LMFDB - Embedded newform 192.8.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,8,Mod(1,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("192.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,0,-54,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			1.2
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)

$f(q)$	$=$	$q-27.0000 q^{3} +366.727 q^{5} +355.029 q^{7} +729.000 q^{9} -7100.47 q^{11} +3599.83 q^{13} -9901.62 q^{15} -21326.7 q^{17} +15138.3 q^{19} -9585.77 q^{21} -95346.3 q^{23} +56363.3 q^{25} -19683.0 q^{27} +194218. q^{29} -16519.2 q^{31} +191713. q^{33} +130198. q^{35} -321381. q^{37} -97195.4 q^{39} +159119. q^{41} +856303. q^{43} +267344. q^{45} -748446. q^{47} -697498. q^{49} +575821. q^{51} +305261. q^{53} -2.60393e6 q^{55} -408735. q^{57} +688982. q^{59} -1.17704e6 q^{61} +258816. q^{63} +1.32015e6 q^{65} -4.25969e6 q^{67} +2.57435e6 q^{69} -4.43415e6 q^{71} -2.12132e6 q^{73} -1.52181e6 q^{75} -2.52087e6 q^{77} -2.37842e6 q^{79} +531441. q^{81} -6.71208e6 q^{83} -7.82107e6 q^{85} -5.24389e6 q^{87} -5.65976e6 q^{89} +1.27804e6 q^{91} +446019. q^{93} +5.55162e6 q^{95} +1.56487e7 q^{97} -5.17625e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$5$	366.727	1.31204	0.656020	−	0.754743i	$-0.272239\pi$
$5$	366.727	1.31204	0.656020	+	0.754743i	$0.272239\pi$
$6$	0	0
$7$	355.029	0.391219	0.195610	−	0.980682i	$-0.437331\pi$
$7$	355.029	0.391219	0.195610	+	0.980682i	$0.437331\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−7100.47	−1.60847	−0.804235	−	0.594312i	$-0.797425\pi$
$11$	−7100.47	−1.60847	−0.804235	+	0.594312i	$0.797425\pi$
$12$	0	0
$13$	3599.83	0.454444	0.227222	−	0.973843i	$-0.427036\pi$
$13$	3599.83	0.454444	0.227222	+	0.973843i	$0.427036\pi$
$14$	0	0
$15$	−9901.62	−0.757507
$16$	0	0
$17$	−21326.7	−1.05282	−0.526408	−	0.850232i	$-0.676462\pi$
$17$	−21326.7	−1.05282	−0.526408	+	0.850232i	$0.676462\pi$
$18$	0	0
$19$	15138.3	0.506337	0.253169	−	0.967422i	$-0.418527\pi$
$19$	15138.3	0.506337	0.253169	+	0.967422i	$0.418527\pi$
$20$	0	0
$21$	−9585.77	−0.225871
$22$	0	0
$23$	−95346.3	−1.63402	−0.817008	−	0.576626i	$-0.804369\pi$
$23$	−95346.3	−1.63402	−0.817008	+	0.576626i	$0.804369\pi$
$24$	0	0
$25$	56363.3	0.721451
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	194218.	1.47876	0.739378	−	0.673291i	$-0.235120\pi$
$29$	194218.	1.47876	0.739378	+	0.673291i	$0.235120\pi$
$30$	0	0
$31$	−16519.2	−0.0995919	−0.0497959	−	0.998759i	$-0.515857\pi$
$31$	−16519.2	−0.0995919	−0.0497959	+	0.998759i	$0.515857\pi$
$32$	0	0
$33$	191713.	0.928650
$34$	0	0
$35$	130198.	0.513295
$36$	0	0
$37$	−321381.	−1.04307	−0.521536	−	0.853229i	$-0.674641\pi$
$37$	−321381.	−1.04307	−0.521536	+	0.853229i	$0.674641\pi$
$38$	0	0
$39$	−97195.4	−0.262373
$40$	0	0
$41$	159119.	0.360561	0.180281	−	0.983615i	$-0.442299\pi$
$41$	159119.	0.360561	0.180281	+	0.983615i	$0.442299\pi$
$42$	0	0
$43$	856303.	1.64243	0.821217	−	0.570616i	$-0.193296\pi$
$43$	856303.	1.64243	0.821217	+	0.570616i	$0.193296\pi$
$44$	0	0
$45$	267344.	0.437347
$46$	0	0
$47$	−748446.	−1.05152	−0.525760	−	0.850633i	$-0.676219\pi$
$47$	−748446.	−1.05152	−0.525760	+	0.850633i	$0.676219\pi$
$48$	0	0
$49$	−697498.	−0.846948
$50$	0	0
$51$	575821.	0.607844
$52$	0	0
$53$	305261.	0.281648	0.140824	−	0.990035i	$-0.455025\pi$
$53$	305261.	0.281648	0.140824	+	0.990035i	$0.455025\pi$
$54$	0	0
$55$	−2.60393e6	−2.11038
$56$	0	0
$57$	−408735.	−0.292334
$58$	0	0
$59$	688982.	0.436743	0.218371	−	0.975866i	$-0.429926\pi$
$59$	688982.	0.436743	0.218371	+	0.975866i	$0.429926\pi$
$60$	0	0
$61$	−1.17704e6	−0.663951	−0.331976	−	0.943288i	$-0.607715\pi$
$61$	−1.17704e6	−0.663951	−0.331976	+	0.943288i	$0.607715\pi$
$62$	0	0
$63$	258816.	0.130406
$64$	0	0
$65$	1.32015e6	0.596249
$66$	0	0
$67$	−4.25969e6	−1.73028	−0.865139	−	0.501532i	$-0.832770\pi$
$67$	−4.25969e6	−1.73028	−0.865139	+	0.501532i	$0.832770\pi$
$68$	0	0
$69$	2.57435e6	0.943400
$70$	0	0
$71$	−4.43415e6	−1.47030	−0.735150	−	0.677905i	$-0.762888\pi$
$71$	−4.43415e6	−1.47030	−0.735150	+	0.677905i	$0.762888\pi$
$72$	0	0
$73$	−2.12132e6	−0.638228	−0.319114	−	0.947716i	$-0.603385\pi$
$73$	−2.12132e6	−0.638228	−0.319114	+	0.947716i	$0.603385\pi$
$74$	0	0
$75$	−1.52181e6	−0.416530
$76$	0	0
$77$	−2.52087e6	−0.629264
$78$	0	0
$79$	−2.37842e6	−0.542742	−0.271371	−	0.962475i	$-0.587477\pi$
$79$	−2.37842e6	−0.542742	−0.271371	+	0.962475i	$0.587477\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−6.71208e6	−1.28850	−0.644249	−	0.764816i	$-0.722830\pi$
$83$	−6.71208e6	−1.28850	−0.644249	+	0.764816i	$0.722830\pi$
$84$	0	0
$85$	−7.82107e6	−1.38134
$86$	0	0
$87$	−5.24389e6	−0.853760
$88$	0	0
