LMFDB - Embedded newform 384.10.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.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:	$197.773761087$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2070x^{2} - 13768x + 561570 $ x^4 - 2070x^2 - 13768x + 561570
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$45.8607$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000 q^{3} +250.262 q^{5} -1655.62 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} +250.262 q^{5} -1655.62 q^{7} +6561.00 q^{9} +848.983 q^{11} +69683.7 q^{13} -20271.3 q^{15} +282069. q^{17} +470922. q^{19} +134105. q^{21} +2.41851e6 q^{23} -1.89049e6 q^{25} -531441. q^{27} -3.90059e6 q^{29} -3.99099e6 q^{31} -68767.7 q^{33} -414339. q^{35} +5.47494e6 q^{37} -5.64438e6 q^{39} +1.25933e7 q^{41} +1.12591e7 q^{43} +1.64197e6 q^{45} +2.89424e7 q^{47} -3.76125e7 q^{49} -2.28476e7 q^{51} -9.61718e6 q^{53} +212469. q^{55} -3.81447e7 q^{57} -1.53642e7 q^{59} -2.89328e7 q^{61} -1.08625e7 q^{63} +1.74392e7 q^{65} +1.43864e8 q^{67} -1.95899e8 q^{69} -1.80186e8 q^{71} +2.57115e8 q^{73} +1.53130e8 q^{75} -1.40559e6 q^{77} -1.36452e8 q^{79} +4.30467e7 q^{81} +2.41915e7 q^{83} +7.05914e7 q^{85} +3.15948e8 q^{87} -8.66242e8 q^{89} -1.15370e8 q^{91} +3.23271e8 q^{93} +1.17854e8 q^{95} -6.24828e8 q^{97} +5.57018e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{3} + 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q - 324 * q^3 + 240 * q^5 + 4840 * q^7 + 26244 * q^9	$ 4 q - 324 q^{3} + 240 q^{5} + 4840 q^{7} + 26244 q^{9} + 99664 q^{11} + 60840 q^{13} - 19440 q^{15} - 434952 q^{17} - 631776 q^{19} - 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} - 2125764 q^{27} + 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} + 25567008 q^{35} + 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} - 14177920 q^{43} + 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} + 35231112 q^{51} - 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} + 115028080 q^{59} - 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} + 231785104 q^{67} - 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} - 485314092 q^{75} + 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} + 607613328 q^{83} - 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} + 847629840 q^{91} - 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} + 653895504 q^{99}+O(q^{100}) $ 4 * q - 324 * q^3 + 240 * q^5 + 4840 * q^7 + 26244 * q^9 + 99664 * q^11 + 60840 * q^13 - 19440 * q^15 - 434952 * q^17 - 631776 * q^19 - 392040 * q^21 + 749392 * q^23 + 5991532 * q^25 - 2125764 * q^27 + 7908544 * q^29 + 11351240 * q^31 - 8072784 * q^33 + 25567008 * q^35 + 13592920 * q^37 - 4928040 * q^39 - 18838888 * q^41 - 14177920 * q^43 + 1574640 * q^45 - 37779120 * q^47 + 9409332 * q^49 + 35231112 * q^51 - 115336512 * q^53 - 184580544 * q^55 + 51173856 * q^57 + 115028080 * q^59 - 173228648 * q^61 + 31755240 * q^63 - 328077984 * q^65 + 231785104 * q^67 - 60700752 * q^69 + 197476208 * q^71 + 44629400 * q^73 - 485314092 * q^75 + 308117920 * q^77 - 355774584 * q^79 + 172186884 * q^81 + 607613328 * q^83 - 1087351392 * q^85 - 640592064 * q^87 - 1157146424 * q^89 + 847629840 * q^91 - 919450440 * q^93 - 329699328 * q^95 - 1599536472 * q^97 + 653895504 * 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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	250.262	0.179073	0.0895366	−	0.995984i	$-0.471461\pi$
$5$	250.262	0.179073	0.0895366	+	0.995984i	$0.471461\pi$
