LMFDB - Embedded newform 882.5.b.a.197.3

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} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.3
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.a.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +3.32941i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} +3.32941i q^{5} -22.6274i q^{8} -9.41699 q^{10} -114.133i q^{11} -74.9150 q^{13} +64.0000 q^{16} +339.432i q^{17} -83.3647 q^{19} -26.6353i q^{20} +322.818 q^{22} -685.465i q^{23} +613.915 q^{25} -211.892i q^{26} -120.260i q^{29} -545.017 q^{31} +181.019i q^{32} -960.057 q^{34} +434.729 q^{37} -235.791i q^{38} +75.3360 q^{40} +700.457i q^{41} -941.725 q^{43} +913.068i q^{44} +1938.79 q^{46} +846.864i q^{47} +1736.41i q^{50} +599.320 q^{52} +221.322i q^{53} +379.997 q^{55} +340.146 q^{58} +5606.68i q^{59} +6531.85 q^{61} -1541.54i q^{62} -512.000 q^{64} -249.423i q^{65} -216.279 q^{67} -2715.45i q^{68} +4438.92i q^{71} +6553.39 q^{73} +1229.60i q^{74} +666.917 q^{76} -11485.1 q^{79} +213.082i q^{80} -1981.19 q^{82} +2463.13i q^{83} -1130.11 q^{85} -2663.60i q^{86} -2582.55 q^{88} -4588.39i q^{89} +5483.72i q^{92} -2395.29 q^{94} -277.555i q^{95} -15647.1 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} - 80 q^{10} - 88 q^{13} + 256 q^{16} + 1000 q^{19} - 656 q^{22} + 2244 q^{25} + 1016 q^{31} - 496 q^{34} - 928 q^{37} + 640 q^{40} - 592 q^{43} + 1744 q^{46} + 704 q^{52} - 4216 q^{55} - 1264 q^{58} + 12560 q^{61} - 2048 q^{64} + 16872 q^{67} + 6000 q^{73} - 8000 q^{76} - 15800 q^{79} - 432 q^{82} + 3184 q^{85} + 5248 q^{88} + 17088 q^{94} - 52704 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 - 80 * q^10 - 88 * q^13 + 256 * q^16 + 1000 * q^19 - 656 * q^22 + 2244 * q^25 + 1016 * q^31 - 496 * q^34 - 928 * q^37 + 640 * q^40 - 592 * q^43 + 1744 * q^46 + 704 * q^52 - 4216 * q^55 - 1264 * q^58 + 12560 * q^61 - 2048 * q^64 + 16872 * q^67 + 6000 * q^73 - 8000 * q^76 - 15800 * q^79 - 432 * q^82 + 3184 * q^85 + 5248 * q^88 + 17088 * q^94 - 52704 * 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
$5$	3.32941i	0.133176i	0.997781	+	0.0665882i	$0.0212114\pi$
$5$	3.32941i	0.133176i	−0.997781	+	0.0665882i	$0.978789\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−9.41699	−0.0941699
$11$	− 114.133i	− 0.943252i	−0.881799	−	0.471626i	$-0.843667\pi$
$11$	− 114.133i	− 0.943252i	0.881799	−	0.471626i	$-0.156333\pi$
$12$	0	0
$13$	−74.9150	−0.443284	−0.221642	−	0.975128i	$-0.571142\pi$
$13$	−74.9150	−0.443284	−0.221642	+	0.975128i	$0.571142\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	339.432i	1.17450i	0.809404	+	0.587252i	$0.199790\pi$
$17$	339.432i	1.17450i	−0.809404	+	0.587252i	$0.800210\pi$
$18$	0	0
$19$	−83.3647	−0.230927	−0.115464	−	0.993312i	$-0.536835\pi$
$19$	−83.3647	−0.230927	−0.115464	+	0.993312i	$0.536835\pi$
$20$	− 26.6353i	− 0.0665882i
$21$	0	0
$22$	322.818	0.666980
$23$	− 685.465i	− 1.29577i	−0.761736	−	0.647887i	$-0.775653\pi$
$23$	− 685.465i	− 1.29577i	0.761736	−	0.647887i	$-0.224347\pi$
$24$	0	0
$25$	613.915	0.982264
$26$	− 211.892i	− 0.313449i
$27$	0	0
$28$	0	0
$29$	− 120.260i	− 0.142996i	−0.997441	−	0.0714981i	$-0.977222\pi$
$29$	− 120.260i	− 0.142996i	0.997441	−	0.0714981i	$-0.0227780\pi$
$30$	0	0
$31$	−545.017	−0.567135	−0.283568	−	0.958952i	$-0.591518\pi$
$31$	−545.017	−0.567135	−0.283568	+	0.958952i	$0.591518\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	−960.057	−0.830499
$35$	0	0
$36$	0	0
$37$	434.729	0.317552	0.158776	−	0.987315i	$-0.449245\pi$
$37$	434.729	0.317552	0.158776	+	0.987315i	$0.449245\pi$
$38$	− 235.791i	− 0.163290i
$39$	0	0
$40$	75.3360	0.0470850
$41$	700.457i	0.416691i	0.978055	+	0.208345i	$0.0668078\pi$
$41$	700.457i	0.416691i	−0.978055	+	0.208345i	$0.933192\pi$
$42$	0	0
$43$	−941.725	−0.509316	−0.254658	−	0.967031i	$-0.581963\pi$
$43$	−941.725	−0.509316	−0.254658	+	0.967031i	$0.581963\pi$
$44$	913.068i	0.471626i
$45$	0	0
$46$	1938.79	0.916251
$47$	846.864i	0.383370i	0.981457	+	0.191685i	$0.0613952\pi$
$47$	846.864i	0.383370i	−0.981457	+	0.191685i	$0.938605\pi$
$48$	0	0
$49$	0	0
$50$	1736.41i	0.694566i
$51$	0	0
$52$	599.320	0.221642
$53$	221.322i	0.0787902i	0.999224	+	0.0393951i	$0.0125431\pi$
$53$	221.322i	0.0787902i	−0.999224	+	0.0393951i	$0.987457\pi$
$54$	0	0
$55$	379.997	0.125619
$56$	0	0
$57$	0	0
$58$	340.146	0.101114
$59$	5606.68i	1.61065i	0.592832	+	0.805326i	$0.298010\pi$
$59$	5606.68i	1.61065i	−0.592832	+	0.805326i	$0.701990\pi$
$60$	0	0
$61$	6531.85	1.75540	0.877701	−	0.479208i	$-0.159076\pi$
$61$	6531.85	1.75540	0.877701	+	0.479208i	$0.159076\pi$
$62$	− 1541.54i	− 0.401025i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 249.423i	− 0.0590350i
$66$	0	0
$67$	−216.279	−0.0481798	−0.0240899	−	0.999710i	$-0.507669\pi$
$67$	−216.279	−0.0481798	−0.0240899	+	0.999710i	$0.507669\pi$
$68$	− 2715.45i	− 0.587252i
$69$	0	0
$70$	0	0
$71$	4438.92i	0.880564i	0.897859	+	0.440282i	$0.145122\pi$
$71$	4438.92i	0.880564i	−0.897859	+	0.440282i	$0.854878\pi$
