LMFDB - Embedded newform 882.5.b.e.197.2

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 13x^{4} - 660x^{3} + 702x^{2} - 5832x + 157464 $ x^6 - 2x^5 + 13x^4 - 660x^3 + 702x^2 - 5832*x + 157464
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.2
Root			$-1.69913 + 7.14933i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.e.197.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -8.92960i q^{5} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} -8.92960i q^{5} +22.6274i q^{8} -25.2567 q^{10} -112.518i q^{11} +86.8767 q^{13} +64.0000 q^{16} +321.029i q^{17} +103.904 q^{19} +71.4368i q^{20} -318.250 q^{22} -120.158i q^{23} +545.262 q^{25} -245.725i q^{26} +1514.81i q^{29} -1367.71 q^{31} -181.019i q^{32} +908.007 q^{34} +68.3392 q^{37} -293.884i q^{38} +202.054 q^{40} -96.3894i q^{41} -273.972 q^{43} +900.148i q^{44} -339.859 q^{46} +2229.27i q^{47} -1542.23i q^{50} -695.014 q^{52} -1807.84i q^{53} -1004.74 q^{55} +4284.53 q^{58} -378.613i q^{59} +3617.69 q^{61} +3868.46i q^{62} -512.000 q^{64} -775.774i q^{65} -5673.59 q^{67} -2568.23i q^{68} +3300.84i q^{71} +9481.81 q^{73} -193.292i q^{74} -831.229 q^{76} +1730.94 q^{79} -571.494i q^{80} -272.630 q^{82} +7610.67i q^{83} +2866.66 q^{85} +774.911i q^{86} +2546.00 q^{88} +1194.37i q^{89} +961.267i q^{92} +6305.34 q^{94} -927.817i q^{95} +10595.6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{4}+O(q^{10}) $ 6 * q - 48 * q^4	$ 6 q - 48 q^{4} - 104 q^{10} - 86 q^{13} + 384 q^{16} - 174 q^{19} + 472 q^{22} - 2170 q^{25} - 1682 q^{31} - 1744 q^{34} + 6018 q^{37} + 832 q^{40} - 2130 q^{43} - 656 q^{46} + 688 q^{52} + 12680 q^{55} + 7040 q^{58} + 20632 q^{61} - 3072 q^{64} - 74 q^{67} + 2942 q^{73} + 1392 q^{76} + 10526 q^{79} + 25464 q^{82} - 33056 q^{85} - 3776 q^{88} + 11640 q^{94} + 46436 q^{97}+O(q^{100}) $ 6 * q - 48 * q^4 - 104 * q^10 - 86 * q^13 + 384 * q^16 - 174 * q^19 + 472 * q^22 - 2170 * q^25 - 1682 * q^31 - 1744 * q^34 + 6018 * q^37 + 832 * q^40 - 2130 * q^43 - 656 * q^46 + 688 * q^52 + 12680 * q^55 + 7040 * q^58 + 20632 * q^61 - 3072 * q^64 - 74 * q^67 + 2942 * q^73 + 1392 * q^76 + 10526 * q^79 + 25464 * q^82 - 33056 * q^85 - 3776 * q^88 + 11640 * q^94 + 46436 * 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
$5$	− 8.92960i	− 0.357184i	−0.983923	−	0.178592i	$-0.942846\pi$
$5$	− 8.92960i	− 0.357184i	0.983923	−	0.178592i	$-0.0571542\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−25.2567	−0.252567
$11$	− 112.518i	− 0.929905i	−0.885336	−	0.464952i	$-0.846071\pi$
$11$	− 112.518i	− 0.929905i	0.885336	−	0.464952i	$-0.153929\pi$
$12$	0	0
$13$	86.8767	0.514064	0.257032	−	0.966403i	$-0.417255\pi$
$13$	86.8767	0.514064	0.257032	+	0.966403i	$0.417255\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	321.029i	1.11083i	0.831574	+	0.555414i	$0.187440\pi$
$17$	321.029i	1.11083i	−0.831574	+	0.555414i	$0.812560\pi$
$18$	0	0
$19$	103.904	0.287822	0.143911	−	0.989591i	$-0.454032\pi$
$19$	103.904	0.287822	0.143911	+	0.989591i	$0.454032\pi$
$20$	71.4368i	0.178592i
$21$	0	0
$22$	−318.250	−0.657542
$23$	− 120.158i	− 0.227143i	−0.993530	−	0.113571i	$-0.963771\pi$
$23$	− 120.158i	− 0.227143i	0.993530	−	0.113571i	$-0.0362290\pi$
$24$	0	0
$25$	545.262	0.872420
$26$	− 245.725i	− 0.363498i
$27$	0	0
$28$	0	0
$29$	1514.81i	1.80120i	0.434646	+	0.900601i	$0.356873\pi$
$29$	1514.81i	1.80120i	−0.434646	+	0.900601i	$0.643127\pi$
$30$	0	0
$31$	−1367.71	−1.42321	−0.711606	−	0.702579i	$-0.752032\pi$
$31$	−1367.71	−1.42321	−0.711606	+	0.702579i	$0.752032\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	908.007	0.785474
$35$	0	0
$36$	0	0
$37$	68.3392	0.0499191	0.0249595	−	0.999688i	$-0.492054\pi$
$37$	68.3392	0.0499191	0.0249595	+	0.999688i	$0.492054\pi$
$38$	− 293.884i	− 0.203521i
$39$	0	0
$40$	202.054	0.126284
$41$	− 96.3894i	− 0.0573405i	−0.999589	−	0.0286702i	$-0.990873\pi$
$41$	− 96.3894i	− 0.0573405i	0.999589	−	0.0286702i	$-0.00912727\pi$
$42$	0	0
$43$	−273.972	−0.148173	−0.0740866	−	0.997252i	$-0.523604\pi$
$43$	−273.972	−0.148173	−0.0740866	+	0.997252i	$0.523604\pi$
$44$	900.148i	0.464952i
$45$	0	0
$46$	−339.859	−0.160614
$47$	2229.27i	1.00918i	0.863360	+	0.504589i	$0.168356\pi$
$47$	2229.27i	1.00918i	−0.863360	+	0.504589i	$0.831644\pi$
$48$	0	0
$49$	0	0
$50$	− 1542.23i	− 0.616894i
$51$	0	0
$52$	−695.014	−0.257032
$53$	− 1807.84i	− 0.643589i	−0.946810	−	0.321794i	$-0.895714\pi$
$53$	− 1807.84i	− 0.643589i	0.946810	−	0.321794i	$-0.104286\pi$
$54$	0	0
$55$	−1004.74	−0.332147
$56$	0	0
$57$	0	0
$58$	4284.53	1.27364
$59$	− 378.613i	− 0.108766i	−0.998520	−	0.0543829i	$-0.982681\pi$
$59$	− 378.613i	− 0.108766i	0.998520	−	0.0543829i	$-0.0173191\pi$
$60$	0	0
$61$	3617.69	0.972236	0.486118	−	0.873893i	$-0.338412\pi$
$61$	3617.69	0.972236	0.486118	+	0.873893i	$0.338412\pi$
$62$	3868.46i	1.00636i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 775.774i	− 0.183615i
$66$	0	0
$67$	−5673.59	−1.26389	−0.631944	−	0.775014i	$-0.717743\pi$
$67$	−5673.59	−1.26389	−0.631944	+	0.775014i	$0.717743\pi$