$89$	−5.65976e6	−0.851007	−0.425504	−	0.904957i	$-0.639903\pi$
$89$	−5.65976e6	−0.851007	−0.425504	+	0.904957i	$0.639903\pi$
$90$	0	0
$91$	1.27804e6	0.177787
$92$	0	0
$93$	446019.	0.0574994
$94$	0	0
$95$	5.55162e6	0.664335
$96$	0	0
$97$	1.56487e7	1.74092	0.870458	−	0.492243i	$-0.163823\pi$
$97$	1.56487e7	1.74092	0.870458	+	0.492243i	$0.163823\pi$
$98$	0	0
$99$	−5.17625e6	−0.536157
$100$	0	0
$101$	1.69363e7	1.63566	0.817830	−	0.575459i	$-0.195177\pi$
$101$	1.69363e7	1.63566	0.817830	+	0.575459i	$0.195177\pi$
$102$	0	0
$103$	1.07025e7	0.965062	0.482531	−	0.875879i	$-0.339718\pi$
$103$	1.07025e7	0.965062	0.482531	+	0.875879i	$0.339718\pi$
$104$	0	0
$105$	−3.51536e6	−0.296351
$106$	0	0
$107$	−1.82697e7	−1.44174	−0.720871	−	0.693069i	$-0.756258\pi$
$107$	−1.82697e7	−1.44174	−0.720871	+	0.693069i	$0.756258\pi$
$108$	0	0
$109$	−936031.	−0.0692305	−0.0346153	−	0.999401i	$-0.511021\pi$
$109$	−936031.	−0.0692305	−0.0346153	+	0.999401i	$0.511021\pi$
$110$	0	0
$111$	8.67730e6	0.602218
$112$	0	0
$113$	−1.92584e7	−1.25558	−0.627791	−	0.778382i	$-0.716041\pi$
$113$	−1.92584e7	−1.25558	−0.627791	+	0.778382i	$0.716041\pi$
$114$	0	0
$115$	−3.49660e7	−2.14390
$116$	0	0
$117$	2.62428e6	0.151481
$118$	0	0
$119$	−7.57159e6	−0.411882
$120$	0	0
$121$	3.09296e7	1.58718
$122$	0	0
$123$	−4.29622e6	−0.208170
$124$	0	0
$125$	−7.98058e6	−0.365468
$126$	0	0
$127$	−3.40542e7	−1.47522	−0.737610	−	0.675227i	$-0.764046\pi$
$127$	−3.40542e7	−1.47522	−0.737610	+	0.675227i	$0.764046\pi$
$128$	0	0
$129$	−2.31202e7	−0.948259
$130$	0	0
$131$	−4.61205e6	−0.179244	−0.0896220	−	0.995976i	$-0.528566\pi$
$131$	−4.61205e6	−0.179244	−0.0896220	+	0.995976i	$0.528566\pi$
$132$	0	0
$133$	5.37454e6	0.198089
$134$	0	0
$135$	−7.21828e6	−0.252502
$136$	0	0
$137$	−1.14188e7	−0.379401	−0.189701	−	0.981842i	$-0.560752\pi$
$137$	−1.14188e7	−0.379401	−0.189701	+	0.981842i	$0.560752\pi$
$138$	0	0
$139$	−9.97814e6	−0.315136	−0.157568	−	0.987508i	$-0.550365\pi$
$139$	−9.97814e6	−0.315136	−0.157568	+	0.987508i	$0.550365\pi$
$140$	0	0
$141$	2.02080e7	0.607096
$142$	0	0
$143$	−2.55605e7	−0.730959
$144$	0	0
$145$	7.12249e7	1.94019
$146$	0	0
$147$	1.88324e7	0.488985
$148$	0	0
$149$	−3.19127e7	−0.790337	−0.395168	−	0.918609i	$-0.629314\pi$
$149$	−3.19127e7	−0.790337	−0.395168	+	0.918609i	$0.629314\pi$
$150$	0	0
$151$	−3.56752e6	−0.0843231	−0.0421616	−	0.999111i	$-0.513424\pi$
$151$	−3.56752e6	−0.0843231	−0.0421616	+	0.999111i	$0.513424\pi$
$152$	0	0
$153$	−1.55472e7	−0.350939
$154$	0	0
$155$	−6.05804e6	−0.130669
$156$	0	0
$157$	−5.56485e7	−1.14764	−0.573818	−	0.818983i	$-0.694538\pi$
$157$	−5.56485e7	−1.14764	−0.573818	+	0.818983i	$0.694538\pi$
$158$	0	0
$159$	−8.24206e6	−0.162609
$160$	0	0
$161$	−3.38506e7	−0.639258
$162$	0	0
$163$	6.68712e7	1.20943	0.604717	−	0.796440i	$-0.293286\pi$
$163$	6.68712e7	1.20943	0.604717	+	0.796440i	$0.293286\pi$
$164$	0	0
$165$	7.03062e7	1.21843
$166$	0	0
$167$	2.58953e7	0.430242	0.215121	−	0.976587i	$-0.430985\pi$
$167$	2.58953e7	0.430242	0.215121	+	0.976587i	$0.430985\pi$
$168$	0	0
$169$	−4.97898e7	−0.793481
$170$	0	0
$171$	1.10358e7	0.168779
$172$	0	0
$173$	−1.85506e7	−0.272394	−0.136197	−	0.990682i	$-0.543488\pi$
$173$	−1.85506e7	−0.272394	−0.136197	+	0.990682i	$0.543488\pi$
$174$	0	0
$175$	2.00106e7	0.282245
$176$	0	0
$177$	−1.86025e7	−0.252154
$178$	0	0
$179$	1.44092e8	1.87783	0.938913	−	0.344155i	$-0.111834\pi$
$179$	1.44092e8	1.87783	0.938913	+	0.344155i	$0.111834\pi$
$180$	0	0
$181$	−7.97401e7	−0.999544	−0.499772	−	0.866157i	$-0.666583\pi$
$181$	−7.97401e7	−0.999544	−0.499772	+	0.866157i	$0.666583\pi$
$182$	0	0
$183$	3.17800e7	0.383333
$184$	0	0
$185$	−1.17859e8	−1.36855
$186$	0	0
$187$	1.51430e8	1.69342
$188$	0	0
$189$	−6.98803e6	−0.0752902
$190$	0	0
$191$	−1.09598e8	−1.13811	−0.569057	−	0.822298i	$-0.692692\pi$
$191$	−1.09598e8	−1.13811	−0.569057	+	0.822298i	$0.692692\pi$
$192$	0	0
$193$	1.79334e8	1.79561	0.897803	−	0.440398i	$-0.145163\pi$
$193$	1.79334e8	1.79561	0.897803	+	0.440398i	$0.145163\pi$
$194$	0	0
$195$	−3.56441e7	−0.344244
$196$	0	0
$197$	1.26410e8	1.17801	0.589005	−	0.808130i	$-0.299520\pi$
$197$	1.26410e8	1.17801	0.589005	+	0.808130i	$0.299520\pi$
$198$	0	0
$199$	5.38271e7	0.484189	0.242095	−	0.970253i	$-0.422166\pi$
$199$	5.38271e7	0.484189	0.242095	+	0.970253i	$0.422166\pi$
$200$	0	0
$201$	1.15012e8	0.998977
$202$	0	0
$203$	6.89529e7	0.578518
$204$	0	0
$205$	5.83533e7	0.473071
$206$	0	0
$207$	−6.95074e7	−0.544672
$208$	0	0
$209$	−1.07489e8	−0.814428
$210$	0	0
$211$	2.40216e7	0.176041	0.0880204	−	0.996119i	$-0.471946\pi$
$211$	2.40216e7	0.176041	0.0880204	+	0.996119i	$0.471946\pi$
$212$	0	0
$213$	1.19722e8	0.848878
$214$	0	0