$72$	0	0
$73$	6553.39	1.22976	0.614879	−	0.788621i	$-0.289205\pi$
$73$	6553.39	1.22976	0.614879	+	0.788621i	$0.289205\pi$
$74$	1229.60i	0.224544i
$75$	0	0
$76$	666.917	0.115464
$77$	0	0
$78$	0	0
$79$	−11485.1	−1.84027	−0.920133	−	0.391606i	$-0.871920\pi$
$79$	−11485.1	−1.84027	−0.920133	+	0.391606i	$0.871920\pi$
$80$	213.082i	0.0332941i
$81$	0	0
$82$	−1981.19	−0.294645
$83$	2463.13i	0.357546i	0.983890	+	0.178773i	$0.0572127\pi$
$83$	2463.13i	0.357546i	−0.983890	+	0.178773i	$0.942787\pi$
$84$	0	0
$85$	−1130.11	−0.156416
$86$	− 2663.60i	− 0.360141i
$87$	0	0
$88$	−2582.55	−0.333490
$89$	− 4588.39i	− 0.579269i	−0.957137	−	0.289635i	$-0.906466\pi$
$89$	− 4588.39i	− 0.579269i	0.957137	−	0.289635i	$-0.0935338\pi$
$90$	0	0
$91$	0	0
$92$	5483.72i	0.647887i
$93$	0	0
$94$	−2395.29	−0.271083
$95$	− 277.555i	− 0.0307540i
$96$	0	0
$97$	−15647.1	−1.66300	−0.831498	−	0.555528i	$-0.812516\pi$
$97$	−15647.1	−1.66300	−0.831498	+	0.555528i	$0.812516\pi$
$98$	0	0
$99$	0	0
$100$	−4911.32	−0.491132
$101$	2662.36i	0.260990i	0.991449	+	0.130495i	$0.0416567\pi$
$101$	2662.36i	0.260990i	−0.991449	+	0.130495i	$0.958343\pi$
$102$	0	0
$103$	483.350	0.0455604	0.0227802	−	0.999740i	$-0.492748\pi$
$103$	483.350	0.0455604	0.0227802	+	0.999740i	$0.492748\pi$
$104$	1695.13i	0.156725i
$105$	0	0
$106$	−625.992	−0.0557131
$107$	5501.70i	0.480539i	0.970706	+	0.240270i	$0.0772359\pi$
$107$	5501.70i	0.480539i	−0.970706	+	0.240270i	$0.922764\pi$
$108$	0	0
$109$	18825.4	1.58450	0.792248	−	0.610199i	$-0.208910\pi$
$109$	18825.4	1.58450	0.792248	+	0.610199i	$0.208910\pi$
$110$	1074.79i	0.0888260i
$111$	0	0
$112$	0	0
$113$	8721.21i	0.682999i	0.939882	+	0.341500i	$0.110935\pi$
$113$	8721.21i	0.682999i	−0.939882	+	0.341500i	$0.889065\pi$
$114$	0	0
$115$	2282.19	0.172567
$116$	962.079i	0.0714981i
$117$	0	0
$118$	−15858.1	−1.13890
$119$	0	0
$120$	0	0
$121$	1614.55	0.110276
$122$	18474.9i	1.24126i
$123$	0	0
$124$	4360.14	0.283568
$125$	4124.86i	0.263991i
$126$	0	0
$127$	1934.63	0.119947	0.0599737	−	0.998200i	$-0.480898\pi$
$127$	1934.63	0.119947	0.0599737	+	0.998200i	$0.480898\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	705.474	0.0417440
$131$	28004.1i	1.63185i	0.578160	+	0.815923i	$0.303771\pi$
$131$	28004.1i	1.63185i	−0.578160	+	0.815923i	$0.696229\pi$
$132$	0	0
$133$	0	0
$134$	− 611.730i	− 0.0340683i
$135$	0	0
$136$	7680.46	0.415250
$137$	8830.35i	0.470475i	0.971938	+	0.235238i	$0.0755868\pi$
$137$	8830.35i	0.470475i	−0.971938	+	0.235238i	$0.924413\pi$
$138$	0	0
$139$	−33009.7	−1.70849	−0.854245	−	0.519871i	$-0.825980\pi$
$139$	−33009.7	−1.70849	−0.854245	+	0.519871i	$0.825980\pi$
$140$	0	0
$141$	0	0
$142$	−12555.2	−0.622653
$143$	8550.31i	0.418129i
$144$	0	0
$145$	400.395	0.0190437
$146$	18535.8i	0.869571i
$147$	0	0
$148$	−3477.83	−0.158776
$149$	15012.4i	0.676202i	0.941110	+	0.338101i	$0.109785\pi$
$149$	15012.4i	0.676202i	−0.941110	+	0.338101i	$0.890215\pi$
$150$	0	0
$151$	16070.7	0.704825	0.352412	−	0.935845i	$-0.385361\pi$
$151$	16070.7	0.704825	0.352412	+	0.935845i	$0.385361\pi$
$152$	1886.33i	0.0816450i
$153$	0	0
$154$	0	0
$155$	− 1814.58i	− 0.0755290i
$156$	0	0
$157$	−38774.3	−1.57306	−0.786529	−	0.617553i	$-0.788124\pi$
$157$	−38774.3	−1.57306	−0.786529	+	0.617553i	$0.788124\pi$
$158$	− 32484.8i	− 1.30126i
$159$	0	0
$160$	−602.688	−0.0235425
$161$	0	0
$162$	0	0
$163$	−32155.9	−1.21028	−0.605139	−	0.796120i	$-0.706883\pi$
$163$	−32155.9	−1.21028	−0.605139	+	0.796120i	$0.706883\pi$
$164$	− 5603.66i	− 0.208345i
$165$	0	0
$166$	−6966.79	−0.252823
$167$	53275.5i	1.91027i	0.296170	+	0.955135i	$0.404290\pi$
$167$	53275.5i	1.91027i	−0.296170	+	0.955135i	$0.595710\pi$
$168$	0	0
$169$	−22948.7	−0.803499
$170$	− 3196.43i	− 0.110603i
$171$	0	0
$172$	7533.80	0.254658
$173$	− 40696.5i	− 1.35977i	−0.733319	−	0.679885i	$-0.762030\pi$
$173$	− 40696.5i	− 1.35977i	0.733319	−	0.679885i	$-0.237970\pi$
$174$	0	0
$175$	0	0
$176$	− 7304.54i	− 0.235813i
$177$	0	0
$178$	12977.9	0.409605
$179$	2647.07i	0.0826151i	0.999146	+	0.0413075i	$0.0131523\pi$
$179$	2647.07i	0.0826151i	−0.999146	+	0.0413075i	$0.986848\pi$
$180$	0	0
$181$	−40869.5	−1.24750	−0.623752	−	0.781622i	$-0.714393\pi$
$181$	−40869.5	−1.24750	−0.623752	+	0.781622i	$0.714393\pi$
$182$	0	0
$183$	0	0
$184$	−15510.3	−0.458125
$185$	1447.39i	0.0422905i
$186$	0	0
$187$	38740.5	1.10785
$188$	− 6774.91i	− 0.191685i
$189$	0	0
$190$	785.045	0.0217464
$191$	36604.1i	1.00337i	0.865049	+	0.501687i	$0.167287\pi$
$191$	36604.1i	1.00337i	−0.865049	+	0.501687i	$0.832713\pi$
$192$	0	0
$193$	18985.0	0.509679	0.254840	−	0.966983i	$-0.417977\pi$
$193$	18985.0	0.509679	0.254840	+	0.966983i	$0.417977\pi$
$194$	− 44256.8i	− 1.17592i
$195$	0	0
$196$	0	0
$197$	60214.3i	1.55155i	0.631007	+	0.775777i	$0.282642\pi$
$197$	60214.3i	1.55155i	−0.631007	+	0.775777i	$0.717358\pi$
$198$	0	0
$199$	5754.54	0.145313	0.0726565	−	0.997357i	$-0.476852\pi$
$199$	5754.54	0.145313	0.0726565	+	0.997357i	$0.476852\pi$
$200$	− 13891.3i	− 0.347283i