$6$	0	0
$7$	−1655.62	−0.260627	−0.130313	−	0.991473i	$-0.541598\pi$
$7$	−1655.62	−0.260627	−0.130313	+	0.991473i	$0.541598\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	848.983	0.0174837	0.00874183	−	0.999962i	$-0.497217\pi$
$11$	848.983	0.0174837	0.00874183	+	0.999962i	$0.497217\pi$
$12$	0	0
$13$	69683.7	0.676684	0.338342	−	0.941023i	$-0.390134\pi$
$13$	69683.7	0.676684	0.338342	+	0.941023i	$0.390134\pi$
$14$	0	0
$15$	−20271.3	−0.103388
$16$	0	0
$17$	282069.	0.819098	0.409549	−	0.912288i	$-0.365686\pi$
$17$	282069.	0.819098	0.409549	+	0.912288i	$0.365686\pi$
$18$	0	0
$19$	470922.	0.829006	0.414503	−	0.910048i	$-0.363955\pi$
$19$	470922.	0.829006	0.414503	+	0.910048i	$0.363955\pi$
$20$	0	0
$21$	134105.	0.150473
$22$	0	0
$23$	2.41851e6	1.80207	0.901036	−	0.433745i	$-0.142808\pi$
$23$	2.41851e6	1.80207	0.901036	+	0.433745i	$0.142808\pi$
$24$	0	0
$25$	−1.89049e6	−0.967933
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	−3.90059e6	−1.02409	−0.512047	−	0.858958i	$-0.671112\pi$
$29$	−3.90059e6	−1.02409	−0.512047	+	0.858958i	$0.671112\pi$
$30$	0	0
$31$	−3.99099e6	−0.776164	−0.388082	−	0.921625i	$-0.626862\pi$
$31$	−3.99099e6	−0.776164	−0.388082	+	0.921625i	$0.626862\pi$
$32$	0	0
$33$	−68767.7	−0.0100942
$34$	0	0
$35$	−414339.	−0.0466713
$36$	0	0
$37$	5.47494e6	0.480254	0.240127	−	0.970741i	$-0.422811\pi$
$37$	5.47494e6	0.480254	0.240127	+	0.970741i	$0.422811\pi$
$38$	0	0
$39$	−5.64438e6	−0.390684
$40$	0	0
$41$	1.25933e7	0.696004	0.348002	−	0.937494i	$-0.386860\pi$
$41$	1.25933e7	0.696004	0.348002	+	0.937494i	$0.386860\pi$
$42$	0	0
$43$	1.12591e7	0.502222	0.251111	−	0.967958i	$-0.419204\pi$
$43$	1.12591e7	0.502222	0.251111	+	0.967958i	$0.419204\pi$
$44$	0	0
$45$	1.64197e6	0.0596911
$46$	0	0
$47$	2.89424e7	0.865155	0.432578	−	0.901597i	$-0.357604\pi$
$47$	2.89424e7	0.865155	0.432578	+	0.901597i	$0.357604\pi$
$48$	0	0
$49$	−3.76125e7	−0.932074
$50$	0	0
$51$	−2.28476e7	−0.472907
$52$	0	0
$53$	−9.61718e6	−0.167420	−0.0837098	−	0.996490i	$-0.526677\pi$
$53$	−9.61718e6	−0.167420	−0.0837098	+	0.996490i	$0.526677\pi$
$54$	0	0
$55$	212469.	0.00313085
$56$	0	0
$57$	−3.81447e7	−0.478627
$58$	0	0
$59$	−1.53642e7	−0.165073	−0.0825364	−	0.996588i	$-0.526302\pi$
$59$	−1.53642e7	−0.165073	−0.0825364	+	0.996588i	$0.526302\pi$
$60$	0	0
$61$	−2.89328e7	−0.267551	−0.133776	−	0.991012i	$-0.542710\pi$
$61$	−2.89328e7	−0.267551	−0.133776	+	0.991012i	$0.542710\pi$
$62$	0	0
$63$	−1.08625e7	−0.0868756
$64$	0	0
$65$	1.74392e7	0.121176
$66$	0	0
$67$	1.43864e8	0.872200	0.436100	−	0.899898i	$-0.356359\pi$
$67$	1.43864e8	0.872200	0.436100	+	0.899898i	$0.356359\pi$
$68$	0	0
$69$	−1.95899e8	−1.04043
$70$	0	0
$71$	−1.80186e8	−0.841506	−0.420753	−	0.907175i	$-0.638234\pi$
$71$	−1.80186e8	−0.841506	−0.420753	+	0.907175i	$0.638234\pi$
$72$	0	0
$73$	2.57115e8	1.05968	0.529839	−	0.848098i	$-0.322252\pi$
$73$	2.57115e8	1.05968	0.529839	+	0.848098i	$0.322252\pi$
$74$	0	0
$75$	1.53130e8	0.558836
$76$	0	0
$77$	−1.40559e6	−0.00455671
$78$	0	0
$79$	−1.36452e8	−0.394148	−0.197074	−	0.980389i	$-0.563144\pi$
$79$	−1.36452e8	−0.394148	−0.197074	+	0.980389i	$0.563144\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	2.41915e7	0.0559514	0.0279757	−	0.999609i	$-0.491094\pi$
$83$	2.41915e7	0.0559514	0.0279757	+	0.999609i	$0.491094\pi$
$84$	0	0
$85$	7.05914e7	0.146679
$86$	0	0
$87$	3.15948e8	0.591260
$88$	0	0
$89$	−8.66242e8	−1.46347	−0.731735	−	0.681589i	$-0.761289\pi$
$89$	−8.66242e8	−1.46347	−0.731735	+	0.681589i	$0.761289\pi$
