LMFDB - Embedded newform 882.5.b.d.197.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(197,882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$1.16372i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.d.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +7.80683i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} +7.80683i q^{5} -22.6274i q^{8} -22.0810 q^{10} -53.7974i q^{11} -192.081 q^{13} +64.0000 q^{16} -46.1625i q^{17} -545.608 q^{19} -62.4546i q^{20} +152.162 q^{22} +260.158i q^{23} +564.053 q^{25} -543.287i q^{26} +469.920i q^{29} +3.04052 q^{31} +181.019i q^{32} +130.567 q^{34} +63.2156 q^{37} -1543.21i q^{38} +176.648 q^{40} +98.0690i q^{41} +2173.76 q^{43} +430.379i q^{44} -735.838 q^{46} -851.567i q^{47} +1595.38i q^{50} +1536.65 q^{52} -4633.53i q^{53} +419.987 q^{55} -1329.13 q^{58} +496.598i q^{59} -1983.93 q^{61} +8.59988i q^{62} -512.000 q^{64} -1499.54i q^{65} -1893.54 q^{67} +369.300i q^{68} -8581.96i q^{71} +8414.36 q^{73} +178.801i q^{74} +4364.86 q^{76} +1985.35 q^{79} +499.637i q^{80} -277.381 q^{82} +12089.9i q^{83} +360.383 q^{85} +6148.31i q^{86} -1217.30 q^{88} -3823.51i q^{89} -2081.26i q^{92} +2408.60 q^{94} -4259.46i q^{95} +7837.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} + 208 q^{10} - 472 q^{13} + 256 q^{16} + 40 q^{19} + 16 q^{22} - 1596 q^{25} - 136 q^{31} - 1552 q^{34} - 4192 q^{37} - 1664 q^{40} + 9584 q^{43} - 3536 q^{46} + 3776 q^{52} + 5384 q^{55} - 1168 q^{58} - 4528 q^{61} - 2048 q^{64} - 1944 q^{67} + 21360 q^{73} - 320 q^{76} + 10312 q^{79} - 18000 q^{82} + 29296 q^{85} - 128 q^{88} + 22080 q^{94} + 44832 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 + 208 * q^10 - 472 * q^13 + 256 * q^16 + 40 * q^19 + 16 * q^22 - 1596 * q^25 - 136 * q^31 - 1552 * q^34 - 4192 * q^37 - 1664 * q^40 + 9584 * q^43 - 3536 * q^46 + 3776 * q^52 + 5384 * q^55 - 1168 * q^58 - 4528 * q^61 - 2048 * q^64 - 1944 * q^67 + 21360 * q^73 - 320 * q^76 + 10312 * q^79 - 18000 * q^82 + 29296 * q^85 - 128 * q^88 + 22080 * q^94 + 44832 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	7.80683i	0.312273i	0.987735	+	0.156137i	$0.0499040\pi$
$5$	7.80683i	0.312273i	−0.987735	+	0.156137i	$0.950096\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−22.0810	−0.220810
$11$	− 53.7974i	− 0.444607i	−0.974978	−	0.222303i	$-0.928642\pi$
$11$	− 53.7974i	− 0.444607i	0.974978	−	0.222303i	$-0.0713575\pi$
$12$	0	0
$13$	−192.081	−1.13657	−0.568287	−	0.822830i	$-0.692394\pi$
$13$	−192.081	−1.13657	−0.568287	+	0.822830i	$0.692394\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 46.1625i	− 0.159732i	−0.996806	−	0.0798659i	$-0.974551\pi$
$17$	− 46.1625i	− 0.159732i	0.996806	−	0.0798659i	$-0.0254492\pi$
$18$	0	0
$19$	−545.608	−1.51138	−0.755689	−	0.654930i	$-0.772698\pi$
$19$	−545.608	−1.51138	−0.755689	+	0.654930i	$0.772698\pi$
$20$	− 62.4546i	− 0.156137i
$21$	0	0
$22$	152.162	0.314384
$23$	260.158i	0.491792i	0.969296	+	0.245896i	$0.0790822\pi$
$23$	260.158i	0.491792i	−0.969296	+	0.245896i	$0.920918\pi$
$24$	0	0
$25$	564.053	0.902486
$26$	− 543.287i	− 0.803679i
$27$	0	0
$28$	0	0
$29$	469.920i	0.558763i	0.960180	+	0.279382i	$0.0901295\pi$
$29$	469.920i	0.558763i	−0.960180	+	0.279382i	$0.909871\pi$
$30$	0	0
$31$	3.04052	0.00316391	0.00158196	−	0.999999i	$-0.499496\pi$
$31$	3.04052	0.00316391	0.00158196	+	0.999999i	$0.499496\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	130.567	0.112947
$35$	0	0
$36$	0	0
$37$	63.2156	0.0461764	0.0230882	−	0.999733i	$-0.492650\pi$
$37$	63.2156	0.0461764	0.0230882	+	0.999733i	$0.492650\pi$
$38$	− 1543.21i	− 1.06871i
$39$	0	0
$40$	176.648	0.110405
$41$	98.0690i	0.0583397i	0.999574	+	0.0291698i	$0.00928636\pi$
$41$	98.0690i	0.0583397i	−0.999574	+	0.0291698i	$0.990714\pi$
$42$	0	0
$43$	2173.76	1.17564	0.587820	−	0.808992i	$-0.299986\pi$
$43$	2173.76	1.17564	0.587820	+	0.808992i	$0.299986\pi$
$44$	430.379i	0.222303i
$45$	0	0
$46$	−735.838	−0.347749
$47$	− 851.567i	− 0.385499i	−0.981248	−	0.192750i	$-0.938260\pi$
$47$	− 851.567i	− 0.385499i	0.981248	−	0.192750i	$-0.0617405\pi$
$48$	0	0
$49$	0	0
$50$	1595.38i	0.638154i
$51$	0	0
$52$	1536.65	0.568287
$53$	− 4633.53i	− 1.64953i	−0.565474	−	0.824766i	$-0.691307\pi$
$53$	− 4633.53i	− 1.64953i	0.565474	−	0.824766i	$-0.308693\pi$
$54$	0	0
$55$	419.987	0.138839
$56$	0	0
$57$	0	0
$58$	−1329.13	−0.395105
$59$	496.598i	0.142659i	0.997453	+	0.0713297i	$0.0227243\pi$
$59$	496.598i	0.142659i	−0.997453	+	0.0713297i	$0.977276\pi$
$60$	0	0
$61$	−1983.93	−0.533172	−0.266586	−	0.963811i	$-0.585896\pi$
$61$	−1983.93	−0.533172	−0.266586	+	0.963811i	$0.585896\pi$
$62$	8.59988i	0.00223722i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 1499.54i	− 0.354921i
$66$	0	0
$67$	−1893.54	−0.421818	−0.210909	−	0.977506i	$-0.567642\pi$
$67$	−1893.54	−0.421818	−0.210909	+	0.977506i	$0.567642\pi$
$68$	369.300i	0.0798659i
$69$	0	0
$70$	0	0
$71$	− 8581.96i	− 1.70243i	−0.524816	−	0.851216i	$-0.675866\pi$
$71$	− 8581.96i	− 1.70243i	0.524816	−	0.851216i	$-0.324134\pi$