$201$	0	0
$202$	−7530.30	−0.184548
$203$	0	0
$204$	0	0
$205$	−2332.11	−0.0554934
$206$	1367.12i	0.0322160i
$207$	0	0
$208$	−4794.56	−0.110821
$209$	9514.70i	0.217822i
$210$	0	0
$211$	−81313.3	−1.82640	−0.913201	−	0.407509i	$-0.866397\pi$
$211$	−81313.3	−1.82640	−0.913201	+	0.407509i	$0.866397\pi$
$212$	− 1770.57i	− 0.0393951i
$213$	0	0
$214$	−15561.1	−0.339793
$215$	− 3135.39i	− 0.0678289i
$216$	0	0
$217$	0	0
$218$	53246.3i	1.12041i
$219$	0	0
$220$	−3039.98	−0.0628095
$221$	− 25428.5i	− 0.520639i
$222$	0	0
$223$	−16211.1	−0.325989	−0.162995	−	0.986627i	$-0.552115\pi$
$223$	−16211.1	−0.325989	−0.162995	+	0.986627i	$0.552115\pi$
$224$	0	0
$225$	0	0
$226$	−24667.3	−0.482953
$227$	− 43540.7i	− 0.844974i	−0.906369	−	0.422487i	$-0.861157\pi$
$227$	− 43540.7i	− 0.844974i	0.906369	−	0.422487i	$-0.138843\pi$
$228$	0	0
$229$	55455.4	1.05748	0.528740	−	0.848784i	$-0.322665\pi$
$229$	55455.4	1.05748	0.528740	+	0.848784i	$0.322665\pi$
$230$	6455.02i	0.122023i
$231$	0	0
$232$	−2721.17	−0.0505568
$233$	97095.1i	1.78849i	0.447582	+	0.894243i	$0.352285\pi$
$233$	97095.1i	1.78849i	−0.447582	+	0.894243i	$0.647715\pi$
$234$	0	0
$235$	−2819.56	−0.0510558
$236$	− 44853.5i	− 0.805326i
$237$	0	0
$238$	0	0
$239$	− 73371.5i	− 1.28449i	−0.766498	−	0.642246i	$-0.778003\pi$
$239$	− 73371.5i	− 1.28449i	0.766498	−	0.642246i	$-0.221997\pi$
$240$	0	0
$241$	−48967.3	−0.843086	−0.421543	−	0.906808i	$-0.638511\pi$
$241$	−48967.3	−0.843086	−0.421543	+	0.906808i	$0.638511\pi$
$242$	4566.63i	0.0779768i
$243$	0	0
$244$	−52254.8	−0.877701
$245$	0	0
$246$	0	0
$247$	6245.27	0.102366
$248$	12332.3i	0.200513i
$249$	0	0
$250$	−11666.9	−0.186670
$251$	83332.1i	1.32271i	0.750073	+	0.661355i	$0.230018\pi$
$251$	83332.1i	1.32271i	−0.750073	+	0.661355i	$0.769982\pi$
$252$	0	0
$253$	−78234.5	−1.22224
$254$	5471.96i	0.0848156i
$255$	0	0
$256$	4096.00	0.0625000
$257$	77352.8i	1.17114i	0.810621	+	0.585571i	$0.199130\pi$
$257$	77352.8i	1.17114i	−0.810621	+	0.585571i	$0.800870\pi$
$258$	0	0
$259$	0	0
$260$	1995.38i	0.0295175i
$261$	0	0
$262$	−79207.6	−1.15389
$263$	− 68475.8i	− 0.989979i	−0.868899	−	0.494989i	$-0.835172\pi$
$263$	− 68475.8i	− 0.989979i	0.868899	−	0.494989i	$-0.164828\pi$
$264$	0	0
$265$	−736.871	−0.0104930
$266$	0	0
$267$	0	0
$268$	1730.23	0.0240899
$269$	− 82685.7i	− 1.14268i	−0.820712	−	0.571342i	$-0.806423\pi$
$269$	− 82685.7i	− 1.14268i	0.820712	−	0.571342i	$-0.193577\pi$
$270$	0	0
$271$	107378.	1.46211	0.731053	−	0.682321i	$-0.239029\pi$
$271$	107378.	1.46211	0.731053	+	0.682321i	$0.239029\pi$
$272$	21723.6i	0.293626i
$273$	0	0
$274$	−24976.0	−0.332676
$275$	− 70068.3i	− 0.926522i
$276$	0	0
$277$	123049.	1.60368	0.801842	−	0.597535i	$-0.203853\pi$
$277$	123049.	1.60368	0.801842	+	0.597535i	$0.203853\pi$
$278$	− 93365.6i	− 1.20808i
$279$	0	0
$280$	0	0
$281$	− 82257.9i	− 1.04175i	−0.853632	−	0.520877i	$-0.825605\pi$
$281$	− 82257.9i	− 1.04175i	0.853632	−	0.520877i	$-0.174395\pi$
$282$	0	0
$283$	−37437.7	−0.467451	−0.233725	−	0.972303i	$-0.575092\pi$
$283$	−37437.7	−0.467451	−0.233725	+	0.972303i	$0.575092\pi$
$284$	− 35511.4i	− 0.440282i
$285$	0	0
$286$	−24183.9	−0.295662
$287$	0	0
$288$	0	0
$289$	−31692.8	−0.379459
$290$	1132.49i	0.0134660i
$291$	0	0
$292$	−52427.1	−0.614879
$293$	92514.3i	1.07764i	0.842421	+	0.538820i	$0.181130\pi$
$293$	92514.3i	1.07764i	−0.842421	+	0.538820i	$0.818870\pi$
$294$	0	0
$295$	−18666.9	−0.214501
$296$	− 9836.80i	− 0.112272i
$297$	0	0
$298$	−42461.4	−0.478147
$299$	51351.6i	0.574396i
$300$	0	0
$301$	0	0
$302$	45454.8i	0.498386i
$303$	0	0
$304$	−5335.34	−0.0577318
$305$	21747.2i	0.233778i
$306$	0	0
$307$	−58450.4	−0.620170	−0.310085	−	0.950709i	$-0.600358\pi$
$307$	−58450.4	−0.620170	−0.310085	+	0.950709i	$0.600358\pi$
$308$	0	0
$309$	0	0
$310$	5132.42	0.0534071
$311$	80536.5i	0.832669i	0.909212	+	0.416334i	$0.136685\pi$
$311$	80536.5i	0.832669i	−0.909212	+	0.416334i	$0.863315\pi$
$312$	0	0
$313$	−47411.0	−0.483939	−0.241969	−	0.970284i	$-0.577793\pi$
$313$	−47411.0	−0.483939	−0.241969	+	0.970284i	$0.577793\pi$
$314$	− 109670.i	− 1.11232i
$315$	0	0
$316$	91880.8	0.920133
$317$	100787.i	1.00296i	0.865169	+	0.501480i	$0.167211\pi$
$317$	100787.i	1.00296i	−0.865169	+	0.501480i	$0.832789\pi$
$318$	0	0
$319$	−13725.7	−0.134882
$320$	− 1704.66i	− 0.0166471i
$321$	0	0
$322$	0	0
$323$	− 28296.6i	− 0.271225i
$324$	0	0
$325$	−45991.5	−0.435422
$326$	− 90950.6i	− 0.855796i
$327$	0	0
$328$	15849.5	0.147322
$329$	0	0
$330$	0	0
$331$	58130.4	0.530575	0.265288	−	0.964169i	$-0.414533\pi$
$331$	58130.4	0.530575	0.265288	+	0.964169i	$0.414533\pi$
$332$	− 19705.0i	− 0.178773i
$333$	0	0
$334$	−150686.	−1.35077
$335$	− 720.082i	− 0.00641642i
$336$	0	0
$337$	121929.	1.07361	0.536807	−	0.843705i	$-0.319630\pi$
$337$	121929.	1.07361	0.536807	+	0.843705i	$0.319630\pi$
$338$	− 64908.8i	− 0.568160i
$339$	0	0
$340$	9040.86	0.0782081
$341$	62204.7i	0.534951i
$342$	0	0
$343$	0	0
$344$	21308.8i	0.180070i
$345$	0	0
$346$	115107.	0.961502
