	0
$829$	−1.19726e7	−0.605065	−0.302532	−	0.953139i	$-0.597832\pi$
$829$	−1.19726e7	−0.605065	−0.302532	+	0.953139i	$0.597832\pi$
$830$	4.26379e7	2.14833
$831$	0	0
$832$	2.70799e6	0.135625
$833$	0	0
$834$	0	0
$835$	−1.39483e6	−0.0692316
$836$	1.65384e6	0.0818422
$837$	0	0
$838$	1.15681e7	0.569053
$839$	1.43453e7	0.703568	0.351784	−	0.936081i	$-0.385575\pi$
$839$	1.43453e7	0.703568	0.351784	+	0.936081i	$0.385575\pi$
$840$	0	0
$841$	−1.26444e7	−0.616463
$842$	−2.68811e7	−1.30667
$843$	0	0
$844$	3.23422e6	0.156284
$845$	−6.98181e6	−0.336377
$846$	0	0
$847$	0	0
$848$	−6.69292e6	−0.319614
$849$	0	0
$850$	1.48327e7	0.704162
$851$	5.09865e6	0.241341
$852$	0	0
$853$	−1.41859e7	−0.667551	−0.333776	−	0.942653i	$-0.608323\pi$
$853$	−1.41859e7	−0.667551	−0.333776	+	0.942653i	$0.608323\pi$
$854$	0	0
$855$	0	0
$856$	−7.76238e6	−0.362085
$857$	−2.41394e7	−1.12273	−0.561363	−	0.827570i	$-0.689723\pi$
$857$	−2.41394e7	−1.12273	−0.561363	+	0.827570i	$0.689723\pi$
$858$	0	0
$859$	1.37343e7	0.635071	0.317536	−	0.948246i	$-0.397145\pi$
$859$	1.37343e7	0.635071	0.317536	+	0.948246i	$0.397145\pi$
$860$	3.38956e7	1.56278
$861$	0	0
$862$	1.88919e7	0.865981
$863$	−1.74905e7	−0.799419	−0.399709	−	0.916642i	$-0.630889\pi$
$863$	−1.74905e7	−0.799419	−0.399709	+	0.916642i	$0.630889\pi$
$864$	0	0
$865$	5.45945e6	0.248090
$866$	2.31254e7	1.04784
$867$	0	0
$868$	0	0
$869$	−675534.	−0.0303458
$870$	0	0
$871$	−3.94418e7	−1.76162
$872$	3.92793e6	0.174933
$873$	0	0
$874$	3.31171e6	0.146647
$875$	0	0
$876$	0	0
$877$	−3.51221e7	−1.54199	−0.770995	−	0.636842i	$-0.780241\pi$
$877$	−3.51221e7	−1.54199	−0.770995	+	0.636842i	$0.780241\pi$
$878$	−2.14749e7	−0.940146
$879$	0	0
$880$	−2.53797e6	−0.110479
$881$	8.78528e6	0.381343	0.190672	−	0.981654i	$-0.438933\pi$
$881$	8.78528e6	0.381343	0.190672	+	0.981654i	$0.438933\pi$
$882$	0	0
$883$	1.87501e7	0.809287	0.404643	−	0.914475i	$-0.367396\pi$
$883$	1.87501e7	0.809287	0.404643	+	0.914475i	$0.367396\pi$
$884$	−4.82276e6	−0.207570
$885$	0	0
$886$	1.72605e7	0.738700
$887$	2.72434e7	1.16266	0.581330	−	0.813668i	$-0.302532\pi$
$887$	2.72434e7	1.16266	0.581330	+	0.813668i	$0.302532\pi$
$888$	0	0
$889$	0	0
$890$	−8.51892e6	−0.360504
$891$	0	0
$892$	1.11094e7	0.467498
$893$	1.57197e7	0.659654
$894$	0	0
$895$	−5.28124e7	−2.20383
$896$	0	0
$897$	0	0
$898$	1.90117e7	0.786737
$899$	1.00775e6	0.0415868
$900$	0	0
$901$	1.19197e7	0.489161
$902$	866847.	0.0354753
$903$	0	0
$904$	−1.55481e7	−0.632785
$905$	2.40987e7	0.978076
$906$	0	0
$907$	2.90420e7	1.17222	0.586110	−	0.810232i	$-0.300659\pi$
$907$	2.90420e7	1.17222	0.586110	+	0.810232i	$0.300659\pi$
$908$	−2.02716e7	−0.815968
$909$	0	0
$910$	0	0
$911$	−4.12057e7	−1.64498	−0.822491	−	0.568778i	$-0.807417\pi$
$911$	−4.12057e7	−1.64498	−0.822491	+	0.568778i	$0.807417\pi$
$912$	0	0
$913$	9.38655e6	0.372674
$914$	−8.54910e6	−0.338497
$915$	0	0
$916$	1.37036e7	0.539631