$68$	− 2568.23i	− 0.555414i
$69$	0	0
$70$	0	0
$71$	3300.84i	0.654799i	0.944886	+	0.327399i	$0.106172\pi$
$71$	3300.84i	0.654799i	−0.944886	+	0.327399i	$0.893828\pi$
$72$	0	0
$73$	9481.81	1.77928	0.889642	−	0.456658i	$-0.150954\pi$
$73$	9481.81	1.77928	0.889642	+	0.456658i	$0.150954\pi$
$74$	− 193.292i	− 0.0352981i
$75$	0	0
$76$	−831.229	−0.143911
$77$	0	0
$78$	0	0
$79$	1730.94	0.277349	0.138675	−	0.990338i	$-0.455716\pi$
$79$	1730.94	0.277349	0.138675	+	0.990338i	$0.455716\pi$
$80$	− 571.494i	− 0.0892960i
$81$	0	0
$82$	−272.630	−0.0405458
$83$	7610.67i	1.10476i	0.833594	+	0.552378i	$0.186280\pi$
$83$	7610.67i	1.10476i	−0.833594	+	0.552378i	$0.813720\pi$
$84$	0	0
$85$	2866.66	0.396770
$86$	774.911i	0.104774i
$87$	0	0
$88$	2546.00	0.328771
$89$	1194.37i	0.150786i	0.997154	+	0.0753928i	$0.0240211\pi$
$89$	1194.37i	0.150786i	−0.997154	+	0.0753928i	$0.975979\pi$
$90$	0	0
$91$	0	0
$92$	961.267i	0.113571i
$93$	0	0
$94$	6305.34	0.713596
$95$	− 927.817i	− 0.102805i
$96$	0	0
$97$	10595.6	1.12611	0.563055	−	0.826420i	$-0.309626\pi$
$97$	10595.6	1.12611	0.563055	+	0.826420i	$0.309626\pi$
$98$	0	0
$99$	0	0
$100$	−4362.10	−0.436210
$101$	1303.52i	0.127783i	0.997957	+	0.0638916i	$0.0203512\pi$
$101$	1303.52i	0.127783i	−0.997957	+	0.0638916i	$0.979649\pi$
$102$	0	0
$103$	16835.8	1.58694	0.793469	−	0.608610i	$-0.208273\pi$
$103$	16835.8	1.58694	0.793469	+	0.608610i	$0.208273\pi$
$104$	1965.80i	0.181749i
$105$	0	0
$106$	−5113.35	−0.455086
$107$	− 4291.73i	− 0.374856i	−0.982278	−	0.187428i	$-0.939985\pi$
$107$	− 4291.73i	− 0.374856i	0.982278	−	0.187428i	$-0.0600151\pi$
$108$	0	0
$109$	2545.66	0.214263	0.107132	−	0.994245i	$-0.465833\pi$
$109$	2545.66	0.214263	0.107132	+	0.994245i	$0.465833\pi$
$110$	2841.85i	0.234863i
$111$	0	0
$112$	0	0
$113$	6580.05i	0.515314i	0.966236	+	0.257657i	$0.0829505\pi$
$113$	6580.05i	0.515314i	−0.966236	+	0.257657i	$0.917049\pi$
$114$	0	0
$115$	−1072.97	−0.0811316
$116$	− 12118.5i	− 0.900601i
$117$	0	0
$118$	−1070.88	−0.0769090
$119$	0	0
$120$	0	0
$121$	1980.60	0.135277
$122$	− 10232.4i	− 0.687475i
$123$	0	0
$124$	10941.6	0.711606
$125$	− 10450.0i	− 0.668798i
$126$	0	0
$127$	−27956.1	−1.73328	−0.866641	−	0.498932i	$-0.833726\pi$
$127$	−27956.1	−1.73328	−0.866641	+	0.498932i	$0.833726\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−2194.22	−0.129836
$131$	− 27130.7i	− 1.58095i	−0.612492	−	0.790476i	$-0.709833\pi$
$131$	− 27130.7i	− 1.58095i	0.612492	−	0.790476i	$-0.290167\pi$
$132$	0	0
$133$	0	0
$134$	16047.3i	0.893704i
$135$	0	0
$136$	−7264.06	−0.392737
$137$	− 34649.0i	− 1.84608i	−0.384706	−	0.923039i	$-0.625697\pi$
$137$	− 34649.0i	− 1.84608i	0.384706	−	0.923039i	$-0.374303\pi$
$138$	0	0
$139$	35399.4	1.83217	0.916086	−	0.400982i	$-0.131331\pi$
$139$	35399.4	1.83217	0.916086	+	0.400982i	$0.131331\pi$
$140$	0	0
$141$	0	0
$142$	9336.19	0.463013
$143$	− 9775.24i	− 0.478030i
$144$	0	0
$145$	13526.7	0.643361
$146$	− 26818.6i	− 1.25814i
$147$	0	0
$148$	−546.714	−0.0249595
$149$	22872.7i	1.03025i	0.857114	+	0.515127i	$0.172255\pi$
$149$	22872.7i	1.03025i	−0.857114	+	0.515127i	$0.827745\pi$
$150$	0	0
$151$	40165.8	1.76158	0.880790	−	0.473507i	$-0.157012\pi$
$151$	40165.8	1.76158	0.880790	+	0.473507i	$0.157012\pi$
$152$	2351.07i	0.101760i
$153$	0	0
$154$	0	0
$155$	12213.1i	0.508348i
$156$	0	0
$157$	44600.0	1.80940	0.904701	−	0.426047i	$-0.140094\pi$
$157$	44600.0	1.80940	0.904701	+	0.426047i	$0.140094\pi$
$158$	− 4895.83i	− 0.196116i
$159$	0	0
$160$	−1616.43	−0.0631418
$161$	0	0
$162$	0	0
$163$	16821.7	0.633131	0.316566	−	0.948571i	$-0.397470\pi$
$163$	16821.7	0.633131	0.316566	+	0.948571i	$0.397470\pi$
$164$	771.115i	0.0286702i
$165$	0	0
$166$	21526.2	0.781181
$167$	48468.6i	1.73791i	0.494889	+	0.868956i	$0.335209\pi$
$167$	48468.6i	1.73791i	−0.494889	+	0.868956i	$0.664791\pi$
$168$	0	0
$169$	−21013.4	−0.735739
$170$	− 8108.14i	− 0.280558i
$171$	0	0
$172$	2191.78	0.0740866
$173$	− 18173.2i	− 0.607210i	−0.952798	−	0.303605i	$-0.901810\pi$
$173$	− 18173.2i	− 0.607210i	0.952798	−	0.303605i	$-0.0981903\pi$
$174$	0	0
$175$	0	0
$176$	− 7201.18i	− 0.232476i
$177$	0	0
$178$	3378.20	0.106621
$179$	26517.9i	0.827624i	0.910362	+	0.413812i	$0.135803\pi$
$179$	26517.9i	0.827624i	−0.910362	+	0.413812i	$0.864197\pi$
$180$	0	0
$181$	−20649.3	−0.630300	−0.315150	−	0.949042i	$-0.602055\pi$
$181$	−20649.3	−0.630300	−0.315150	+	0.949042i	$0.602055\pi$
$182$	0	0
$183$	0	0
$184$	2718.87	0.0803070
$185$	− 610.242i	− 0.0178303i
$186$	0	0
$187$	36121.7	1.03296
$188$	− 17834.2i	− 0.504589i
$189$	0	0
$190$	−2624.26	−0.0726943
$191$	− 56787.3i	− 1.55663i	−0.627875	−	0.778314i	$-0.716075\pi$
$191$	− 56787.3i	− 1.55663i	0.627875	−	0.778314i	$-0.283925\pi$
$192$	0	0
$193$	745.136	0.0200042	0.0100021	−	0.999950i	$-0.496816\pi$
$193$	745.136	0.0200042	0.0100021	+	0.999950i	$0.496816\pi$
$194$	− 29968.8i	− 0.796280i
$195$	0	0
$196$	0	0
$197$	56200.0i	1.44812i	0.689738	+	0.724059i	$0.257726\pi$
$197$	56200.0i	1.44812i	−0.689738	+	0.724059i	$0.742274\pi$
$198$	0	0