$72$	0	0
$73$	8414.36	1.57898	0.789488	−	0.613766i	$-0.210346\pi$
$73$	8414.36	1.57898	0.789488	+	0.613766i	$0.210346\pi$
$74$	178.801i	0.0326517i
$75$	0	0
$76$	4364.86	0.755689
$77$	0	0
$78$	0	0
$79$	1985.35	0.318114	0.159057	−	0.987269i	$-0.449155\pi$
$79$	1985.35	0.318114	0.159057	+	0.987269i	$0.449155\pi$
$80$	499.637i	0.0780683i
$81$	0	0
$82$	−277.381	−0.0412524
$83$	12089.9i	1.75496i	0.479611	+	0.877481i	$0.340778\pi$
$83$	12089.9i	1.75496i	−0.479611	+	0.877481i	$0.659222\pi$
$84$	0	0
$85$	360.383	0.0498799
$86$	6148.31i	0.831302i
$87$	0	0
$88$	−1217.30	−0.157192
$89$	− 3823.51i	− 0.482705i	−0.970438	−	0.241352i	$-0.922409\pi$
$89$	− 3823.51i	− 0.482705i	0.970438	−	0.241352i	$-0.0775910\pi$
$90$	0	0
$91$	0	0
$92$	− 2081.26i	− 0.245896i
$93$	0	0
$94$	2408.60	0.272589
$95$	− 4259.46i	− 0.471963i
$96$	0	0
$97$	7837.31	0.832959	0.416480	−	0.909145i	$-0.363264\pi$
$97$	7837.31	0.832959	0.416480	+	0.909145i	$0.363264\pi$
$98$	0	0
$99$	0	0
$100$	−4512.43	−0.451243
$101$	− 13854.1i	− 1.35811i	−0.734086	−	0.679056i	$-0.762389\pi$
$101$	− 13854.1i	− 1.35811i	0.734086	−	0.679056i	$-0.237611\pi$
$102$	0	0
$103$	4984.93	0.469878	0.234939	−	0.972010i	$-0.424511\pi$
$103$	4984.93	0.469878	0.234939	+	0.972010i	$0.424511\pi$
$104$	4346.30i	0.401840i
$105$	0	0
$106$	13105.6	1.16639
$107$	13528.8i	1.18166i	0.806797	+	0.590829i	$0.201199\pi$
$107$	13528.8i	1.18166i	−0.806797	+	0.590829i	$0.798801\pi$
$108$	0	0
$109$	−3981.45	−0.335111	−0.167555	−	0.985863i	$-0.553587\pi$
$109$	−3981.45	−0.335111	−0.167555	+	0.985863i	$0.553587\pi$
$110$	1187.90i	0.0981738i
$111$	0	0
$112$	0	0
$113$	17386.3i	1.36160i	0.732468	+	0.680801i	$0.238368\pi$
$113$	17386.3i	1.36160i	−0.732468	+	0.680801i	$0.761632\pi$
$114$	0	0
$115$	−2031.01	−0.153573
$116$	− 3759.36i	− 0.279382i
$117$	0	0
$118$	−1404.59	−0.100875
$119$	0	0
$120$	0	0
$121$	11746.8	0.802325
$122$	− 5611.41i	− 0.377009i
$123$	0	0
$124$	−24.3241	−0.00158196
$125$	9282.73i	0.594095i
$126$	0	0
$127$	18568.9	1.15128	0.575638	−	0.817704i	$-0.304754\pi$
$127$	18568.9	1.15128	0.575638	+	0.817704i	$0.304754\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	4241.35	0.250967
$131$	− 18429.5i	− 1.07392i	−0.843608	−	0.536960i	$-0.819573\pi$
$131$	− 18429.5i	− 1.07392i	0.843608	−	0.536960i	$-0.180427\pi$
$132$	0	0
$133$	0	0
$134$	− 5355.74i	− 0.298270i
$135$	0	0
$136$	−1044.54	−0.0564737
$137$	− 3482.74i	− 0.185558i	−0.995687	−	0.0927789i	$-0.970425\pi$
$137$	− 3482.74i	− 0.185558i	0.995687	−	0.0927789i	$-0.0295750\pi$
$138$	0	0
$139$	25721.9	1.33129	0.665646	−	0.746268i	$-0.268156\pi$
$139$	25721.9	1.33129	0.665646	+	0.746268i	$0.268156\pi$
$140$	0	0
$141$	0	0
$142$	24273.4	1.20380
$143$	10333.5i	0.505329i
$144$	0	0
$145$	−3668.58	−0.174487
$146$	23799.4i	1.11650i
$147$	0	0
$148$	−505.724	−0.0230882
$149$	37651.7i	1.69595i	0.530039	+	0.847973i	$0.322177\pi$
$149$	37651.7i	1.69595i	−0.530039	+	0.847973i	$0.677823\pi$
$150$	0	0
$151$	18488.4	0.810861	0.405430	−	0.914126i	$-0.367122\pi$
$151$	18488.4	0.810861	0.405430	+	0.914126i	$0.367122\pi$
$152$	12345.7i	0.534353i
$153$	0	0
$154$	0	0
$155$	23.7368i	0 0.000988004i
$156$	0	0
$157$	14465.3	0.586853	0.293426	−	0.955982i	$-0.405204\pi$
$157$	14465.3	0.586853	0.293426	+	0.955982i	$0.405204\pi$
$158$	5615.42i	0.224941i
$159$	0	0
$160$	−1413.19	−0.0552026
$161$	0	0
$162$	0	0
$163$	31749.9	1.19500	0.597498	−	0.801870i	$-0.296162\pi$
$163$	31749.9	1.19500	0.597498	+	0.801870i	$0.296162\pi$
$164$	− 784.552i	− 0.0291698i
$165$	0	0
$166$	−34195.5	−1.24095
$167$	5530.56i	0.198306i	0.995072	+	0.0991530i	$0.0316133\pi$
$167$	5530.56i	0.198306i	−0.995072	+	0.0991530i	$0.968387\pi$
$168$	0	0
$169$	8334.12	0.291801
$170$	1019.32i	0.0352704i
$171$	0	0
$172$	−17390.1	−0.587820
$173$	− 28129.6i	− 0.939878i	−0.882699	−	0.469939i	$-0.844276\pi$
$173$	− 28129.6i	− 0.939878i	0.882699	−	0.469939i	$-0.155724\pi$
$174$	0	0
$175$	0	0
$176$	− 3443.03i	− 0.111152i
$177$	0	0
$178$	10814.5	0.341324
$179$	− 40142.9i	− 1.25286i	−0.779478	−	0.626430i	$-0.784516\pi$
$179$	− 40142.9i	− 1.25286i	0.779478	−	0.626430i	$-0.215484\pi$
$180$	0	0
$181$	60742.2	1.85410	0.927051	−	0.374936i	$-0.122335\pi$
$181$	60742.2	1.85410	0.927051	+	0.374936i	$0.122335\pi$
$182$	0	0
$183$	0	0
$184$	5886.70	0.173875
$185$	493.513i	0.0144197i
$186$	0	0
$187$	−2483.42	−0.0710178
$188$	6812.54i	0.192750i
$189$	0	0
$190$	12047.6	0.333728
$191$	− 29387.9i	− 0.805567i	−0.915295	−	0.402784i	$-0.868043\pi$
$191$	− 29387.9i	− 0.805567i	0.915295	−	0.402784i	$-0.131957\pi$
$192$	0	0
$193$	−74183.3	−1.99155	−0.995775	−	0.0918227i	$-0.970731\pi$
$193$	−74183.3	−1.99155	−0.995775	+	0.0918227i	$0.970731\pi$
$194$	22167.3i	0.588991i
$195$	0	0
$196$	0	0
$197$	56876.9i	1.46556i	0.680466	+	0.732779i	$0.261777\pi$
$197$	56876.9i	1.46556i	−0.680466	+	0.732779i	$0.738223\pi$
$198$	0	0
$199$	14469.6	0.365384	0.182692	−	0.983170i	$-0.441519\pi$
$199$	14469.6	0.365384	0.182692	+	0.983170i	$0.441519\pi$
$200$	− 12763.1i	− 0.319077i
