LMFDB - Embedded newform 192.8.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.a (trivial)

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:	yes
Analytic conductor:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} -338.727 q^{5} -1291.03 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} -338.727 q^{5} -1291.03 q^{7} +729.000 q^{9} +3716.47 q^{11} +13476.2 q^{13} +9145.62 q^{15} +2658.70 q^{17} +29717.7 q^{19} +34857.8 q^{21} +51858.3 q^{23} +36610.7 q^{25} -19683.0 q^{27} -174734. q^{29} -316337. q^{31} -100345. q^{33} +437306. q^{35} +330457. q^{37} -363857. q^{39} +332661. q^{41} -345319. q^{43} -246932. q^{45} -1.03298e6 q^{47} +843212. q^{49} -71784.9 q^{51} +1.09043e6 q^{53} -1.25887e6 q^{55} -802377. q^{57} +845122. q^{59} -415149. q^{61} -941160. q^{63} -4.56474e6 q^{65} +3.09019e6 q^{67} -1.40017e6 q^{69} -1.28265e6 q^{71} +3.30220e6 q^{73} -988488. q^{75} -4.79807e6 q^{77} -4.15969e6 q^{79} +531441. q^{81} -1.00931e7 q^{83} -900573. q^{85} +4.71782e6 q^{87} -459162. q^{89} -1.73981e7 q^{91} +8.54109e6 q^{93} -1.00662e7 q^{95} +8.07215e6 q^{97} +2.70931e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} + 28 q^{5} - 936 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q - 54 * q^3 + 28 * q^5 - 936 * q^7 + 1458 * q^9	$ 2 q - 54 q^{3} + 28 q^{5} - 936 q^{7} + 1458 q^{9} - 3384 q^{11} + 17076 q^{13} - 756 q^{15} - 18668 q^{17} + 44856 q^{19} + 25272 q^{21} - 43488 q^{23} + 92974 q^{25} - 39366 q^{27} + 19484 q^{29} - 332856 q^{31} + 91368 q^{33} + 567504 q^{35} + 9076 q^{37} - 461052 q^{39} + 491780 q^{41} + 510984 q^{43} + 20412 q^{45} - 1781424 q^{47} + 145714 q^{49} + 504036 q^{51} + 1395692 q^{53} - 3862800 q^{55} - 1211112 q^{57} + 1534104 q^{59} - 1592188 q^{61} - 682344 q^{63} - 3244584 q^{65} - 1169496 q^{67} + 1174176 q^{69} - 5716800 q^{71} + 1180884 q^{73} - 2510298 q^{75} - 7318944 q^{77} - 6538104 q^{79} + 1062882 q^{81} - 16805160 q^{83} - 8721640 q^{85} - 526068 q^{87} - 6118924 q^{89} - 16120080 q^{91} + 8987112 q^{93} - 4514544 q^{95} + 23720868 q^{97} - 2466936 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 + 28 * q^5 - 936 * q^7 + 1458 * q^9 - 3384 * q^11 + 17076 * q^13 - 756 * q^15 - 18668 * q^17 + 44856 * q^19 + 25272 * q^21 - 43488 * q^23 + 92974 * q^25 - 39366 * q^27 + 19484 * q^29 - 332856 * q^31 + 91368 * q^33 + 567504 * q^35 + 9076 * q^37 - 461052 * q^39 + 491780 * q^41 + 510984 * q^43 + 20412 * q^45 - 1781424 * q^47 + 145714 * q^49 + 504036 * q^51 + 1395692 * q^53 - 3862800 * q^55 - 1211112 * q^57 + 1534104 * q^59 - 1592188 * q^61 - 682344 * q^63 - 3244584 * q^65 - 1169496 * q^67 + 1174176 * q^69 - 5716800 * q^71 + 1180884 * q^73 - 2510298 * q^75 - 7318944 * q^77 - 6538104 * q^79 + 1062882 * q^81 - 16805160 * q^83 - 8721640 * q^85 - 526068 * q^87 - 6118924 * q^89 - 16120080 * q^91 + 8987112 * q^93 - 4514544 * q^95 + 23720868 * q^97 - 2466936 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−338.727	−1.21186	−0.605932	−	0.795516i	$-0.707200\pi$
$5$	−338.727	−1.21186	−0.605932	+	0.795516i	$0.707200\pi$
$6$	0	0
$7$	−1291.03	−1.42263	−0.711316	−	0.702872i	$-0.751901\pi$
$7$	−1291.03	−1.42263	−0.711316	+	0.702872i	$0.751901\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	3716.47	0.841892	0.420946	−	0.907086i	$-0.361698\pi$
$11$	3716.47	0.841892	0.420946	+	0.907086i	$0.361698\pi$
$12$	0	0
$13$	13476.2	1.70124	0.850618	−	0.525783i	$-0.176228\pi$
$13$	13476.2	1.70124	0.850618	+	0.525783i	$0.176228\pi$
$14$	0	0
$15$	9145.62	0.699670
$16$	0	0
$17$	2658.70	0.131250	0.0656249	−	0.997844i	$-0.479096\pi$
$17$	2658.70	0.131250	0.0656249	+	0.997844i	$0.479096\pi$
$18$	0	0
$19$	29717.7	0.993979	0.496990	−	0.867756i	$-0.334439\pi$
$19$	29717.7	0.993979	0.496990	+	0.867756i	$0.334439\pi$
$20$	0	0
$21$	34857.8	0.821357
$22$	0	0
$23$	51858.3	0.888732	0.444366	−	0.895845i	$-0.353429\pi$
$23$	51858.3	0.888732	0.444366	+	0.895845i	$0.353429\pi$
$24$	0	0
$25$	36610.7	0.468616
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−174734.	−1.33041	−0.665203	−	0.746662i	$-0.731655\pi$
$29$	−174734.	−1.33041	−0.665203	+	0.746662i	$0.731655\pi$
$30$	0	0
$31$	−316337.	−1.90715	−0.953573	−	0.301163i	$-0.902625\pi$
$31$	−316337.	−1.90715	−0.953573	+	0.301163i	$0.902625\pi$
$32$	0	0
$33$	−100345.	−0.486067
$34$	0	0
$35$	437306.	1.72404
$36$	0	0
$37$	330457.	1.07253	0.536265	−	0.844050i	$-0.319835\pi$
$37$	330457.	1.07253	0.536265	+	0.844050i	$0.319835\pi$
$38$	0	0
$39$	−363857.	−0.982210
$40$	0	0
$41$	332661.	0.753803	0.376902	−	0.926253i	$-0.376989\pi$
$41$	332661.	0.753803	0.376902	+	0.926253i	$0.376989\pi$
$42$	0	0
$43$	−345319.	−0.662340	−0.331170	−	0.943571i	$-0.607443\pi$
$43$	−345319.	−0.662340	−0.331170	+	0.943571i	$0.607443\pi$
$44$	0	0
$45$	−246932.	−0.403955
$46$	0	0
$47$	−1.03298e6	−1.45127	−0.725636	−	0.688079i	$-0.758454\pi$
$47$	−1.03298e6	−1.45127	−0.725636	+	0.688079i	$0.758454\pi$
$48$	0	0
$49$	843212.	1.02388
$50$	0	0
$51$	−71784.9	−0.0757771
$52$	0	0
$53$	1.09043e6	1.00608	0.503040	−	0.864263i	$-0.332215\pi$
$53$	1.09043e6	1.00608	0.503040	+	0.864263i	$0.332215\pi$
$54$	0	0