$199$	20589.3	0.519920	0.259960	−	0.965619i	$-0.416291\pi$
$199$	20589.3	0.519920	0.259960	+	0.965619i	$0.416291\pi$
$200$	12337.9i	0.308447i
$201$	0	0
$202$	3686.90	0.0903564
$203$	0	0
$204$	0	0
$205$	−860.718	−0.0204811
$206$	− 47618.9i	− 1.12214i
$207$	0	0
$208$	5560.11	0.128516
$209$	− 11691.1i	− 0.267647i
$210$	0	0
$211$	−19310.6	−0.433742	−0.216871	−	0.976200i	$-0.569585\pi$
$211$	−19310.6	−0.433742	−0.216871	+	0.976200i	$0.569585\pi$
$212$	14462.7i	0.321794i
$213$	0	0
$214$	−12138.8	−0.265063
$215$	2446.46i	0.0529251i
$216$	0	0
$217$	0	0
$218$	− 7200.22i	− 0.151507i
$219$	0	0
$220$	8037.96	0.166073
$221$	27890.0i	0.571036i
$222$	0	0
$223$	−31601.9	−0.635483	−0.317741	−	0.948177i	$-0.602924\pi$
$223$	−31601.9	−0.635483	−0.317741	+	0.948177i	$0.602924\pi$
$224$	0	0
$225$	0	0
$226$	18611.2	0.364382
$227$	− 34703.5i	− 0.673476i	−0.941598	−	0.336738i	$-0.890676\pi$
$227$	− 34703.5i	− 0.673476i	0.941598	−	0.336738i	$-0.109324\pi$
$228$	0	0
$229$	71984.5	1.37268	0.686338	−	0.727283i	$-0.259217\pi$
$229$	71984.5	1.37268	0.686338	+	0.727283i	$0.259217\pi$
$230$	3034.81i	0.0573687i
$231$	0	0
$232$	−34276.3	−0.636821
$233$	− 70868.9i	− 1.30540i	−0.757616	−	0.652701i	$-0.773636\pi$
$233$	− 70868.9i	− 1.30540i	0.757616	−	0.652701i	$-0.226364\pi$
$234$	0	0
$235$	19906.5	0.360462
$236$	3028.91i	0.0543829i
$237$	0	0
$238$	0	0
$239$	− 43161.5i	− 0.755616i	−0.925884	−	0.377808i	$-0.876678\pi$
$239$	− 43161.5i	− 0.755616i	0.925884	−	0.377808i	$-0.123322\pi$
$240$	0	0
$241$	74995.9	1.29123	0.645615	−	0.763663i	$-0.276601\pi$
$241$	74995.9	1.29123	0.645615	+	0.763663i	$0.276601\pi$
$242$	− 5601.98i	− 0.0956556i
$243$	0	0
$244$	−28941.5	−0.486118
$245$	0	0
$246$	0	0
$247$	9026.81	0.147959
$248$	− 30947.7i	− 0.503181i
$249$	0	0
$250$	−29557.0	−0.472912
$251$	− 78804.6i	− 1.25085i	−0.780285	−	0.625424i	$-0.784926\pi$
$251$	− 78804.6i	− 1.25085i	0.780285	−	0.625424i	$-0.215074\pi$
$252$	0	0
$253$	−13520.0	−0.211221
$254$	79071.8i	1.22562i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 108179.i	− 1.63785i	−0.573897	−	0.818927i	$-0.694569\pi$
$257$	− 108179.i	− 1.63785i	0.573897	−	0.818927i	$-0.305431\pi$
$258$	0	0
$259$	0	0
$260$	6206.19i	0.0918076i
$261$	0	0
$262$	−76737.3	−1.11790
$263$	2723.70i	0.0393775i	0.999806	+	0.0196888i	$0.00626753\pi$
$263$	2723.70i	0.0393775i	−0.999806	+	0.0196888i	$0.993732\pi$
$264$	0	0
$265$	−16143.3	−0.229880
$266$	0	0
$267$	0	0
$268$	45388.8	0.631944
$269$	77897.4i	1.07651i	0.842782	+	0.538255i	$0.180916\pi$
$269$	77897.4i	1.07651i	−0.842782	+	0.538255i	$0.819084\pi$
$270$	0	0
$271$	−5738.75	−0.0781410	−0.0390705	−	0.999236i	$-0.512440\pi$
$271$	−5738.75	−0.0781410	−0.0390705	+	0.999236i	$0.512440\pi$
$272$	20545.9i	0.277707i
$273$	0	0
$274$	−98002.3	−1.30537
$275$	− 61352.1i	− 0.811267i
$276$	0	0
$277$	35951.5	0.468552	0.234276	−	0.972170i	$-0.424728\pi$
$277$	35951.5	0.468552	0.234276	+	0.972170i	$0.424728\pi$
$278$	− 100125.i	− 1.29554i
$279$	0	0
$280$	0	0
$281$	142606.i	1.80603i	0.429606	+	0.903016i	$0.358652\pi$
$281$	142606.i	1.80603i	−0.429606	+	0.903016i	$0.641348\pi$
$282$	0	0
$283$	−33189.5	−0.414407	−0.207204	−	0.978298i	$-0.566436\pi$
$283$	−33189.5	−0.414407	−0.207204	+	0.978298i	$0.566436\pi$
$284$	− 26406.7i	− 0.327399i
$285$	0	0
$286$	−27648.5	−0.338018
$287$	0	0
$288$	0	0
$289$	−19538.7	−0.233937
$290$	− 38259.2i	− 0.454925i
$291$	0	0
$292$	−75854.5	−0.889642
$293$	− 46620.5i	− 0.543053i	−0.962431	−	0.271526i	$-0.912472\pi$
$293$	− 46620.5i	− 0.543053i	0.962431	−	0.271526i	$-0.0875284\pi$
$294$	0	0
$295$	−3380.87	−0.0388494
$296$	1546.34i	0.0176491i
$297$	0	0
$298$	64693.6	0.728499
$299$	− 10439.0i	− 0.116766i
$300$	0	0
$301$	0	0
$302$	− 113606.i	− 1.24563i
$303$	0	0
$304$	6649.83	0.0719554
$305$	− 32304.5i	− 0.347267i
$306$	0	0
$307$	103384.	1.09692	0.548462	−	0.836175i	$-0.315213\pi$
$307$	103384.	1.09692	0.548462	+	0.836175i	$0.315213\pi$
$308$	0	0
$309$	0	0
$310$	34543.8	0.359456
$311$	42595.6i	0.440396i	0.975455	+	0.220198i	$0.0706704\pi$
$311$	42595.6i	0.440396i	−0.975455	+	0.220198i	$0.929330\pi$
$312$	0	0
$313$	40598.7	0.414404	0.207202	−	0.978298i	$-0.433564\pi$
$313$	40598.7	0.414404	0.207202	+	0.978298i	$0.433564\pi$
$314$	− 126148.i	− 1.27944i
$315$	0	0
$316$	−13847.5	−0.138675
$317$	− 116222.i	− 1.15657i	−0.815836	−	0.578284i	$-0.803723\pi$
$317$	− 116222.i	− 1.15657i	0.815836	−	0.578284i	$-0.196277\pi$
$318$	0	0
$319$	170444.	1.67495
$320$	4571.95i	0.0446480i
$321$	0	0
$322$	0	0
$323$	33356.1i	0.319720i
$324$	0	0
$325$	47370.6	0.448479
$326$	− 47578.8i	− 0.447691i
$327$	0	0
$328$	2181.04	0.0202729
$329$	0	0
$330$	0	0
$331$	−140654.	−1.28379	−0.641897	−	0.766791i	$-0.721852\pi$
$331$	−140654.	−1.28379	−0.641897	+	0.766791i	$0.721852\pi$
$332$	− 60885.3i	− 0.552378i
$333$	0	0
$334$	137090.	1.22889
$335$	50662.9i	0.451440i
$336$	0	0
$337$	−88836.7	−0.782226	−0.391113	−	0.920343i	$-0.627910\pi$
$337$	−88836.7	−0.782226	−0.391113	+	0.920343i	$0.627910\pi$
$338$	59435.0i	0.520246i
$339$	0	0
$340$	−22933.3	−0.198385
$341$	153892.i	1.32345i