$347$	46144.2i	0.383229i	0.981470	+	0.191614i	$0.0613723\pi$
$347$	46144.2i	0.383229i	−0.981470	+	0.191614i	$0.938628\pi$
$348$	0	0
$349$	−36285.8	−0.297911	−0.148955	−	0.988844i	$-0.547591\pi$
$349$	−36285.8	−0.297911	−0.148955	+	0.988844i	$0.547591\pi$
$350$	0	0
$351$	0	0
$352$	20660.4	0.166745
$353$	− 116054.i	− 0.931348i	−0.884956	−	0.465674i	$-0.845812\pi$
$353$	− 116054.i	− 0.931348i	0.884956	−	0.465674i	$-0.154188\pi$
$354$	0	0
$355$	−14779.0	−0.117270
$356$	36707.1i	0.289635i
$357$	0	0
$358$	−7487.04	−0.0584177
$359$	82065.0i	0.636750i	0.947965	+	0.318375i	$0.103137\pi$
$359$	82065.0i	0.636750i	−0.947965	+	0.318375i	$0.896863\pi$
$360$	0	0
$361$	−123371.	−0.946673
$362$	− 115596.i	− 0.882119i
$363$	0	0
$364$	0	0
$365$	21818.9i	0.163775i
$366$	0	0
$367$	−99050.4	−0.735400	−0.367700	−	0.929944i	$-0.619855\pi$
$367$	−99050.4	−0.735400	−0.367700	+	0.929944i	$0.619855\pi$
$368$	− 43869.7i	− 0.323944i
$369$	0	0
$370$	−4093.84	−0.0299039
$371$	0	0
$372$	0	0
$373$	87608.2	0.629690	0.314845	−	0.949143i	$-0.398047\pi$
$373$	87608.2	0.629690	0.314845	+	0.949143i	$0.398047\pi$
$374$	109575.i	0.783370i
$375$	0	0
$376$	19162.3	0.135542
$377$	9009.27i	0.0633880i
$378$	0	0
$379$	−153496.	−1.06861	−0.534305	−	0.845292i	$-0.679427\pi$
$379$	−153496.	−1.06861	−0.534305	+	0.845292i	$0.679427\pi$
$380$	2220.44i	0.0153770i
$381$	0	0
$382$	−103532.	−0.709493
$383$	64323.7i	0.438504i	0.975668	+	0.219252i	$0.0703617\pi$
$383$	64323.7i	0.438504i	−0.975668	+	0.219252i	$0.929638\pi$
$384$	0	0
$385$	0	0
$386$	53697.8i	0.360398i
$387$	0	0
$388$	125177.	0.831498
$389$	207284.i	1.36983i	0.728624	+	0.684914i	$0.240160\pi$
$389$	207284.i	1.36983i	−0.728624	+	0.684914i	$0.759840\pi$
$390$	0	0
$391$	232668.	1.52189
$392$	0	0
$393$	0	0
$394$	−170312.	−1.09711
$395$	− 38238.6i	− 0.245080i
$396$	0	0
$397$	38041.1	0.241364	0.120682	−	0.992691i	$-0.461492\pi$
$397$	38041.1	0.241364	0.120682	+	0.992691i	$0.461492\pi$
$398$	16276.3i	0.102752i
$399$	0	0
$400$	39290.6	0.245566
$401$	− 68026.6i	− 0.423049i	−0.977373	−	0.211524i	$-0.932157\pi$
$401$	− 68026.6i	− 0.423049i	0.977373	−	0.211524i	$-0.0678428\pi$
$402$	0	0
$403$	40830.0	0.251402
$404$	− 21298.9i	− 0.130495i
$405$	0	0
$406$	0	0
$407$	− 49617.2i	− 0.299532i
$408$	0	0
$409$	239270.	1.43035	0.715175	−	0.698946i	$-0.246347\pi$
$409$	239270.	1.43035	0.715175	+	0.698946i	$0.246347\pi$
$410$	− 6596.20i	− 0.0392397i
$411$	0	0
$412$	−3866.80	−0.0227802
$413$	0	0
$414$	0	0
$415$	−8200.77	−0.0476166
$416$	− 13561.1i	− 0.0783623i
$417$	0	0
$418$	−26911.6	−0.154024
$419$	− 330216.i	− 1.88092i	−0.339903	−	0.940461i	$-0.610394\pi$
$419$	− 330216.i	− 1.88092i	0.339903	−	0.940461i	$-0.389606\pi$
$420$	0	0
$421$	105862.	0.597276	0.298638	−	0.954366i	$-0.403468\pi$
$421$	105862.	0.597276	0.298638	+	0.954366i	$0.403468\pi$
$422$	− 229989.i	− 1.29146i
$423$	0	0
$424$	5007.94	0.0278565
$425$	208382.i	1.15367i
$426$	0	0
$427$	0	0
$428$	− 44013.6i	− 0.240270i
$429$	0	0
$430$	8868.22	0.0479623
$431$	343430.i	1.84877i	0.381459	+	0.924386i	$0.375422\pi$
$431$	343430.i	1.84877i	−0.381459	+	0.924386i	$0.624578\pi$
$432$	0	0
$433$	337875.	1.80211	0.901054	−	0.433708i	$-0.142795\pi$
$433$	337875.	1.80211	0.901054	+	0.433708i	$0.142795\pi$
$434$	0	0
$435$	0	0
$436$	−150603.	−0.792248
$437$	57143.5i	0.299229i
$438$	0	0
$439$	117947.	0.612010	0.306005	−	0.952030i	$-0.401008\pi$
$439$	117947.	0.612010	0.306005	+	0.952030i	$0.401008\pi$
$440$	− 8598.36i	− 0.0444130i
$441$	0	0
$442$	71922.7	0.368147
$443$	61063.2i	0.311152i	0.987824	+	0.155576i	$0.0497233\pi$
$443$	61063.2i	0.311152i	−0.987824	+	0.155576i	$0.950277\pi$
$444$	0	0
$445$	15276.6	0.0771450
$446$	− 45851.9i	− 0.230509i
$447$	0	0
$448$	0	0
$449$	− 93055.4i	− 0.461582i	−0.973003	−	0.230791i	$-0.925869\pi$
$449$	− 93055.4i	− 0.461582i	0.973003	−	0.230791i	$-0.0741314\pi$
$450$	0	0
$451$	79945.6	0.393044
$452$	− 69769.7i	− 0.341500i
$453$	0	0
$454$	123152.	0.597487
$455$	0	0
$456$	0	0
$457$	209876.	1.00492	0.502459	−	0.864601i	$-0.332429\pi$
$457$	209876.	1.00492	0.502459	+	0.864601i	$0.332429\pi$
$458$	156851.i	0.747752i
$459$	0	0
$460$	−18257.5	−0.0862833
$461$	− 147880.i	− 0.695837i	−0.937525	−	0.347919i	$-0.886889\pi$
$461$	− 147880.i	− 0.695837i	0.937525	−	0.347919i	$-0.113111\pi$
$462$	0	0
$463$	256712.	1.19752	0.598761	−	0.800928i	$-0.295660\pi$
$463$	256712.	1.19752	0.598761	+	0.800928i	$0.295660\pi$
$464$	− 7696.63i	− 0.0357491i
$465$	0	0
$466$	−274626.	−1.26465
$467$	280459.i	1.28598i	0.765873	+	0.642991i	$0.222307\pi$
$467$	280459.i	1.28598i	−0.765873	+	0.642991i	$0.777693\pi$
$468$	0	0
$469$	0	0
$470$	− 7974.91i	− 0.0361019i
$471$	0	0
$472$	126865.	0.569452
$473$	107482.i	0.480413i
$474$	0	0
$475$	−51178.8	−0.226831
$476$	0	0
$477$	0	0
$478$	207526.	0.908273
$479$	− 101962.i	− 0.444392i	−0.975002	−	0.222196i	$-0.928677\pi$
$479$	− 101962.i	− 0.444392i	0.975002	−	0.222196i	$-0.0713225\pi$
$480$	0	0
$481$	−32567.8	−0.140766
$482$	− 138500.i	− 0.596152i
$483$	0	0