$55$	−1.25887e6	−1.02026
$56$	0	0
$57$	−802377.	−0.573874
$58$	0	0
$59$	845122.	0.535719	0.267860	−	0.963458i	$-0.413684\pi$
$59$	845122.	0.535719	0.267860	+	0.963458i	$0.413684\pi$
$60$	0	0
$61$	−415149.	−0.234180	−0.117090	−	0.993121i	$-0.537357\pi$
$61$	−415149.	−0.234180	−0.117090	+	0.993121i	$0.537357\pi$
$62$	0	0
$63$	−941160.	−0.474211
$64$	0	0
$65$	−4.56474e6	−2.06167
$66$	0	0
$67$	3.09019e6	1.25523	0.627616	−	0.778523i	$-0.284031\pi$
$67$	3.09019e6	1.25523	0.627616	+	0.778523i	$0.284031\pi$
$68$	0	0
$69$	−1.40017e6	−0.513109
$70$	0	0
$71$	−1.28265e6	−0.425309	−0.212655	−	0.977127i	$-0.568211\pi$
$71$	−1.28265e6	−0.425309	−0.212655	+	0.977127i	$0.568211\pi$
$72$	0	0
$73$	3.30220e6	0.993514	0.496757	−	0.867890i	$-0.334524\pi$
$73$	3.30220e6	0.993514	0.496757	+	0.867890i	$0.334524\pi$
$74$	0	0
$75$	−988488.	−0.270556
$76$	0	0
$77$	−4.79807e6	−1.19770
$78$	0	0
$79$	−4.15969e6	−0.949218	−0.474609	−	0.880197i	$-0.657410\pi$
$79$	−4.15969e6	−0.949218	−0.474609	+	0.880197i	$0.657410\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−1.00931e7	−1.93754	−0.968769	−	0.247963i	$-0.920239\pi$
$83$	−1.00931e7	−1.93754	−0.968769	+	0.247963i	$0.920239\pi$
$84$	0	0
$85$	−900573.	−0.159057
$86$	0	0
$87$	4.71782e6	0.768111
$88$	0	0
$89$	−459162.	−0.0690400	−0.0345200	−	0.999404i	$-0.510990\pi$
$89$	−459162.	−0.0690400	−0.0345200	+	0.999404i	$0.510990\pi$
$90$	0	0
$91$	−1.73981e7	−2.42023
$92$	0	0
$93$	8.54109e6	1.10109
$94$	0	0
$95$	−1.00662e7	−1.20457
$96$	0	0
$97$	8.07215e6	0.898025	0.449012	−	0.893526i	$-0.351776\pi$
$97$	8.07215e6	0.898025	0.449012	+	0.893526i	$0.351776\pi$
$98$	0	0
$99$	2.70931e6	0.280631
$100$	0	0
$101$	9.27295e6	0.895556	0.447778	−	0.894145i	$-0.352215\pi$
$101$	9.27295e6	0.895556	0.447778	+	0.894145i	$0.352215\pi$
$102$	0	0
$103$	5.73776e6	0.517383	0.258691	−	0.965960i	$-0.416709\pi$
$103$	5.73776e6	0.517383	0.258691	+	0.965960i	$0.416709\pi$
$104$	0	0
$105$	−1.18073e7	−0.995374
$106$	0	0
$107$	−1.99581e7	−1.57498	−0.787490	−	0.616327i	$-0.788620\pi$
$107$	−1.99581e7	−1.57498	−0.787490	+	0.616327i	$0.788620\pi$
$108$	0	0
$109$	−1.36666e7	−1.01081	−0.505404	−	0.862883i	$-0.668657\pi$
$109$	−1.36666e7	−1.01081	−0.505404	+	0.862883i	$0.668657\pi$
$110$	0	0
$111$	−8.92235e6	−0.619225
$112$	0	0
$113$	1.22641e7	0.799577	0.399789	−	0.916607i	$-0.369084\pi$
$113$	1.22641e7	0.799577	0.399789	+	0.916607i	$0.369084\pi$
$114$	0	0
$115$	−1.75658e7	−1.07702
$116$	0	0
$117$	9.82413e6	0.567079
$118$	0	0
$119$	−3.43246e6	−0.186720
$120$	0	0
$121$	−5.67500e6	−0.291217
$122$	0	0
$123$	−8.98184e6	−0.435209
$124$	0	0
$125$	1.40620e7	0.643965
$126$	0	0
$127$	−1.70890e6	−0.0740291	−0.0370145	−	0.999315i	$-0.511785\pi$
$127$	−1.70890e6	−0.0740291	−0.0370145	+	0.999315i	$0.511785\pi$
$128$	0	0
$129$	9.32361e6	0.382402
$130$	0	0
$131$	−1.25921e7	−0.489384	−0.244692	−	0.969601i	$-0.578687\pi$
$131$	−1.25921e7	−0.489384	−0.244692	+	0.969601i	$0.578687\pi$
$132$	0	0
$133$	−3.83664e7	−1.41407
$134$	0	0
$135$	6.66715e6	0.233223
$136$	0	0
$137$	−4.63627e7	−1.54045	−0.770224	−	0.637774i	$-0.779856\pi$
$137$	−4.63627e7	−1.54045	−0.770224	+	0.637774i	$0.779856\pi$
$138$	0	0
$139$	1.39922e7	0.441911	0.220955	−	0.975284i	$-0.429082\pi$
$139$	1.39922e7	0.441911	0.220955	+	0.975284i	$0.429082\pi$
$140$	0	0
$141$	2.78904e7	0.837892
$142$	0	0
$143$	5.00838e7	1.43226
$144$	0	0
$145$	5.91870e7	1.61227
$146$	0	0
$147$	−2.27667e7	−0.591139
$148$	0	0
$149$	1.14296e7	0.283060	0.141530	−	0.989934i	$-0.454798\pi$
$149$	1.14296e7	0.283060	0.141530	+	0.989934i	$0.454798\pi$
$150$	0	0
$151$	−4.47037e7	−1.05663	−0.528316	−	0.849048i	$-0.677176\pi$
$151$	−4.47037e7	−1.05663	−0.528316	+	0.849048i	$0.677176\pi$
$152$	0	0
$153$	1.93819e6	0.0437499
$154$	0	0
$155$	1.07152e8	2.31120
$156$	0	0
$157$	−8.13749e7	−1.67819	−0.839096	−	0.543983i	$-0.816916\pi$
$157$	−8.13749e7	−1.67819	−0.839096	+	0.543983i	$0.816916\pi$
$158$	0	0
$159$	−2.94416e7	−0.580861
$160$	0	0
$161$	−6.69505e7	−1.26434
$162$	0	0
$163$	3.78427e7	0.684424	0.342212	−	0.939623i	$-0.388824\pi$
$163$	3.78427e7	0.684424	0.342212	+	0.939623i	$0.388824\pi$
$164$	0	0
$165$	3.39894e7	0.589047
$166$	0	0
$167$	−1.00629e8	−1.67192	−0.835961	−	0.548789i	$-0.815089\pi$
$167$	−1.00629e8	−1.67192	−0.835961	+	0.548789i	$0.815089\pi$
$168$	0	0
$169$	1.18859e8	1.89421
$170$	0	0
$171$	2.16642e7	0.331326
$172$	0	0
$173$	−9.02205e6	−0.132478	−0.0662390	−	0.997804i	$-0.521100\pi$
$173$	−9.02205e6	−0.132478	−0.0662390	+	0.997804i	$0.521100\pi$
$174$	0	0
$175$	−4.72654e7	−0.666669
$176$	0	0
$177$	−2.28183e7	−0.309298
$178$	0	0
$179$	−7.29972e7	−0.951307	−0.475653	−	0.879633i	$-0.657788\pi$
$179$	−7.29972e7	−0.951307	−0.475653	+	0.879633i	$0.657788\pi$
$180$	0	0
$181$	1.02804e8	1.28865	0.644327	−	0.764750i	$-0.277138\pi$
$181$	1.02804e8	1.28865	0.644327	+	0.764750i	$0.277138\pi$
$182$	0	0
$183$	1.12090e7	0.135204
$184$	0	0
$185$	−1.11935e8	−1.29976
$186$	0	0
$187$	9.88099e6	0.110498
$188$	0	0
$189$	2.54113e7	0.273786
$190$	0	0
$191$	3.62766e7	0.376712	0.188356	−	0.982101i	$-0.439684\pi$
$191$	3.62766e7	0.376712	0.188356	+	0.982101i	$0.439684\pi$