$215$	3.14029e8	2.15494
$216$	0	0
$217$	−5.86480e6	−0.0389622
$218$	0	0
$219$	5.72756e7	0.368481
$220$	0	0
$221$	−7.67725e7	−0.478446
$222$	0	0
$223$	7.70788e7	0.465445	0.232722	−	0.972543i	$-0.425237\pi$
$223$	7.70788e7	0.465445	0.232722	+	0.972543i	$0.425237\pi$
$224$	0	0
$225$	4.10889e7	0.240484
$226$	0	0
$227$	−2.46115e8	−1.39652	−0.698261	−	0.715844i	$-0.746042\pi$
$227$	−2.46115e8	−1.39652	−0.698261	+	0.715844i	$0.746042\pi$
$228$	0	0
$229$	−3.29557e8	−1.81345	−0.906725	−	0.421722i	$-0.861426\pi$
$229$	−3.29557e8	−1.81345	−0.906725	+	0.421722i	$0.861426\pi$
$230$	0	0
$231$	6.80635e7	0.363306
$232$	0	0
$233$	1.14579e8	0.593416	0.296708	−	0.954968i	$-0.404111\pi$
$233$	1.14579e8	0.593416	0.296708	+	0.954968i	$0.404111\pi$
$234$	0	0
$235$	−2.74475e8	−1.37964
$236$	0	0
$237$	6.42173e7	0.313352
$238$	0	0
$239$	1.39074e8	0.658953	0.329476	−	0.944164i	$-0.393128\pi$
$239$	1.39074e8	0.658953	0.329476	+	0.944164i	$0.393128\pi$
$240$	0	0
$241$	−1.21938e8	−0.561150	−0.280575	−	0.959832i	$-0.590525\pi$
$241$	−1.21938e8	−0.561150	−0.280575	+	0.959832i	$0.590525\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.55791e8	−1.11123
$246$	0	0
$247$	5.44954e7	0.230102
$248$	0	0
$249$	1.81226e8	0.743915
$250$	0	0
$251$	4.59091e8	1.83249	0.916243	−	0.400622i	$-0.131206\pi$
$251$	4.59091e8	1.83249	0.916243	+	0.400622i	$0.131206\pi$
$252$	0	0
$253$	6.77004e8	2.62827
$254$	0	0
$255$	2.11169e8	0.797516
$256$	0	0
$257$	−2.44461e8	−0.898347	−0.449173	−	0.893445i	$-0.648281\pi$
$257$	−2.44461e8	−0.898347	−0.449173	+	0.893445i	$0.648281\pi$
$258$	0	0
$259$	−1.14100e8	−0.408070
$260$	0	0
$261$	1.41585e8	0.492919
$262$	0	0
$263$	4.17922e7	0.141661	0.0708305	−	0.997488i	$-0.477435\pi$
$263$	4.17922e7	0.141661	0.0708305	+	0.997488i	$0.477435\pi$
$264$	0	0
$265$	1.11947e8	0.369533
$266$	0	0
$267$	1.52814e8	0.491329
$268$	0	0
$269$	−5.67457e8	−1.77746	−0.888730	−	0.458431i	$-0.848412\pi$
$269$	−5.67457e8	−1.77746	−0.888730	+	0.458431i	$0.848412\pi$
$270$	0	0
$271$	3.17865e8	0.970176	0.485088	−	0.874465i	$-0.338788\pi$
$271$	3.17865e8	0.970176	0.485088	+	0.874465i	$0.338788\pi$
$272$	0	0
$273$	−3.45071e7	−0.102645
$274$	0	0
$275$	−4.00206e8	−1.16043
$276$	0	0
$277$	6.09683e8	1.72355	0.861777	−	0.507287i	$-0.169351\pi$
$277$	6.09683e8	1.72355	0.861777	+	0.507287i	$0.169351\pi$
$278$	0	0
$279$	−1.20425e7	−0.0331973
$280$	0	0
$281$	−1.74608e8	−0.469453	−0.234726	−	0.972061i	$-0.575419\pi$
$281$	−1.74608e8	−0.469453	−0.234726	+	0.972061i	$0.575419\pi$
$282$	0	0
$283$	5.36563e8	1.40724	0.703620	−	0.710576i	$-0.251566\pi$
$283$	5.36563e8	1.40724	0.703620	+	0.710576i	$0.251566\pi$
$284$	0	0
$285$	−1.49894e8	−0.383554
$286$	0	0
$287$	5.64919e7	0.141059
$288$	0	0
$289$	4.44895e7	0.108422
$290$	0	0
$291$	−4.22515e8	−1.00512
$292$	0	0
$293$	−6.35688e8	−1.47641	−0.738206	−	0.674576i	$-0.764327\pi$
$293$	−6.35688e8	−1.47641	−0.738206	+	0.674576i	$0.764327\pi$
$294$	0	0
$295$	2.52668e8	0.573024
$296$	0	0
$297$	1.39759e8	0.309550
$298$	0	0
$299$	−3.43230e8	−0.742568
$300$	0	0
$301$	3.04012e8	0.642551
$302$	0	0
$303$	−4.57280e8	−0.944349
$304$	0	0
$305$	−4.31651e8	−0.871131
$306$	0	0
$307$	−3.05086e8	−0.601781	−0.300890	−	0.953659i	$-0.597284\pi$
$307$	−3.05086e8	−0.601781	−0.300890	+	0.953659i	$0.597284\pi$
$308$	0	0
$309$	−2.88968e8	−0.557179
$310$	0	0
$311$	−4.78228e8	−0.901517	−0.450758	−	0.892646i	$-0.648846\pi$
$311$	−4.78228e8	−0.901517	−0.450758	+	0.892646i	$0.648846\pi$
$312$	0	0
$313$	−3.87232e8	−0.713783	−0.356891	−	0.934146i	$-0.616163\pi$
$313$	−3.87232e8	−0.713783	−0.356891	+	0.934146i	$0.616163\pi$
$314$	0	0
$315$	9.49146e7	0.171098
$316$	0	0
$317$	−4.58088e8	−0.807683	−0.403842	−	0.914829i	$-0.632325\pi$
$317$	−4.58088e8	−0.807683	−0.403842	+	0.914829i	$0.632325\pi$
$318$	0	0
$319$	−1.37904e9	−2.37853
$320$	0	0
$321$	4.93281e8	0.832391
$322$	0	0
$323$	−3.22850e8	−0.533080
$324$	0	0
$325$	2.02898e8	0.327859
$326$	0	0
$327$	2.52728e7	0.0399703
$328$	0	0
$329$	−2.65720e8	−0.411375
$330$	0	0
$331$	3.25648e8	0.493572	0.246786	−	0.969070i	$-0.420626\pi$
$331$	3.25648e8	0.493572	0.246786	+	0.969070i	$0.420626\pi$
$332$	0	0
$333$	−2.34287e8	−0.347691
$334$	0	0
$335$	−1.56214e9	−2.27020
$336$	0	0
$337$	7.38971e8	1.05177	0.525887	−	0.850554i	$-0.323733\pi$
$337$	7.38971e8	1.05177	0.525887	+	0.850554i	$0.323733\pi$
$338$	0	0
$339$	5.19976e8	0.724911
$340$	0	0
$341$	1.17294e8	0.160190
$342$	0	0
$343$	−5.40013e8	−0.722561
$344$	0	0
$345$	9.44082e8	1.23778
$346$	0	0
$347$	9.79778e8	1.25885	0.629426	−	0.777060i	$-0.283290\pi$
$347$	9.79778e8	1.25885	0.629426	+	0.777060i	$0.283290\pi$
$348$	0	0
$349$	1.24643e9	1.56956	0.784780	−	0.619775i	$-0.212776\pi$