$201$	0	0
$202$	39185.3	0.960330
$203$	0	0
$204$	0	0
$205$	−765.607	−0.0182179
$206$	14099.5i	0.332254i
$207$	0	0
$208$	−12293.2	−0.284144
$209$	29352.3i	0.671969i
$210$	0	0
$211$	13204.8	0.296596	0.148298	−	0.988943i	$-0.452620\pi$
$211$	13204.8	0.296596	0.148298	+	0.988943i	$0.452620\pi$
$212$	37068.3i	0.824766i
$213$	0	0
$214$	−38265.2	−0.835559
$215$	16970.1i	0.367120i
$216$	0	0
$217$	0	0
$218$	− 11261.2i	− 0.236959i
$219$	0	0
$220$	−3359.90	−0.0694193
$221$	8866.94i	0.181547i
$222$	0	0
$223$	82650.3	1.66201	0.831007	−	0.556262i	$-0.187765\pi$
$223$	82650.3	1.66201	0.831007	+	0.556262i	$0.187765\pi$
$224$	0	0
$225$	0	0
$226$	−49175.9	−0.962798
$227$	72846.6i	1.41370i	0.707363	+	0.706851i	$0.249885\pi$
$227$	72846.6i	1.41370i	−0.707363	+	0.706851i	$0.750115\pi$
$228$	0	0
$229$	13326.1	0.254116	0.127058	−	0.991895i	$-0.459447\pi$
$229$	13326.1	0.254116	0.127058	+	0.991895i	$0.459447\pi$
$230$	− 5744.56i	− 0.108593i
$231$	0	0
$232$	10633.1	0.197553
$233$	− 53449.8i	− 0.984542i	−0.870442	−	0.492271i	$-0.836167\pi$
$233$	− 53449.8i	− 0.984542i	0.870442	−	0.492271i	$-0.163833\pi$
$234$	0	0
$235$	6648.04	0.120381
$236$	− 3972.78i	− 0.0713297i
$237$	0	0
$238$	0	0
$239$	74977.6i	1.31261i	0.754496	+	0.656305i	$0.227882\pi$
$239$	74977.6i	1.31261i	−0.754496	+	0.656305i	$0.772118\pi$
$240$	0	0
$241$	−21947.7	−0.377882	−0.188941	−	0.981988i	$-0.560505\pi$
$241$	−21947.7	−0.377882	−0.188941	+	0.981988i	$0.560505\pi$
$242$	33225.1i	0.567329i
$243$	0	0
$244$	15871.5	0.266586
$245$	0	0
$246$	0	0
$247$	104801.	1.71779
$248$	− 68.7991i	− 0.00111861i
$249$	0	0
$250$	−26255.5	−0.420089
$251$	107992.i	1.71413i	0.515205	+	0.857067i	$0.327716\pi$
$251$	107992.i	1.71413i	−0.515205	+	0.857067i	$0.672284\pi$
$252$	0	0
$253$	13995.8	0.218654
$254$	52520.9i	0.814076i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 118075.i	− 1.78769i	−0.448380	−	0.893843i	$-0.647999\pi$
$257$	− 118075.i	− 1.78769i	0.448380	−	0.893843i	$-0.352001\pi$
$258$	0	0
$259$	0	0
$260$	11996.3i	0.177461i
$261$	0	0
$262$	52126.6	0.759376
$263$	− 92558.4i	− 1.33815i	−0.743195	−	0.669075i	$-0.766691\pi$
$263$	− 92558.4i	− 1.33815i	0.743195	−	0.669075i	$-0.233309\pi$
$264$	0	0
$265$	36173.2	0.515104
$266$	0	0
$267$	0	0
$268$	15148.3	0.210909
$269$	102466.i	1.41603i	0.706196	+	0.708016i	$0.250410\pi$
$269$	102466.i	1.41603i	−0.706196	+	0.708016i	$0.749590\pi$
$270$	0	0
$271$	−93396.0	−1.27171	−0.635857	−	0.771807i	$-0.719353\pi$
$271$	−93396.0	−1.27171	−0.635857	+	0.771807i	$0.719353\pi$
$272$	− 2954.40i	− 0.0399330i
$273$	0	0
$274$	9850.66	0.131209
$275$	− 30344.6i	− 0.401251i
$276$	0	0
$277$	−17721.5	−0.230962	−0.115481	−	0.993310i	$-0.536841\pi$
$277$	−17721.5	−0.230962	−0.115481	+	0.993310i	$0.536841\pi$
$278$	72752.5i	0.941366i
$279$	0	0
$280$	0	0
$281$	− 1791.42i	− 0.0226874i	−0.999936	−	0.0113437i	$-0.996389\pi$
$281$	− 1791.42i	− 0.0226874i	0.999936	−	0.0113437i	$-0.00361090\pi$
$282$	0	0
$283$	−82690.0	−1.03248	−0.516238	−	0.856445i	$-0.672668\pi$
$283$	−82690.0	−1.03248	−0.516238	+	0.856445i	$0.672668\pi$
$284$	68655.7i	0.851216i
$285$	0	0
$286$	−29227.4	−0.357321
$287$	0	0
$288$	0	0
$289$	81390.0	0.974486
$290$	− 10376.3i	− 0.123381i
$291$	0	0
$292$	−67314.9	−0.789488
$293$	− 115695.i	− 1.34766i	−0.738886	−	0.673831i	$-0.764648\pi$
$293$	− 115695.i	− 1.34766i	0.738886	−	0.673831i	$-0.235352\pi$
$294$	0	0
$295$	−3876.85	−0.0445487
$296$	− 1430.40i	− 0.0163258i
$297$	0	0
$298$	−106495.	−1.19922
$299$	− 49971.4i	− 0.558958i
$300$	0	0
$301$	0	0
$302$	52293.2i	0.573365i
$303$	0	0
$304$	−34918.9	−0.377845
$305$	− 15488.2i	− 0.166495i
$306$	0	0
$307$	−87509.3	−0.928490	−0.464245	−	0.885707i	$-0.653674\pi$
$307$	−87509.3	−0.928490	−0.464245	+	0.885707i	$0.653674\pi$
$308$	0	0
$309$	0	0
$310$	−67.1378	−0.000698624 0
$311$	51764.9i	0.535198i	0.963530	+	0.267599i	$0.0862302\pi$
$311$	51764.9i	0.535198i	−0.963530	+	0.267599i	$0.913770\pi$
$312$	0	0
$313$	105089.	1.07268	0.536339	−	0.844002i	$-0.319807\pi$
$313$	105089.	1.07268	0.536339	+	0.844002i	$0.319807\pi$
$314$	40914.1i	0.414968i
$315$	0	0
$316$	−15882.8	−0.159057
$317$	84813.0i	0.844002i	0.906595	+	0.422001i	$0.138672\pi$
$317$	84813.0i	0.844002i	−0.906595	+	0.422001i	$0.861328\pi$
$318$	0	0
$319$	25280.5	0.248430
$320$	− 3997.09i	− 0.0390341i
$321$	0	0
$322$	0	0
$323$	25186.6i	0.241415i
$324$	0	0
$325$	−108344.	−1.02574
$326$	89802.2i	0.844990i
$327$	0	0
$328$	2219.05	0.0206262
$329$	0	0
$330$	0	0
$331$	−6749.11	−0.0616014	−0.0308007	−	0.999526i	$-0.509806\pi$
$331$	−6749.11	−0.0616014	−0.0308007	+	0.999526i	$0.509806\pi$
$332$	− 96719.5i	− 0.877481i
$333$	0	0
$334$	−15642.8	−0.140224
$335$	− 14782.5i	− 0.131722i
$336$	0	0
$337$	13160.7	0.115883	0.0579415	−	0.998320i	$-0.481546\pi$
$337$	13160.7	0.115883	0.0579415	+	0.998320i	$0.481546\pi$
$338$	23572.5i	0.206334i
$339$	0	0
$340$	−2883.06	−0.0249400
$341$	− 163.572i	− 0.00140670i
$342$	0	0
$343$	0	0
$344$	− 49186.5i	− 0.415651i
$345$	0	0
$346$	79562.6	0.664594