$192$	0	0
$193$	−1.21477e8	−1.21631	−0.608155	−	0.793818i	$-0.708090\pi$
$193$	−1.21477e8	−1.21631	−0.608155	+	0.793818i	$0.708090\pi$
$194$	0	0
$195$	1.23248e8	1.19031
$196$	0	0
$197$	−1.56885e8	−1.46201	−0.731006	−	0.682372i	$-0.760949\pi$
$197$	−1.56885e8	−1.46201	−0.731006	+	0.682372i	$0.760949\pi$
$198$	0	0
$199$	1.91897e8	1.72617	0.863084	−	0.505061i	$-0.168530\pi$
$199$	1.91897e8	1.72617	0.863084	+	0.505061i	$0.168530\pi$
$200$	0	0
$201$	−8.34352e7	−0.724708
$202$	0	0
$203$	2.25587e8	1.89268
$204$	0	0
$205$	−1.12681e8	−0.913508
$206$	0	0
$207$	3.78047e7	0.296244
$208$	0	0
$209$	1.10445e8	0.836824
$210$	0	0
$211$	8.11595e7	0.594773	0.297386	−	0.954757i	$-0.403885\pi$
$211$	8.11595e7	0.594773	0.297386	+	0.954757i	$0.403885\pi$
$212$	0	0
$213$	3.46316e7	0.245552
$214$	0	0
$215$	1.16969e8	0.802666
$216$	0	0
$217$	4.08400e8	2.71317
$218$	0	0
$219$	−8.91595e7	−0.573605
$220$	0	0
$221$	3.58291e7	0.223287
$222$	0	0
$223$	2.37153e7	0.143206	0.0716031	−	0.997433i	$-0.477188\pi$
$223$	2.37153e7	0.143206	0.0716031	+	0.997433i	$0.477188\pi$
$224$	0	0
$225$	2.66892e7	0.156205
$226$	0	0
$227$	−1.00486e8	−0.570187	−0.285093	−	0.958500i	$-0.592025\pi$
$227$	−1.00486e8	−0.570187	−0.285093	+	0.958500i	$0.592025\pi$
$228$	0	0
$229$	3.44951e8	1.89816	0.949081	−	0.315032i	$-0.102015\pi$
$229$	3.44951e8	1.89816	0.949081	+	0.315032i	$0.102015\pi$
$230$	0	0
$231$	1.29548e8	0.691494
$232$	0	0
$233$	−2.14628e8	−1.11158	−0.555789	−	0.831323i	$-0.687584\pi$
$233$	−2.14628e8	−1.11158	−0.555789	+	0.831323i	$0.687584\pi$
$234$	0	0
$235$	3.49897e8	1.75875
$236$	0	0
$237$	1.12312e8	0.548031
$238$	0	0
$239$	6.35918e7	0.301307	0.150653	−	0.988587i	$-0.451862\pi$
$239$	6.35918e7	0.301307	0.150653	+	0.988587i	$0.451862\pi$
$240$	0	0
$241$	9.84542e7	0.453080	0.226540	−	0.974002i	$-0.427259\pi$
$241$	9.84542e7	0.453080	0.226540	+	0.974002i	$0.427259\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.85618e8	−1.24081
$246$	0	0
$247$	4.00481e8	1.69099
$248$	0	0
$249$	2.72513e8	1.11864
$250$	0	0
$251$	−3.85019e8	−1.53682	−0.768412	−	0.639956i	$-0.778953\pi$
$251$	−3.85019e8	−1.53682	−0.768412	+	0.639956i	$0.778953\pi$
$252$	0	0
$253$	1.92730e8	0.748216
$254$	0	0
$255$	2.43155e7	0.0918316
$256$	0	0
$257$	−3.60720e8	−1.32557	−0.662787	−	0.748808i	$-0.730627\pi$
$257$	−3.60720e8	−1.32557	−0.662787	+	0.748808i	$0.730627\pi$
$258$	0	0
$259$	−4.26630e8	−1.52582
$260$	0	0
$261$	−1.27381e8	−0.443469
$262$	0	0
$263$	4.20803e8	1.42637	0.713187	−	0.700973i	$-0.247251\pi$
$263$	4.20803e8	1.42637	0.713187	+	0.700973i	$0.247251\pi$
$264$	0	0
$265$	−3.69358e8	−1.21923
$266$	0	0
$267$	1.23974e7	0.0398603
$268$	0	0
$269$	−2.60293e8	−0.815324	−0.407662	−	0.913133i	$-0.633656\pi$
$269$	−2.60293e8	−0.815324	−0.407662	+	0.913133i	$0.633656\pi$
$270$	0	0
$271$	−1.72083e8	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$271$	−1.72083e8	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$272$	0	0
$273$	4.69749e8	1.39732
$274$	0	0
$275$	1.36063e8	0.394525
$276$	0	0
$277$	−2.27383e8	−0.642805	−0.321403	−	0.946943i	$-0.604154\pi$
$277$	−2.27383e8	−0.642805	−0.321403	+	0.946943i	$0.604154\pi$
$278$	0	0
$279$	−2.30610e8	−0.635715
$280$	0	0
$281$	−3.45181e8	−0.928057	−0.464028	−	0.885820i	$-0.653596\pi$
$281$	−3.45181e8	−0.928057	−0.464028	+	0.885820i	$0.653596\pi$
$282$	0	0
$283$	−4.19452e8	−1.10009	−0.550047	−	0.835134i	$-0.685390\pi$
$283$	−4.19452e8	−1.10009	−0.550047	+	0.835134i	$0.685390\pi$
$284$	0	0
$285$	2.71787e8	0.695458
$286$	0	0
$287$	−4.29474e8	−1.07239
$288$	0	0
$289$	−4.03270e8	−0.982774
$290$	0	0
$291$	−2.17948e8	−0.518475
$292$	0	0
$293$	−2.37355e8	−0.551266	−0.275633	−	0.961263i	$-0.588887\pi$
$293$	−2.37355e8	−0.551266	−0.275633	+	0.961263i	$0.588887\pi$
$294$	0	0
$295$	−2.86265e8	−0.649220
$296$	0	0
$297$	−7.31513e7	−0.162022
$298$	0	0
$299$	6.98851e8	1.51194
$300$	0	0
$301$	4.45816e8	0.942266
$302$	0	0
$303$	−2.50370e8	−0.517050
$304$	0	0
$305$	1.40622e8	0.283795
$306$	0	0
$307$	−2.48200e8	−0.489574	−0.244787	−	0.969577i	$-0.578718\pi$
$307$	−2.48200e8	−0.489574	−0.244787	+	0.969577i	$0.578718\pi$
$308$	0	0
$309$	−1.54920e8	−0.298711
$310$	0	0
$311$	3.11572e8	0.587349	0.293675	−	0.955905i	$-0.405122\pi$
$311$	3.11572e8	0.587349	0.293675	+	0.955905i	$0.405122\pi$
$312$	0	0
$313$	1.01525e7	0.0187141	0.00935704	−	0.999956i	$-0.497022\pi$
$313$	1.01525e7	0.0187141	0.00935704	+	0.999956i	$0.497022\pi$
$314$	0	0
$315$	3.18796e8	0.574679
$316$	0	0
$317$	−3.95059e8	−0.696553	−0.348277	−	0.937392i	$-0.613233\pi$
$317$	−3.95059e8	−0.696553	−0.348277	+	0.937392i	$0.613233\pi$
$318$	0	0
$319$	−6.49394e8	−1.12006
$320$	0	0
$321$	5.38868e8	0.909316
$322$	0	0
$323$	7.90105e7	0.130460
$324$	0	0
$325$	4.93371e8	0.797228
$326$	0	0
$327$	3.68999e8	0.583591
$328$	0	0
$329$	1.33360e9	2.06463
$330$	0	0
$331$	6.28384e8	0.952417	0.476209	−	0.879332i	$-0.342011\pi$
$331$	6.28384e8	0.952417	0.476209	+	0.879332i	$0.342011\pi$
$332$	0	0
$333$	2.40903e8	0.357510
$334$	0	0
$335$	−1.04673e9	−1.52117
$336$	0	0
$337$	−9.86640e8	−1.40428	−0.702141	−	0.712038i	$-0.747772\pi$