$349$	1.24643e9	1.56956	0.784780	+	0.619775i	$0.212776\pi$
$350$	0	0
$351$	−7.08554e7	−0.0874577
$352$	0	0
$353$	−6.58065e8	−0.796264	−0.398132	−	0.917328i	$-0.630341\pi$
$353$	−6.58065e8	−0.796264	−0.398132	+	0.917328i	$0.630341\pi$
$354$	0	0
$355$	−1.62612e9	−1.92909
$356$	0	0
$357$	2.04433e8	0.237800
$358$	0	0
$359$	4.52591e7	0.0516269	0.0258134	−	0.999667i	$-0.491782\pi$
$359$	4.52591e7	0.0516269	0.0258134	+	0.999667i	$0.491782\pi$
$360$	0	0
$361$	−6.64703e8	−0.743622
$362$	0	0
$363$	−8.35098e8	−0.916356
$364$	0	0
$365$	−7.77944e8	−0.837382
$366$	0	0
$367$	−1.03217e9	−1.08998	−0.544991	−	0.838442i	$-0.683467\pi$
$367$	−1.03217e9	−1.08998	−0.544991	+	0.838442i	$0.683467\pi$
$368$	0	0
$369$	1.15998e8	0.120187
$370$	0	0
$371$	1.08377e8	0.110186
$372$	0	0
$373$	5.25005e8	0.523821	0.261910	−	0.965092i	$-0.415647\pi$
$373$	5.25005e8	0.523821	0.261910	+	0.965092i	$0.415647\pi$
$374$	0	0
$375$	2.15476e8	0.211003
$376$	0	0
$377$	6.99151e8	0.672011
$378$	0	0
$379$	8.22050e8	0.775642	0.387821	−	0.921735i	$-0.373228\pi$
$379$	8.22050e8	0.775642	0.387821	+	0.921735i	$0.373228\pi$
$380$	0	0
$381$	9.19462e8	0.851719
$382$	0	0
$383$	2.99387e8	0.272293	0.136147	−	0.990689i	$-0.456528\pi$
$383$	2.99387e8	0.272293	0.136147	+	0.990689i	$0.456528\pi$
$384$	0	0
$385$	−9.24470e8	−0.825620
$386$	0	0
$387$	6.24245e8	0.547478
$388$	0	0
$389$	−7.12130e8	−0.613389	−0.306694	−	0.951808i	$-0.599223\pi$
$389$	−7.12130e8	−0.613389	−0.306694	+	0.951808i	$0.599223\pi$
$390$	0	0
$391$	2.03342e9	1.72032
$392$	0	0
$393$	1.24525e8	0.103487
$394$	0	0
$395$	−8.72229e8	−0.712099
$396$	0	0
$397$	−5.53020e8	−0.443582	−0.221791	−	0.975094i	$-0.571190\pi$
$397$	−5.53020e8	−0.443582	−0.221791	+	0.975094i	$0.571190\pi$
$398$	0	0
$399$	−1.45112e8	−0.114367
$400$	0	0
$401$	5.24965e8	0.406561	0.203280	−	0.979121i	$-0.434840\pi$
$401$	5.24965e8	0.406561	0.203280	+	0.979121i	$0.434840\pi$
$402$	0	0
$403$	−5.94664e7	−0.0452589
$404$	0	0
$405$	1.94894e8	0.145782
$406$	0	0
$407$	2.28196e9	1.67775
$408$	0	0
$409$	2.85772e8	0.206533	0.103266	−	0.994654i	$-0.467071\pi$
$409$	2.85772e8	0.206533	0.103266	+	0.994654i	$0.467071\pi$
$410$	0	0
$411$	3.08308e8	0.219047
$412$	0	0
$413$	2.44608e8	0.170862
$414$	0	0
$415$	−2.46150e9	−1.69056
$416$	0	0
$417$	2.69410e8	0.181944
$418$	0	0
$419$	−1.14977e9	−0.763591	−0.381796	−	0.924247i	$-0.624694\pi$
$419$	−1.14977e9	−0.763591	−0.381796	+	0.924247i	$0.624694\pi$
$420$	0	0
$421$	−2.46430e7	−0.0160956	−0.00804778	−	0.999968i	$-0.502562\pi$
$421$	−2.46430e7	−0.0160956	−0.00804778	+	0.999968i	$0.502562\pi$
$422$	0	0
$423$	−5.45617e8	−0.350507
$424$	0	0
$425$	−1.20204e9	−0.759555
$426$	0	0
$427$	−4.17882e8	−0.259751
$428$	0	0
$429$	6.90133e8	0.422019
$430$	0	0
$431$	−1.58335e9	−0.952593	−0.476297	−	0.879285i	$-0.658021\pi$
$431$	−1.58335e9	−0.952593	−0.476297	+	0.879285i	$0.658021\pi$
$432$	0	0
$433$	−8.36866e8	−0.495391	−0.247695	−	0.968838i	$-0.579673\pi$
$433$	−8.36866e8	−0.495391	−0.247695	+	0.968838i	$0.579673\pi$
$434$	0	0
$435$	−1.92307e9	−1.12017
$436$	0	0
$437$	−1.44338e9	−0.827363
$438$	0	0
$439$	3.08460e9	1.74010	0.870048	−	0.492967i	$-0.164088\pi$
$439$	3.08460e9	1.74010	0.870048	+	0.492967i	$0.164088\pi$
$440$	0	0
$441$	−5.08476e8	−0.282316
$442$	0	0
$443$	9.35708e8	0.511360	0.255680	−	0.966761i	$-0.417701\pi$
$443$	9.35708e8	0.511360	0.255680	+	0.966761i	$0.417701\pi$
$444$	0	0
$445$	−2.07558e9	−1.11656
$446$	0	0
$447$	8.61644e8	0.456301
$448$	0	0
$449$	3.11014e9	1.62150	0.810752	−	0.585390i	$-0.199058\pi$
$449$	3.11014e9	1.62150	0.810752	+	0.585390i	$0.199058\pi$
$450$	0	0
$451$	−1.12982e9	−0.579952
$452$	0	0
$453$	9.63229e7	0.0486840
$454$	0	0
$455$	4.68692e8	0.233264
$456$	0	0
$457$	2.01909e9	0.989577	0.494789	−	0.869013i	$-0.335246\pi$
$457$	2.01909e9	0.989577	0.494789	+	0.869013i	$0.335246\pi$
$458$	0	0
$459$	4.19773e8	0.202615
$460$	0	0
$461$	7.35152e8	0.349481	0.174741	−	0.984614i	$-0.444091\pi$
$461$	7.35152e8	0.349481	0.174741	+	0.984614i	$0.444091\pi$
$462$	0	0
$463$	2.98518e9	1.39777	0.698887	−	0.715232i	$-0.253679\pi$
$463$	2.98518e9	1.39777	0.698887	+	0.715232i	$0.253679\pi$
$464$	0	0
$465$	1.63567e8	0.0754415
$466$	0	0
$467$	−3.07591e9	−1.39754	−0.698769	−	0.715347i	$-0.746269\pi$
$467$	−3.07591e9	−1.39754	−0.698769	+	0.715347i	$0.746269\pi$
$468$	0	0
$469$	−1.51231e9	−0.676918
$470$	0	0
$471$	1.50251e9	0.662588
$472$	0	0
$473$	−6.08016e9	−2.64180
$474$	0	0
$475$	8.53246e8	0.365298
$476$	0	0
$477$	2.22536e8	0.0938826
$478$	0	0
$479$	1.60321e9	0.666523	0.333262	−	0.942834i	$-0.391851\pi$
$479$	1.60321e9	0.666523	0.333262	+	0.942834i	$0.391851\pi$
$480$	0	0
$481$	−1.15692e9	−0.474018
$482$	0	0