$484$	−12916.4	−0.0551379
$485$	− 52095.7i	− 0.221472i
$486$	0	0
$487$	318704.	1.34378	0.671892	−	0.740650i	$-0.265482\pi$
$487$	318704.	1.34378	0.671892	+	0.740650i	$0.265482\pi$
$488$	− 147799.i	− 0.620629i
$489$	0	0
$490$	0	0
$491$	94125.6i	0.390431i	0.980760	+	0.195216i	$0.0625407\pi$
$491$	94125.6i	0.390431i	−0.980760	+	0.195216i	$0.937459\pi$
$492$	0	0
$493$	40820.0	0.167950
$494$	17664.3i	0.0723839i
$495$	0	0
$496$	−34881.1	−0.141784
$497$	0	0
$498$	0	0
$499$	238499.	0.957824	0.478912	−	0.877863i	$-0.341031\pi$
$499$	238499.	0.957824	0.478912	+	0.877863i	$0.341031\pi$
$500$	− 32998.9i	− 0.131995i
$501$	0	0
$502$	−235699.	−0.935297
$503$	− 222565.i	− 0.879674i	−0.898078	−	0.439837i	$-0.855036\pi$
$503$	− 222565.i	− 0.879674i	0.898078	−	0.439837i	$-0.144964\pi$
$504$	0	0
$505$	−8864.10	−0.0347578
$506$	− 221280.i	− 0.864255i
$507$	0	0
$508$	−15477.1	−0.0599737
$509$	− 83501.6i	− 0.322299i	−0.986930	−	0.161150i	$-0.948480\pi$
$509$	− 83501.6i	− 0.322299i	0.986930	−	0.161150i	$-0.0515202\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−218787.	−0.828123
$515$	1609.27i	0.00606757i
$516$	0	0
$517$	96655.5	0.361614
$518$	0	0
$519$	0	0
$520$	−5643.80	−0.0208720
$521$	345105.i	1.27138i	0.771944	+	0.635690i	$0.219284\pi$
$521$	345105.i	1.27138i	−0.771944	+	0.635690i	$0.780716\pi$
$522$	0	0
$523$	52135.8	0.190604	0.0953022	−	0.995448i	$-0.469618\pi$
$523$	52135.8	0.190604	0.0953022	+	0.995448i	$0.469618\pi$
$524$	− 224033.i	− 0.815923i
$525$	0	0
$526$	193679.	0.700021
$527$	− 184996.i	− 0.666102i
$528$	0	0
$529$	−190021.	−0.679031
$530$	− 2084.18i	− 0.00741967i
$531$	0	0
$532$	0	0
$533$	− 52474.8i	− 0.184712i
$534$	0	0
$535$	−18317.4	−0.0639965
$536$	4893.84i	0.0170341i
$537$	0	0
$538$	233870.	0.807999
$539$	0	0
$540$	0	0
$541$	111921.	0.382401	0.191200	−	0.981551i	$-0.438762\pi$
$541$	111921.	0.382401	0.191200	+	0.981551i	$0.438762\pi$
$542$	303712.i	1.03386i
$543$	0	0
$544$	−61443.7	−0.207625
$545$	62677.5i	0.211018i
$546$	0	0
$547$	333825.	1.11569	0.557846	−	0.829944i	$-0.311628\pi$
$547$	333825.	1.11569	0.557846	+	0.829944i	$0.311628\pi$
$548$	− 70642.8i	− 0.235238i
$549$	0	0
$550$	198183.	0.655150
$551$	10025.4i	0.0330217i
$552$	0	0
$553$	0	0
$554$	348036.i	1.13398i
$555$	0	0
$556$	264078.	0.854245
$557$	− 128171.i	− 0.413121i	−0.978434	−	0.206561i	$-0.933773\pi$
$557$	− 128171.i	− 0.413121i	0.978434	−	0.206561i	$-0.0662271\pi$
$558$	0	0
$559$	70549.4	0.225772
$560$	0	0
$561$	0	0
$562$	232660.	0.736631
$563$	− 218560.i	− 0.689531i	−0.938689	−	0.344765i	$-0.887958\pi$
$563$	− 218560.i	− 0.689531i	0.938689	−	0.344765i	$-0.112042\pi$
$564$	0	0
$565$	−29036.5	−0.0909594
$566$	− 105890.i	− 0.330538i
$567$	0	0
$568$	100441.	0.311326
$569$	− 542902.i	− 1.67686i	−0.545009	−	0.838430i	$-0.683474\pi$
$569$	− 542902.i	− 1.67686i	0.545009	−	0.838430i	$-0.316526\pi$
$570$	0	0
$571$	−145319.	−0.445709	−0.222854	−	0.974852i	$-0.571537\pi$
$571$	−145319.	−0.445709	−0.222854	+	0.974852i	$0.571537\pi$
$572$	− 68402.5i	− 0.209064i
$573$	0	0
$574$	0	0
$575$	− 420817.i	− 1.27279i
$576$	0	0
$577$	98992.2	0.297337	0.148669	−	0.988887i	$-0.452501\pi$
$577$	98992.2	0.297337	0.148669	+	0.988887i	$0.452501\pi$
$578$	− 89640.7i	− 0.268318i
$579$	0	0
$580$	−3203.16	−0.00952187
$581$	0	0
$582$	0	0
$583$	25260.2	0.0743190
$584$	− 148286.i	− 0.434785i
$585$	0	0
$586$	−261670.	−0.762006
$587$	40286.5i	0.116919i	0.998290	+	0.0584593i	$0.0186188\pi$
$587$	40286.5i	0.116919i	−0.998290	+	0.0584593i	$0.981381\pi$
$588$	0	0
$589$	45435.2	0.130967
$590$	− 52798.1i	− 0.151675i
$591$	0	0
$592$	27822.7	0.0793881
$593$	552114.i	1.57007i	0.619450	+	0.785036i	$0.287356\pi$
$593$	552114.i	1.57007i	−0.619450	+	0.785036i	$0.712644\pi$
$594$	0	0
$595$	0	0
$596$	− 120099.i	− 0.338101i
$597$	0	0
$598$	−145244.	−0.406159
$599$	405734.i	1.13081i	0.824815	+	0.565403i	$0.191279\pi$
$599$	405734.i	1.13081i	−0.824815	+	0.565403i	$0.808721\pi$
$600$	0	0
$601$	683398.	1.89202	0.946008	−	0.324144i	$-0.105076\pi$
$601$	683398.	1.89202	0.946008	+	0.324144i	$0.105076\pi$
$602$	0	0
$603$	0	0
$604$	−128566.	−0.352412
$605$	5375.49i	0.0146861i
$606$	0	0
$607$	−243766.	−0.661600	−0.330800	−	0.943701i	$-0.607318\pi$
$607$	−243766.	−0.661600	−0.330800	+	0.943701i	$0.607318\pi$
$608$	− 15090.6i	− 0.0408225i
$609$	0	0
$610$	−61510.4	−0.165306
$611$	− 63442.8i	− 0.169942i
$612$	0	0
$613$	−211556.	−0.562996	−0.281498	−	0.959562i	$-0.590831\pi$
$613$	−211556.	−0.562996	−0.281498	+	0.959562i	$0.590831\pi$
$614$	− 165323.i	− 0.438527i
$615$	0	0
$616$	0	0
$617$	− 293226.i	− 0.770249i	−0.922865	−	0.385125i	$-0.874158\pi$
$617$	− 293226.i	− 0.770249i	0.922865	−	0.385125i	$-0.125842\pi$
$618$	0	0
$619$	108683.	0.283649	0.141825	−	0.989892i	$-0.454703\pi$
$619$	108683.	0.283649	0.141825	+	0.989892i	$0.454703\pi$
$620$	14516.7i	0.0377645i
$621$	0	0
$622$	−227792.	−0.588786
$623$	0	0
$624$	0	0
$625$	369964.	0.947107
$626$	− 134099.i	− 0.342196i
$627$	0	0
$628$	310195.	0.786529