$917$	0	0
$918$	0	0
$919$	3.04768e7	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$919$	3.04768e7	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$920$	−5.08214e6	−0.197960
$921$	0	0
$922$	−6.91679e6	−0.267964
$923$	−5.88831e6	−0.227503
$924$	0	0
$925$	−5.54114e7	−2.12934
$926$	2.61747e7	1.00312
$927$	0	0
$928$	2.87210e6	0.109479
$929$	3.60640e7	1.37099	0.685495	−	0.728077i	$-0.259586\pi$
$929$	3.60640e7	1.37099	0.685495	+	0.728077i	$0.259586\pi$
$930$	0	0
$931$	0	0
$932$	4.54972e6	0.171571
$933$	0	0
$934$	−7.17676e6	−0.269191
$935$	4.51996e6	0.169085
$936$	0	0
$937$	4.80602e7	1.78828	0.894141	−	0.447785i	$-0.147787\pi$
$937$	4.80602e7	1.78828	0.894141	+	0.447785i	$0.147787\pi$
$938$	0	0
$939$	0	0
$940$	−2.41234e7	−0.890469
$941$	4.28577e7	1.57781	0.788906	−	0.614514i	$-0.210648\pi$
$941$	4.28577e7	1.57781	0.788906	+	0.614514i	$0.210648\pi$
$942$	0	0
$943$	1.73581e6	0.0635657
$944$	−1.25551e6	−0.0458555
$945$	0	0
$946$	7.46197e6	0.271098
$947$	−4.32432e7	−1.56690	−0.783452	−	0.621452i	$-0.786543\pi$
$947$	−4.32432e7	−1.56690	−0.783452	+	0.621452i	$0.786543\pi$
$948$	0	0
$949$	−6.92904e6	−0.249751
$950$	−3.59912e7	−1.29386
$951$	0	0
$952$	0	0
$953$	2.81115e7	1.00265	0.501327	−	0.865258i	$-0.332845\pi$
$953$	2.81115e7	1.00265	0.501327	+	0.865258i	$0.332845\pi$
$954$	0	0
$955$	5.33476e7	1.89281
$956$	6.84706e6	0.242303
$957$	0	0
$958$	1.63761e7	0.576497
$959$	0	0
$960$	0	0
$961$	−2.85001e7	−0.995491
$962$	1.80167e7	0.627678
$963$	0	0
$964$	1.29533e7	0.448940
$965$	−1.48890e7	−0.514691
$966$	0	0
$967$	3.70437e7	1.27394	0.636968	−	0.770890i	$-0.280188\pi$
$967$	3.70437e7	1.27394	0.636968	+	0.770890i	$0.280188\pi$
$968$	9.74854e6	0.334388
$969$	0	0
$970$	−9.89793e6	−0.337765
$971$	4.58140e7	1.55937	0.779686	−	0.626170i	$-0.215379\pi$
$971$	4.58140e7	1.55937	0.779686	+	0.626170i	$0.215379\pi$
$972$	0	0
$973$	0	0
$974$	1.78084e7	0.601489
$975$	0	0
$976$	−3.38004e6	−0.113579
$977$	−1.17497e6	−0.0393812	−0.0196906	−	0.999806i	$-0.506268\pi$
$977$	−1.17497e6	−0.0393812	−0.0196906	+	0.999806i	$0.506268\pi$
$978$	0	0
$979$	−1.87540e6	−0.0625372
$980$	0	0
$981$	0	0
$982$	2.83870e7	0.939377
$983$	2.63520e7	0.869822	0.434911	−	0.900474i	$-0.356780\pi$
$983$	2.63520e7	0.869822	0.434911	+	0.900474i	$0.356780\pi$
$984$	0	0
$985$	4.02132e7	1.32062
$986$	−5.11502e6	−0.167554
$987$	0	0
$988$	1.17023e7	0.381399
$989$	1.49422e7	0.485761
$990$	0	0
$991$	3.82424e7	1.23697	0.618487	−	0.785795i	$-0.287746\pi$
$991$	3.82424e7	1.23697	0.618487	+	0.785795i	$0.287746\pi$
$992$	367922.	0.0118707
$993$	0	0
$994$	0	0
$995$	4.89170e7	1.56640
$996$	0	0
$997$	4.36919e7	1.39208	0.696038	−	0.718005i	$-0.254945\pi$
$997$	4.36919e7	1.39208	0.696038	+	0.718005i	$0.254945\pi$
$998$	−3.76761e7	−1.19740
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.ba.1.1		2
3.2	odd	2		98.6.a.h.1.1		2
7.2	even	3		126.6.g.j.109.2		4