$483$	9.13967e8	0.369076
$484$	0	0
$485$	5.73880e9	2.28415
$486$	0	0
$487$	1.70605e9	0.669331	0.334665	−	0.942337i	$-0.391377\pi$
$487$	1.70605e9	0.669331	0.334665	+	0.942337i	$0.391377\pi$
$488$	0	0
$489$	−1.80552e9	−0.698267
$490$	0	0
$491$	3.03396e9	1.15671	0.578355	−	0.815785i	$-0.303695\pi$
$491$	3.03396e9	1.15671	0.578355	+	0.815785i	$0.303695\pi$
$492$	0	0
$493$	−4.14203e9	−1.55686
$494$	0	0
$495$	−1.89827e9	−0.703459
$496$	0	0
$497$	−1.57425e9	−0.575209
$498$	0	0
$499$	2.91071e9	1.04869	0.524344	−	0.851506i	$-0.324310\pi$
$499$	2.91071e9	1.04869	0.524344	+	0.851506i	$0.324310\pi$
$500$	0	0
$501$	−6.99173e8	−0.248400
$502$	0	0
$503$	1.61700e9	0.566528	0.283264	−	0.959042i	$-0.408583\pi$
$503$	1.61700e9	0.566528	0.283264	+	0.959042i	$0.408583\pi$
$504$	0	0
$505$	6.21098e9	2.14605
$506$	0	0
$507$	1.34432e9	0.458116
$508$	0	0
$509$	−1.47914e8	−0.0497163	−0.0248581	−	0.999691i	$-0.507913\pi$
$509$	−1.47914e8	−0.0497163	−0.0248581	+	0.999691i	$0.507913\pi$
$510$	0	0
$511$	−7.53129e8	−0.249687
$512$	0	0
$513$	−2.97968e8	−0.0974447
$514$	0	0
$515$	3.92489e9	1.26620
$516$	0	0
$517$	5.31432e9	1.69134
$518$	0	0
$519$	5.00866e8	0.157266
$520$	0	0
$521$	2.28069e9	0.706536	0.353268	−	0.935522i	$-0.385070\pi$
$521$	2.28069e9	0.706536	0.353268	+	0.935522i	$0.385070\pi$
$522$	0	0
$523$	3.70860e9	1.13359	0.566793	−	0.823860i	$-0.308184\pi$
$523$	3.70860e9	1.13359	0.566793	+	0.823860i	$0.308184\pi$
$524$	0	0
$525$	−5.40286e8	−0.162954
$526$	0	0
$527$	3.52301e8	0.104852
$528$	0	0
$529$	5.68609e9	1.67001
$530$	0	0
$531$	5.02268e8	0.145581
$532$	0	0
$533$	5.72802e8	0.163855
$534$	0	0
$535$	−6.69998e9	−1.89163
$536$	0	0
$537$	−3.89049e9	−1.08416
$538$	0	0
$539$	4.95256e9	1.36229
$540$	0	0
$541$	−2.43003e8	−0.0659813	−0.0329907	−	0.999456i	$-0.510503\pi$
$541$	−2.43003e8	−0.0659813	−0.0329907	+	0.999456i	$0.510503\pi$
$542$	0	0
$543$	2.15298e9	0.577087
$544$	0	0
$545$	−3.43267e8	−0.0908333
$546$	0	0
$547$	−1.23514e9	−0.322672	−0.161336	−	0.986900i	$-0.551580\pi$
$547$	−1.23514e9	−0.322672	−0.161336	+	0.986900i	$0.551580\pi$
$548$	0	0
$549$	−8.58061e8	−0.221317
$550$	0	0
$551$	2.94013e9	0.748749
$552$	0	0
$553$	−8.44406e8	−0.212331
$554$	0	0
$555$	3.18219e9	0.790135
$556$	0	0
$557$	4.03766e8	0.0990002	0.0495001	−	0.998774i	$-0.484237\pi$
$557$	4.03766e8	0.0990002	0.0495001	+	0.998774i	$0.484237\pi$
$558$	0	0
$559$	3.08254e9	0.746394
$560$	0	0
$561$	−4.08860e9	−0.977698
$562$	0	0
$563$	−1.17239e9	−0.276882	−0.138441	−	0.990371i	$-0.544209\pi$
$563$	−1.17239e9	−0.276882	−0.138441	+	0.990371i	$0.544209\pi$
$564$	0	0
$565$	−7.06256e9	−1.64738
$566$	0	0
$567$	1.88677e8	0.0434688
$568$	0	0
$569$	3.08771e9	0.702658	0.351329	−	0.936252i	$-0.385730\pi$
$569$	3.08771e9	0.702658	0.351329	+	0.936252i	$0.385730\pi$
$570$	0	0
$571$	1.26406e9	0.284147	0.142073	−	0.989856i	$-0.454623\pi$
$571$	1.26406e9	0.284147	0.142073	+	0.989856i	$0.454623\pi$
$572$	0	0
$573$	2.95914e9	0.657090
$574$	0	0
$575$	−5.37403e9	−1.17886
$576$	0	0
$577$	−4.12237e9	−0.893371	−0.446685	−	0.894691i	$-0.647396\pi$
$577$	−4.12237e9	−0.893371	−0.446685	+	0.894691i	$0.647396\pi$
$578$	0	0
$579$	−4.84201e9	−1.03669
$580$	0	0
$581$	−2.38298e9	−0.504085
$582$	0	0
$583$	−2.16750e9	−0.453022
$584$	0	0
$585$	9.62391e8	0.198750
$586$	0	0
$587$	1.43332e9	0.292489	0.146245	−	0.989248i	$-0.453281\pi$
$587$	1.43332e9	0.292489	0.146245	+	0.989248i	$0.453281\pi$
$588$	0	0
$589$	−2.50073e8	−0.0504271
$590$	0	0
$591$	−3.41306e9	−0.680124
$592$	0	0
$593$	9.06949e9	1.78604	0.893021	−	0.450015i	$-0.148581\pi$
$593$	9.06949e9	1.78604	0.893021	+	0.450015i	$0.148581\pi$
$594$	0	0
$595$	−2.77670e9	−0.540406
$596$	0	0
$597$	−1.45333e9	−0.279547
$598$	0	0
$599$	−6.87038e9	−1.30613	−0.653065	−	0.757302i	$-0.726517\pi$
$599$	−6.87038e9	−1.30613	−0.653065	+	0.757302i	$0.726517\pi$
$600$	0	0
$601$	1.21863e9	0.228988	0.114494	−	0.993424i	$-0.463475\pi$
$601$	1.21863e9	0.228988	0.114494	+	0.993424i	$0.463475\pi$
$602$	0	0
$603$	−3.10531e9	−0.576759
$604$	0	0
$605$	1.13427e10	2.08244
$606$	0	0
$607$	2.23889e8	0.0406324	0.0203162	−	0.999794i	$-0.493533\pi$
$607$	2.23889e8	0.0406324	0.0203162	+	0.999794i	$0.493533\pi$
$608$	0	0
$609$	−1.86173e9	−0.334007
$610$	0	0
$611$	−2.69428e9	−0.477857
$612$	0	0
$613$	−2.74524e9	−0.481359	−0.240680	−	0.970605i	$-0.577370\pi$
$613$	−2.74524e9	−0.481359	−0.240680	+	0.970605i	$0.577370\pi$
$614$	0	0
$615$	−1.57554e9	−0.273128
$616$	0	0
$617$	2.52382e7	0.00432574	0.00216287	−	0.999998i	$-0.499312\pi$
$617$	2.52382e7	0.00432574	0.00216287	+	0.999998i	$0.499312\pi$
$618$	0	0
$619$	4.17398e9	0.707347	0.353674	−	0.935369i	$-0.384932\pi$
$619$	4.17398e9	0.707347	0.353674	+	0.935369i	$0.384932\pi$