$347$	116200.i	0.965045i	0.875884	+	0.482522i	$0.160279\pi$
$347$	116200.i	0.965045i	−0.875884	+	0.482522i	$0.839721\pi$
$348$	0	0
$349$	115833.	0.951001	0.475501	−	0.879715i	$-0.342267\pi$
$349$	115833.	0.951001	0.475501	+	0.879715i	$0.342267\pi$
$350$	0	0
$351$	0	0
$352$	9738.37	0.0785961
$353$	− 52825.7i	− 0.423932i	−0.977277	−	0.211966i	$-0.932013\pi$
$353$	− 52825.7i	− 0.423932i	0.977277	−	0.211966i	$-0.0679866\pi$
$354$	0	0
$355$	66997.9	0.531624
$356$	30588.1i	0.241352i
$357$	0	0
$358$	113541.	0.885905
$359$	− 47769.0i	− 0.370644i	−0.982678	−	0.185322i	$-0.940667\pi$
$359$	− 47769.0i	− 0.370644i	0.982678	−	0.185322i	$-0.0593329\pi$
$360$	0	0
$361$	167367.	1.28427
$362$	171805.i	1.31105i
$363$	0	0
$364$	0	0
$365$	65689.5i	0.493072i
$366$	0	0
$367$	59119.6	0.438934	0.219467	−	0.975620i	$-0.429568\pi$
$367$	59119.6	0.438934	0.219467	+	0.975620i	$0.429568\pi$
$368$	16650.1i	0.122948i
$369$	0	0
$370$	−1395.86	−0.0101962
$371$	0	0
$372$	0	0
$373$	−28940.5	−0.208012	−0.104006	−	0.994577i	$-0.533166\pi$
$373$	−28940.5	−0.208012	−0.104006	+	0.994577i	$0.533166\pi$
$374$	− 7024.18i	− 0.0502172i
$375$	0	0
$376$	−19268.8	−0.136295
$377$	− 90262.7i	− 0.635076i
$378$	0	0
$379$	−173021.	−1.20454	−0.602270	−	0.798293i	$-0.705737\pi$
$379$	−173021.	−1.20454	−0.602270	+	0.798293i	$0.705737\pi$
$380$	34075.7i	0.235981i
$381$	0	0
$382$	83121.5	0.569622
$383$	− 63039.8i	− 0.429751i	−0.976641	−	0.214876i	$-0.931065\pi$
$383$	− 63039.8i	− 0.429751i	0.976641	−	0.214876i	$-0.0689346\pi$
$384$	0	0
$385$	0	0
$386$	− 209822.i	− 1.40824i
$387$	0	0
$388$	−62698.5	−0.416480
$389$	− 156385.i	− 1.03346i	−0.856148	−	0.516731i	$-0.827149\pi$
$389$	− 156385.i	− 1.03346i	0.856148	−	0.516731i	$-0.172851\pi$
$390$	0	0
$391$	12009.5	0.0785548
$392$	0	0
$393$	0	0
$394$	−160872.	−1.03631
$395$	15499.3i	0.0993385i
$396$	0	0
$397$	169829.	1.07753	0.538765	−	0.842456i	$-0.318891\pi$
$397$	169829.	1.07753	0.538765	+	0.842456i	$0.318891\pi$
$398$	40926.2i	0.258366i
$399$	0	0
$400$	36099.4	0.225621
$401$	− 255013.i	− 1.58589i	−0.609291	−	0.792947i	$-0.708546\pi$
$401$	− 255013.i	− 1.58589i	0.609291	−	0.792947i	$-0.291454\pi$
$402$	0	0
$403$	−584.026	−0.00359602
$404$	110833.i	0.679056i
$405$	0	0
$406$	0	0
$407$	− 3400.83i	− 0.0205304i
$408$	0	0
$409$	160053.	0.956794	0.478397	−	0.878144i	$-0.341218\pi$
$409$	160053.	0.956794	0.478397	+	0.878144i	$0.341218\pi$
$410$	− 2165.46i	− 0.0128820i
$411$	0	0
$412$	−39879.5	−0.234939
$413$	0	0
$414$	0	0
$415$	−94384.0	−0.548027
$416$	− 34770.4i	− 0.200920i
$417$	0	0
$418$	−83020.8	−0.475154
$419$	− 186.800i	− 0.00106402i	−1.00000		0.000532009i	$-0.999831\pi$
$419$	− 186.800i	− 0.00106402i	1.00000		0.000532009i	$-0.000169344\pi$
$420$	0	0
$421$	−210918.	−1.19001	−0.595003	−	0.803723i	$-0.702849\pi$
$421$	−210918.	−1.19001	−0.595003	+	0.803723i	$0.702849\pi$
$422$	37348.7i	0.209725i
$423$	0	0
$424$	−104845.	−0.583197
$425$	− 26038.1i	− 0.144156i
$426$	0	0
$427$	0	0
$428$	− 108230.i	− 0.590829i
$429$	0	0
$430$	−47998.8	−0.259593
$431$	− 73931.1i	− 0.397991i	−0.980000	−	0.198995i	$-0.936232\pi$
$431$	− 73931.1i	− 0.397991i	0.980000	−	0.198995i	$-0.0637679\pi$
$432$	0	0
$433$	−278116.	−1.48337	−0.741687	−	0.670746i	$-0.765974\pi$
$433$	−278116.	−1.48337	−0.741687	+	0.670746i	$0.765974\pi$
$434$	0	0
$435$	0	0
$436$	31851.6	0.167555
$437$	− 141944.i	− 0.743284i
$438$	0	0
$439$	330310.	1.71393	0.856964	−	0.515377i	$-0.172348\pi$
$439$	330310.	1.71393	0.856964	+	0.515377i	$0.172348\pi$
$440$	− 9503.22i	− 0.0490869i
$441$	0	0
$442$	−25079.5	−0.128373
$443$	− 20577.4i	− 0.104854i	−0.998625	−	0.0524268i	$-0.983304\pi$
$443$	− 20577.4i	− 0.104854i	0.998625	−	0.0524268i	$-0.0166956\pi$
$444$	0	0
$445$	29849.4	0.150736
$446$	233770.i	1.17522i
$447$	0	0
$448$	0	0
$449$	− 236093.i	− 1.17109i	−0.810640	−	0.585545i	$-0.800881\pi$
$449$	− 236093.i	− 1.17109i	0.810640	−	0.585545i	$-0.199119\pi$
$450$	0	0
$451$	5275.86	0.0259382
$452$	− 139090.i	− 0.680801i
$453$	0	0
$454$	−206041.	−0.999638
$455$	0	0
$456$	0	0
$457$	−36204.6	−0.173353	−0.0866765	−	0.996237i	$-0.527625\pi$
$457$	−36204.6	−0.173353	−0.0866765	+	0.996237i	$0.527625\pi$
$458$	37691.9i	0.179687i
$459$	0	0
$460$	16248.1	0.0767867
$461$	− 200930.i	− 0.945461i	−0.881207	−	0.472731i	$-0.843268\pi$
$461$	− 200930.i	− 0.945461i	0.881207	−	0.472731i	$-0.156732\pi$
$462$	0	0
$463$	−139843.	−0.652349	−0.326175	−	0.945310i	$-0.605760\pi$
$463$	−139843.	−0.652349	−0.326175	+	0.945310i	$0.605760\pi$
$464$	30074.9i	0.139691i
$465$	0	0
$466$	151179.	0.696177
$467$	− 354448.i	− 1.62525i	−0.582790	−	0.812623i	$-0.698039\pi$
$467$	− 354448.i	− 1.62525i	0.582790	−	0.812623i	$-0.301961\pi$
$468$	0	0
$469$	0	0
$470$	18803.5i	0.0851222i
$471$	0	0
$472$	11236.7	0.0504377
$473$	− 116943.i	− 0.522697i
$474$	0	0
$475$	−307752.	−1.36400
$476$	0	0
$477$	0	0
$478$	−212069.	−0.928156
$479$	− 257165.i	− 1.12083i	−0.828210	−	0.560417i	$-0.810641\pi$
$479$	− 257165.i	− 1.12083i	0.828210	−	0.560417i	$-0.189359\pi$