$337$	−9.86640e8	−1.40428	−0.702141	+	0.712038i	$0.747772\pi$
$338$	0	0
$339$	−3.31130e8	−0.461636
$340$	0	0
$341$	−1.17566e9	−1.60561
$342$	0	0
$343$	−2.53929e7	−0.0339768
$344$	0	0
$345$	4.74276e8	0.621819
$346$	0	0
$347$	−1.02248e9	−1.31372	−0.656858	−	0.754014i	$-0.728115\pi$
$347$	−1.02248e9	−1.31372	−0.656858	+	0.754014i	$0.728115\pi$
$348$	0	0
$349$	9.08029e8	1.14343	0.571717	−	0.820451i	$-0.306278\pi$
$349$	9.08029e8	1.14343	0.571717	+	0.820451i	$0.306278\pi$
$350$	0	0
$351$	−2.65251e8	−0.327403
$352$	0	0
$353$	−8.94411e8	−1.08225	−0.541123	−	0.840944i	$-0.682001\pi$
$353$	−8.94411e8	−1.08225	−0.541123	+	0.840944i	$0.682001\pi$
$354$	0	0
$355$	4.34469e8	0.515418
$356$	0	0
$357$	9.26764e7	0.107803
$358$	0	0
$359$	−6.16830e8	−0.703614	−0.351807	−	0.936072i	$-0.614433\pi$
$359$	−6.16830e8	−0.703614	−0.351807	+	0.936072i	$0.614433\pi$
$360$	0	0
$361$	−1.07311e7	−0.0120052
$362$	0	0
$363$	1.53225e8	0.168134
$364$	0	0
$365$	−1.11854e9	−1.20400
$366$	0	0
$367$	−1.54501e9	−1.63155	−0.815774	−	0.578371i	$-0.803689\pi$
$367$	−1.54501e9	−1.63155	−0.815774	+	0.578371i	$0.803689\pi$
$368$	0	0
$369$	2.42510e8	0.251268
$370$	0	0
$371$	−1.40778e9	−1.43128
$372$	0	0
$373$	1.62602e9	1.62235	0.811176	−	0.584802i	$-0.198828\pi$
$373$	1.62602e9	1.62235	0.811176	+	0.584802i	$0.198828\pi$
$374$	0	0
$375$	−3.79674e8	−0.371793
$376$	0	0
$377$	−2.35474e9	−2.26334
$378$	0	0
$379$	1.05548e9	0.995891	0.497945	−	0.867208i	$-0.334088\pi$
$379$	1.05548e9	0.995891	0.497945	+	0.867208i	$0.334088\pi$
$380$	0	0
$381$	4.61402e7	0.0427407
$382$	0	0
$383$	−1.94022e9	−1.76463	−0.882317	−	0.470655i	$-0.844018\pi$
$383$	−1.94022e9	−1.76463	−0.882317	+	0.470655i	$0.844018\pi$
$384$	0	0
$385$	1.62523e9	1.45145
$386$	0	0
$387$	−2.51737e8	−0.220780
$388$	0	0
$389$	8.25509e8	0.711046	0.355523	−	0.934667i	$-0.384303\pi$
$389$	8.25509e8	0.711046	0.355523	+	0.934667i	$0.384303\pi$
$390$	0	0
$391$	1.37876e8	0.116646
$392$	0	0
$393$	3.39988e8	0.282546
$394$	0	0
$395$	1.40900e9	1.15032
$396$	0	0
$397$	−4.35960e8	−0.349688	−0.174844	−	0.984596i	$-0.555942\pi$
$397$	−4.35960e8	−0.349688	−0.174844	+	0.984596i	$0.555942\pi$
$398$	0	0
$399$	1.03589e9	0.816412
$400$	0	0
$401$	1.09745e8	0.0849927	0.0424963	−	0.999097i	$-0.486469\pi$
$401$	1.09745e8	0.0849927	0.0424963	+	0.999097i	$0.486469\pi$
$402$	0	0
$403$	−4.26301e9	−3.24451
$404$	0	0
$405$	−1.80013e8	−0.134652
$406$	0	0
$407$	1.22814e9	0.902955
$408$	0	0
$409$	1.60382e9	1.15911	0.579555	−	0.814933i	$-0.303227\pi$
$409$	1.60382e9	1.15911	0.579555	+	0.814933i	$0.303227\pi$
$410$	0	0
$411$	1.25179e9	0.889378
$412$	0	0
$413$	−1.09108e9	−0.762132
$414$	0	0
$415$	3.41879e9	2.34804
$416$	0	0
$417$	−3.77790e8	−0.255137
$418$	0	0
$419$	−2.00817e9	−1.33368	−0.666839	−	0.745202i	$-0.732353\pi$
$419$	−2.00817e9	−1.33368	−0.666839	+	0.745202i	$0.732353\pi$
$420$	0	0
$421$	1.23746e8	0.0808248	0.0404124	−	0.999183i	$-0.487133\pi$
$421$	1.23746e8	0.0808248	0.0404124	+	0.999183i	$0.487133\pi$
$422$	0	0
$423$	−7.53041e8	−0.483757
$424$	0	0
$425$	9.73368e7	0.0615058
$426$	0	0
$427$	5.35970e8	0.333152
$428$	0	0
$429$	−1.35226e9	−0.826915
$430$	0	0
$431$	4.19131e8	0.252162	0.126081	−	0.992020i	$-0.459760\pi$
$431$	4.19131e8	0.252162	0.126081	+	0.992020i	$0.459760\pi$
$432$	0	0
$433$	−4.67552e8	−0.276772	−0.138386	−	0.990378i	$-0.544192\pi$
$433$	−4.67552e8	−0.276772	−0.138386	+	0.990378i	$0.544192\pi$
$434$	0	0
$435$	−1.59805e9	−0.930846
$436$	0	0
$437$	1.54111e9	0.883381
$438$	0	0
$439$	−1.13919e8	−0.0642645	−0.0321322	−	0.999484i	$-0.510230\pi$
$439$	−1.13919e8	−0.0642645	−0.0321322	+	0.999484i	$0.510230\pi$
$440$	0	0
$441$	6.14701e8	0.341294
$442$	0	0
$443$	7.79226e8	0.425844	0.212922	−	0.977069i	$-0.431702\pi$
$443$	7.79226e8	0.425844	0.212922	+	0.977069i	$0.431702\pi$
$444$	0	0
$445$	1.55530e8	0.0836672
$446$	0	0
$447$	−3.08599e8	−0.163425
$448$	0	0
$449$	3.12248e7	0.0162794	0.00813969	−	0.999967i	$-0.497409\pi$
$449$	3.12248e7	0.0162794	0.00813969	+	0.999967i	$0.497409\pi$
$450$	0	0
$451$	1.23632e9	0.634621
$452$	0	0
$453$	1.20700e9	0.610047
$454$	0	0
$455$	5.89321e9	2.93300
$456$	0	0
$457$	−1.47975e9	−0.725242	−0.362621	−	0.931937i	$-0.618118\pi$
$457$	−1.47975e9	−0.725242	−0.362621	+	0.931937i	$0.618118\pi$
$458$	0	0
$459$	−5.23312e7	−0.0252590
$460$	0	0
$461$	−2.79373e7	−0.0132810	−0.00664051	−	0.999978i	$-0.502114\pi$
$461$	−2.79373e7	−0.0132810	−0.00664051	+	0.999978i	$0.502114\pi$
$462$	0	0
$463$	−8.11614e8	−0.380029	−0.190014	−	0.981781i	$-0.560853\pi$
$463$	−8.11614e8	−0.380029	−0.190014	+	0.981781i	$0.560853\pi$
$464$	0	0
$465$	−2.89309e9	−1.33437
$466$	0	0
$467$	2.57691e9	1.17082	0.585410	−	0.810737i	$-0.300934\pi$
$467$	2.57691e9	1.17082	0.585410	+	0.810737i	$0.300934\pi$
$468$	0	0
$469$	−3.98953e9	−1.78573
$470$	0	0
$471$	2.19712e9	0.968905
$472$	0	0
$473$	−1.28337e9	−0.557619
$474$	0	0
$475$	1.08798e9	0.465795
$476$	0	0
$477$	7.94924e8	0.335360
$478$	0	0
$479$	8.65193e8	0.359699	0.179849	−	0.983694i	$-0.442439\pi$
$479$	8.65193e8	0.359699	0.179849	+	0.983694i	$0.442439\pi$
$480$	0	0
$481$	4.45330e9	1.82463
$482$	0	0
$483$	1.80766e9	0.729966
$484$	0	0
$485$	−2.73425e9	−1.08828