$620$	0	0
$621$	1.87670e9	0.314467
$622$	0	0
$623$	−2.00938e9	−0.332930
$624$	0	0
$625$	−7.33008e9	−1.20096
$626$	0	0
$627$	2.90221e9	0.470210
$628$	0	0
$629$	6.85400e9	1.09816
$630$	0	0
$631$	1.64464e9	0.260596	0.130298	−	0.991475i	$-0.458407\pi$
$631$	1.64464e9	0.260596	0.130298	+	0.991475i	$0.458407\pi$
$632$	0	0
$633$	−6.48583e8	−0.101637
$634$	0	0
$635$	−1.24886e10	−1.93555
$636$	0	0
$637$	−2.51087e9	−0.384890
$638$	0	0
$639$	−3.23249e9	−0.490100
$640$	0	0
$641$	7.21295e8	0.108171	0.0540854	−	0.998536i	$-0.482776\pi$
$641$	7.21295e8	0.108171	0.0540854	+	0.998536i	$0.482776\pi$
$642$	0	0
$643$	9.90611e8	0.146948	0.0734741	−	0.997297i	$-0.476591\pi$
$643$	9.90611e8	0.146948	0.0734741	+	0.997297i	$0.476591\pi$
$644$	0	0
$645$	−8.47878e9	−1.24415
$646$	0	0
$647$	6.87612e9	0.998110	0.499055	−	0.866570i	$-0.333681\pi$
$647$	6.87612e9	0.998110	0.499055	+	0.866570i	$0.333681\pi$
$648$	0	0
$649$	−4.89210e9	−0.702488
$650$	0	0
$651$	1.58350e8	0.0224949
$652$	0	0
$653$	−6.58692e9	−0.925734	−0.462867	−	0.886428i	$-0.653179\pi$
$653$	−6.58692e9	−0.925734	−0.462867	+	0.886428i	$0.653179\pi$
$654$	0	0
$655$	−1.69136e9	−0.235175
$656$	0	0
$657$	−1.54644e9	−0.212743
$658$	0	0
$659$	−1.21929e9	−0.165961	−0.0829806	−	0.996551i	$-0.526444\pi$
$659$	−1.21929e9	−0.165961	−0.0829806	+	0.996551i	$0.526444\pi$
$660$	0	0
$661$	−1.93235e9	−0.260244	−0.130122	−	0.991498i	$-0.541537\pi$
$661$	−1.93235e9	−0.260244	−0.130122	+	0.991498i	$0.541537\pi$
$662$	0	0
$663$	2.07286e9	0.276231
$664$	0	0
$665$	1.97098e9	0.259901
$666$	0	0
$667$	−1.85180e10	−2.41631
$668$	0	0
$669$	−2.08113e9	−0.268725
$670$	0	0
$671$	8.35753e9	1.06795
$672$	0	0
$673$	−9.47851e9	−1.19864	−0.599318	−	0.800511i	$-0.704561\pi$
$673$	−9.47851e9	−1.19864	−0.599318	+	0.800511i	$0.704561\pi$
$674$	0	0
$675$	−1.10940e9	−0.138843
$676$	0	0
$677$	7.46353e9	0.924452	0.462226	−	0.886762i	$-0.347051\pi$
$677$	7.46353e9	0.924452	0.462226	+	0.886762i	$0.347051\pi$
$678$	0	0
$679$	5.55574e9	0.681079
$680$	0	0
$681$	6.64511e9	0.806282
$682$	0	0
$683$	−1.79020e9	−0.214995	−0.107498	−	0.994205i	$-0.534284\pi$
$683$	−1.79020e9	−0.214995	−0.107498	+	0.994205i	$0.534284\pi$
$684$	0	0
$685$	−4.18758e9	−0.497790
$686$	0	0
$687$	8.89803e9	1.04700
$688$	0	0
$689$	1.09889e9	0.127993
$690$	0	0
$691$	−7.44397e9	−0.858285	−0.429142	−	0.903237i	$-0.641184\pi$
$691$	−7.44397e9	−0.858285	−0.429142	+	0.903237i	$0.641184\pi$
$692$	0	0
$693$	−1.83771e9	−0.209755
$694$	0	0
$695$	−3.65925e9	−0.413471
$696$	0	0
$697$	−3.39349e9	−0.379605
$698$	0	0
$699$	−3.09363e9	−0.342609
$700$	0	0
$701$	1.28971e9	0.141410	0.0707050	−	0.997497i	$-0.477475\pi$
$701$	1.28971e9	0.141410	0.0707050	+	0.997497i	$0.477475\pi$
$702$	0	0
$703$	−4.86517e9	−0.528147
$704$	0	0
$705$	7.41082e9	0.796534
$706$	0	0
$707$	6.01286e9	0.639902
$708$	0	0
$709$	−3.13464e9	−0.330313	−0.165157	−	0.986267i	$-0.552813\pi$
$709$	−3.13464e9	−0.330313	−0.165157	+	0.986267i	$0.552813\pi$
$710$	0	0
$711$	−1.73387e9	−0.180914
$712$	0	0
$713$	1.57505e9	0.162735
$714$	0	0
$715$	−9.37371e9	−0.959048
$716$	0	0
$717$	−3.75501e9	−0.380447
$718$	0	0
$719$	−7.98384e9	−0.801052	−0.400526	−	0.916285i	$-0.631173\pi$
$719$	−7.98384e9	−0.801052	−0.400526	+	0.916285i	$0.631173\pi$
$720$	0	0
$721$	3.79969e9	0.377551
$722$	0	0
$723$	3.29232e9	0.323980
$724$	0	0
$725$	1.09468e10	1.06685
$726$	0	0
$727$	−3.62828e9	−0.350212	−0.175106	−	0.984550i	$-0.556027\pi$
$727$	−3.62828e9	−0.350212	−0.175106	+	0.984550i	$0.556027\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.82621e10	−1.72918
$732$	0	0
$733$	1.42705e10	1.33837	0.669183	−	0.743098i	$-0.266644\pi$
$733$	1.42705e10	1.33837	0.669183	+	0.743098i	$0.266644\pi$
$734$	0	0
$735$	6.90635e9	0.641569
$736$	0	0
$737$	3.02458e10	2.78310
$738$	0	0
$739$	1.65044e10	1.50434	0.752168	−	0.658971i	$-0.229008\pi$
$739$	1.65044e10	1.50434	0.752168	+	0.658971i	$0.229008\pi$
$740$	0	0
$741$	−1.47137e9	−0.132849
$742$	0	0
$743$	−9.34147e9	−0.835516	−0.417758	−	0.908558i	$-0.637184\pi$
$743$	−9.34147e9	−0.835516	−0.417758	+	0.908558i	$0.637184\pi$
$744$	0	0
$745$	−1.17032e10	−1.03695
$746$	0	0
$747$	−4.89311e9	−0.429499
$748$	0	0
$749$	−6.48626e9	−0.564037
$750$	0	0
$751$	−1.36465e10	−1.17566	−0.587829	−	0.808986i	$-0.700017\pi$
$751$	−1.36465e10	−1.17566	−0.587829	+	0.808986i	$0.700017\pi$
$752$	0	0
$753$	−1.23955e10	−1.05799
$754$	0	0
$755$	−1.30830e9	−0.110635
$756$	0	0
$757$	6.63373e9	0.555804	0.277902	−	0.960609i	$-0.410361\pi$
$757$	6.63373e9	0.555804	0.277902	+	0.960609i	$0.410361\pi$
$758$	0	0
$759$	−1.82791e10	−1.51743
$760$	0	0
$761$	1.32467e10	1.08958	0.544792	−	0.838571i	$-0.316609\pi$