$629$	147561.i	0.372967i
$630$	0	0
$631$	−195365.	−0.490669	−0.245334	−	0.969439i	$-0.578898\pi$
$631$	−195365.	−0.490669	−0.245334	+	0.969439i	$0.578898\pi$
$632$	259878.i	0.650632i
$633$	0	0
$634$	−285067.	−0.709201
$635$	6441.18i	0.0159742i
$636$	0	0
$637$	0	0
$638$	− 38822.1i	− 0.0953756i
$639$	0	0
$640$	4821.50	0.0117712
$641$	− 451280.i	− 1.09832i	−0.835716	−	0.549162i	$-0.814947\pi$
$641$	− 451280.i	− 1.09832i	0.835716	−	0.549162i	$-0.185053\pi$
$642$	0	0
$643$	495459.	1.19835	0.599177	−	0.800616i	$-0.295494\pi$
$643$	495459.	1.19835	0.599177	+	0.800616i	$0.295494\pi$
$644$	0	0
$645$	0	0
$646$	80034.9	0.191785
$647$	− 215529.i	− 0.514870i	−0.966296	−	0.257435i	$-0.917123\pi$
$647$	− 215529.i	− 0.514870i	0.966296	−	0.257435i	$-0.0828774\pi$
$648$	0	0
$649$	639910.	1.51925
$650$	− 130083.i	− 0.307890i
$651$	0	0
$652$	257247.	0.605139
$653$	740949.i	1.73765i	0.495120	+	0.868824i	$0.335124\pi$
$653$	740949.i	1.73765i	−0.495120	+	0.868824i	$0.664876\pi$
$654$	0	0
$655$	−93237.2	−0.217323
$656$	44829.3i	0.104173i
$657$	0	0
$658$	0	0
$659$	123691.i	0.284817i	0.989808	+	0.142409i	$0.0454847\pi$
$659$	123691.i	0.284817i	−0.989808	+	0.142409i	$0.954515\pi$
$660$	0	0
$661$	−91044.7	−0.208378	−0.104189	−	0.994558i	$-0.533225\pi$
$661$	−91044.7	−0.208378	−0.104189	+	0.994558i	$0.533225\pi$
$662$	164418.i	0.375173i
$663$	0	0
$664$	55734.3	0.126411
$665$	0	0
$666$	0	0
$667$	−82433.9	−0.185291
$668$	− 426204.i	− 0.955135i
$669$	0	0
$670$	2036.70	0.00453709
$671$	− 745503.i	− 1.65579i
$672$	0	0
$673$	−380207.	−0.839441	−0.419721	−	0.907653i	$-0.637872\pi$
$673$	−380207.	−0.839441	−0.419721	+	0.907653i	$0.637872\pi$
$674$	344868.i	0.759160i
$675$	0	0
$676$	183590.	0.401750
$677$	− 231093.i	− 0.504207i	−0.967700	−	0.252103i	$-0.918878\pi$
$677$	− 231093.i	− 0.504207i	0.967700	−	0.252103i	$-0.0811223\pi$
$678$	0	0
$679$	0	0
$680$	25571.4i	0.0553015i
$681$	0	0
$682$	−175941.	−0.378268
$683$	381979.i	0.818838i	0.912347	+	0.409419i	$0.134269\pi$
$683$	381979.i	0.818838i	−0.912347	+	0.409419i	$0.865731\pi$
$684$	0	0
$685$	−29399.8	−0.0626562
$686$	0	0
$687$	0	0
$688$	−60270.4	−0.127329
$689$	− 16580.3i	− 0.0349264i
$690$	0	0
$691$	−260296.	−0.545144	−0.272572	−	0.962135i	$-0.587874\pi$
$691$	−260296.	−0.545144	−0.272572	+	0.962135i	$0.587874\pi$
$692$	325572.i	0.679885i
$693$	0	0
$694$	−130515.	−0.270983
$695$	− 109903.i	− 0.227531i
$696$	0	0
$697$	−237757.	−0.489405
$698$	− 102632.i	− 0.210655i
$699$	0	0
$700$	0	0
$701$	97451.7i	0.198314i	0.995072	+	0.0991570i	$0.0316146\pi$
$701$	97451.7i	0.198314i	−0.995072	+	0.0991570i	$0.968385\pi$
$702$	0	0
$703$	−36241.1	−0.0733315
$704$	58436.3i	0.117906i
$705$	0	0
$706$	328251.	0.658562
$707$	0	0
$708$	0	0
$709$	−450109.	−0.895417	−0.447708	−	0.894180i	$-0.647760\pi$
$709$	−450109.	−0.895417	−0.447708	+	0.894180i	$0.647760\pi$
$710$	− 41801.3i	− 0.0829227i
$711$	0	0
$712$	−103823.	−0.204803
$713$	373590.i	0.734879i
$714$	0	0
$715$	−28467.5	−0.0556849
$716$	− 21176.6i	− 0.0413075i
$717$	0	0
$718$	−232115.	−0.450250
$719$	281309.i	0.544159i	0.962275	+	0.272079i	$0.0877113\pi$
$719$	281309.i	0.544159i	−0.962275	+	0.272079i	$0.912289\pi$
$720$	0	0
$721$	0	0
$722$	− 348947.i	− 0.669399i
$723$	0	0
$724$	326956.	0.623752
$725$	− 73829.4i	− 0.140460i
$726$	0	0
$727$	−989653.	−1.87247	−0.936233	−	0.351379i	$-0.885713\pi$
$727$	−989653.	−1.87247	−0.936233	+	0.351379i	$0.885713\pi$
$728$	0	0
$729$	0	0
$730$	−61713.2	−0.115806
$731$	− 319651.i	− 0.598194i
$732$	0	0
$733$	−757559.	−1.40997	−0.704983	−	0.709224i	$-0.749045\pi$
$733$	−757559.	−1.40997	−0.704983	+	0.709224i	$0.749045\pi$
$734$	− 280157.i	− 0.520007i
$735$	0	0
$736$	124082.	0.229063
$737$	24684.7i	0.0454457i
$738$	0	0
$739$	−364398.	−0.667247	−0.333623	−	0.942706i	$-0.608271\pi$
$739$	−364398.	−0.667247	−0.333623	+	0.942706i	$0.608271\pi$
$740$	− 11579.1i	− 0.0211453i
$741$	0	0
$742$	0	0
$743$	299785.i	0.543040i	0.962433	+	0.271520i	$0.0875263\pi$
$743$	299785.i	0.543040i	−0.962433	+	0.271520i	$0.912474\pi$
$744$	0	0
$745$	−49982.3	−0.0900542
$746$	247793.i	0.445258i
$747$	0	0
$748$	−309924.	−0.553926
$749$	0	0
$750$	0	0
$751$	258424.	0.458198	0.229099	−	0.973403i	$-0.426422\pi$
$751$	258424.	0.458198	0.229099	+	0.973403i	$0.426422\pi$
$752$	54199.3i	0.0958425i
$753$	0	0
$754$	−25482.1	−0.0448221
$755$	53506.0i	0.0938660i
$756$	0	0
$757$	23059.3	0.0402396	0.0201198	−	0.999798i	$-0.493595\pi$
$757$	23059.3	0.0402396	0.0201198	+	0.999798i	$0.493595\pi$
$758$	− 434153.i	− 0.755622i
$759$	0	0
$760$	−6280.36	−0.0108732
$761$	− 711299.i	− 1.22824i	−0.789213	−	0.614119i	$-0.789511\pi$
$761$	− 711299.i	− 1.22824i	0.789213	−	0.614119i	$-0.210489\pi$
$762$	0	0
$763$	0	0
$764$	− 292833.i	− 0.501687i
$765$	0	0
$766$	−181935.	−0.310069
$767$	− 420025.i	− 0.713977i
$768$	0	0
$769$	746268.	1.26195	0.630975	−	0.775803i	$-0.282655\pi$
$769$	746268.	1.26195	0.630975	+	0.775803i	$0.282655\pi$
$770$	0	0
$771$	0	0
$772$	−151880.	−0.254840