$90$	0	0
$91$	−1.15370e8	−0.176362
$92$	0	0
$93$	3.23271e8	0.448118
$94$	0	0
$95$	1.17854e8	0.148453
$96$	0	0
$97$	−6.24828e8	−0.716618	−0.358309	−	0.933603i	$-0.616647\pi$
$97$	−6.24828e8	−0.716618	−0.358309	+	0.933603i	$0.616647\pi$
$98$	0	0
$99$	5.57018e6	0.00582788
$100$	0	0
$101$	−4.63587e8	−0.443288	−0.221644	−	0.975128i	$-0.571142\pi$
$101$	−4.63587e8	−0.443288	−0.221644	+	0.975128i	$0.571142\pi$
$102$	0	0
$103$	−1.18683e9	−1.03901	−0.519507	−	0.854466i	$-0.673884\pi$
$103$	−1.18683e9	−1.03901	−0.519507	+	0.854466i	$0.673884\pi$
$104$	0	0
$105$	3.35615e7	0.0269457
$106$	0	0
$107$	1.61716e9	1.19269	0.596343	−	0.802730i	$-0.296620\pi$
$107$	1.61716e9	1.19269	0.596343	+	0.802730i	$0.296620\pi$
$108$	0	0
$109$	−2.78924e8	−0.189264	−0.0946318	−	0.995512i	$-0.530167\pi$
$109$	−2.78924e8	−0.189264	−0.0946318	+	0.995512i	$0.530167\pi$
$110$	0	0
$111$	−4.43470e8	−0.277275
$112$	0	0
$113$	7.83311e8	0.451940	0.225970	−	0.974134i	$-0.427445\pi$
$113$	7.83311e8	0.451940	0.225970	+	0.974134i	$0.427445\pi$
$114$	0	0
$115$	6.05261e8	0.322703
$116$	0	0
$117$	4.57195e8	0.225561
$118$	0	0
$119$	−4.66999e8	−0.213479
$120$	0	0
$121$	−2.35723e9	−0.999694
$122$	0	0
$123$	−1.02006e9	−0.401838
$124$	0	0
$125$	−9.61913e8	−0.352404
$126$	0	0
$127$	2.05194e9	0.699919	0.349960	−	0.936765i	$-0.386195\pi$
$127$	2.05194e9	0.699919	0.349960	+	0.936765i	$0.386195\pi$
$128$	0	0
$129$	−9.11988e8	−0.289958
$130$	0	0
$131$	1.10603e9	0.328131	0.164065	−	0.986449i	$-0.447539\pi$
$131$	1.10603e9	0.328131	0.164065	+	0.986449i	$0.447539\pi$
$132$	0	0
$133$	−7.79667e8	−0.216061
$134$	0	0
$135$	−1.33000e8	−0.0344627
$136$	0	0
$137$	1.04415e9	0.253233	0.126617	−	0.991952i	$-0.459588\pi$
$137$	1.04415e9	0.253233	0.126617	+	0.991952i	$0.459588\pi$
$138$	0	0
$139$	−2.93771e9	−0.667487	−0.333743	−	0.942664i	$-0.608312\pi$
$139$	−2.93771e9	−0.667487	−0.333743	+	0.942664i	$0.608312\pi$
$140$	0	0
$141$	−2.34433e9	−0.499498
$142$	0	0
$143$	5.91603e7	0.0118309
$144$	0	0
$145$	−9.76171e8	−0.183388
$146$	0	0
$147$	3.04662e9	0.538133
$148$	0	0
$149$	3.75086e9	0.623437	0.311719	−	0.950174i	$-0.399095\pi$
$149$	3.75086e9	0.623437	0.311719	+	0.950174i	$0.399095\pi$
$150$	0	0
$151$	2.48184e9	0.388487	0.194244	−	0.980953i	$-0.437775\pi$
$151$	2.48184e9	0.388487	0.194244	+	0.980953i	$0.437775\pi$
$152$	0	0
$153$	1.85066e9	0.273033
$154$	0	0
$155$	−9.98796e8	−0.138990
$156$	0	0
$157$	5.65770e9	0.743175	0.371588	−	0.928398i	$-0.378814\pi$
$157$	5.65770e9	0.743175	0.371588	+	0.928398i	$0.378814\pi$
$158$	0	0
$159$	7.78991e8	0.0966597
$160$	0	0
$161$	−4.00412e9	−0.469668
$162$	0	0
$163$	6.28827e9	0.697729	0.348865	−	0.937173i	$-0.386567\pi$
$163$	6.28827e9	0.697729	0.348865	+	0.937173i	$0.386567\pi$
$164$	0	0
$165$	−1.72100e7	−0.00180760
$166$	0	0
$167$	−1.43509e9	−0.142776	−0.0713882	−	0.997449i	$-0.522743\pi$
$167$	−1.43509e9	−0.142776	−0.0713882	+	0.997449i	$0.522743\pi$
$168$	0	0
$169$	−5.74868e9	−0.542098
$170$	0	0
$171$	3.08972e9	0.276335
$172$	0	0
$173$	1.06084e10	0.900416	0.450208	−	0.892924i	$-0.351350\pi$
$173$	1.06084e10	0.900416	0.450208	+	0.892924i	$0.351350\pi$
$174$	0	0
$175$	3.12994e9	0.252269
$176$	0	0
$177$	1.24450e9	0.0953048
$178$	0	0
$179$	−3.84540e8	−0.0279964	−0.0139982	−	0.999902i	$-0.504456\pi$
$179$	−3.84540e8	−0.0279964	−0.0139982	+	0.999902i	$0.504456\pi$
$180$	0	0
$181$	3.54919e9	0.245797	0.122898	−	0.992419i	$-0.460781\pi$
$181$	3.54919e9	0.245797	0.122898	+	0.992419i	$0.460781\pi$
$182$	0	0