$480$	0	0
$481$	−12142.5	−0.0524830
$482$	− 62077.6i	− 0.267203i
$483$	0	0
$484$	−93974.7	−0.401162
$485$	61184.5i	0.260111i
$486$	0	0
$487$	379057.	1.59826	0.799128	−	0.601161i	$-0.205295\pi$
$487$	379057.	1.59826	0.799128	+	0.601161i	$0.205295\pi$
$488$	44891.3i	0.188505i
$489$	0	0
$490$	0	0
$491$	− 35227.0i	− 0.146121i	−0.997328	−	0.0730606i	$-0.976723\pi$
$491$	− 35227.0i	− 0.146121i	0.997328	−	0.0730606i	$-0.0232766\pi$
$492$	0	0
$493$	21692.7	0.0892523
$494$	296422.i	1.21466i
$495$	0	0
$496$	194.593	0.000790978 0
$497$	0	0
$498$	0	0
$499$	−272237.	−1.09332	−0.546659	−	0.837355i	$-0.684101\pi$
$499$	−272237.	−1.09332	−0.546659	+	0.837355i	$0.684101\pi$
$500$	− 74261.9i	− 0.297047i
$501$	0	0
$502$	−305448.	−1.21208
$503$	− 297487.i	− 1.17580i	−0.808935	−	0.587898i	$-0.799955\pi$
$503$	− 297487.i	− 1.17580i	0.808935	−	0.587898i	$-0.200045\pi$
$504$	0	0
$505$	108157.	0.424102
$506$	39586.2i	0.154612i
$507$	0	0
$508$	−148552.	−0.575638
$509$	− 16020.2i	− 0.0618347i	−0.999522	−	0.0309174i	$-0.990157\pi$
$509$	− 16020.2i	− 0.0618347i	0.999522	−	0.0309174i	$-0.00984287\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	333966.	1.26408
$515$	38916.5i	0.146730i
$516$	0	0
$517$	−45812.1	−0.171395
$518$	0	0
$519$	0	0
$520$	−33930.8	−0.125484
$521$	− 93176.5i	− 0.343266i	−0.985161	−	0.171633i	$-0.945096\pi$
$521$	− 93176.5i	− 0.343266i	0.985161	−	0.171633i	$-0.0549044\pi$
$522$	0	0
$523$	139430.	0.509745	0.254872	−	0.966975i	$-0.417967\pi$
$523$	139430.	0.509745	0.254872	+	0.966975i	$0.417967\pi$
$524$	147436.i	0.536960i
$525$	0	0
$526$	261795.	0.946214
$527$	− 140.358i	0 0.000505377i
$528$	0	0
$529$	212159.	0.758141
$530$	102313.i	0.364234i
$531$	0	0
$532$	0	0
$533$	− 18837.2i	− 0.0663073i
$534$	0	0
$535$	−105617.	−0.369000
$536$	42845.9i	0.149135i
$537$	0	0
$538$	−289816.	−1.00129
$539$	0	0
$540$	0	0
$541$	−74011.4	−0.252874	−0.126437	−	0.991975i	$-0.540354\pi$
$541$	−74011.4	−0.252874	−0.126437	+	0.991975i	$0.540354\pi$
$542$	− 264164.i	− 0.899238i
$543$	0	0
$544$	8356.30	0.0282369
$545$	− 31082.5i	− 0.104646i
$546$	0	0
$547$	157115.	0.525102	0.262551	−	0.964918i	$-0.415436\pi$
$547$	157115.	0.525102	0.262551	+	0.964918i	$0.415436\pi$
$548$	27861.9i	0.0927789i
$549$	0	0
$550$	85827.5	0.283727
$551$	− 256392.i	− 0.844503i
$552$	0	0
$553$	0	0
$554$	− 50123.9i	− 0.163315i
$555$	0	0
$556$	−205775.	−0.665646
$557$	326988.i	1.05395i	0.849880	+	0.526977i	$0.176675\pi$
$557$	326988.i	1.05395i	−0.849880	+	0.526977i	$0.823325\pi$
$558$	0	0
$559$	−417537.	−1.33620
$560$	0	0
$561$	0	0
$562$	5066.91	0.0160424
$563$	171824.i	0.542083i	0.962568	+	0.271042i	$0.0873681\pi$
$563$	171824.i	0.542083i	−0.962568	+	0.271042i	$0.912632\pi$
$564$	0	0
$565$	−135732.	−0.425192
$566$	− 233883.i	− 0.730071i
$567$	0	0
$568$	−194188.	−0.601901
$569$	− 519575.i	− 1.60481i	−0.596781	−	0.802404i	$-0.703554\pi$
$569$	− 519575.i	− 1.60481i	0.596781	−	0.802404i	$-0.296446\pi$
$570$	0	0
$571$	18520.0	0.0568026	0.0284013	−	0.999597i	$-0.490958\pi$
$571$	18520.0	0.0568026	0.0284013	+	0.999597i	$0.490958\pi$
$572$	− 82667.7i	− 0.252664i
$573$	0	0
$574$	0	0
$575$	146743.i	0.443835i
$576$	0	0
$577$	−635931.	−1.91011	−0.955055	−	0.296428i	$-0.904204\pi$
$577$	−635931.	−1.91011	−0.955055	+	0.296428i	$0.904204\pi$
$578$	230206.i	0.689065i
$579$	0	0
$580$	29348.7	0.0872434
$581$	0	0
$582$	0	0
$583$	−249272.	−0.733393
$584$	− 190395.i	− 0.558252i
$585$	0	0
$586$	327236.	0.952940
$587$	− 196224.i	− 0.569477i	−0.958605	−	0.284738i	$-0.908093\pi$
$587$	− 196224.i	− 0.569477i	0.958605	−	0.284738i	$-0.0919067\pi$
$588$	0	0
$589$	−1658.93	−0.00478187
$590$	− 10965.4i	− 0.0315007i
$591$	0	0
$592$	4045.80	0.0115441
$593$	600512.i	1.70770i	0.520516	+	0.853852i	$0.325740\pi$
$593$	600512.i	1.70770i	−0.520516	+	0.853852i	$0.674260\pi$
$594$	0	0
$595$	0	0
$596$	− 301214.i	− 0.847973i
$597$	0	0
$598$	141341.	0.395243
$599$	− 419120.i	− 1.16811i	−0.811713	−	0.584056i	$-0.801465\pi$
$599$	− 419120.i	− 1.16811i	0.811713	−	0.584056i	$-0.198535\pi$
$600$	0	0
$601$	−593084.	−1.64198	−0.820989	−	0.570944i	$-0.806577\pi$
$601$	−593084.	−1.64198	−0.820989	+	0.570944i	$0.806577\pi$
$602$	0	0
$603$	0	0
$604$	−147908.	−0.405430
$605$	91705.5i	0.250544i
$606$	0	0
$607$	−7739.72	−0.0210062	−0.0105031	−	0.999945i	$-0.503343\pi$
$607$	−7739.72	−0.0210062	−0.0105031	+	0.999945i	$0.503343\pi$
$608$	− 98765.6i	− 0.267177i
$609$	0	0
$610$	43807.3	0.117730
$611$	163570.i	0.438148i
$612$	0	0
$613$	−36029.1	−0.0958810	−0.0479405	−	0.998850i	$-0.515266\pi$
$613$	−36029.1	−0.0958810	−0.0479405	+	0.998850i	$0.515266\pi$
$614$	− 247514.i	− 0.656542i
$615$	0	0
$616$	0	0
$617$	− 318456.i	− 0.836524i	−0.908326	−	0.418262i	$-0.862639\pi$
$617$	− 318456.i	− 0.836524i	0.908326	−	0.418262i	$-0.137361\pi$
$618$	0	0
$619$	44669.6	0.116582	0.0582909	−	0.998300i	$-0.481435\pi$
$619$	44669.6	0.116582	0.0582909	+	0.998300i	$0.481435\pi$
$620$	− 189.894i	0 0.000494002i
$621$	0	0
$622$	−146413.	−0.378442
$623$	0	0
$624$	0	0