$486$	0	0
$487$	2.30548e9	0.904505	0.452252	−	0.891890i	$-0.350621\pi$
$487$	2.30548e9	0.904505	0.452252	+	0.891890i	$0.350621\pi$
$488$	0	0
$489$	−1.02175e9	−0.395153
$490$	0	0
$491$	1.69008e9	0.644349	0.322175	−	0.946680i	$-0.395586\pi$
$491$	1.69008e9	0.644349	0.322175	+	0.946680i	$0.395586\pi$
$492$	0	0
$493$	−4.64566e8	−0.174615
$494$	0	0
$495$	−9.17715e8	−0.340087
$496$	0	0
$497$	1.65594e9	0.605059
$498$	0	0
$499$	6.54349e8	0.235753	0.117877	−	0.993028i	$-0.462391\pi$
$499$	6.54349e8	0.235753	0.117877	+	0.993028i	$0.462391\pi$
$500$	0	0
$501$	2.71699e9	0.965285
$502$	0	0
$503$	−9.69660e8	−0.339728	−0.169864	−	0.985467i	$-0.554333\pi$
$503$	−9.69660e8	−0.339728	−0.169864	+	0.985467i	$0.554333\pi$
$504$	0	0
$505$	−3.14099e9	−1.08529
$506$	0	0
$507$	−3.20918e9	−1.09362
$508$	0	0
$509$	4.87416e9	1.63828	0.819139	−	0.573595i	$-0.194452\pi$
$509$	4.87416e9	1.63828	0.819139	+	0.573595i	$0.194452\pi$
$510$	0	0
$511$	−4.26324e9	−1.41340
$512$	0	0
$513$	−5.84933e8	−0.191291
$514$	0	0
$515$	−1.94353e9	−0.626998
$516$	0	0
$517$	−3.83904e9	−1.22181
$518$	0	0
$519$	2.43595e8	0.0764863
$520$	0	0
$521$	−1.34212e9	−0.415776	−0.207888	−	0.978153i	$-0.566659\pi$
$521$	−1.34212e9	−0.415776	−0.207888	+	0.978153i	$0.566659\pi$
$522$	0	0
$523$	1.06786e9	0.326405	0.163202	−	0.986593i	$-0.447818\pi$
$523$	1.06786e9	0.326405	0.163202	+	0.986593i	$0.447818\pi$
$524$	0	0
$525$	1.27617e9	0.384901
$526$	0	0
$527$	−8.41045e8	−0.250312
$528$	0	0
$529$	−7.15546e8	−0.210156
$530$	0	0
$531$	6.16094e8	0.178573
$532$	0	0
$533$	4.48299e9	1.28240
$534$	0	0
$535$	6.76033e9	1.90866
$536$	0	0
$537$	1.97092e9	0.549237
$538$	0	0
$539$	3.13377e9	0.861999
$540$	0	0
$541$	3.84869e8	0.104501	0.0522507	−	0.998634i	$-0.483361\pi$
$541$	3.84869e8	0.104501	0.0522507	+	0.998634i	$0.483361\pi$
$542$	0	0
$543$	−2.77572e9	−0.744005
$544$	0	0
$545$	4.62925e9	1.22496
$546$	0	0
$547$	2.39106e9	0.624646	0.312323	−	0.949976i	$-0.398893\pi$
$547$	2.39106e9	0.624646	0.312323	+	0.949976i	$0.398893\pi$
$548$	0	0
$549$	−3.02644e8	−0.0780600
$550$	0	0
$551$	−5.19269e9	−1.32240
$552$	0	0
$553$	5.37027e9	1.35039
$554$	0	0
$555$	3.02224e9	0.750417
$556$	0	0
$557$	−1.80314e9	−0.442116	−0.221058	−	0.975261i	$-0.570951\pi$
$557$	−1.80314e9	−0.442116	−0.221058	+	0.975261i	$0.570951\pi$
$558$	0	0
$559$	−4.65358e9	−1.12680
$560$	0	0
$561$	−2.66787e8	−0.0637961
$562$	0	0
$563$	6.17666e9	1.45873	0.729364	−	0.684125i	$-0.239816\pi$
$563$	6.17666e9	1.45873	0.729364	+	0.684125i	$0.239816\pi$
$564$	0	0
$565$	−4.15417e9	−0.968980
$566$	0	0
$567$	−6.86106e8	−0.158070
$568$	0	0
$569$	2.12619e9	0.483848	0.241924	−	0.970295i	$-0.422222\pi$
$569$	2.12619e9	0.483848	0.241924	+	0.970295i	$0.422222\pi$
$570$	0	0
$571$	4.46529e9	1.00374	0.501872	−	0.864942i	$-0.332645\pi$
$571$	4.46529e9	1.00374	0.501872	+	0.864942i	$0.332645\pi$
$572$	0	0
$573$	−9.79468e8	−0.217495
$574$	0	0
$575$	1.89857e9	0.416474
$576$	0	0
$577$	3.77410e9	0.817897	0.408948	−	0.912557i	$-0.365896\pi$
$577$	3.77410e9	0.817897	0.408948	+	0.912557i	$0.365896\pi$
$578$	0	0
$579$	3.27989e9	0.702237
$580$	0	0
$581$	1.30305e10	2.75641
$582$	0	0
$583$	4.05256e9	0.847011
$584$	0	0
$585$	−3.32769e9	−0.687223
$586$	0	0
$587$	−8.31813e8	−0.169743	−0.0848716	−	0.996392i	$-0.527048\pi$
$587$	−8.31813e8	−0.169743	−0.0848716	+	0.996392i	$0.527048\pi$
$588$	0	0
$589$	−9.40080e9	−1.89566
$590$	0	0
$591$	4.23590e9	0.844092
$592$	0	0
$593$	−5.01933e9	−0.988450	−0.494225	−	0.869334i	$-0.664548\pi$
$593$	−5.01933e9	−0.988450	−0.494225	+	0.869334i	$0.664548\pi$
$594$	0	0
$595$	1.16267e9	0.226280
$596$	0	0
$597$	−5.18122e9	−0.996603
$598$	0	0
$599$	−5.96043e9	−1.13314	−0.566570	−	0.824014i	$-0.691730\pi$
$599$	−5.96043e9	−1.13314	−0.566570	+	0.824014i	$0.691730\pi$
$600$	0	0
$601$	−5.90198e9	−1.10902	−0.554508	−	0.832179i	$-0.687093\pi$
$601$	−5.90198e9	−1.10902	−0.554508	+	0.832179i	$0.687093\pi$
$602$	0	0
$603$	2.25275e9	0.418410
$604$	0	0
$605$	1.92227e9	0.352916
$606$	0	0
$607$	−1.62314e9	−0.294575	−0.147287	−	0.989094i	$-0.547054\pi$
$607$	−1.62314e9	−0.294575	−0.147287	+	0.989094i	$0.547054\pi$
$608$	0	0
$609$	−6.09084e9	−1.09274
$610$	0	0
$611$	−1.39206e10	−2.46896
$612$	0	0
$613$	−2.67412e9	−0.468887	−0.234444	−	0.972130i	$-0.575327\pi$
$613$	−2.67412e9	−0.468887	−0.234444	+	0.972130i	$0.575327\pi$
$614$	0	0
$615$	3.04239e9	0.527414
$616$	0	0
$617$	6.48033e9	1.11071	0.555353	−	0.831615i	$-0.312583\pi$
$617$	6.48033e9	1.11071	0.555353	+	0.831615i	$0.312583\pi$
$618$	0	0
$619$	−4.34137e9	−0.735714	−0.367857	−	0.929882i	$-0.619908\pi$
$619$	−4.34137e9	−0.735714	−0.367857	+	0.929882i	$0.619908\pi$
$620$	0	0
$621$	−1.02073e9	−0.171036
$622$	0	0
$623$	5.92791e8	0.0982186
$624$	0	0
$625$	−7.62338e9	−1.24902
$626$	0	0
$627$	−2.98201e9	−0.483140
$628$	0	0
$629$	8.78587e8	0.140769
$630$	0	0
$631$	−9.02970e9	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$631$	−9.02970e9	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$632$	0	0
$633$	−2.19131e9	−0.343392
$634$	0	0
$635$	5.78848e8	0.0897133
$636$	0	0
$637$	1.13633e10	1.74187
$638$	0	0
$639$	−9.35054e8	−0.141770