$342$	0	0
$343$	0	0
$344$	− 6199.29i	− 0.0523872i
$345$	0	0
$346$	−51401.6	−0.429362
$347$	119666.i	0.993827i	0.867800	+	0.496914i	$0.165533\pi$
$347$	119666.i	0.993827i	−0.867800	+	0.496914i	$0.834467\pi$
$348$	0	0
$349$	8621.89	0.0707867	0.0353933	−	0.999373i	$-0.488732\pi$
$349$	8621.89	0.0707867	0.0353933	+	0.999373i	$0.488732\pi$
$350$	0	0
$351$	0	0
$352$	−20368.0	−0.164385
$353$	150238.i	1.20568i	0.797863	+	0.602838i	$0.205964\pi$
$353$	150238.i	1.20568i	−0.797863	+	0.602838i	$0.794036\pi$
$354$	0	0
$355$	29475.2	0.233884
$356$	− 9554.98i	− 0.0753928i
$357$	0	0
$358$	75004.0	0.585219
$359$	38588.1i	0.299409i	0.988731	+	0.149704i	$0.0478322\pi$
$359$	38588.1i	0.299409i	−0.988731	+	0.149704i	$0.952168\pi$
$360$	0	0
$361$	−119525.	−0.917159
$362$	58404.9i	0.445689i
$363$	0	0
$364$	0	0
$365$	− 84668.7i	− 0.635532i
$366$	0	0
$367$	126569.	0.939713	0.469857	−	0.882743i	$-0.344306\pi$
$367$	126569.	0.939713	0.469857	+	0.882743i	$0.344306\pi$
$368$	− 7690.14i	− 0.0567856i
$369$	0	0
$370$	−1726.02	−0.0126079
$371$	0	0
$372$	0	0
$373$	−140774.	−1.01182	−0.505911	−	0.862585i	$-0.668844\pi$
$373$	−140774.	−1.01182	−0.505911	+	0.862585i	$0.668844\pi$
$374$	− 102168.i	− 0.730415i
$375$	0	0
$376$	−50442.7	−0.356798
$377$	131602.i	0.925933i
$378$	0	0
$379$	155176.	1.08030	0.540152	−	0.841568i	$-0.318367\pi$
$379$	155176.	1.08030	0.540152	+	0.841568i	$0.318367\pi$
$380$	7422.54i	0.0514026i
$381$	0	0
$382$	−160619.	−1.10070
$383$	− 72926.8i	− 0.497153i	−0.968612	−	0.248576i	$-0.920037\pi$
$383$	− 72926.8i	− 0.497153i	0.968612	−	0.248576i	$-0.0799627\pi$
$384$	0	0
$385$	0	0
$386$	− 2107.56i	− 0.0141451i
$387$	0	0
$388$	−84764.5	−0.563055
$389$	65989.8i	0.436091i	0.975939	+	0.218046i	$0.0699682\pi$
$389$	65989.8i	0.436091i	−0.975939	+	0.218046i	$0.930032\pi$
$390$	0	0
$391$	38574.3	0.252316
$392$	0	0
$393$	0	0
$394$	158958.	1.02397
$395$	− 15456.6i	− 0.0990647i
$396$	0	0
$397$	−279796.	−1.77526	−0.887628	−	0.460561i	$-0.847648\pi$
$397$	−279796.	−1.77526	−0.887628	+	0.460561i	$0.847648\pi$
$398$	− 58235.5i	− 0.367639i
$399$	0	0
$400$	34896.8	0.218105
$401$	134654.i	0.837397i	0.908125	+	0.418698i	$0.137514\pi$
$401$	134654.i	0.837397i	−0.908125	+	0.418698i	$0.862486\pi$
$402$	0	0
$403$	−118822.	−0.731621
$404$	− 10428.1i	− 0.0638916i
$405$	0	0
$406$	0	0
$407$	− 7689.42i	− 0.0464200i
$408$	0	0
$409$	58559.8	0.350068	0.175034	−	0.984562i	$-0.443996\pi$
$409$	58559.8	0.350068	0.175034	+	0.984562i	$0.443996\pi$
$410$	2434.48i	0.0144823i
$411$	0	0
$412$	−134687.	−0.793469
$413$	0	0
$414$	0	0
$415$	67960.2	0.394601
$416$	− 15726.4i	− 0.0908745i
$417$	0	0
$418$	−33067.3	−0.189255
$419$	204121.i	1.16268i	0.813661	+	0.581339i	$0.197471\pi$
$419$	204121.i	1.16268i	−0.813661	+	0.581339i	$0.802529\pi$
$420$	0	0
$421$	254823.	1.43772	0.718859	−	0.695156i	$-0.244665\pi$
$421$	254823.	1.43772	0.718859	+	0.695156i	$0.244665\pi$
$422$	54618.6i	0.306702i
$423$	0	0
$424$	40906.8	0.227543
$425$	175045.i	0.969108i
$426$	0	0
$427$	0	0
$428$	34333.8i	0.187428i
$429$	0	0
$430$	6919.64	0.0374237
$431$	− 63107.2i	− 0.339722i	−0.985468	−	0.169861i	$-0.945668\pi$
$431$	− 63107.2i	− 0.339722i	0.985468	−	0.169861i	$-0.0543319\pi$
$432$	0	0
$433$	71889.3	0.383432	0.191716	−	0.981450i	$-0.438595\pi$
$433$	71889.3	0.383432	0.191716	+	0.981450i	$0.438595\pi$
$434$	0	0
$435$	0	0
$436$	−20365.3	−0.107132
$437$	− 12484.9i	− 0.0653765i
$438$	0	0
$439$	−11227.1	−0.0582555	−0.0291277	−	0.999576i	$-0.509273\pi$
$439$	−11227.1	−0.0582555	−0.0291277	+	0.999576i	$0.509273\pi$
$440$	− 22734.8i	− 0.117432i
$441$	0	0
$442$	78884.7	0.403783
$443$	128525.i	0.654907i	0.944867	+	0.327453i	$0.106190\pi$
$443$	128525.i	0.654907i	−0.944867	+	0.327453i	$0.893810\pi$
$444$	0	0
$445$	10665.3	0.0538582
$446$	89383.7i	0.449354i
$447$	0	0
$448$	0	0
$449$	− 288439.i	− 1.43074i	−0.698745	−	0.715371i	$-0.746258\pi$
$449$	− 288439.i	− 1.43074i	0.698745	−	0.715371i	$-0.253742\pi$
$450$	0	0
$451$	−10845.6	−0.0533212
$452$	− 52640.4i	− 0.257657i
$453$	0	0
$454$	−98156.4	−0.476219
$455$	0	0
$456$	0	0
$457$	141946.	0.679657	0.339828	−	0.940487i	$-0.389631\pi$
$457$	141946.	0.679657	0.339828	+	0.940487i	$0.389631\pi$
$458$	− 203603.i	− 0.970629i
$459$	0	0
$460$	8583.73	0.0405658
$461$	387036.i	1.82117i	0.413326	+	0.910583i	$0.364367\pi$
$461$	387036.i	1.82117i	−0.413326	+	0.910583i	$0.635633\pi$
$462$	0	0
$463$	194710.	0.908293	0.454147	−	0.890927i	$-0.349944\pi$
$463$	194710.	0.908293	0.454147	+	0.890927i	$0.349944\pi$
$464$	96947.9i	0.450301i
$465$	0	0
$466$	−200448.	−0.923058
$467$	118589.i	0.543766i	0.962330	+	0.271883i	$0.0876464\pi$
$467$	118589.i	0.543766i	−0.962330	+	0.271883i	$0.912354\pi$
$468$	0	0
$469$	0	0
$470$	− 56304.1i	− 0.254885i
$471$	0	0
$472$	8567.04	0.0384545
$473$	30826.9i	0.137787i
$474$	0	0
$475$	56654.7	0.251101
$476$	0	0
$477$	0	0
$478$	−122079.	−0.534301
$479$	337078.i	1.46913i	0.678540	+	0.734563i	$0.262613\pi$
$479$	337078.i	1.46913i	−0.678540	+	0.734563i	$0.737387\pi$
$480$	0	0
$481$	5937.09	0.0256616