$625$	280065.	0.716966
$626$	297237.i	0.758498i
$627$	0	0
$628$	−115723.	−0.293426
$629$	− 2918.19i	− 0.00737585i
$630$	0	0
$631$	−493904.	−1.24046	−0.620232	−	0.784419i	$-0.712961\pi$
$631$	−493904.	−1.24046	−0.620232	+	0.784419i	$0.712961\pi$
$632$	− 44923.4i	− 0.112470i
$633$	0	0
$634$	−239887.	−0.596800
$635$	144964.i	0.359513i
$636$	0	0
$637$	0	0
$638$	71504.0i	0.175667i
$639$	0	0
$640$	11305.5	0.0276013
$641$	− 62345.1i	− 0.151735i	−0.997118	−	0.0758676i	$-0.975827\pi$
$641$	− 62345.1i	− 0.151735i	0.997118	−	0.0758676i	$-0.0241726\pi$
$642$	0	0
$643$	560242.	1.35505	0.677523	−	0.735502i	$-0.263054\pi$
$643$	560242.	1.35505	0.677523	+	0.735502i	$0.263054\pi$
$644$	0	0
$645$	0	0
$646$	−71238.5	−0.170706
$647$	− 72598.9i	− 0.173429i	−0.996233	−	0.0867144i	$-0.972363\pi$
$647$	− 72598.9i	− 0.173429i	0.996233	−	0.0867144i	$-0.0276368\pi$
$648$	0	0
$649$	26715.7	0.0634274
$650$	− 306443.i	− 0.725309i
$651$	0	0
$652$	−253999.	−0.597498
$653$	− 601820.i	− 1.41137i	−0.708526	−	0.705684i	$-0.750640\pi$
$653$	− 601820.i	− 1.41137i	0.708526	−	0.705684i	$-0.249360\pi$
$654$	0	0
$655$	143876.	0.335356
$656$	6276.41i	0.0145849i
$657$	0	0
$658$	0	0
$659$	− 70037.1i	− 0.161271i	−0.996744	−	0.0806357i	$-0.974305\pi$
$659$	− 70037.1i	− 0.161271i	0.996744	−	0.0806357i	$-0.0256950\pi$
$660$	0	0
$661$	449536.	1.02887	0.514436	−	0.857529i	$-0.328001\pi$
$661$	449536.	1.02887	0.514436	+	0.857529i	$0.328001\pi$
$662$	− 19089.4i	− 0.0435587i
$663$	0	0
$664$	273564.	0.620473
$665$	0	0
$666$	0	0
$667$	−122253.	−0.274795
$668$	− 44244.4i	− 0.0991530i
$669$	0	0
$670$	41811.3	0.0931417
$671$	106730.i	0.237052i
$672$	0	0
$673$	−300629.	−0.663744	−0.331872	−	0.943324i	$-0.607680\pi$
$673$	−300629.	−0.663744	−0.331872	+	0.943324i	$0.607680\pi$
$674$	37224.1i	0.0819416i
$675$	0	0
$676$	−66673.0	−0.145900
$677$	− 505612.i	− 1.10316i	−0.834121	−	0.551581i	$-0.814025\pi$
$677$	− 505612.i	− 1.10316i	0.834121	−	0.551581i	$-0.185975\pi$
$678$	0	0
$679$	0	0
$680$	− 8154.53i	− 0.0176352i
$681$	0	0
$682$	462.652	0.000994684 0
$683$	474026.i	1.01616i	0.861311	+	0.508078i	$0.169644\pi$
$683$	474026.i	1.01616i	−0.861311	+	0.508078i	$0.830356\pi$
$684$	0	0
$685$	27189.1	0.0579447
$686$	0	0
$687$	0	0
$688$	139120.	0.293910
$689$	890014.i	1.87482i
$690$	0	0
$691$	612925.	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$691$	612925.	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$692$	225037.i	0.469939i
$693$	0	0
$694$	−328663.	−0.682390
$695$	200806.i	0.415727i
$696$	0	0
$697$	4527.11	0.00931870
$698$	327625.i	0.672459i
$699$	0	0
$700$	0	0
$701$	114932.i	0.233886i	0.993139	+	0.116943i	$0.0373095\pi$
$701$	114932.i	0.233886i	−0.993139	+	0.116943i	$0.962690\pi$
$702$	0	0
$703$	−34490.9	−0.0697901
$704$	27544.3i	0.0555758i
$705$	0	0
$706$	149414.	0.299765
$707$	0	0
$708$	0	0
$709$	−157699.	−0.313716	−0.156858	−	0.987621i	$-0.550137\pi$
$709$	−157699.	−0.313716	−0.156858	+	0.987621i	$0.550137\pi$
$710$	189499.i	0.375915i
$711$	0	0
$712$	−86516.1	−0.170662
$713$	791.015i	0.00155599i
$714$	0	0
$715$	−80671.5	−0.157800
$716$	321143.i	0.626430i
$717$	0	0
$718$	135111.	0.262085
$719$	748032.i	1.44698i	0.690335	+	0.723490i	$0.257463\pi$
$719$	748032.i	1.44698i	−0.690335	+	0.723490i	$0.742537\pi$
$720$	0	0
$721$	0	0
$722$	473385.i	0.908113i
$723$	0	0
$724$	−485938.	−0.927051
$725$	265060.i	0.504276i
$726$	0	0
$727$	−129454.	−0.244933	−0.122466	−	0.992473i	$-0.539080\pi$
$727$	−129454.	−0.244933	−0.122466	+	0.992473i	$0.539080\pi$
$728$	0	0
$729$	0	0
$730$	−185798.	−0.348654
$731$	− 100346.i	− 0.187787i
$732$	0	0
$733$	28743.7	0.0534977	0.0267489	−	0.999642i	$-0.491485\pi$
$733$	28743.7	0.0534977	0.0267489	+	0.999642i	$0.491485\pi$
$734$	167215.i	0.310373i
$735$	0	0
$736$	−47093.6	−0.0869374
$737$	101868.i	0.187543i
$738$	0	0
$739$	−180676.	−0.330835	−0.165418	−	0.986224i	$-0.552897\pi$
$739$	−180676.	−0.330835	−0.165418	+	0.986224i	$0.552897\pi$
$740$	− 3948.10i	− 0.00720983i
$741$	0	0
$742$	0	0
$743$	283695.i	0.513895i	0.966425	+	0.256948i	$0.0827168\pi$
$743$	283695.i	0.513895i	−0.966425	+	0.256948i	$0.917283\pi$
$744$	0	0
$745$	−293940.	−0.529598
$746$	− 81856.0i	− 0.147086i
$747$	0	0
$748$	19867.4	0.0355089
$749$	0	0
$750$	0	0
$751$	423303.	0.750536	0.375268	−	0.926916i	$-0.377551\pi$
$751$	423303.	0.750536	0.375268	+	0.926916i	$0.377551\pi$
$752$	− 54500.3i	− 0.0963748i
$753$	0	0
$754$	255302.	0.449067
$755$	144336.i	0.253210i
$756$	0	0
$757$	574735.	1.00294	0.501471	−	0.865175i	$-0.332792\pi$
$757$	574735.	1.00294	0.501471	+	0.865175i	$0.332792\pi$
$758$	− 489378.i	− 0.851738i
$759$	0	0
$760$	−96380.7	−0.166864
$761$	443249.i	0.765382i	0.923876	+	0.382691i	$0.125003\pi$
$761$	443249.i	0.765382i	−0.923876	+	0.382691i	$0.874997\pi$
$762$	0	0
$763$	0	0
$764$	235103.i	0.402784i
$765$	0	0
$766$	178303.	0.303880
$767$	− 95387.0i	− 0.162143i
$768$	0	0
$769$	1.11324e6	1.88251	0.941254	−	0.337699i	$-0.109649\pi$
$769$	1.11324e6	1.88251	0.941254	+	0.337699i	$0.109649\pi$