$183$	2.34356e9	0.154471
$184$	0	0
$185$	1.37017e9	0.0860007
$186$	0	0
$187$	2.39472e8	0.0143208
$188$	0	0
$189$	8.79863e8	0.0501576
$190$	0	0
$191$	2.80130e10	1.52304	0.761518	−	0.648144i	$-0.224455\pi$
$191$	2.80130e10	1.52304	0.761518	+	0.648144i	$0.224455\pi$
$192$	0	0
$193$	1.01773e10	0.527986	0.263993	−	0.964525i	$-0.414960\pi$
$193$	1.01773e10	0.527986	0.263993	+	0.964525i	$0.414960\pi$
$194$	0	0
$195$	−1.41258e9	−0.0699610
$196$	0	0
$197$	7.40263e9	0.350177	0.175089	−	0.984553i	$-0.443979\pi$
$197$	7.40263e9	0.350177	0.175089	+	0.984553i	$0.443979\pi$
$198$	0	0
$199$	1.14131e10	0.515899	0.257950	−	0.966158i	$-0.416953\pi$
$199$	1.14131e10	0.515899	0.257950	+	0.966158i	$0.416953\pi$
$200$	0	0
$201$	−1.16530e10	−0.503565
$202$	0	0
$203$	6.45789e9	0.266906
$204$	0	0
$205$	3.15162e9	0.124636
$206$	0	0
$207$	1.58678e10	0.600690
$208$	0	0
$209$	3.99805e8	0.0144941
$210$	0	0
$211$	1.50794e10	0.523735	0.261868	−	0.965104i	$-0.415662\pi$
$211$	1.50794e10	0.523735	0.261868	+	0.965104i	$0.415662\pi$
$212$	0	0
$213$	1.45950e10	0.485844
$214$	0	0
$215$	2.81773e9	0.0899345
$216$	0	0
$217$	6.60756e9	0.202289
$218$	0	0
$219$	−2.08263e10	−0.611806
$220$	0	0
$221$	1.96556e10	0.554271
$222$	0	0
$223$	1.05027e10	0.284399	0.142200	−	0.989838i	$-0.454583\pi$
$223$	1.05027e10	0.284399	0.142200	+	0.989838i	$0.454583\pi$
$224$	0	0
$225$	−1.24035e10	−0.322644
$226$	0	0
$227$	6.86346e10	1.71564	0.857821	−	0.513948i	$-0.171818\pi$
$227$	6.86346e10	1.71564	0.857821	+	0.513948i	$0.171818\pi$
$228$	0	0
$229$	5.87491e10	1.41170	0.705848	−	0.708363i	$-0.250566\pi$
$229$	5.87491e10	1.41170	0.705848	+	0.708363i	$0.250566\pi$
$230$	0	0
$231$	1.13853e8	0.00263082
$232$	0	0
$233$	−3.66986e10	−0.815734	−0.407867	−	0.913041i	$-0.633727\pi$
$233$	−3.66986e10	−0.815734	−0.407867	+	0.913041i	$0.633727\pi$
$234$	0	0
$235$	7.24319e9	0.154926
$236$	0	0
$237$	1.10526e10	0.227561
$238$	0	0
$239$	2.46017e10	0.487724	0.243862	−	0.969810i	$-0.421586\pi$
$239$	2.46017e10	0.487724	0.243862	+	0.969810i	$0.421586\pi$
$240$	0	0
$241$	−7.56409e10	−1.44438	−0.722188	−	0.691697i	$-0.756863\pi$
$241$	−7.56409e10	−1.44438	−0.722188	+	0.691697i	$0.756863\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−9.41301e9	−0.166909
$246$	0	0
$247$	3.28156e10	0.560975
$248$	0	0
$249$	−1.95951e9	−0.0323036
$250$	0	0
$251$	−2.92636e10	−0.465367	−0.232683	−	0.972553i	$-0.574751\pi$
$251$	−2.92636e10	−0.465367	−0.232683	+	0.972553i	$0.574751\pi$
$252$	0	0
$253$	2.05327e9	0.0315068
$254$	0	0
$255$	−5.71790e9	−0.0846849
$256$	0	0
$257$	−2.61646e10	−0.374123	−0.187062	−	0.982348i	$-0.559896\pi$
$257$	−2.61646e10	−0.374123	−0.187062	+	0.982348i	$0.559896\pi$
$258$	0	0
$259$	−9.06440e9	−0.125167
$260$	0	0
$261$	−2.55918e10	−0.341364
$262$	0	0
$263$	4.64847e9	0.0599114	0.0299557	−	0.999551i	$-0.490463\pi$
$263$	4.64847e9	0.0599114	0.0299557	+	0.999551i	$0.490463\pi$
$264$	0	0
$265$	−2.40682e9	−0.0299804
$266$	0	0
$267$	7.01656e10	0.844935
$268$	0	0
$269$	5.15725e10	0.600527	0.300264	−	0.953856i	$-0.402925\pi$
$269$	5.15725e10	0.600527	0.300264	+	0.953856i	$0.402925\pi$
$270$	0	0
$271$	1.43382e11	1.61485	0.807424	−	0.589971i	$-0.200861\pi$
$271$	1.43382e11	1.61485	0.807424	+	0.589971i	$0.200861\pi$
$272$	0	0
$273$	9.34494e9	0.101823
$274$	0	0
$275$	−1.60500e9	−0.0169230
$276$	0	0
$277$	7.25340e10	0.740257	0.370128	−	0.928981i	$-0.379314\pi$
$277$	7.25340e10	0.740257	0.370128	+	0.928981i	$0.379314\pi$
$278$	0	0
$279$	−2.61849e10	−0.258721
$280$	0	0