$761$	1.32467e10	1.08958	0.544792	+	0.838571i	$0.316609\pi$
$762$	0	0
$763$	−3.32318e8	−0.0270843
$764$	0	0
$765$	−5.70156e9	−0.460446
$766$	0	0
$767$	2.48022e9	0.198475
$768$	0	0
$769$	−1.73702e10	−1.37741	−0.688706	−	0.725041i	$-0.741821\pi$
$769$	−1.73702e10	−1.37741	−0.688706	+	0.725041i	$0.741821\pi$
$770$	0	0
$771$	6.60045e9	0.518661
$772$	0	0
$773$	−8.20862e9	−0.639207	−0.319604	−	0.947551i	$-0.603550\pi$
$773$	−8.20862e9	−0.639207	−0.319604	+	0.947551i	$0.603550\pi$
$774$	0	0
$775$	−9.31079e8	−0.0718506
$776$	0	0
$777$	3.08069e9	0.235599
$778$	0	0
$779$	2.40880e9	0.182566
$780$	0	0
$781$	3.14845e10	2.36493
$782$	0	0
$783$	−3.82279e9	−0.284587
$784$	0	0
$785$	−2.04078e10	−1.50575
$786$	0	0
$787$	−5.20336e9	−0.380516	−0.190258	−	0.981734i	$-0.560932\pi$
$787$	−5.20336e9	−0.380516	−0.190258	+	0.981734i	$0.560932\pi$
$788$	0	0
$789$	−1.12839e9	−0.0817881
$790$	0	0
$791$	−6.83728e9	−0.491208
$792$	0	0
$793$	−4.23714e9	−0.301729
$794$	0	0
$795$	−3.02258e9	−0.213350
$796$	0	0
$797$	−1.74511e10	−1.22101	−0.610504	−	0.792013i	$-0.709033\pi$
$797$	−1.74511e10	−1.22101	−0.610504	+	0.792013i	$0.709033\pi$
$798$	0	0
$799$	1.59619e10	1.10706
$800$	0	0
$801$	−4.12597e9	−0.283669
$802$	0	0
$803$	1.50624e10	1.02657
$804$	0	0
$805$	−1.24139e10	−0.838733
$806$	0	0
$807$	1.53213e10	1.02622
$808$	0	0
$809$	5.67781e8	0.0377017	0.0188509	−	0.999822i	$-0.493999\pi$
$809$	5.67781e8	0.0377017	0.0188509	+	0.999822i	$0.493999\pi$
$810$	0	0
$811$	2.26569e10	1.49151	0.745756	−	0.666219i	$-0.232088\pi$
$811$	2.26569e10	1.49151	0.745756	+	0.666219i	$0.232088\pi$
$812$	0	0
$813$	−8.58236e9	−0.560131
$814$	0	0
$815$	2.45234e10	1.58683
$816$	0	0
$817$	1.29630e10	0.831626
$818$	0	0
$819$	9.31693e8	0.0592624
$820$	0	0
$821$	5.43924e9	0.343034	0.171517	−	0.985181i	$-0.445133\pi$
$821$	5.43924e9	0.343034	0.171517	+	0.985181i	$0.445133\pi$
$822$	0	0
$823$	−3.81390e9	−0.238490	−0.119245	−	0.992865i	$-0.538047\pi$
$823$	−3.81390e9	−0.238490	−0.119245	+	0.992865i	$0.538047\pi$
$824$	0	0
$825$	1.08056e10	0.669976
$826$	0	0
$827$	2.04027e10	1.25435	0.627176	−	0.778878i	$-0.284211\pi$
$827$	2.04027e10	1.25435	0.627176	+	0.778878i	$0.284211\pi$
$828$	0	0
$829$	2.09133e10	1.27492	0.637458	−	0.770485i	$-0.279986\pi$
$829$	2.09133e10	1.27492	0.637458	+	0.770485i	$0.279986\pi$
$830$	0	0
$831$	−1.64614e10	−0.995095
$832$	0	0
$833$	1.48753e10	0.891680
$834$	0	0
$835$	9.49649e9	0.564495
$836$	0	0
$837$	3.25148e8	0.0191665
$838$	0	0
$839$	−2.29188e10	−1.33975	−0.669876	−	0.742473i	$-0.733653\pi$
$839$	−2.29188e10	−1.33975	−0.669876	+	0.742473i	$0.733653\pi$
$840$	0	0
$841$	2.04707e10	1.18672
$842$	0	0
$843$	4.71442e9	0.271039
$844$	0	0
$845$	−1.82592e10	−1.04108
$846$	0	0
$847$	1.09809e10	0.620933
$848$	0	0
$849$	−1.44872e10	−0.812471
$850$	0	0
$851$	3.06425e10	1.70440
$852$	0	0
$853$	−3.12228e9	−0.172247	−0.0861233	−	0.996284i	$-0.527448\pi$
$853$	−3.12228e9	−0.172247	−0.0861233	+	0.996284i	$0.527448\pi$
$854$	0	0
$855$	4.04713e9	0.221445
$856$	0	0
$857$	−3.39835e10	−1.84431	−0.922157	−	0.386817i	$-0.873575\pi$
$857$	−3.39835e10	−1.84431	−0.922157	+	0.386817i	$0.873575\pi$
$858$	0	0
$859$	2.07390e10	1.11638	0.558190	−	0.829713i	$-0.311496\pi$
$859$	2.07390e10	1.11638	0.558190	+	0.829713i	$0.311496\pi$
$860$	0	0
$861$	−1.52528e9	−0.0814402
$862$	0	0
$863$	1.13047e10	0.598719	0.299360	−	0.954140i	$-0.403227\pi$
$863$	1.13047e10	0.598719	0.299360	+	0.954140i	$0.403227\pi$
$864$	0	0
$865$	−6.80300e9	−0.357391
$866$	0	0
$867$	−1.20122e9	−0.0625972
$868$	0	0
$869$	1.68879e10	0.872984
$870$	0	0
$871$	−1.53341e10	−0.786314
$872$	0	0
$873$	1.14079e10	0.580305
$874$	0	0
$875$	−2.83333e9	−0.142978
$876$	0	0
$877$	−1.41724e10	−0.709486	−0.354743	−	0.934964i	$-0.615432\pi$
$877$	−1.41724e10	−0.709486	−0.354743	+	0.934964i	$0.615432\pi$
$878$	0	0
$879$	1.71636e10	0.852407
$880$	0	0
$881$	3.71670e8	0.0183123	0.00915614	−	0.999958i	$-0.497085\pi$
$881$	3.71670e8	0.0183123	0.00915614	+	0.999958i	$0.497085\pi$
$882$	0	0
$883$	2.28024e10	1.11460	0.557299	−	0.830312i	$-0.311838\pi$
$883$	2.28024e10	1.11460	0.557299	+	0.830312i	$0.311838\pi$
$884$	0	0
$885$	−6.82203e9	−0.330836
$886$	0	0
$887$	−4.09283e9	−0.196920	−0.0984602	−	0.995141i	$-0.531392\pi$
$887$	−4.09283e9	−0.196920	−0.0984602	+	0.995141i	$0.531392\pi$
$888$	0	0
$889$	−1.20902e10	−0.577135
$890$	0	0
$891$	−3.77348e9	−0.178719
$892$	0	0
$893$	−1.13302e10	−0.532424
$894$	0	0
$895$	5.28425e10	2.46378
$896$	0	0
$897$	9.26722e9	0.428722
$898$	0	0
$899$	−3.20833e9	−0.147272
$900$	0	0
$901$	−6.51022e9	−0.296523
$902$	0	0
$903$	−8.20832e9	−0.370977
$904$	0	0
$905$	−2.92428e10	−1.31144
$906$	0	0