$482$	− 212121.i	− 0.913037i
$483$	0	0
$484$	−15844.8	−0.0676387
$485$	− 94614.1i	− 0.402228i
$486$	0	0
$487$	−404826.	−1.70691	−0.853454	−	0.521167i	$-0.825497\pi$
$487$	−404826.	−1.70691	−0.853454	+	0.521167i	$0.825497\pi$
$488$	81859.0i	0.343737i
$489$	0	0
$490$	0	0
$491$	282619.i	1.17230i	0.810203	+	0.586149i	$0.199357\pi$
$491$	282619.i	1.17230i	−0.810203	+	0.586149i	$0.800643\pi$
$492$	0	0
$493$	−486299.	−2.00083
$494$	− 25531.7i	− 0.104623i
$495$	0	0
$496$	−87533.2	−0.355803
$497$	0	0
$498$	0	0
$499$	−213524.	−0.857523	−0.428762	−	0.903418i	$-0.641050\pi$
$499$	−213524.	−0.857523	−0.428762	+	0.903418i	$0.641050\pi$
$500$	83599.8i	0.334399i
$501$	0	0
$502$	−222893.	−0.884482
$503$	− 47839.3i	− 0.189081i	−0.995521	−	0.0945407i	$-0.969862\pi$
$503$	− 47839.3i	− 0.189081i	0.995521	−	0.0945407i	$-0.0301382\pi$
$504$	0	0
$505$	11639.9	0.0456421
$506$	38240.4i	0.149356i
$507$	0	0
$508$	223649.	0.866641
$509$	418791.i	1.61645i	0.588875	+	0.808224i	$0.299571\pi$
$509$	418791.i	1.61645i	−0.588875	+	0.808224i	$0.700429\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−305975.	−1.15814
$515$	− 150337.i	− 0.566829i
$516$	0	0
$517$	250834.	0.938439
$518$	0	0
$519$	0	0
$520$	17553.8	0.0649178
$521$	275519.i	1.01502i	0.861645	+	0.507511i	$0.169434\pi$
$521$	275519.i	1.01502i	−0.861645	+	0.507511i	$0.830566\pi$
$522$	0	0
$523$	−392490.	−1.43491	−0.717456	−	0.696604i	$-0.754694\pi$
$523$	−392490.	−1.43491	−0.717456	+	0.696604i	$0.754694\pi$
$524$	217046.i	0.790476i
$525$	0	0
$526$	7703.79	0.0278441
$527$	− 439074.i	− 1.58094i
$528$	0	0
$529$	265403.	0.948406
$530$	45660.1i	0.162549i
$531$	0	0
$532$	0	0
$533$	− 8373.99i	− 0.0294767i
$534$	0	0
$535$	−38323.4	−0.133893
$536$	− 128379.i	− 0.446852i
$537$	0	0
$538$	220327.	0.761208
$539$	0	0
$540$	0	0
$541$	−217314.	−0.742493	−0.371247	−	0.928534i	$-0.621070\pi$
$541$	−217314.	−0.742493	−0.371247	+	0.928534i	$0.621070\pi$
$542$	16231.6i	0.0552540i
$543$	0	0
$544$	58112.5	0.196368
$545$	− 22731.7i	− 0.0765314i
$546$	0	0
$547$	179744.	0.600730	0.300365	−	0.953824i	$-0.402891\pi$
$547$	179744.	0.600730	0.300365	+	0.953824i	$0.402891\pi$
$548$	277192.i	0.923039i
$549$	0	0
$550$	−173530.	−0.573652
$551$	157394.i	0.518425i
$552$	0	0
$553$	0	0
$554$	− 101686.i	− 0.331316i
$555$	0	0
$556$	−283195.	−0.916086
$557$	424124.i	1.36704i	0.729930	+	0.683522i	$0.239552\pi$
$557$	424124.i	1.36704i	−0.729930	+	0.683522i	$0.760448\pi$
$558$	0	0
$559$	−23801.8	−0.0761705
$560$	0	0
$561$	0	0
$562$	403351.	1.27706
$563$	593898.i	1.87368i	0.349761	+	0.936839i	$0.386263\pi$
$563$	593898.i	1.87368i	−0.349761	+	0.936839i	$0.613737\pi$
$564$	0	0
$565$	58757.2	0.184062
$566$	93874.0i	0.293030i
$567$	0	0
$568$	−74689.5	−0.231506
$569$	265516.i	0.820098i	0.912064	+	0.410049i	$0.134488\pi$
$569$	265516.i	0.820098i	−0.912064	+	0.410049i	$0.865512\pi$
$570$	0	0
$571$	−123293.	−0.378153	−0.189076	−	0.981962i	$-0.560549\pi$
$571$	−123293.	−0.378153	−0.189076	+	0.981962i	$0.560549\pi$
$572$	78201.9i	0.239015i
$573$	0	0
$574$	0	0
$575$	− 65517.8i	− 0.198164i
$576$	0	0
$577$	345662.	1.03825	0.519123	−	0.854699i	$-0.326259\pi$
$577$	345662.	1.03825	0.519123	+	0.854699i	$0.326259\pi$
$578$	55263.8i	0.165419i
$579$	0	0
$580$	−108213.	−0.321680
$581$	0	0
$582$	0	0
$583$	−203415.	−0.598476
$584$	214549.i	0.629072i
$585$	0	0
$586$	−131863.	−0.383996
$587$	13288.5i	0.0385655i	0.999814	+	0.0192828i	$0.00613827\pi$
$587$	13288.5i	0.0385655i	−0.999814	+	0.0192828i	$0.993862\pi$
$588$	0	0
$589$	−142110.	−0.409631
$590$	9562.53i	0.0274706i
$591$	0	0
$592$	4373.71	0.0124798
$593$	− 236953.i	− 0.673835i	−0.941534	−	0.336917i	$-0.890616\pi$
$593$	− 236953.i	− 0.673835i	0.941534	−	0.336917i	$-0.109384\pi$
$594$	0	0
$595$	0	0
$596$	− 182981.i	− 0.515127i
$597$	0	0
$598$	−29525.9	−0.0825658
$599$	− 169331.i	− 0.471935i	−0.971761	−	0.235967i	$-0.924174\pi$
$599$	− 169331.i	− 0.471935i	0.971761	−	0.235967i	$-0.0758259\pi$
$600$	0	0
$601$	−79140.3	−0.219103	−0.109552	−	0.993981i	$-0.534942\pi$
$601$	−79140.3	−0.219103	−0.109552	+	0.993981i	$0.534942\pi$
$602$	0	0
$603$	0	0
$604$	−321326.	−0.880790
$605$	− 17685.9i	− 0.0483189i
$606$	0	0
$607$	−171036.	−0.464207	−0.232103	−	0.972691i	$-0.574561\pi$
$607$	−171036.	−0.464207	−0.232103	+	0.972691i	$0.574561\pi$
$608$	− 18808.6i	− 0.0508802i
$609$	0	0
$610$	−91371.0	−0.245555
$611$	193672.i	0.518782i
$612$	0	0
$613$	24686.3	0.0656955	0.0328478	−	0.999460i	$-0.489542\pi$
$613$	24686.3	0.0656955	0.0328478	+	0.999460i	$0.489542\pi$
$614$	− 292414.i	− 0.775643i
$615$	0	0
$616$	0	0
$617$	33719.1i	0.0885739i	0.999019	+	0.0442869i	$0.0141016\pi$
$617$	33719.1i	0.0885739i	−0.999019	+	0.0442869i	$0.985898\pi$
$618$	0	0
$619$	−346054.	−0.903157	−0.451578	−	0.892231i	$-0.649139\pi$
$619$	−346054.	−0.903157	−0.451578	+	0.892231i	$0.649139\pi$
$620$	− 97704.5i	− 0.254174i
$621$	0	0
$622$	120478.	0.311407
$623$	0	0
$624$	0	0
$625$	247475.	0.633536
$626$	− 114831.i	− 0.293028i
$627$	0	0
$628$	−356800.	−0.904701
$629$	21938.9i	0.0554515i
$630$	0	0