$281$	−1.62285e11	−1.55275	−0.776374	−	0.630272i	$-0.782943\pi$
$281$	−1.62285e11	−1.55275	−0.776374	+	0.630272i	$0.782943\pi$
$282$	0	0
$283$	1.79317e11	1.66182	0.830908	−	0.556410i	$-0.187822\pi$
$283$	1.79317e11	1.66182	0.830908	+	0.556410i	$0.187822\pi$
$284$	0	0
$285$	−9.54618e9	−0.0857093
$286$	0	0
$287$	−2.08497e10	−0.181397
$288$	0	0
$289$	−3.90247e10	−0.329078
$290$	0	0
$291$	5.06111e10	0.413740
$292$	0	0
$293$	2.30277e11	1.82535	0.912676	−	0.408684i	$-0.134012\pi$
$293$	2.30277e11	1.82535	0.912676	+	0.408684i	$0.134012\pi$
$294$	0	0
$295$	−3.84508e9	−0.0295601
$296$	0	0
$297$	−4.51185e8	−0.00336473
$298$	0	0
$299$	1.68530e11	1.21943
$300$	0	0
$301$	−1.86408e10	−0.130893
$302$	0	0
$303$	3.75506e10	0.255932
$304$	0	0
$305$	−7.24081e9	−0.0479113
$306$	0	0
$307$	2.81856e11	1.81094	0.905471	−	0.424409i	$-0.139518\pi$
$307$	2.81856e11	1.81094	0.905471	+	0.424409i	$0.139518\pi$
$308$	0	0
$309$	9.61333e10	0.599875
$310$	0	0
$311$	2.33240e11	1.41378	0.706889	−	0.707325i	$-0.250098\pi$
$311$	2.33240e11	1.41378	0.706889	+	0.707325i	$0.250098\pi$
$312$	0	0
$313$	−3.29142e11	−1.93836	−0.969179	−	0.246360i	$-0.920766\pi$
$313$	−3.29142e11	−1.93836	−0.969179	+	0.246360i	$0.920766\pi$
$314$	0	0
$315$	−2.71848e9	−0.0155571
$316$	0	0
$317$	−9.06182e10	−0.504021	−0.252011	−	0.967724i	$-0.581092\pi$
$317$	−9.06182e10	−0.504021	−0.252011	+	0.967724i	$0.581092\pi$
$318$	0	0
$319$	−3.31154e9	−0.0179049
$320$	0	0
$321$	−1.30990e11	−0.688597
$322$	0	0
$323$	1.32833e11	0.679037
$324$	0	0
$325$	−1.31737e11	−0.654985
$326$	0	0
$327$	2.25929e10	0.109271
$328$	0	0
$329$	−4.79175e10	−0.225483
$330$	0	0
$331$	−3.07245e11	−1.40689	−0.703443	−	0.710752i	$-0.748355\pi$
$331$	−3.07245e11	−1.40689	−0.703443	+	0.710752i	$0.748355\pi$
$332$	0	0
$333$	3.59211e10	0.160085
$334$	0	0
$335$	3.60038e10	0.156188
$336$	0	0
$337$	1.39132e11	0.587615	0.293807	−	0.955865i	$-0.405078\pi$
$337$	1.39132e11	0.587615	0.293807	+	0.955865i	$0.405078\pi$
$338$	0	0
$339$	−6.34482e10	−0.260928
$340$	0	0
$341$	−3.38829e9	−0.0135702
$342$	0	0
$343$	1.29082e11	0.503550
$344$	0	0
$345$	−4.90262e10	−0.186313
$346$	0	0
$347$	2.79900e11	1.03638	0.518192	−	0.855264i	$-0.326605\pi$
$347$	2.79900e11	1.03638	0.518192	+	0.855264i	$0.326605\pi$
$348$	0	0
$349$	4.32395e11	1.56015	0.780075	−	0.625686i	$-0.215181\pi$
$349$	4.32395e11	1.56015	0.780075	+	0.625686i	$0.215181\pi$
$350$	0	0
$351$	−3.70328e10	−0.130228
$352$	0	0
$353$	−1.11287e11	−0.381467	−0.190734	−	0.981642i	$-0.561087\pi$
$353$	−1.11287e11	−0.381467	−0.190734	+	0.981642i	$0.561087\pi$
$354$	0	0
$355$	−4.50937e10	−0.150691
$356$	0	0
$357$	3.78269e10	0.123252
$358$	0	0
$359$	1.49311e10	0.0474423	0.0237212	−	0.999719i	$-0.492449\pi$
$359$	1.49311e10	0.0474423	0.0237212	+	0.999719i	$0.492449\pi$
$360$	0	0
$361$	−1.00920e11	−0.312749
$362$	0	0
$363$	1.90935e11	0.577174
$364$	0	0
$365$	6.43462e10	0.189760
$366$	0	0
$367$	−9.39582e10	−0.270357	−0.135178	−	0.990821i	$-0.543161\pi$
$367$	−9.39582e10	−0.270357	−0.135178	+	0.990821i	$0.543161\pi$
$368$	0	0
$369$	8.26245e10	0.232001
$370$	0	0
$371$	1.59224e10	0.0436340
$372$	0	0
$373$	6.00502e11	1.60629	0.803146	−	0.595783i	$-0.203158\pi$
$373$	6.00502e11	1.60629	0.803146	+	0.595783i	$0.203158\pi$
$374$	0	0
$375$	7.79150e10	0.203461
$376$	0	0
$377$	−2.71808e11	−0.692988
$378$	0	0
$379$	5.20120e11	1.29487	0.647437	−	0.762119i	$-0.275841\pi$
$379$	5.20120e11	1.29487	0.647437	+	0.762119i	$0.275841\pi$
$380$	0	0
$381$	−1.66207e11	−0.404099
$382$	0	0