$640$	0	0
$641$	−3.57322e9	−0.535867	−0.267933	−	0.963437i	$-0.586341\pi$
$641$	−3.57322e9	−0.535867	−0.267933	+	0.963437i	$0.586341\pi$
$642$	0	0
$643$	−6.25863e9	−0.928412	−0.464206	−	0.885727i	$-0.653660\pi$
$643$	−6.25863e9	−0.928412	−0.464206	+	0.885727i	$0.653660\pi$
$644$	0	0
$645$	−3.15815e9	−0.463419
$646$	0	0
$647$	7.86732e9	1.14199	0.570994	−	0.820954i	$-0.306558\pi$
$647$	7.86732e9	1.14199	0.570994	+	0.820954i	$0.306558\pi$
$648$	0	0
$649$	3.14087e9	0.451018
$650$	0	0
$651$	−1.10268e10	−1.56645
$652$	0	0
$653$	−9.94794e9	−1.39810	−0.699048	−	0.715075i	$-0.746393\pi$
$653$	−9.94794e9	−1.39810	−0.699048	+	0.715075i	$0.746393\pi$
$654$	0	0
$655$	4.26529e9	0.593067
$656$	0	0
$657$	2.40731e9	0.331171
$658$	0	0
$659$	2.58338e9	0.351633	0.175817	−	0.984423i	$-0.443743\pi$
$659$	2.58338e9	0.351633	0.175817	+	0.984423i	$0.443743\pi$
$660$	0	0
$661$	3.71275e8	0.0500024	0.0250012	−	0.999687i	$-0.492041\pi$
$661$	3.71275e8	0.0500024	0.0250012	+	0.999687i	$0.492041\pi$
$662$	0	0
$663$	−9.67386e8	−0.128915
$664$	0	0
$665$	1.29957e10	1.71366
$666$	0	0
$667$	−9.06140e9	−1.18237
$668$	0	0
$669$	−6.40314e8	−0.0826802
$670$	0	0
$671$	−1.54289e9	−0.197154
$672$	0	0
$673$	−3.44072e9	−0.435108	−0.217554	−	0.976048i	$-0.569808\pi$
$673$	−3.44072e9	−0.435108	−0.217554	+	0.976048i	$0.569808\pi$
$674$	0	0
$675$	−7.20608e8	−0.0901853
$676$	0	0
$677$	4.54125e9	0.562491	0.281245	−	0.959636i	$-0.409253\pi$
$677$	4.54125e9	0.562491	0.281245	+	0.959636i	$0.409253\pi$
$678$	0	0
$679$	−1.04214e10	−1.27756
$680$	0	0
$681$	2.71313e9	0.329197
$682$	0	0
$683$	−6.33333e9	−0.760606	−0.380303	−	0.924862i	$-0.624180\pi$
$683$	−6.33333e9	−0.760606	−0.380303	+	0.924862i	$0.624180\pi$
$684$	0	0
$685$	1.57043e10	1.86681
$686$	0	0
$687$	−9.31368e9	−1.09590
$688$	0	0
$689$	1.46948e10	1.71158
$690$	0	0
$691$	−1.15035e9	−0.132635	−0.0663175	−	0.997799i	$-0.521125\pi$
$691$	−1.15035e9	−0.132635	−0.0663175	+	0.997799i	$0.521125\pi$
$692$	0	0
$693$	−3.49780e9	−0.399235
$694$	0	0
$695$	−4.73953e9	−0.535536
$696$	0	0
$697$	8.84446e8	0.0989365
$698$	0	0
$699$	5.79495e9	0.641770
$700$	0	0
$701$	−8.33342e9	−0.913714	−0.456857	−	0.889540i	$-0.651025\pi$
$701$	−8.33342e9	−0.913714	−0.456857	+	0.889540i	$0.651025\pi$
$702$	0	0
$703$	9.82042e9	1.06607
$704$	0	0
$705$	−9.44722e9	−1.01541
$706$	0	0
$707$	−1.19716e10	−1.27405
$708$	0	0
$709$	−9.98188e8	−0.105184	−0.0525921	−	0.998616i	$-0.516748\pi$
$709$	−9.98188e8	−0.105184	−0.0525921	+	0.998616i	$0.516748\pi$
$710$	0	0
$711$	−3.03241e9	−0.316406
$712$	0	0
$713$	−1.64047e10	−1.69494
$714$	0	0
$715$	−1.69647e10	−1.73570
$716$	0	0
$717$	−1.71698e9	−0.173959
$718$	0	0
$719$	6.32202e9	0.634315	0.317157	−	0.948373i	$-0.397272\pi$
$719$	6.32202e9	0.634315	0.317157	+	0.948373i	$0.397272\pi$
$720$	0	0
$721$	−7.40761e9	−0.736046
$722$	0	0
$723$	−2.65826e9	−0.261586
$724$	0	0
$725$	−6.39713e9	−0.623450
$726$	0	0
$727$	2.99865e9	0.289438	0.144719	−	0.989473i	$-0.453772\pi$
$727$	2.99865e9	0.289438	0.144719	+	0.989473i	$0.453772\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−9.18100e8	−0.0869319
$732$	0	0
$733$	−8.09232e9	−0.758943	−0.379471	−	0.925204i	$-0.623894\pi$
$733$	−8.09232e9	−0.758943	−0.379471	+	0.925204i	$0.623894\pi$
$734$	0	0
$735$	7.71169e9	0.716381
$736$	0	0
$737$	1.14846e10	1.05677
$738$	0	0
$739$	−2.55720e8	−0.0233082	−0.0116541	−	0.999932i	$-0.503710\pi$
$739$	−2.55720e8	−0.0233082	−0.0116541	+	0.999932i	$0.503710\pi$
$740$	0	0
$741$	−1.08130e10	−0.976296
$742$	0	0
$743$	−7.53694e9	−0.674115	−0.337058	−	0.941484i	$-0.609432\pi$
$743$	−7.53694e9	−0.674115	−0.337058	+	0.941484i	$0.609432\pi$
$744$	0	0
$745$	−3.87151e9	−0.343031
$746$	0	0
$747$	−7.35786e9	−0.645846
$748$	0	0
$749$	2.57664e10	2.24062
$750$	0	0
$751$	5.99443e9	0.516426	0.258213	−	0.966088i	$-0.416866\pi$
$751$	5.99443e9	0.516426	0.258213	+	0.966088i	$0.416866\pi$
$752$	0	0
$753$	1.03955e10	0.887285
$754$	0	0
$755$	1.51423e10	1.28050
$756$	0	0
$757$	6.75008e9	0.565553	0.282777	−	0.959186i	$-0.408745\pi$
$757$	6.75008e9	0.565553	0.282777	+	0.959186i	$0.408745\pi$
$758$	0	0
$759$	−5.20371e9	−0.431983
$760$	0	0
$761$	6.78823e9	0.558354	0.279177	−	0.960240i	$-0.409938\pi$
$761$	6.78823e9	0.558354	0.279177	+	0.960240i	$0.409938\pi$
$762$	0	0
$763$	1.76440e10	1.43801
$764$	0	0
$765$	−6.56518e8	−0.0530190
$766$	0	0
$767$	1.13890e10	0.911386
$768$	0	0
$769$	−2.38264e10	−1.88937	−0.944683	−	0.327986i	$-0.893630\pi$
$769$	−2.38264e10	−1.88937	−0.944683	+	0.327986i	$0.893630\pi$
$770$	0	0
$771$	9.73943e9	0.765321
$772$	0	0
$773$	−1.43625e9	−0.111841	−0.0559205	−	0.998435i	$-0.517809\pi$
$773$	−1.43625e9	−0.111841	−0.0559205	+	0.998435i	$0.517809\pi$
$774$	0	0
$775$	−1.15813e10	−0.893720
$776$	0	0
$777$	1.15190e10	0.880930
$778$	0	0
$779$	9.88591e9	0.749265
$780$	0	0
$781$	−4.76695e9	−0.358065
$782$	0	0
$783$	3.43929e9	0.256037
$784$	0	0
$785$	2.75638e10	2.03374
$786$	0	0
$787$	1.05790e10	0.773630	0.386815	−	0.922157i	$-0.373575\pi$
$787$	1.05790e10	0.773630	0.386815	+	0.922157i	$0.373575\pi$
$788$	0	0
$789$	−1.13617e10	−0.823518
$790$	0	0
$791$	−1.58333e10	−1.13750
$792$	0	0