$631$	634074.	1.59251	0.796253	−	0.604964i	$-0.206812\pi$
$631$	634074.	1.59251	0.796253	+	0.604964i	$0.206812\pi$
$632$	39166.6i	0.0980578i
$633$	0	0
$634$	−328726.	−0.817817
$635$	249637.i	0.619100i
$636$	0	0
$637$	0	0
$638$	− 482089.i	− 1.18437i
$639$	0	0
$640$	12931.4	0.0315709
$641$	143616.i	0.349533i	0.984610	+	0.174766i	$0.0559170\pi$
$641$	143616.i	0.349533i	−0.984610	+	0.174766i	$0.944083\pi$
$642$	0	0
$643$	−770556.	−1.86373	−0.931863	−	0.362810i	$-0.881817\pi$
$643$	−770556.	−1.86373	−0.931863	+	0.362810i	$0.881817\pi$
$644$	0	0
$645$	0	0
$646$	94345.2	0.226076
$647$	− 510348.i	− 1.21915i	−0.792728	−	0.609576i	$-0.791340\pi$
$647$	− 510348.i	− 1.21915i	0.792728	−	0.609576i	$-0.208660\pi$
$648$	0	0
$649$	−42601.0	−0.101142
$650$	− 133984.i	− 0.317123i
$651$	0	0
$652$	−134573.	−0.316566
$653$	− 491359.i	− 1.15232i	−0.817337	−	0.576160i	$-0.804551\pi$
$653$	− 491359.i	− 1.15232i	0.817337	−	0.576160i	$-0.195449\pi$
$654$	0	0
$655$	−242266.	−0.564691
$656$	− 6168.92i	− 0.0143351i
$657$	0	0
$658$	0	0
$659$	− 84282.5i	− 0.194074i	−0.995281	−	0.0970368i	$-0.969064\pi$
$659$	− 84282.5i	− 0.194074i	0.995281	−	0.0970368i	$-0.0309365\pi$
$660$	0	0
$661$	79159.6	0.181176	0.0905879	−	0.995888i	$-0.471125\pi$
$661$	79159.6	0.181176	0.0905879	+	0.995888i	$0.471125\pi$
$662$	397829.i	0.907779i
$663$	0	0
$664$	−172210.	−0.390590
$665$	0	0
$666$	0	0
$667$	182017.	0.409130
$668$	− 387749.i	− 0.868956i
$669$	0	0
$670$	143296.	0.319217
$671$	− 407057.i	− 0.904087i
$672$	0	0
$673$	145295.	0.320789	0.160395	−	0.987053i	$-0.448723\pi$
$673$	145295.	0.320789	0.160395	+	0.987053i	$0.448723\pi$
$674$	251268.i	0.553117i
$675$	0	0
$676$	168107.	0.367869
$677$	731984.i	1.59707i	0.601947	+	0.798536i	$0.294392\pi$
$677$	731984.i	1.59707i	−0.601947	+	0.798536i	$0.705608\pi$
$678$	0	0
$679$	0	0
$680$	64865.1i	0.140279i
$681$	0	0
$682$	435273.	0.935821
$683$	− 212269.i	− 0.455036i	−0.973774	−	0.227518i	$-0.926939\pi$
$683$	− 212269.i	− 0.455036i	0.973774	−	0.227518i	$-0.0730611\pi$
$684$	0	0
$685$	−309402.	−0.659389
$686$	0	0
$687$	0	0
$688$	−17534.2	−0.0370433
$689$	− 157059.i	− 0.330846i
$690$	0	0
$691$	−414702.	−0.868521	−0.434260	−	0.900787i	$-0.642990\pi$
$691$	−414702.	−0.868521	−0.434260	+	0.900787i	$0.642990\pi$
$692$	145386.i	0.303605i
$693$	0	0
$694$	338466.	0.702742
$695$	− 316102.i	− 0.654422i
$696$	0	0
$697$	30943.8	0.0636954
$698$	− 24386.4i	− 0.0500537i
$699$	0	0
$700$	0	0
$701$	− 834753.i	− 1.69872i	−0.527813	−	0.849360i	$-0.676988\pi$
$701$	− 834753.i	− 1.69872i	0.527813	−	0.849360i	$-0.323012\pi$
$702$	0	0
$703$	7100.69	0.0143678
$704$	57609.4i	0.116238i
$705$	0	0
$706$	424938.	0.852542
$707$	0	0
$708$	0	0
$709$	−726862.	−1.44597	−0.722985	−	0.690864i	$-0.757230\pi$
$709$	−726862.	−1.44597	−0.722985	+	0.690864i	$0.757230\pi$
$710$	− 83368.4i	− 0.165381i
$711$	0	0
$712$	−27025.6	−0.0533107
$713$	164341.i	0.323272i
$714$	0	0
$715$	−87288.9	−0.170745
$716$	− 212143.i	− 0.413812i
$717$	0	0
$718$	109144.	0.211714
$719$	− 807855.i	− 1.56270i	−0.624093	−	0.781350i	$-0.714531\pi$
$719$	− 807855.i	− 1.56270i	0.624093	−	0.781350i	$-0.285469\pi$
$720$	0	0
$721$	0	0
$722$	338068.i	0.648529i
$723$	0	0
$724$	165194.	0.315150
$725$	825970.i	1.57140i
$726$	0	0
$727$	35185.3	0.0665721	0.0332860	−	0.999446i	$-0.489403\pi$
$727$	35185.3	0.0665721	0.0332860	+	0.999446i	$0.489403\pi$
$728$	0	0
$729$	0	0
$730$	−239479.	−0.449389
$731$	− 87953.1i	− 0.164595i
$732$	0	0
$733$	750177.	1.39623	0.698113	−	0.715987i	$-0.254023\pi$
$733$	750177.	1.39623	0.698113	+	0.715987i	$0.254023\pi$
$734$	− 357991.i	− 0.664478i
$735$	0	0
$736$	−21751.0	−0.0401535
$737$	638384.i	1.17530i
$738$	0	0
$739$	329895.	0.604070	0.302035	−	0.953297i	$-0.402334\pi$
$739$	329895.	0.604070	0.302035	+	0.953297i	$0.402334\pi$
$740$	4881.93i	0.00891514i
$741$	0	0
$742$	0	0
$743$	743521.i	1.34684i	0.739260	+	0.673420i	$0.235175\pi$
$743$	743521.i	1.34684i	−0.739260	+	0.673420i	$0.764825\pi$
$744$	0	0
$745$	204244.	0.367990
$746$	398169.i	0.715467i
$747$	0	0
$748$	−288974.	−0.516482
$749$	0	0
$750$	0	0
$751$	−61124.3	−0.108376	−0.0541881	−	0.998531i	$-0.517257\pi$
$751$	−61124.3	−0.108376	−0.0541881	+	0.998531i	$0.517257\pi$
$752$	142674.i	0.252294i
$753$	0	0
$754$	372226.	0.654733
$755$	− 358664.i	− 0.629208i
$756$	0	0
$757$	20104.5	0.0350834	0.0175417	−	0.999846i	$-0.494416\pi$
$757$	20104.5	0.0350834	0.0175417	+	0.999846i	$0.494416\pi$
$758$	− 438903.i	− 0.763890i
$759$	0	0
$760$	20994.1	0.0363471
$761$	− 517675.i	− 0.893898i	−0.894559	−	0.446949i	$-0.852511\pi$
$761$	− 517675.i	− 0.893898i	0.894559	−	0.446949i	$-0.147489\pi$
$762$	0	0
$763$	0	0
$764$	454299.i	0.778314i
$765$	0	0
$766$	−206268.	−0.351540
$767$	− 32892.7i	− 0.0559125i
$768$	0	0
$769$	734173.	1.24150	0.620748	−	0.784010i	$-0.286829\pi$
$769$	734173.	1.24150	0.620748	+	0.784010i	$0.286829\pi$
$770$	0	0
$771$	0	0
$772$	−5961.09	−0.0100021
$773$	− 664138.i	− 1.11147i	−0.831358	−	0.555737i	$-0.812436\pi$
$773$	− 664138.i	− 1.11147i	0.831358	−	0.555737i	$-0.187564\pi$