$383$	4.99135e11	1.18529	0.592643	−	0.805465i	$-0.298084\pi$
$383$	4.99135e11	1.18529	0.592643	+	0.805465i	$0.298084\pi$
$384$	0	0
$385$	−3.51767e8	−0.000815984 0
$386$	0	0
$387$	7.38710e10	0.167407
$388$	0	0
$389$	−1.73874e11	−0.385001	−0.192500	−	0.981297i	$-0.561660\pi$
$389$	−1.73874e11	−0.385001	−0.192500	+	0.981297i	$0.561660\pi$
$390$	0	0
$391$	6.82187e11	1.47607
$392$	0	0
$393$	−8.95886e10	−0.189446
$394$	0	0
$395$	−3.41489e10	−0.0705813
$396$	0	0
$397$	−4.03254e11	−0.814745	−0.407373	−	0.913262i	$-0.633555\pi$
$397$	−4.03254e11	−0.814745	−0.407373	+	0.913262i	$0.633555\pi$
$398$	0	0
$399$	6.31530e10	0.124743
$400$	0	0
$401$	−3.41251e11	−0.659059	−0.329529	−	0.944145i	$-0.606890\pi$
$401$	−3.41251e11	−0.659059	−0.329529	+	0.944145i	$0.606890\pi$
$402$	0	0
$403$	−2.78107e11	−0.525218
$404$	0	0
$405$	1.07730e10	0.0198970
$406$	0	0
$407$	4.64813e9	0.00839660
$408$	0	0
$409$	2.45414e11	0.433655	0.216828	−	0.976210i	$-0.430429\pi$
$409$	2.45414e11	0.433655	0.216828	+	0.976210i	$0.430429\pi$
$410$	0	0
$411$	−8.45762e10	−0.146204
$412$	0	0
$413$	2.54372e10	0.0430224
$414$	0	0
$415$	6.05422e9	0.0100194
$416$	0	0
$417$	2.37955e11	0.385374
$418$	0	0
$419$	−3.19132e11	−0.505833	−0.252917	−	0.967488i	$-0.581390\pi$
$419$	−3.19132e11	−0.505833	−0.252917	+	0.967488i	$0.581390\pi$
$420$	0	0
$421$	1.08208e12	1.67877	0.839386	−	0.543535i	$-0.182915\pi$
$421$	1.08208e12	1.67877	0.839386	+	0.543535i	$0.182915\pi$
$422$	0	0
$423$	1.89891e11	0.288385
$424$	0	0
$425$	−5.33251e11	−0.792832
$426$	0	0
$427$	4.79017e10	0.0697310
$428$	0	0
$429$	−4.79199e9	−0.00683058
$430$	0	0
$431$	−6.02827e11	−0.841482	−0.420741	−	0.907181i	$-0.638230\pi$
$431$	−6.02827e11	−0.841482	−0.420741	+	0.907181i	$0.638230\pi$
$432$	0	0
$433$	3.98033e11	0.544157	0.272078	−	0.962275i	$-0.412289\pi$
$433$	3.98033e11	0.544157	0.272078	+	0.962275i	$0.412289\pi$
$434$	0	0
$435$	7.90699e10	0.105879
$436$	0	0
$437$	1.13893e12	1.49393
$438$	0	0
$439$	1.67851e11	0.215692	0.107846	−	0.994168i	$-0.465605\pi$
$439$	1.67851e11	0.215692	0.107846	+	0.994168i	$0.465605\pi$
$440$	0	0
$441$	−2.46776e11	−0.310691
$442$	0	0
$443$	−2.78026e11	−0.342980	−0.171490	−	0.985186i	$-0.554858\pi$
$443$	−2.78026e11	−0.342980	−0.171490	+	0.985186i	$0.554858\pi$
$444$	0	0
$445$	−2.16788e11	−0.262068
$446$	0	0
$447$	−3.03820e11	−0.359942
$448$	0	0
$449$	1.66356e12	1.93165	0.965826	−	0.259191i	$-0.0834560\pi$
$449$	1.66356e12	1.93165	0.965826	+	0.259191i	$0.0834560\pi$
$450$	0	0
$451$	1.06915e10	0.0121687
$452$	0	0
$453$	−2.01029e11	−0.224293
$454$	0	0
$455$	−2.88727e10	−0.0315817
$456$	0	0
$457$	−6.38333e11	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$457$	−6.38333e11	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$458$	0	0
$459$	−1.49903e11	−0.157636
$460$	0	0
$461$	−2.76291e11	−0.284913	−0.142456	−	0.989801i	$-0.545500\pi$
$461$	−2.76291e11	−0.284913	−0.142456	+	0.989801i	$0.545500\pi$
$462$	0	0
$463$	1.55833e11	0.157596	0.0787980	−	0.996891i	$-0.474892\pi$
$463$	1.55833e11	0.157596	0.0787980	+	0.996891i	$0.474892\pi$
$464$	0	0
$465$	8.09025e10	0.0802460
$466$	0	0
$467$	1.11029e11	0.108022	0.0540108	−	0.998540i	$-0.482799\pi$
$467$	1.11029e11	0.108022	0.0540108	+	0.998540i	$0.482799\pi$
$468$	0	0
$469$	−2.38184e11	−0.227319
$470$	0	0
$471$	−4.58273e11	−0.429072
$472$	0	0
$473$	9.55879e9	0.00878068
$474$	0	0
$475$	−8.90275e11	−0.802422
$476$	0	0
$477$	−6.30983e10	−0.0558065
$478$	0	0
$479$	6.41651e11	0.556915	0.278458	−	0.960449i	$-0.410177\pi$