$773$	− 505278.i	− 0.845612i	−0.906220	−	0.422806i	$-0.861045\pi$
$773$	− 505278.i	− 0.845612i	0.906220	−	0.422806i	$-0.138955\pi$
$774$	0	0
$775$	−334594.	−0.557076
$776$	354054.i	0.587958i
$777$	0	0
$778$	−586287.	−0.968614
$779$	− 58393.4i	− 0.0962252i
$780$	0	0
$781$	506630.	0.830594
$782$	658085.i	1.07614i
$783$	0	0
$784$	0	0
$785$	− 129096.i	− 0.209494i
$786$	0	0
$787$	−671183.	−1.08366	−0.541828	−	0.840489i	$-0.682268\pi$
$787$	−671183.	−1.08366	−0.541828	+	0.840489i	$0.682268\pi$
$788$	− 481714.i	− 0.775777i
$789$	0	0
$790$	108155.	0.173298
$791$	0	0
$792$	0	0
$793$	−489334.	−0.778142
$794$	107597.i	0.170670i
$795$	0	0
$796$	−46036.3	−0.0726565
$797$	808727.i	1.27317i	0.771208	+	0.636584i	$0.219653\pi$
$797$	808727.i	1.27317i	−0.771208	+	0.636584i	$0.780347\pi$
$798$	0	0
$799$	−287452.	−0.450269
$800$	111130.i	0.173641i
$801$	0	0
$802$	192408.	0.299141
$803$	− 747961.i	− 1.15997i
$804$	0	0
$805$	0	0
$806$	115485.i	0.177768i
$807$	0	0
$808$	60242.4	0.0922740
$809$	− 648685.i	− 0.991144i	−0.868567	−	0.495572i	$-0.834958\pi$
$809$	− 648685.i	− 0.991144i	0.868567	−	0.495572i	$-0.165042\pi$
$810$	0	0
$811$	−1.13453e6	−1.72495	−0.862473	−	0.506103i	$-0.831086\pi$
$811$	−1.13453e6	−1.72495	−0.862473	+	0.506103i	$0.831086\pi$
$812$	0	0
$813$	0	0
$814$	140339.	0.211801
$815$	− 107060.i	− 0.161181i
$816$	0	0
$817$	78506.6	0.117615
$818$	676758.i	1.01141i
$819$	0	0
$820$	18656.9	0.0277467
$821$	594350.i	0.881771i	0.897563	+	0.440885i	$0.145335\pi$
$821$	594350.i	0.881771i	−0.897563	+	0.440885i	$0.854665\pi$
$822$	0	0
$823$	−61274.3	−0.0904646	−0.0452323	−	0.998976i	$-0.514403\pi$
$823$	−61274.3	−0.0904646	−0.0452323	+	0.998976i	$0.514403\pi$
$824$	− 10937.0i	− 0.0161080i
$825$	0	0
$826$	0	0
$827$	− 1.00033e6i	− 1.46262i	−0.682045	−	0.731310i	$-0.738909\pi$
$827$	− 1.00033e6i	− 1.46262i	0.682045	−	0.731310i	$-0.261091\pi$
$828$	0	0
$829$	−359332.	−0.522862	−0.261431	−	0.965222i	$-0.584194\pi$
$829$	−359332.	−0.522862	−0.261431	+	0.965222i	$0.584194\pi$
$830$	− 23195.3i	− 0.0336700i
$831$	0	0
$832$	38356.5	0.0554105
$833$	0	0
$834$	0	0
$835$	−177376.	−0.254403
$836$	− 76117.6i	− 0.108911i
$837$	0	0
$838$	933993.	1.33001
$839$	180280.i	0.256109i	0.991767	+	0.128054i	$0.0408732\pi$
$839$	180280.i	0.256109i	−0.991767	+	0.128054i	$0.959127\pi$
$840$	0	0
$841$	692819.	0.979552
$842$	299423.i	0.422338i
$843$	0	0
$844$	650506.	0.913201
$845$	− 76405.8i	− 0.107007i
$846$	0	0
$847$	0	0
$848$	14164.6i	0.0196975i
$849$	0	0
$850$	−589394.	−0.815770
$851$	− 297992.i	− 0.411476i
$852$	0	0
$853$	−1.28410e6	−1.76483	−0.882413	−	0.470475i	$-0.844083\pi$
$853$	−1.28410e6	−1.76483	−0.882413	+	0.470475i	$0.844083\pi$
$854$	0	0
$855$	0	0
$856$	124489.	0.169896
$857$	1.03888e6i	1.41451i	0.706960	+	0.707254i	$0.250066\pi$
$857$	1.03888e6i	1.41451i	−0.706960	+	0.707254i	$0.749934\pi$
$858$	0	0
$859$	401198.	0.543716	0.271858	−	0.962337i	$-0.412362\pi$
$859$	401198.	0.543716	0.271858	+	0.962337i	$0.412362\pi$
$860$	25083.1i	0.0339144i
$861$	0	0
$862$	−971366.	−1.30728
$863$	315863.i	0.424108i	0.977258	+	0.212054i	$0.0680153\pi$
$863$	315863.i	0.424108i	−0.977258	+	0.212054i	$0.931985\pi$
$864$	0	0
$865$	135496.	0.181089
$866$	955656.i	1.27428i
$867$	0	0
$868$	0	0
$869$	1.31083e6i	1.73583i
$870$	0	0
$871$	16202.6	0.0213574
$872$	− 425970.i	− 0.560204i
$873$	0	0
$874$	−161626.	−0.211587
$875$	0	0
$876$	0	0
$877$	876233.	1.13925	0.569627	−	0.821903i	$-0.307088\pi$
$877$	876233.	1.13925	0.569627	+	0.821903i	$0.307088\pi$
$878$	333605.i	0.432756i
$879$	0	0
$880$	24319.8	0.0314047
$881$	575681.i	0.741704i	0.928692	+	0.370852i	$0.120934\pi$
$881$	575681.i	0.741704i	−0.928692	+	0.370852i	$0.879066\pi$
$882$	0	0
$883$	−57626.0	−0.0739090	−0.0369545	−	0.999317i	$-0.511766\pi$
$883$	−57626.0	−0.0739090	−0.0369545	+	0.999317i	$0.511766\pi$
$884$	203428.i	0.260319i
$885$	0	0
$886$	−172713.	−0.220017
$887$	− 91923.2i	− 0.116836i	−0.998292	−	0.0584182i	$-0.981394\pi$
$887$	− 91923.2i	− 0.116836i	0.998292	−	0.0584182i	$-0.0186057\pi$
$888$	0	0
$889$	0	0
$890$	43208.8i	0.0545497i
$891$	0	0
$892$	129689.	0.162995
$893$	− 70598.5i	− 0.0885305i
$894$	0	0
$895$	−8813.18	−0.0110024
$896$	0	0
$897$	0	0
$898$	263200.	0.326388
$899$	65543.7i	0.0810982i
$900$	0	0
$901$	−75123.5	−0.0925394
$902$	226120.i	0.277924i
$903$	0	0
$904$	197339.	0.241477
$905$	− 136071.i	− 0.166138i
$906$	0	0
$907$	1.44940e6	1.76187	0.880934	−	0.473239i	$-0.156915\pi$
$907$	1.44940e6	1.76187	0.880934	+	0.473239i	$0.156915\pi$
$908$	348325.i	0.422487i
$909$	0	0
$910$	0	0
$911$	633417.i	0.763225i	0.924322	+	0.381613i	$0.124631\pi$
$911$	633417.i	0.763225i	−0.924322	+	0.381613i	$0.875369\pi$
$912$	0	0
$913$	281126.	0.337256
$914$	593620.i	0.710585i
$915$	0	0
$916$	−443643.	−0.528740
$917$	0	0
$918$	0	0
$919$	−1.51364e6	−1.79222	−0.896108	−	0.443837i	$-0.853617\pi$
$919$	−1.51364e6	−1.79222	−0.896108	+	0.443837i	$0.853617\pi$
$920$	− 51640.1i	− 0.0610115i
$921$	0	0
$922$	418268.	0.492031
$923$	− 332542.i	− 0.390340i