$774$	0	0
$775$	−745759.	−1.24164
$776$	239750.i	0.398140i
$777$	0	0
$778$	186647.	0.308363
$779$	− 10015.2i	− 0.0165038i
$780$	0	0
$781$	371406.	0.608900
$782$	− 109105.i	− 0.178414i
$783$	0	0
$784$	0	0
$785$	− 398260.i	− 0.646289i
$786$	0	0
$787$	583342.	0.941833	0.470917	−	0.882178i	$-0.343923\pi$
$787$	583342.	0.941833	0.470917	+	0.882178i	$0.343923\pi$
$788$	− 449600.i	− 0.724059i
$789$	0	0
$790$	−43717.8	−0.0700493
$791$	0	0
$792$	0	0
$793$	314293.	0.499791
$794$	791383.i	1.25530i
$795$	0	0
$796$	−164715.	−0.259960
$797$	− 506231.i	− 0.796952i	−0.917179	−	0.398476i	$-0.869539\pi$
$797$	− 506231.i	− 0.796952i	0.917179	−	0.398476i	$-0.130461\pi$
$798$	0	0
$799$	−715662.	−1.12102
$800$	− 98703.0i	− 0.154223i
$801$	0	0
$802$	380860.	0.592129
$803$	− 1.06688e6i	− 1.65457i
$804$	0	0
$805$	0	0
$806$	336079.i	0.517334i
$807$	0	0
$808$	−29495.2	−0.0451782
$809$	− 856750.i	− 1.30905i	−0.756039	−	0.654526i	$-0.772868\pi$
$809$	− 856750.i	− 1.30905i	0.756039	−	0.654526i	$-0.227132\pi$
$810$	0	0
$811$	78192.8	0.118884	0.0594422	−	0.998232i	$-0.481068\pi$
$811$	78192.8	0.118884	0.0594422	+	0.998232i	$0.481068\pi$
$812$	0	0
$813$	0	0
$814$	−21749.0	−0.0328239
$815$	− 150211.i	− 0.226144i
$816$	0	0
$817$	−28466.7	−0.0426475
$818$	− 165632.i	− 0.247536i
$819$	0	0
$820$	6885.74	0.0102405
$821$	87472.6i	0.129773i	0.997893	+	0.0648867i	$0.0206686\pi$
$821$	87472.6i	0.129773i	−0.997893	+	0.0648867i	$0.979331\pi$
$822$	0	0
$823$	658056.	0.971545	0.485773	−	0.874085i	$-0.338538\pi$
$823$	658056.	0.971545	0.485773	+	0.874085i	$0.338538\pi$
$824$	380951.i	0.561068i
$825$	0	0
$826$	0	0
$827$	612045.i	0.894895i	0.894310	+	0.447448i	$0.147667\pi$
$827$	612045.i	0.894895i	−0.894310	+	0.447448i	$0.852333\pi$
$828$	0	0
$829$	−1.05889e6	−1.54078	−0.770391	−	0.637572i	$-0.779939\pi$
$829$	−1.05889e6	−1.54078	−0.770391	+	0.637572i	$0.779939\pi$
$830$	− 192220.i	− 0.279025i
$831$	0	0
$832$	−44480.9	−0.0642579
$833$	0	0
$834$	0	0
$835$	432805.	0.620754
$836$	93528.6i	0.133823i
$837$	0	0
$838$	577342.	0.822138
$839$	− 162821.i	− 0.231305i	−0.993290	−	0.115653i	$-0.963104\pi$
$839$	− 162821.i	− 0.231305i	0.993290	−	0.115653i	$-0.0368959\pi$
$840$	0	0
$841$	−1.58737e6	−2.24433
$842$	− 720747.i	− 1.01662i
$843$	0	0
$844$	154485.	0.216871
$845$	187641.i	0.262794i
$846$	0	0
$847$	0	0
$848$	− 115702.i	− 0.160897i
$849$	0	0
$850$	495102.	0.685263
$851$	− 8211.53i	− 0.0113387i
$852$	0	0
$853$	−19530.5	−0.0268421	−0.0134210	−	0.999910i	$-0.504272\pi$
$853$	−19530.5	−0.0268421	−0.0134210	+	0.999910i	$0.504272\pi$
$854$	0	0
$855$	0	0
$856$	97110.7	0.132532
$857$	722174.i	0.983287i	0.870797	+	0.491643i	$0.163604\pi$
$857$	722174.i	0.983287i	−0.870797	+	0.491643i	$0.836396\pi$
$858$	0	0
$859$	1.42212e6	1.92730	0.963649	−	0.267173i	$-0.0860896\pi$
$859$	1.42212e6	1.92730	0.963649	+	0.267173i	$0.0860896\pi$
$860$	− 19571.7i	− 0.0264625i
$861$	0	0
$862$	−178494.	−0.240220
$863$	995911.i	1.33721i	0.743619	+	0.668604i	$0.233108\pi$
$863$	995911.i	1.33721i	−0.743619	+	0.668604i	$0.766892\pi$
$864$	0	0
$865$	−162279.	−0.216886
$866$	− 203334.i	− 0.271128i
$867$	0	0
$868$	0	0
$869$	− 194762.i	− 0.257908i
$870$	0	0
$871$	−492903.	−0.649719
$872$	57601.7i	0.0757535i
$873$	0	0
$874$	−35312.6	−0.0462282
$875$	0	0
$876$	0	0
$877$	1.01951e6	1.32553	0.662766	−	0.748826i	$-0.269382\pi$
$877$	1.01951e6	1.32553	0.662766	+	0.748826i	$0.269382\pi$
$878$	31754.9i	0.0411929i
$879$	0	0
$880$	−64303.6	−0.0830367
$881$	− 474634.i	− 0.611515i	−0.952109	−	0.305757i	$-0.901090\pi$
$881$	− 474634.i	− 0.611515i	0.952109	−	0.305757i	$-0.0989096\pi$
$882$	0	0
$883$	−916167.	−1.17504	−0.587521	−	0.809209i	$-0.699896\pi$
$883$	−916167.	−1.17504	−0.587521	+	0.809209i	$0.699896\pi$
$884$	− 223120.i	− 0.285518i
$885$	0	0
$886$	363523.	0.463089
$887$	− 139541.i	− 0.177359i	−0.996060	−	0.0886796i	$-0.971735\pi$
$887$	− 139541.i	− 0.177359i	0.996060	−	0.0886796i	$-0.0282647\pi$
$888$	0	0
$889$	0	0
$890$	− 30165.9i	− 0.0380835i
$891$	0	0
$892$	252815.	0.317741
$893$	231630.i	0.290463i
$894$	0	0
$895$	236794.	0.295614
$896$	0	0
$897$	0	0
$898$	−815829.	−1.01169
$899$	− 2.07182e6i	− 2.56349i
$900$	0	0
$901$	580369.	0.714916
$902$	30675.9i	0.0377038i
$903$	0	0
$904$	−148890.	−0.182191
$905$	184389.i	0.225133i
$906$	0	0
$907$	−653823.	−0.794778	−0.397389	−	0.917650i	$-0.630084\pi$
$907$	−653823.	−0.794778	−0.397389	+	0.917650i	$0.630084\pi$
$908$	277628.i	0.336738i
$909$	0	0
$910$	0	0
$911$	− 545.357i	0 0.000657119i	−1.00000		0.000328560i	$-0.999895\pi$
$911$	− 545.357i	0 0.000657119i	1.00000		0.000328560i	$-0.000104584\pi$
$912$	0	0
$913$	856341.	1.02732
$914$	− 401483.i	− 0.480590i
$915$	0	0
$916$	−575876.	−0.686338
$917$	0	0
$918$	0	0
$919$	50339.2	0.0596040	0.0298020	−	0.999556i	$-0.490512\pi$
$919$	50339.2	0.0596040	0.0298020	+	0.999556i	$0.490512\pi$
$920$	− 24278.4i	− 0.0286844i
$921$	0	0
$922$	1.09470e6	1.28776
$923$	286766.i	0.336608i
$924$	0	0
$925$	37262.8	0.0435504
$926$	− 550723.i	− 0.642260i
$927$	0	0
$928$	274210.	0.318411