$793$	−5.59462e9	−0.398396
$794$	0	0
$795$	9.97266e9	0.703924
$796$	0	0
$797$	1.69547e10	1.18628	0.593139	−	0.805100i	$-0.297889\pi$
$797$	1.69547e10	1.18628	0.593139	+	0.805100i	$0.297889\pi$
$798$	0	0
$799$	−2.74638e9	−0.190479
$800$	0	0
$801$	−3.34729e8	−0.0230133
$802$	0	0
$803$	1.22726e10	0.836432
$804$	0	0
$805$	2.26779e10	1.53221
$806$	0	0
$807$	7.02792e9	0.470727
$808$	0	0
$809$	−1.97032e10	−1.30833	−0.654165	−	0.756352i	$-0.726980\pi$
$809$	−1.97032e10	−1.30833	−0.654165	+	0.756352i	$0.726980\pi$
$810$	0	0
$811$	−7.91210e8	−0.0520858	−0.0260429	−	0.999661i	$-0.508291\pi$
$811$	−7.91210e8	−0.0520858	−0.0260429	+	0.999661i	$0.508291\pi$
$812$	0	0
$813$	4.64625e9	0.303240
$814$	0	0
$815$	−1.28183e10	−0.829430
$816$	0	0
$817$	−1.02621e10	−0.658352
$818$	0	0
$819$	−1.26832e10	−0.806745
$820$	0	0
$821$	9.58275e9	0.604351	0.302175	−	0.953252i	$-0.402287\pi$
$821$	9.58275e9	0.604351	0.302175	+	0.953252i	$0.402287\pi$
$822$	0	0
$823$	−3.07363e10	−1.92199	−0.960997	−	0.276558i	$-0.910806\pi$
$823$	−3.07363e10	−1.92199	−0.960997	+	0.276558i	$0.910806\pi$
$824$	0	0
$825$	−3.67369e9	−0.227779
$826$	0	0
$827$	−2.84345e10	−1.74814	−0.874070	−	0.485799i	$-0.838529\pi$
$827$	−2.84345e10	−1.74814	−0.874070	+	0.485799i	$0.838529\pi$
$828$	0	0
$829$	−1.13447e10	−0.691596	−0.345798	−	0.938309i	$-0.612392\pi$
$829$	−1.13447e10	−0.691596	−0.345798	+	0.938309i	$0.612392\pi$
$830$	0	0
$831$	6.13935e9	0.371124
$832$	0	0
$833$	2.24185e9	0.134384
$834$	0	0
$835$	3.40858e10	2.02614
$836$	0	0
$837$	6.22646e9	0.367030
$838$	0	0
$839$	8.05383e9	0.470799	0.235399	−	0.971899i	$-0.424360\pi$
$839$	8.05383e9	0.470799	0.235399	+	0.971899i	$0.424360\pi$
$840$	0	0
$841$	1.32821e10	0.769981
$842$	0	0
$843$	9.31988e9	0.535814
$844$	0	0
$845$	−4.02606e10	−2.29552
$846$	0	0
$847$	7.32658e9	0.414295
$848$	0	0
$849$	1.13252e10	0.635139
$850$	0	0
$851$	1.71369e10	0.953191
$852$	0	0
$853$	−2.00004e10	−1.10336	−0.551680	−	0.834056i	$-0.686013\pi$
$853$	−2.00004e10	−1.10336	−0.551680	+	0.834056i	$0.686013\pi$
$854$	0	0
$855$	−7.33824e9	−0.401523
$856$	0	0
$857$	−2.08256e10	−1.13022	−0.565112	−	0.825014i	$-0.691167\pi$
$857$	−2.08256e10	−1.13022	−0.565112	+	0.825014i	$0.691167\pi$
$858$	0	0
$859$	5.03656e9	0.271118	0.135559	−	0.990769i	$-0.456717\pi$
$859$	5.03656e9	0.271118	0.135559	+	0.990769i	$0.456717\pi$
$860$	0	0
$861$	1.15958e10	0.619142
$862$	0	0
$863$	−9.79867e9	−0.518955	−0.259477	−	0.965749i	$-0.583550\pi$
$863$	−9.79867e9	−0.518955	−0.259477	+	0.965749i	$0.583550\pi$
$864$	0	0
$865$	3.05601e9	0.160546
$866$	0	0
$867$	1.08883e10	0.567405
$868$	0	0
$869$	−1.54594e10	−0.799139
$870$	0	0
$871$	4.16440e10	2.13545
$872$	0	0
$873$	5.88460e9	0.299342
$874$	0	0
$875$	−1.81545e10	−0.916126
$876$	0	0
$877$	2.62336e9	0.131329	0.0656644	−	0.997842i	$-0.479083\pi$
$877$	2.62336e9	0.131329	0.0656644	+	0.997842i	$0.479083\pi$
$878$	0	0
$879$	6.40858e9	0.318274
$880$	0	0
$881$	3.31106e10	1.63137	0.815683	−	0.578499i	$-0.196361\pi$
$881$	3.31106e10	1.63137	0.815683	+	0.578499i	$0.196361\pi$
$882$	0	0
$883$	−1.57707e10	−0.770881	−0.385440	−	0.922733i	$-0.625950\pi$
$883$	−1.57707e10	−0.770881	−0.385440	+	0.922733i	$0.625950\pi$
$884$	0	0
$885$	7.72916e9	0.374827
$886$	0	0
$887$	2.97445e10	1.43111	0.715556	−	0.698555i	$-0.246173\pi$
$887$	2.97445e10	1.43111	0.715556	+	0.698555i	$0.246173\pi$
$888$	0	0
$889$	2.20623e9	0.105316
$890$	0	0
$891$	1.97509e9	0.0935436
$892$	0	0
$893$	−3.06977e10	−1.44253
$894$	0	0
$895$	2.47261e10	1.15285
$896$	0	0
$897$	−1.88690e10	−0.872921
$898$	0	0
$899$	5.52748e10	2.53728
$900$	0	0
$901$	2.89913e9	0.132048
$902$	0	0
$903$	−1.20370e10	−0.544017
$904$	0	0
$905$	−3.48226e10	−1.56167
$906$	0	0
$907$	−1.72763e10	−0.768820	−0.384410	−	0.923162i	$-0.625595\pi$
$907$	−1.72763e10	−0.768820	−0.384410	+	0.923162i	$0.625595\pi$
$908$	0	0
$909$	6.75998e9	0.298519
$910$	0	0
$911$	−2.42904e9	−0.106444	−0.0532218	−	0.998583i	$-0.516949\pi$
$911$	−2.42904e9	−0.106444	−0.0532218	+	0.998583i	$0.516949\pi$
$912$	0	0
$913$	−3.75107e10	−1.63120
$914$	0	0
$915$	−3.79680e9	−0.163849
$916$	0	0
$917$	1.62568e10	0.696214
$918$	0	0
$919$	−1.41310e10	−0.600576	−0.300288	−	0.953849i	$-0.597083\pi$
$919$	−1.41310e10	−0.600576	−0.300288	+	0.953849i	$0.597083\pi$
$920$	0	0
$921$	6.70141e9	0.282655
$922$	0	0
$923$	−1.72853e10	−0.723552
$924$	0	0
$925$	1.20983e10	0.502605
$926$	0	0
$927$	4.18283e9	0.172461
$928$	0	0
$929$	−3.17488e10	−1.29919	−0.649594	−	0.760281i	$-0.725061\pi$
$929$	−3.17488e10	−1.29919	−0.649594	+	0.760281i	$0.725061\pi$
$930$	0	0
$931$	2.50583e10	1.01772
$932$	0	0
$933$	−8.41243e9	−0.339106
$934$	0	0
$935$	−3.34695e9	−0.133909
$936$	0	0
$937$	3.87049e10	1.53701	0.768505	−	0.639843i	$-0.221001\pi$
$937$	3.87049e10	1.53701	0.768505	+	0.639843i	$0.221001\pi$
$938$	0	0
$939$	−2.74118e8	−0.0108046
$940$	0	0
$941$	−1.39438e10	−0.545527	−0.272763	−	0.962081i	$-0.587938\pi$
$941$	−1.39438e10	−0.545527	−0.272763	+	0.962081i	$0.587938\pi$
$942$	0	0
$943$	1.72512e10	0.669929
$944$	0	0
$945$	−8.60749e9	−0.331791
$946$	0	0
$947$	2.72965e10	1.04444	0.522219	−	0.852812i	$-0.325104\pi$