$924$	0	0
$925$	266887.	0.311920
$926$	726090.i	0.846776i
$927$	0	0
$928$	21769.4	0.0252784
$929$	− 1.55984e6i	− 1.80738i	−0.428188	−	0.903690i	$-0.640848\pi$
$929$	− 1.55984e6i	− 1.80738i	0.428188	−	0.903690i	$-0.359152\pi$
$930$	0	0
$931$	0	0
$932$	− 776761.i	− 0.894243i
$933$	0	0
$934$	−793257.	−0.909327
$935$	128983.i	0.147540i
$936$	0	0
$937$	−872523.	−0.993796	−0.496898	−	0.867809i	$-0.665528\pi$
$937$	−872523.	−0.993796	−0.496898	+	0.867809i	$0.665528\pi$
$938$	0	0
$939$	0	0
$940$	22556.5	0.0255279
$941$	− 430745.i	− 0.486454i	−0.969969	−	0.243227i	$-0.921794\pi$
$941$	− 430745.i	− 0.486454i	0.969969	−	0.243227i	$-0.0782059\pi$
$942$	0	0
$943$	480139.	0.539937
$944$	358828.i	0.402663i
$945$	0	0
$946$	−304006.	−0.339704
$947$	− 494014.i	− 0.550858i	−0.961321	−	0.275429i	$-0.911180\pi$
$947$	− 494014.i	− 0.550858i	0.961321	−	0.275429i	$-0.0888198\pi$
$948$	0	0
$949$	−490947.	−0.545133
$950$	− 144756.i	− 0.160394i
$951$	0	0
$952$	0	0
$953$	1.01986e6i	1.12293i	0.827499	+	0.561467i	$0.189763\pi$
$953$	1.01986e6i	1.12293i	−0.827499	+	0.561467i	$0.810237\pi$
$954$	0	0
$955$	−121870.	−0.133626
$956$	586972.i	0.642246i
$957$	0	0
$958$	288391.	0.314233
$959$	0	0
$960$	0	0
$961$	−626478.	−0.678358
$962$	− 92115.5i	− 0.0995366i
$963$	0	0
$964$	391738.	0.421543
$965$	63209.0i	0.0678772i
$966$	0	0
$967$	−1.54770e6	−1.65513	−0.827567	−	0.561367i	$-0.810275\pi$
$967$	−1.54770e6	−1.65513	−0.827567	+	0.561367i	$0.810275\pi$
$968$	− 36533.0i	− 0.0389884i
$969$	0	0
$970$	147349.	0.156604
$971$	116422.i	0.123480i	0.998092	+	0.0617398i	$0.0196649\pi$
$971$	116422.i	0.123480i	−0.998092	+	0.0617398i	$0.980335\pi$
$972$	0	0
$973$	0	0
$974$	901430.i	0.950198i
$975$	0	0
$976$	418039.	0.438851
$977$	− 1.18320e6i	− 1.23956i	−0.784775	−	0.619781i	$-0.787221\pi$
$977$	− 1.18320e6i	− 1.23956i	0.784775	−	0.619781i	$-0.212779\pi$
$978$	0	0
$979$	−523689.	−0.546397
$980$	0	0
$981$	0	0
$982$	−266227.	−0.276077
$983$	987556.i	1.02201i	0.859578	+	0.511004i	$0.170726\pi$
$983$	987556.i	1.02201i	−0.859578	+	0.511004i	$0.829274\pi$
$984$	0	0
$985$	−200478.	−0.206631
$986$	115456.i	0.118758i
$987$	0	0
$988$	−49962.1	−0.0511832
$989$	645519.i	0.659959i
$990$	0	0
$991$	847504.	0.862968	0.431484	−	0.902121i	$-0.357990\pi$
$991$	847504.	0.862968	0.431484	+	0.902121i	$0.357990\pi$
$992$	− 98658.6i	− 0.100256i
$993$	0	0
$994$	0	0
$995$	19159.2i	0.0193523i
$996$	0	0
$997$	77142.3	0.0776072	0.0388036	−	0.999247i	$-0.487645\pi$
$997$	77142.3	0.0776072	0.0388036	+	0.999247i	$0.487645\pi$
$998$	674578.i	0.677284i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.a.197.3		4
3.2	odd	2	inner	882.5.b.a.197.2		4
7.6	odd	2		126.5.b.b.71.4	yes	4
21.20	even	2		126.5.b.b.71.1	✓	4
28.27	even	2		1008.5.d.a.449.2		4
84.83	odd	2		1008.5.d.a.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.b.b.71.1	✓	4	21.20	even	2
126.5.b.b.71.4	yes	4	7.6	odd	2
882.5.b.a.197.2		4	3.2	odd	2	inner
882.5.b.a.197.3		4	1.1	even	1	trivial
1008.5.d.a.449.2		4	28.27	even	2
1008.5.d.a.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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +3.32941i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +3.32941i q^{5} -22.6274i q^{8} -9.41699 q^{10} -114.133i q^{11} -74.9150 q^{13} +64.0000 q^{16} +339.432i q^{17} -83.3647 q^{19} -26.6353i q^{20} +322.818 q^{22} -685.465i q^{23} +613.915 q^{25} -211.892i q^{26} -120.260i q^{29} -545.017 q^{31} +181.019i q^{32} -960.057 q^{34} +434.729 q^{37} -235.791i q^{38} +75.3360 q^{40} +700.457i q^{41} -941.725 q^{43} +913.068i q^{44} +1938.79 q^{46} +846.864i q^{47} +1736.41i q^{50} +599.320 q^{52} +221.322i q^{53} +379.997 q^{55} +340.146 q^{58} +5606.68i q^{59} +6531.85 q^{61} -1541.54i q^{62} -512.000 q^{64} -249.423i q^{65} -216.279 q^{67} -2715.45i q^{68} +4438.92i q^{71} +6553.39 q^{73} +1229.60i q^{74} +666.917 q^{76} -11485.1 q^{79} +213.082i q^{80} -1981.19 q^{82} +2463.13i q^{83} -1130.11 q^{85} -2663.60i q^{86} -2582.55 q^{88} -4588.39i q^{89} +5483.72i q^{92} -2395.29 q^{94} -277.555i q^{95} -15647.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} - 80 q^{10} - 88 q^{13} + 256 q^{16} + 1000 q^{19} - 656 q^{22} + 2244 q^{25} + 1016 q^{31} - 496 q^{34} - 928 q^{37} + 640 q^{40} - 592 q^{43} + 1744 q^{46} + 704 q^{52} - 4216 q^{55} - 1264 q^{58} + 12560 q^{61} - 2048 q^{64} + 16872 q^{67} + 6000 q^{73} - 8000 q^{76} - 15800 q^{79} - 432 q^{82} + 3184 q^{85} + 5248 q^{88} + 17088 q^{94} - 52704 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 - 80 * q^10 - 88 * q^13 + 256 * q^16 + 1000 * q^19 - 656 * q^22 + 2244 * q^25 + 1016 * q^31 - 496 * q^34 - 928 * q^37 + 640 * q^40 - 592 * q^43 + 1744 * q^46 + 704 * q^52 - 4216 * q^55 - 1264 * q^58 + 12560 * q^61 - 2048 * q^64 + 16872 * q^67 + 6000 * q^73 - 8000 * q^76 - 15800 * q^79 - 432 * q^82 + 3184 * q^85 + 5248 * q^88 + 17088 * q^94 - 52704 * q^97

Introduction

L-functions

Modular forms

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