$770$	0	0
$771$	0	0
$772$	593466.	0.995775
$773$	− 596924.i	− 0.998988i	−0.866317	−	0.499494i	$-0.833519\pi$
$773$	− 596924.i	− 0.998988i	0.866317	−	0.499494i	$-0.166481\pi$
$774$	0	0
$775$	1715.01	0.00285538
$776$	− 177338.i	− 0.294496i
$777$	0	0
$778$	442322.	0.730768
$779$	− 53507.2i	− 0.0881733i
$780$	0	0
$781$	−461687.	−0.756913
$782$	33968.1i	0.0555467i
$783$	0	0
$784$	0	0
$785$	112928.i	0.183258i
$786$	0	0
$787$	−635636.	−1.02626	−0.513132	−	0.858310i	$-0.671515\pi$
$787$	−635636.	−1.02626	−0.513132	+	0.858310i	$0.671515\pi$
$788$	− 455015.i	− 0.732779i
$789$	0	0
$790$	−43838.6	−0.0702429
$791$	0	0
$792$	0	0
$793$	381076.	0.605989
$794$	480348.i	0.761929i
$795$	0	0
$796$	−115757.	−0.182692
$797$	− 418525.i	− 0.658878i	−0.944177	−	0.329439i	$-0.893140\pi$
$797$	− 418525.i	− 0.658878i	0.944177	−	0.329439i	$-0.106860\pi$
$798$	0	0
$799$	−39310.5	−0.0615765
$800$	102105.i	0.159538i
$801$	0	0
$802$	721286.	1.12140
$803$	− 452671.i	− 0.702023i
$804$	0	0
$805$	0	0
$806$	− 1651.87i	− 0.00254277i
$807$	0	0
$808$	−313483.	−0.480165
$809$	302698.i	0.462502i	0.972894	+	0.231251i	$0.0742818\pi$
$809$	302698.i	0.462502i	−0.972894	+	0.231251i	$0.925718\pi$
$810$	0	0
$811$	−684158.	−1.04019	−0.520097	−	0.854107i	$-0.674104\pi$
$811$	−684158.	−1.04019	−0.520097	+	0.854107i	$0.674104\pi$
$812$	0	0
$813$	0	0
$814$	9619.01	0.0145172
$815$	247866.i	0.373165i
$816$	0	0
$817$	−1.18602e6	−1.77684
$818$	452699.i	0.676555i
$819$	0	0
$820$	6124.86	0.00910895
$821$	325422.i	0.482792i	0.970427	+	0.241396i	$0.0776052\pi$
$821$	325422.i	0.482792i	−0.970427	+	0.241396i	$0.922395\pi$
$822$	0	0
$823$	−572245.	−0.844855	−0.422427	−	0.906397i	$-0.638822\pi$
$823$	−572245.	−0.844855	−0.422427	+	0.906397i	$0.638822\pi$
$824$	− 112796.i	− 0.166127i
$825$	0	0
$826$	0	0
$827$	515061.i	0.753091i	0.926398	+	0.376546i	$0.122888\pi$
$827$	515061.i	0.753091i	−0.926398	+	0.376546i	$0.877112\pi$
$828$	0	0
$829$	426059.	0.619956	0.309978	−	0.950744i	$-0.399678\pi$
$829$	426059.	0.619956	0.309978	+	0.950744i	$0.399678\pi$
$830$	− 266958.i	− 0.387514i
$831$	0	0
$832$	98345.5	0.142072
$833$	0	0
$834$	0	0
$835$	−43176.1	−0.0619256
$836$	− 234818.i	− 0.335985i
$837$	0	0
$838$	528.350	0.000752374 0
$839$	796763.i	1.13189i	0.824442	+	0.565946i	$0.191489\pi$
$839$	796763.i	1.13189i	−0.824442	+	0.565946i	$0.808511\pi$
$840$	0	0
$841$	486456.	0.687783
$842$	− 596566.i	− 0.841462i
$843$	0	0
$844$	−105638.	−0.148298
$845$	65063.1i	0.0911215i
$846$	0	0
$847$	0	0
$848$	− 296546.i	− 0.412383i
$849$	0	0
$850$	73646.9	0.101933
$851$	16446.0i	0.0227092i
$852$	0	0
$853$	1.18712e6	1.63154	0.815769	−	0.578377i	$-0.196314\pi$
$853$	1.18712e6	1.63154	0.815769	+	0.578377i	$0.196314\pi$
$854$	0	0
$855$	0	0
$856$	306122.	0.417779
$857$	396944.i	0.540465i	0.962795	+	0.270232i	$0.0871006\pi$
$857$	396944.i	0.540465i	−0.962795	+	0.270232i	$0.912899\pi$
$858$	0	0
$859$	−1.25257e6	−1.69753	−0.848765	−	0.528771i	$-0.822653\pi$
$859$	−1.25257e6	−1.69753	−0.848765	+	0.528771i	$0.822653\pi$
$860$	− 135761.i	− 0.183560i
$861$	0	0
$862$	209109.	0.281422
$863$	1.39922e6i	1.87873i	0.342922	+	0.939364i	$0.388583\pi$
$863$	1.39922e6i	1.87873i	−0.342922	+	0.939364i	$0.611417\pi$
$864$	0	0
$865$	219603.	0.293499
$866$	− 786632.i	− 1.04890i
$867$	0	0
$868$	0	0
$869$	− 106807.i	− 0.141436i
$870$	0	0
$871$	363713.	0.479427
$872$	90090.0i	0.118480i
$873$	0	0
$874$	401479.	0.525581
$875$	0	0
$876$	0	0
$877$	1.22832e6	1.59703	0.798513	−	0.601977i	$-0.205620\pi$
$877$	1.22832e6	1.59703	0.798513	+	0.601977i	$0.205620\pi$
$878$	934257.i	1.21193i
$879$	0	0
$880$	26879.2	0.0347097
$881$	− 547453.i	− 0.705334i	−0.935749	−	0.352667i	$-0.885275\pi$
$881$	− 547453.i	− 0.705334i	0.935749	−	0.352667i	$-0.114725\pi$
$882$	0	0
$883$	−441760.	−0.566585	−0.283292	−	0.959034i	$-0.591427\pi$
$883$	−441760.	−0.566585	−0.283292	+	0.959034i	$0.591427\pi$
$884$	− 70935.5i	− 0.0907735i
$885$	0	0
$886$	58201.7	0.0741426
$887$	− 969187.i	− 1.23186i	−0.787802	−	0.615928i	$-0.788781\pi$
$887$	− 969187.i	− 1.23186i	0.787802	−	0.615928i	$-0.211219\pi$
$888$	0	0
$889$	0	0
$890$	84427.0i	0.106586i
$891$	0	0
$892$	−661203.	−0.831007
$893$	464622.i	0.582635i
$894$	0	0
$895$	313388.	0.391234
$896$	0	0
$897$	0	0
$898$	667771.	0.828085
$899$	1428.80i	0.00176788i
$900$	0	0
$901$	−213896.	−0.263483
$902$	14922.4i	0.0183411i
$903$	0	0
$904$	393407.	0.481399
$905$	474204.i	0.578986i
$906$	0	0
$907$	−446699.	−0.543000	−0.271500	−	0.962438i	$-0.587520\pi$
$907$	−446699.	−0.543000	−0.271500	+	0.962438i	$0.587520\pi$
$908$	− 582773.i	− 0.706851i
$909$	0	0
$910$	0	0
$911$	1.07232e6i	1.29208i	0.763305	+	0.646038i	$0.223575\pi$
$911$	1.07232e6i	1.29208i	−0.763305	+	0.646038i	$0.776425\pi$
$912$	0	0
$913$	650407.	0.780268
$914$	− 102402.i	− 0.122579i
$915$	0	0
$916$	−106609.	−0.127058
$917$	0	0
$918$	0	0
$919$	−521777.	−0.617809	−0.308904	−	0.951093i	$-0.599962\pi$
$919$	−521777.	−0.617809	−0.308904	+	0.951093i	$0.599962\pi$
$920$	45956.5i	0.0542964i
$921$	0	0