$947$	2.72965e10	1.04444	0.522219	+	0.852812i	$0.325104\pi$
$948$	0	0
$949$	4.45011e10	1.69020
$950$	0	0
$951$	1.06666e10	0.402155
$952$	0	0
$953$	2.24555e10	0.840423	0.420211	−	0.907426i	$-0.361956\pi$
$953$	2.24555e10	0.840423	0.420211	+	0.907426i	$0.361956\pi$
$954$	0	0
$955$	−1.22878e10	−0.456524
$956$	0	0
$957$	1.75336e10	0.646666
$958$	0	0
$959$	5.98556e10	2.19149
$960$	0	0
$961$	7.25563e10	2.63720
$962$	0	0
$963$	−1.45494e10	−0.524994
$964$	0	0
$965$	4.11476e10	1.47400
$966$	0	0
$967$	4.80797e10	1.70989	0.854947	−	0.518715i	$-0.173589\pi$
$967$	4.80797e10	1.70989	0.854947	+	0.518715i	$0.173589\pi$
$968$	0	0
$969$	−2.13328e9	−0.0753208
$970$	0	0
$971$	−7.35332e9	−0.257760	−0.128880	−	0.991660i	$-0.541138\pi$
$971$	−7.35332e9	−0.257760	−0.128880	+	0.991660i	$0.541138\pi$
$972$	0	0
$973$	−1.80643e10	−0.628677
$974$	0	0
$975$	−1.33210e10	−0.460280
$976$	0	0
$977$	−2.49016e10	−0.854272	−0.427136	−	0.904187i	$-0.640478\pi$
$977$	−2.49016e10	−0.854272	−0.427136	+	0.904187i	$0.640478\pi$
$978$	0	0
$979$	−1.70646e9	−0.0581243
$980$	0	0
$981$	−9.96298e9	−0.336936
$982$	0	0
$983$	1.53138e10	0.514215	0.257107	−	0.966383i	$-0.417231\pi$
$983$	1.53138e10	0.514215	0.257107	+	0.966383i	$0.417231\pi$
$984$	0	0
$985$	5.31412e10	1.77176
$986$	0	0
$987$	−3.60073e10	−1.19201
$988$	0	0
$989$	−1.79076e10	−0.588642
$990$	0	0
$991$	−1.09699e10	−0.358052	−0.179026	−	0.983844i	$-0.557295\pi$
$991$	−1.09699e10	−0.358052	−0.179026	+	0.983844i	$0.557295\pi$
$992$	0	0
$993$	−1.69664e10	−0.549878
$994$	0	0
$995$	−6.50007e10	−2.09188
$996$	0	0
$997$	−5.99325e10	−1.91527	−0.957633	−	0.287990i	$-0.907013\pi$
$997$	−5.99325e10	−1.91527	−0.957633	+	0.287990i	$0.907013\pi$
$998$	0	0
$999$	−6.50439e9	−0.206408

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.s.1.1		2
3.2	odd	2		576.8.a.bg.1.2		2
4.3	odd	2		192.8.a.v.1.1		2
8.3	odd	2		96.8.a.c.1.2	✓	2
8.5	even	2		96.8.a.f.1.2	yes	2
12.11	even	2		576.8.a.bh.1.2		2
24.5	odd	2		288.8.a.l.1.1		2
24.11	even	2		288.8.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.8.a.c.1.2	✓	2	8.3	odd	2
96.8.a.f.1.2	yes	2	8.5	even	2
192.8.a.s.1.1		2	1.1	even	1	trivial
192.8.a.v.1.1		2	4.3	odd	2
288.8.a.l.1.1		2	24.5	odd	2
288.8.a.m.1.1		2	24.11	even	2
576.8.a.bg.1.2		2	3.2	odd	2
576.8.a.bh.1.2		2	12.11	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -338.727 q^{5} -1291.03 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} -338.727 q^{5} -1291.03 q^{7} +729.000 q^{9} +3716.47 q^{11} +13476.2 q^{13} +9145.62 q^{15} +2658.70 q^{17} +29717.7 q^{19} +34857.8 q^{21} +51858.3 q^{23} +36610.7 q^{25} -19683.0 q^{27} -174734. q^{29} -316337. q^{31} -100345. q^{33} +437306. q^{35} +330457. q^{37} -363857. q^{39} +332661. q^{41} -345319. q^{43} -246932. q^{45} -1.03298e6 q^{47} +843212. q^{49} -71784.9 q^{51} +1.09043e6 q^{53} -1.25887e6 q^{55} -802377. q^{57} +845122. q^{59} -415149. q^{61} -941160. q^{63} -4.56474e6 q^{65} +3.09019e6 q^{67} -1.40017e6 q^{69} -1.28265e6 q^{71} +3.30220e6 q^{73} -988488. q^{75} -4.79807e6 q^{77} -4.15969e6 q^{79} +531441. q^{81} -1.00931e7 q^{83} -900573. q^{85} +4.71782e6 q^{87} -459162. q^{89} -1.73981e7 q^{91} +8.54109e6 q^{93} -1.00662e7 q^{95} +8.07215e6 q^{97} +2.70931e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} + 28 q^{5} - 936 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q - 54 * q^3 + 28 * q^5 - 936 * q^7 + 1458 * q^9	\( 2 q - 54 q^{3} + 28 q^{5} - 936 q^{7} + 1458 q^{9} - 3384 q^{11} + 17076 q^{13} - 756 q^{15} - 18668 q^{17} + 44856 q^{19} + 25272 q^{21} - 43488 q^{23} + 92974 q^{25} - 39366 q^{27} + 19484 q^{29} - 332856 q^{31} + 91368 q^{33} + 567504 q^{35} + 9076 q^{37} - 461052 q^{39} + 491780 q^{41} + 510984 q^{43} + 20412 q^{45} - 1781424 q^{47} + 145714 q^{49} + 504036 q^{51} + 1395692 q^{53} - 3862800 q^{55} - 1211112 q^{57} + 1534104 q^{59} - 1592188 q^{61} - 682344 q^{63} - 3244584 q^{65} - 1169496 q^{67} + 1174176 q^{69} - 5716800 q^{71} + 1180884 q^{73} - 2510298 q^{75} - 7318944 q^{77} - 6538104 q^{79} + 1062882 q^{81} - 16805160 q^{83} - 8721640 q^{85} - 526068 q^{87} - 6118924 q^{89} - 16120080 q^{91} + 8987112 q^{93} - 4514544 q^{95} + 23720868 q^{97} - 2466936 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 + 28 * q^5 - 936 * q^7 + 1458 * q^9 - 3384 * q^11 + 17076 * q^13 - 756 * q^15 - 18668 * q^17 + 44856 * q^19 + 25272 * q^21 - 43488 * q^23 + 92974 * q^25 - 39366 * q^27 + 19484 * q^29 - 332856 * q^31 + 91368 * q^33 + 567504 * q^35 + 9076 * q^37 - 461052 * q^39 + 491780 * q^41 + 510984 * q^43 + 20412 * q^45 - 1781424 * q^47 + 145714 * q^49 + 504036 * q^51 + 1395692 * q^53 - 3862800 * q^55 - 1211112 * q^57 + 1534104 * q^59 - 1592188 * q^61 - 682344 * q^63 - 3244584 * q^65 - 1169496 * q^67 + 1174176 * q^69 - 5716800 * q^71 + 1180884 * q^73 - 2510298 * q^75 - 7318944 * q^77 - 6538104 * q^79 + 1062882 * q^81 - 16805160 * q^83 - 8721640 * q^85 - 526068 * q^87 - 6118924 * q^89 - 16120080 * q^91 + 8987112 * q^93 - 4514544 * q^95 + 23720868 * q^97 - 2466936 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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