$929$	− 962126.i	− 1.11481i	−0.830241	−	0.557405i	$-0.811797\pi$
$929$	− 962126.i	− 1.11481i	0.830241	−	0.557405i	$-0.188203\pi$
$930$	0	0
$931$	0	0
$932$	566952.i	0.652701i
$933$	0	0
$934$	335422.	0.384501
$935$	− 322552.i	− 0.368958i
$936$	0	0
$937$	−305667.	−0.348152	−0.174076	−	0.984732i	$-0.555694\pi$
$937$	−305667.	−0.348152	−0.174076	+	0.984732i	$0.555694\pi$
$938$	0	0
$939$	0	0
$940$	−159252.	−0.180231
$941$	253654.i	0.286459i	0.989690	+	0.143229i	$0.0457487\pi$
$941$	253654.i	0.286459i	−0.989690	+	0.143229i	$0.954251\pi$
$942$	0	0
$943$	−11582.0	−0.0130245
$944$	− 24231.3i	− 0.0271914i
$945$	0	0
$946$	87191.8	0.0974301
$947$	− 1.51787e6i	− 1.69252i	−0.532771	−	0.846259i	$-0.678849\pi$
$947$	− 1.51787e6i	− 1.69252i	0.532771	−	0.846259i	$-0.321151\pi$
$948$	0	0
$949$	823749.	0.914666
$950$	− 160244.i	− 0.177555i
$951$	0	0
$952$	0	0
$953$	− 21086.0i	− 0.0232171i	−0.999933	−	0.0116085i	$-0.996305\pi$
$953$	− 21086.0i	− 0.0232171i	0.999933	−	0.0116085i	$-0.00369520\pi$
$954$	0	0
$955$	−507088.	−0.556002
$956$	345292.i	0.377808i
$957$	0	0
$958$	953400.	1.03883
$959$	0	0
$960$	0	0
$961$	947099.	1.02553
$962$	− 16792.6i	− 0.0181455i
$963$	0	0
$964$	−599967.	−0.645615
$965$	− 6653.76i	− 0.00714517i
$966$	0	0
$967$	−114301.	−0.122236	−0.0611179	−	0.998131i	$-0.519467\pi$
$967$	−114301.	−0.122236	−0.0611179	+	0.998131i	$0.519467\pi$
$968$	44815.8i	0.0478278i
$969$	0	0
$970$	−267609.	−0.284418
$971$	− 759076.i	− 0.805095i	−0.915399	−	0.402547i	$-0.868125\pi$
$971$	− 759076.i	− 0.805095i	0.915399	−	0.402547i	$-0.131875\pi$
$972$	0	0
$973$	0	0
$974$	1.14502e6i	1.20697i
$975$	0	0
$976$	231532.	0.243059
$977$	− 775094.i	− 0.812017i	−0.913869	−	0.406008i	$-0.866920\pi$
$977$	− 775094.i	− 0.812017i	0.913869	−	0.406008i	$-0.133080\pi$
$978$	0	0
$979$	134389.	0.140216
$980$	0	0
$981$	0	0
$982$	799367.	0.828941
$983$	− 302783.i	− 0.313346i	−0.987651	−	0.156673i	$-0.949923\pi$
$983$	− 302783.i	− 0.313346i	0.987651	−	0.156673i	$-0.0500769\pi$
$984$	0	0
$985$	501843.	0.517244
$986$	1.37546e6i	1.41480i
$987$	0	0
$988$	−72214.4	−0.0739793
$989$	32920.1i	0.0336564i
$990$	0	0
$991$	−392180.	−0.399336	−0.199668	−	0.979864i	$-0.563986\pi$
$991$	−392180.	−0.399336	−0.199668	+	0.979864i	$0.563986\pi$
$992$	247581.i	0.251591i
$993$	0	0
$994$	0	0
$995$	− 183855.i	− 0.185707i
$996$	0	0
$997$	1.56054e6	1.56994	0.784971	−	0.619532i	$-0.212678\pi$
$997$	1.56054e6	1.56994	0.784971	+	0.619532i	$0.212678\pi$
$998$	603937.i	0.606361i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.e.197.2		6
3.2	odd	2	inner	882.5.b.e.197.5		6
7.3	odd	6		126.5.s.b.107.5	yes	12
7.5	odd	6		126.5.s.b.53.2	✓	12
7.6	odd	2		882.5.b.h.197.2		6
21.5	even	6		126.5.s.b.53.5	yes	12
21.17	even	6		126.5.s.b.107.2	yes	12
21.20	even	2		882.5.b.h.197.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.s.b.53.2	✓	12	7.5	odd	6
126.5.s.b.53.5	yes	12	21.5	even	6
126.5.s.b.107.2	yes	12	21.17	even	6
126.5.s.b.107.5	yes	12	7.3	odd	6
882.5.b.e.197.2		6	1.1	even	1	trivial
882.5.b.e.197.5		6	3.2	odd	2	inner
882.5.b.h.197.2		6	7.6	odd	2
882.5.b.h.197.5		6	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -8.92960i q^{5} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -8.92960i q^{5} +22.6274i q^{8} -25.2567 q^{10} -112.518i q^{11} +86.8767 q^{13} +64.0000 q^{16} +321.029i q^{17} +103.904 q^{19} +71.4368i q^{20} -318.250 q^{22} -120.158i q^{23} +545.262 q^{25} -245.725i q^{26} +1514.81i q^{29} -1367.71 q^{31} -181.019i q^{32} +908.007 q^{34} +68.3392 q^{37} -293.884i q^{38} +202.054 q^{40} -96.3894i q^{41} -273.972 q^{43} +900.148i q^{44} -339.859 q^{46} +2229.27i q^{47} -1542.23i q^{50} -695.014 q^{52} -1807.84i q^{53} -1004.74 q^{55} +4284.53 q^{58} -378.613i q^{59} +3617.69 q^{61} +3868.46i q^{62} -512.000 q^{64} -775.774i q^{65} -5673.59 q^{67} -2568.23i q^{68} +3300.84i q^{71} +9481.81 q^{73} -193.292i q^{74} -831.229 q^{76} +1730.94 q^{79} -571.494i q^{80} -272.630 q^{82} +7610.67i q^{83} +2866.66 q^{85} +774.911i q^{86} +2546.00 q^{88} +1194.37i q^{89} +961.267i q^{92} +6305.34 q^{94} -927.817i q^{95} +10595.6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{4}+O(q^{10}) \) 6 * q - 48 * q^4	\( 6 q - 48 q^{4} - 104 q^{10} - 86 q^{13} + 384 q^{16} - 174 q^{19} + 472 q^{22} - 2170 q^{25} - 1682 q^{31} - 1744 q^{34} + 6018 q^{37} + 832 q^{40} - 2130 q^{43} - 656 q^{46} + 688 q^{52} + 12680 q^{55} + 7040 q^{58} + 20632 q^{61} - 3072 q^{64} - 74 q^{67} + 2942 q^{73} + 1392 q^{76} + 10526 q^{79} + 25464 q^{82} - 33056 q^{85} - 3776 q^{88} + 11640 q^{94} + 46436 q^{97}+O(q^{100}) \) 6 * q - 48 * q^4 - 104 * q^10 - 86 * q^13 + 384 * q^16 - 174 * q^19 + 472 * q^22 - 2170 * q^25 - 1682 * q^31 - 1744 * q^34 + 6018 * q^37 + 832 * q^40 - 2130 * q^43 - 656 * q^46 + 688 * q^52 + 12680 * q^55 + 7040 * q^58 + 20632 * q^61 - 3072 * q^64 - 74 * q^67 + 2942 * q^73 + 1392 * q^76 + 10526 * q^79 + 25464 * q^82 - 33056 * q^85 - 3776 * q^88 + 11640 * q^94 + 46436 * q^97

Introduction

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