7.4	even	3		126.6.g.j.37.2		4
7.6	odd	2		882.6.a.bi.1.2		2
12.11	even	2		784.6.a.s.1.2		2
21.2	odd	6		14.6.c.a.11.2	yes	4
21.5	even	6		98.6.c.e.67.1		4
21.11	odd	6		14.6.c.a.9.2	✓	4
21.17	even	6		98.6.c.e.79.1		4
21.20	even	2		98.6.a.g.1.2		2
84.11	even	6		112.6.i.d.65.1		4
84.23	even	6		112.6.i.d.81.1		4
84.83	odd	2		784.6.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.a.9.2	✓	4	21.11	odd	6
14.6.c.a.11.2	yes	4	21.2	odd	6
98.6.a.g.1.2		2	21.20	even	2
98.6.a.h.1.1		2	3.2	odd	2
98.6.c.e.67.1		4	21.5	even	6
98.6.c.e.79.1		4	21.17	even	6
112.6.i.d.65.1		4	84.11	even	6
112.6.i.d.81.1		4	84.23	even	6
126.6.g.j.37.2		4	7.4	even	3
126.6.g.j.109.2		4	7.2	even	3
784.6.a.s.1.2		2	12.11	even	2
784.6.a.bb.1.1		2	84.83	odd	2
882.6.a.ba.1.1		2	1.1	even	1	trivial
882.6.a.bi.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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -106.106 q^{5} -64.0000 q^{8} +424.422 q^{10} +93.4347 q^{11} +661.131 q^{13} +256.000 q^{16} -455.919 q^{17} +1106.28 q^{19} -1697.69 q^{20} -373.739 q^{22} -748.390 q^{23} +8133.39 q^{25} -2644.52 q^{26} -2804.78 q^{29} -359.299 q^{31} -1024.00 q^{32} +1823.68 q^{34} -6812.82 q^{37} -4425.12 q^{38} +6790.76 q^{40} -2319.39 q^{41} -19965.7 q^{43} +1494.96 q^{44} +2993.56 q^{46} +14209.5 q^{47} -32533.6 q^{50} +10578.1 q^{52} -26144.2 q^{53} -9913.94 q^{55} +11219.1 q^{58} -4904.35 q^{59} -13203.3 q^{61} +1437.19 q^{62} +4096.00 q^{64} -70149.6 q^{65} -59658.1 q^{67} -7294.71 q^{68} -8906.43 q^{71} -10480.6 q^{73} +27251.3 q^{74} +17700.5 q^{76} -7230.02 q^{79} -27163.0 q^{80} +9277.57 q^{82} +100461. q^{83} +48375.6 q^{85} +79862.9 q^{86} -5979.82 q^{88} -20071.8 q^{89} -11974.2 q^{92} -56838.2 q^{94} -117382. q^{95} -23320.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 70 q^{5} - 128 q^{8} + 280 q^{10} - 62 q^{11} + 1820 q^{13} + 512 q^{16} - 1694 q^{17} + 826 q^{19} - 1120 q^{20} + 248 q^{22} + 2734 q^{23} + 6312 q^{25} - 7280 q^{26} + 2852 q^{29}+ \cdots - 8316 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 70 * q^5 - 128 * q^8 + 280 * q^10 - 62 * q^11 + 1820 * q^13 + 512 * q^16 - 1694 * q^17 + 826 * q^19 - 1120 * q^20 + 248 * q^22 + 2734 * q^23 + 6312 * q^25 - 7280 * q^26 + 2852 * q^29 - 2674 * q^31 - 2048 * q^32 + 6776 * q^34 - 9146 * q^37 - 3304 * q^38 + 4480 * q^40 - 6132 * q^41 - 16040 * q^43 - 992 * q^44 - 10936 * q^46 + 25326 * q^47 - 25248 * q^50 + 29120 * q^52 - 14958 * q^53 - 15526 * q^55 - 11408 * q^58 + 1106 * q^59 - 28042 * q^61 + 10696 * q^62 + 8192 * q^64 - 28308 * q^65 - 102642 * q^67 - 27104 * q^68 + 11056 * q^71 + 35070 * q^73 + 36584 * q^74 + 13216 * q^76 + 101762 * q^79 - 17920 * q^80 + 24528 * q^82 + 44632 * q^83 + 3674 * q^85 + 64160 * q^86 + 3968 * q^88 + 75474 * q^89 + 43744 * q^92 - 101304 * q^94 - 127502 * q^95 - 8316 * q^97

Introduction

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