$479$	6.41651e11	0.556915	0.278458	+	0.960449i	$0.410177\pi$
$480$	0	0
$481$	3.81514e11	0.324981
$482$	0	0
$483$	3.24334e11	0.271163
$484$	0	0
$485$	−1.56371e11	−0.128327
$486$	0	0
$487$	−5.00844e11	−0.403480	−0.201740	−	0.979439i	$-0.564660\pi$
$487$	−5.00844e11	−0.403480	−0.201740	+	0.979439i	$0.564660\pi$
$488$	0	0
$489$	−5.09350e11	−0.402834
$490$	0	0
$491$	1.02945e12	0.799353	0.399677	−	0.916656i	$-0.369122\pi$
$491$	1.02945e12	0.799353	0.399677	+	0.916656i	$0.369122\pi$
$492$	0	0
$493$	−1.10024e12	−0.838833
$494$	0	0
$495$	1.39401e9	0.00104362
$496$	0	0
$497$	2.98318e11	0.219319
$498$	0	0
$499$	−1.93280e12	−1.39552	−0.697758	−	0.716333i	$-0.745819\pi$
$499$	−1.93280e12	−1.39552	−0.697758	+	0.716333i	$0.745819\pi$
$500$	0	0
$501$	1.16243e11	0.0824320
$502$	0	0
$503$	−6.06855e11	−0.422697	−0.211348	−	0.977411i	$-0.567785\pi$
$503$	−6.06855e11	−0.422697	−0.211348	+	0.977411i	$0.567785\pi$
$504$	0	0
$505$	−1.16019e11	−0.0793809
$506$	0	0
$507$	4.65643e11	0.312981
$508$	0	0
$509$	−2.82186e12	−1.86340	−0.931698	−	0.363235i	$-0.881672\pi$
$509$	−2.82186e12	−1.86340	−0.931698	+	0.363235i	$0.881672\pi$
$510$	0	0
$511$	−4.25684e11	−0.276181
$512$	0	0
$513$	−2.50267e11	−0.159542
$514$	0	0
$515$	−2.97019e11	−0.186060
$516$	0	0
$517$	2.45716e10	0.0151261
$518$	0	0
$519$	−8.59281e11	−0.519855
$520$	0	0
$521$	3.19165e11	0.189778	0.0948888	−	0.995488i	$-0.469750\pi$
$521$	3.19165e11	0.189778	0.0948888	+	0.995488i	$0.469750\pi$
$522$	0	0
$523$	−7.03090e11	−0.410916	−0.205458	−	0.978666i	$-0.565868\pi$
$523$	−7.03090e11	−0.410916	−0.205458	+	0.978666i	$0.565868\pi$
$524$	0	0
$525$	−2.53525e11	−0.145648
$526$	0	0
$527$	−1.12574e12	−0.635754
$528$	0	0
$529$	4.04802e12	2.24746
$530$	0	0
$531$	−1.00804e11	−0.0550243
$532$	0	0
$533$	8.77546e11	0.470975
$534$	0	0
$535$	4.04714e11	0.213578
$536$	0	0
$537$	3.11477e10	0.0161637
$538$	0	0
$539$	−3.19324e10	−0.0162961
$540$	0	0
$541$	−2.63274e11	−0.132136	−0.0660679	−	0.997815i	$-0.521045\pi$
$541$	−2.63274e11	−0.132136	−0.0660679	+	0.997815i	$0.521045\pi$
$542$	0	0
$543$	−2.87485e11	−0.141911
$544$	0	0
$545$	−6.98043e10	−0.0338920
$546$	0	0
$547$	9.43116e11	0.450425	0.225212	−	0.974310i	$-0.427692\pi$
$547$	9.43116e11	0.450425	0.225212	+	0.974310i	$0.427692\pi$
$548$	0	0
$549$	−1.89828e11	−0.0891837
$550$	0	0
$551$	−1.83687e12	−0.848979
$552$	0	0
$553$	2.25913e11	0.102725
$554$	0	0
$555$	−1.10984e11	−0.0496525
$556$	0	0
$557$	−3.73948e12	−1.64613	−0.823063	−	0.567950i	$-0.807737\pi$
$557$	−3.73948e12	−1.64613	−0.823063	+	0.567950i	$0.807737\pi$
$558$	0	0
$559$	7.84576e11	0.339846
$560$	0	0
$561$	−1.93973e10	−0.00826813
$562$	0	0
$563$	3.81433e11	0.160004	0.0800019	−	0.996795i	$-0.474507\pi$
$563$	3.81433e11	0.160004	0.0800019	+	0.996795i	$0.474507\pi$
$564$	0	0
$565$	1.96033e11	0.0809304
$566$	0	0
$567$	−7.12689e10	−0.0289585
$568$	0	0
$569$	−3.58954e12	−1.43560	−0.717800	−	0.696249i	$-0.754851\pi$
$569$	−3.58954e12	−1.43560	−0.717800	+	0.696249i	$0.754851\pi$
$570$	0	0
$571$	−3.19723e12	−1.25867	−0.629334	−	0.777135i	$-0.716672\pi$
$571$	−3.19723e12	−1.25867	−0.629334	+	0.777135i	$0.716672\pi$
$572$	0	0
$573$	−2.26906e12	−0.879325
$574$	0	0
$575$	−4.57217e12	−1.74428
$576$	0	0
$577$	1.67840e12	0.630381	0.315191	−	0.949028i	$-0.397932\pi$
$577$	1.67840e12	0.630381	0.315191	+	0.949028i	$0.397932\pi$
$578$	0	0
$579$	−8.24357e11	−0.304833
$580$	0	0
$581$	−4.00518e10	−0.0145824
$582$	0	0
$583$	−8.16482e9	−0.00292711
$584$	0	0
$585$	1.14419e11	0.0403920
$586$	0	0