$922$	568317.	0.668542
$923$	1.64843e6i	1.93494i
$924$	0	0
$925$	35657.0	0.0416736
$926$	− 395537.i	− 0.461281i
$927$	0	0
$928$	−85064.6	−0.0987763
$929$	718452.i	0.832466i	0.909258	+	0.416233i	$0.136650\pi$
$929$	718452.i	0.832466i	−0.909258	+	0.416233i	$0.863350\pi$
$930$	0	0
$931$	0	0
$932$	427599.i	0.492271i
$933$	0	0
$934$	1.00253e6	1.14922
$935$	− 19387.7i	− 0.0221770i
$936$	0	0
$937$	799725.	0.910881	0.455440	−	0.890266i	$-0.349482\pi$
$937$	799725.	0.910881	0.455440	+	0.890266i	$0.349482\pi$
$938$	0	0
$939$	0	0
$940$	−53184.3	−0.0601905
$941$	− 1.51870e6i	− 1.71511i	−0.514393	−	0.857555i	$-0.671983\pi$
$941$	− 1.51870e6i	− 1.71511i	0.514393	−	0.857555i	$-0.328017\pi$
$942$	0	0
$943$	−25513.4	−0.0286910
$944$	31782.2i	0.0356649i
$945$	0	0
$946$	330763.	0.369603
$947$	147181.i	0.164116i	0.996628	+	0.0820582i	$0.0261493\pi$
$947$	147181.i	0.164116i	−0.996628	+	0.0820582i	$0.973851\pi$
$948$	0	0
$949$	−1.61624e6	−1.79462
$950$	− 870454.i	− 0.964492i
$951$	0	0
$952$	0	0
$953$	− 1.55290e6i	− 1.70985i	−0.518754	−	0.854924i	$-0.673604\pi$
$953$	− 1.55290e6i	− 1.70985i	0.518754	−	0.854924i	$-0.326396\pi$
$954$	0	0
$955$	229426.	0.251557
$956$	− 599821.i	− 0.656305i
$957$	0	0
$958$	727374.	0.792550
$959$	0	0
$960$	0	0
$961$	−923512.	−0.999990
$962$	− 34344.2i	− 0.0371111i
$963$	0	0
$964$	175582.	0.188941
$965$	− 579136.i	− 0.621908i
$966$	0	0
$967$	1.82071e6	1.94710	0.973548	−	0.228482i	$-0.0733762\pi$
$967$	1.82071e6	1.94710	0.973548	+	0.228482i	$0.0733762\pi$
$968$	− 265801.i	− 0.283665i
$969$	0	0
$970$	−173056.	−0.183926
$971$	1.29084e6i	1.36909i	0.728970	+	0.684546i	$0.240000\pi$
$971$	1.29084e6i	1.36909i	−0.728970	+	0.684546i	$0.760000\pi$
$972$	0	0
$973$	0	0
$974$	1.07213e6i	1.13014i
$975$	0	0
$976$	−126972.	−0.133293
$977$	− 902246.i	− 0.945226i	−0.881270	−	0.472613i	$-0.843311\pi$
$977$	− 902246.i	− 0.945226i	0.881270	−	0.472613i	$-0.156689\pi$
$978$	0	0
$979$	−205695.	−0.214614
$980$	0	0
$981$	0	0
$982$	99637.1	0.103323
$983$	1.15645e6i	1.19679i	0.801200	+	0.598397i	$0.204195\pi$
$983$	1.15645e6i	1.19679i	−0.801200	+	0.598397i	$0.795805\pi$
$984$	0	0
$985$	−444028.	−0.457654
$986$	61356.2i	0.0631109i
$987$	0	0
$988$	−838407.	−0.858897
$989$	565520.i	0.578170i
$990$	0	0
$991$	−934696.	−0.951750	−0.475875	−	0.879513i	$-0.657869\pi$
$991$	−934696.	−0.951750	−0.475875	+	0.879513i	$0.657869\pi$
$992$	550.393i	0 0.000559306i
$993$	0	0
$994$	0	0
$995$	112961.i	0.114100i
$996$	0	0
$997$	961897.	0.967694	0.483847	−	0.875153i	$-0.339239\pi$
$997$	961897.	0.967694	0.483847	+	0.875153i	$0.339239\pi$
$998$	− 770003.i	− 0.773093i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.d.197.4		4
3.2	odd	2	inner	882.5.b.d.197.1		4
7.6	odd	2		126.5.b.a.71.3	yes	4
21.20	even	2		126.5.b.a.71.2	✓	4
28.27	even	2		1008.5.d.b.449.2		4
84.83	odd	2		1008.5.d.b.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.b.a.71.2	✓	4	21.20	even	2
126.5.b.a.71.3	yes	4	7.6	odd	2
882.5.b.d.197.1		4	3.2	odd	2	inner
882.5.b.d.197.4		4	1.1	even	1	trivial
1008.5.d.b.449.2		4	28.27	even	2
1008.5.d.b.449.3		4	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +7.80683i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +7.80683i q^{5} -22.6274i q^{8} -22.0810 q^{10} -53.7974i q^{11} -192.081 q^{13} +64.0000 q^{16} -46.1625i q^{17} -545.608 q^{19} -62.4546i q^{20} +152.162 q^{22} +260.158i q^{23} +564.053 q^{25} -543.287i q^{26} +469.920i q^{29} +3.04052 q^{31} +181.019i q^{32} +130.567 q^{34} +63.2156 q^{37} -1543.21i q^{38} +176.648 q^{40} +98.0690i q^{41} +2173.76 q^{43} +430.379i q^{44} -735.838 q^{46} -851.567i q^{47} +1595.38i q^{50} +1536.65 q^{52} -4633.53i q^{53} +419.987 q^{55} -1329.13 q^{58} +496.598i q^{59} -1983.93 q^{61} +8.59988i q^{62} -512.000 q^{64} -1499.54i q^{65} -1893.54 q^{67} +369.300i q^{68} -8581.96i q^{71} +8414.36 q^{73} +178.801i q^{74} +4364.86 q^{76} +1985.35 q^{79} +499.637i q^{80} -277.381 q^{82} +12089.9i q^{83} +360.383 q^{85} +6148.31i q^{86} -1217.30 q^{88} -3823.51i q^{89} -2081.26i q^{92} +2408.60 q^{94} -4259.46i q^{95} +7837.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 208 q^{10} - 472 q^{13} + 256 q^{16} + 40 q^{19} + 16 q^{22} - 1596 q^{25} - 136 q^{31} - 1552 q^{34} - 4192 q^{37} - 1664 q^{40} + 9584 q^{43} - 3536 q^{46} + 3776 q^{52} + 5384 q^{55} - 1168 q^{58} - 4528 q^{61} - 2048 q^{64} - 1944 q^{67} + 21360 q^{73} - 320 q^{76} + 10312 q^{79} - 18000 q^{82} + 29296 q^{85} - 128 q^{88} + 22080 q^{94} + 44832 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 + 208 * q^10 - 472 * q^13 + 256 * q^16 + 40 * q^19 + 16 * q^22 - 1596 * q^25 - 136 * q^31 - 1552 * q^34 - 4192 * q^37 - 1664 * q^40 + 9584 * q^43 - 3536 * q^46 + 3776 * q^52 + 5384 * q^55 - 1168 * q^58 - 4528 * q^61 - 2048 * q^64 - 1944 * q^67 + 21360 * q^73 - 320 * q^76 + 10312 * q^79 - 18000 * q^82 + 29296 * q^85 - 128 * q^88 + 22080 * q^94 + 44832 * q^97

Introduction

L-functions

Modular forms

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