$907$	−3.93238e10	−1.74997	−0.874983	−	0.484153i	$-0.839128\pi$
$907$	−3.93238e10	−1.74997	−0.874983	+	0.484153i	$0.839128\pi$
$908$	0	0
$909$	1.23465e10	0.545220
$910$	0	0
$911$	2.39548e10	1.04973	0.524866	−	0.851185i	$-0.324115\pi$
$911$	2.39548e10	1.04973	0.524866	+	0.851185i	$0.324115\pi$
$912$	0	0
$913$	4.76589e10	2.07251
$914$	0	0
$915$	1.16546e10	0.502948
$916$	0	0
$917$	−1.63741e9	−0.0701237
$918$	0	0
$919$	−3.48503e10	−1.48116	−0.740580	−	0.671968i	$-0.765449\pi$
$919$	−3.48503e10	−1.48116	−0.740580	+	0.671968i	$0.765449\pi$
$920$	0	0
$921$	8.23733e9	0.347438
$922$	0	0
$923$	−1.59622e10	−0.668168
$924$	0	0
$925$	−1.81141e10	−0.752526
$926$	0	0
$927$	7.80213e9	0.321687
$928$	0	0
$929$	−1.68507e9	−0.0689547	−0.0344774	−	0.999405i	$-0.510977\pi$
$929$	−1.68507e9	−0.0689547	−0.0344774	+	0.999405i	$0.510977\pi$
$930$	0	0
$931$	−1.05589e10	−0.428841
$932$	0	0
$933$	1.29122e10	0.520491
$934$	0	0
$935$	5.55333e10	2.22184
$936$	0	0
$937$	−1.83372e10	−0.728189	−0.364094	−	0.931362i	$-0.618621\pi$
$937$	−1.83372e10	−0.728189	−0.364094	+	0.931362i	$0.618621\pi$
$938$	0	0
$939$	1.04553e10	0.412103
$940$	0	0
$941$	3.09477e9	0.121078	0.0605390	−	0.998166i	$-0.480718\pi$
$941$	3.09477e9	0.121078	0.0605390	+	0.998166i	$0.480718\pi$
$942$	0	0
$943$	−1.51714e10	−0.589163
$944$	0	0
$945$	−2.56269e9	−0.0987838
$946$	0	0
$947$	−4.46943e10	−1.71012	−0.855061	−	0.518527i	$-0.826481\pi$
$947$	−4.46943e10	−1.71012	−0.855061	+	0.518527i	$0.826481\pi$
$948$	0	0
$949$	−7.63639e9	−0.290039
$950$	0	0
$951$	1.23684e10	0.466316
$952$	0	0
$953$	−2.13357e10	−0.798514	−0.399257	−	0.916839i	$-0.630732\pi$
$953$	−2.13357e10	−0.798514	−0.399257	+	0.916839i	$0.630732\pi$
$954$	0	0
$955$	−4.01925e10	−1.49325
$956$	0	0
$957$	3.72341e10	1.37325
$958$	0	0
$959$	−4.05400e9	−0.148429
$960$	0	0
$961$	−2.72397e10	−0.990081
$962$	0	0
$963$	−1.33186e10	−0.480581
$964$	0	0
$965$	6.57664e10	2.35591
$966$	0	0
$967$	1.55906e10	0.554461	0.277231	−	0.960803i	$-0.410583\pi$
$967$	1.55906e10	0.554461	0.277231	+	0.960803i	$0.410583\pi$
$968$	0	0
$969$	8.71696e9	0.307774
$970$	0	0
$971$	7.00913e9	0.245695	0.122848	−	0.992426i	$-0.460797\pi$
$971$	7.00913e9	0.245695	0.122848	+	0.992426i	$0.460797\pi$
$972$	0	0
$973$	−3.54252e9	−0.123287
$974$	0	0
$975$	−5.47826e9	−0.189289
$976$	0	0
$977$	1.94015e10	0.665587	0.332794	−	0.943000i	$-0.392009\pi$
$977$	1.94015e10	0.665587	0.332794	+	0.943000i	$0.392009\pi$
$978$	0	0
$979$	4.01870e10	1.36882
$980$	0	0
$981$	−6.82367e8	−0.0230768
$982$	0	0
$983$	−5.00668e10	−1.68117	−0.840587	−	0.541677i	$-0.817790\pi$
$983$	−5.00668e10	−1.68117	−0.840587	+	0.541677i	$0.817790\pi$
$984$	0	0
$985$	4.63578e10	1.54560
$986$	0	0
$987$	7.17443e9	0.237507
$988$	0	0
$989$	−8.16453e10	−2.68376
$990$	0	0
$991$	5.16083e10	1.68446	0.842232	−	0.539115i	$-0.181241\pi$
$991$	5.16083e10	1.68446	0.842232	+	0.539115i	$0.181241\pi$
$992$	0	0
$993$	−8.79249e9	−0.284964
$994$	0	0
$995$	1.97398e10	0.635276
$996$	0	0
$997$	−5.60890e9	−0.179244	−0.0896221	−	0.995976i	$-0.528566\pi$
$997$	−5.60890e9	−0.179244	−0.0896221	+	0.995976i	$0.528566\pi$
$998$	0	0
$999$	6.32575e9	0.200739

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.8.a.c.1.1	✓	2	8.3	odd	2
96.8.a.f.1.1	yes	2	8.5	even	2
192.8.a.s.1.2		2	1.1	even	1	trivial
192.8.a.v.1.2		2	4.3	odd	2
288.8.a.l.1.2		2	24.5	odd	2
288.8.a.m.1.2		2	24.11	even	2
576.8.a.bg.1.1		2	3.2	odd	2
576.8.a.bh.1.1		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +366.727 q^{5} +355.029 q^{7} +729.000 q^{9} -7100.47 q^{11} +3599.83 q^{13} -9901.62 q^{15} -21326.7 q^{17} +15138.3 q^{19} -9585.77 q^{21} -95346.3 q^{23} +56363.3 q^{25} -19683.0 q^{27} +194218. q^{29} -16519.2 q^{31} +191713. q^{33} +130198. q^{35} -321381. q^{37} -97195.4 q^{39} +159119. q^{41} +856303. q^{43} +267344. q^{45} -748446. q^{47} -697498. q^{49} +575821. q^{51} +305261. q^{53} -2.60393e6 q^{55} -408735. q^{57} +688982. q^{59} -1.17704e6 q^{61} +258816. q^{63} +1.32015e6 q^{65} -4.25969e6 q^{67} +2.57435e6 q^{69} -4.43415e6 q^{71} -2.12132e6 q^{73} -1.52181e6 q^{75} -2.52087e6 q^{77} -2.37842e6 q^{79} +531441. q^{81} -6.71208e6 q^{83} -7.82107e6 q^{85} -5.24389e6 q^{87} -5.65976e6 q^{89} +1.27804e6 q^{91} +446019. q^{93} +5.55162e6 q^{95} +1.56487e7 q^{97} -5.17625e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} + 28 q^{5} - 936 q^{7} + 1458 q^{9} - 3384 q^{11} + 17076 q^{13} - 756 q^{15} - 18668 q^{17} + 44856 q^{19} + 25272 q^{21} - 43488 q^{23} + 92974 q^{25} - 39366 q^{27} + 19484 q^{29} - 332856 q^{31}+ \cdots - 2466936 q^{99}+O(q^{100}) \) 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

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