$587$	−4.78170e12	−1.66231	−0.831153	−	0.556043i	$-0.812319\pi$
$587$	−4.78170e12	−1.66231	−0.831153	+	0.556043i	$0.812319\pi$
$588$	0	0
$589$	−1.87945e12	−0.643445
$590$	0	0
$591$	−5.99613e11	−0.202175
$592$	0	0
$593$	1.42074e12	0.471811	0.235906	−	0.971776i	$-0.424194\pi$
$593$	1.42074e12	0.471811	0.235906	+	0.971776i	$0.424194\pi$
$594$	0	0
$595$	−1.16872e11	−0.0382284
$596$	0	0
$597$	−9.24461e11	−0.297854
$598$	0	0
$599$	−3.72919e12	−1.18357	−0.591784	−	0.806096i	$-0.701576\pi$
$599$	−3.72919e12	−1.18357	−0.591784	+	0.806096i	$0.701576\pi$
$600$	0	0
$601$	3.32876e11	0.104075	0.0520375	−	0.998645i	$-0.483428\pi$
$601$	3.32876e11	0.104075	0.0520375	+	0.998645i	$0.483428\pi$
$602$	0	0
$603$	9.43893e11	0.290733
$604$	0	0
$605$	−5.89925e11	−0.179018
$606$	0	0
$607$	1.38715e12	0.414739	0.207369	−	0.978263i	$-0.433510\pi$
$607$	1.38715e12	0.414739	0.207369	+	0.978263i	$0.433510\pi$
$608$	0	0
$609$	−5.23089e11	−0.154098
$610$	0	0
$611$	2.01681e12	0.585437
$612$	0	0
$613$	1.54790e12	0.442763	0.221382	−	0.975187i	$-0.428943\pi$
$613$	1.54790e12	0.442763	0.221382	+	0.975187i	$0.428943\pi$
$614$	0	0
$615$	−2.55282e11	−0.0719584
$616$	0	0
$617$	1.17696e12	0.326947	0.163473	−	0.986548i	$-0.447730\pi$
$617$	1.17696e12	0.326947	0.163473	+	0.986548i	$0.447730\pi$
$618$	0	0
$619$	1.72922e12	0.473415	0.236708	−	0.971581i	$-0.423932\pi$
$619$	1.72922e12	0.473415	0.236708	+	0.971581i	$0.423932\pi$
$620$	0	0
$621$	−1.28529e12	−0.346809
$622$	0	0
$623$	1.43417e12	0.381420
$624$	0	0
$625$	3.45164e12	0.904827
$626$	0	0
$627$	−3.23842e10	−0.00836815
$628$	0	0
$629$	1.54431e12	0.393376
$630$	0	0
$631$	−5.73742e12	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$631$	−5.73742e12	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$632$	0	0
$633$	−1.22143e12	−0.302379
$634$	0	0
$635$	5.13524e11	0.125337
$636$	0	0
$637$	−2.62098e12	−0.630720
$638$	0	0
$639$	−1.18220e12	−0.280502
$640$	0	0
$641$	7.57753e12	1.77283	0.886414	−	0.462894i	$-0.153189\pi$
$641$	7.57753e12	1.77283	0.886414	+	0.462894i	$0.153189\pi$
$642$	0	0
$643$	−4.80446e12	−1.10840	−0.554199	−	0.832384i	$-0.686975\pi$
$643$	−4.80446e12	−1.10840	−0.554199	+	0.832384i	$0.686975\pi$
$644$	0	0
$645$	−2.28236e11	−0.0519237
$646$	0	0
$647$	1.43505e12	0.321958	0.160979	−	0.986958i	$-0.448535\pi$
$647$	1.43505e12	0.321958	0.160979	+	0.986958i	$0.448535\pi$
$648$	0	0
$649$	−1.30439e10	−0.00288608
$650$	0	0
$651$	−5.35213e11	−0.116792
$652$	0	0
$653$	−6.17024e12	−1.32798	−0.663991	−	0.747740i	$-0.731139\pi$
$653$	−6.17024e12	−1.32798	−0.663991	+	0.747740i	$0.731139\pi$
$654$	0	0
$655$	2.76798e11	0.0587594
$656$	0	0
$657$	1.68693e12	0.353226
$658$	0	0
$659$	−2.01866e11	−0.0416945	−0.0208472	−	0.999783i	$-0.506636\pi$
$659$	−2.01866e11	−0.0416945	−0.0208472	+	0.999783i	$0.506636\pi$
$660$	0	0
$661$	8.37300e12	1.70598	0.852991	−	0.521925i	$-0.174786\pi$
$661$	8.37300e12	1.70598	0.852991	+	0.521925i	$0.174786\pi$
$662$	0	0
$663$	−1.59211e12	−0.320008
$664$	0	0
$665$	−1.95121e11	−0.0386908
$666$	0	0
$667$	−9.43360e12	−1.84549
$668$	0	0
$669$	−8.50717e11	−0.164198
$670$	0	0
$671$	−2.45635e10	−0.00467777
$672$	0	0
$673$	9.11671e12	1.71305	0.856525	−	0.516105i	$-0.172619\pi$
$673$	9.11671e12	1.71305	0.856525	+	0.516105i	$0.172619\pi$
$674$	0	0
$675$	1.00469e12	0.186279
$676$	0	0
$677$	−8.62986e12	−1.57890	−0.789450	−	0.613815i	$-0.789634\pi$
$677$	−8.62986e12	−1.57890	−0.789450	+	0.613815i	$0.789634\pi$
$678$	0	0
$679$	1.03448e12	0.186770
$680$	0	0
$681$	−5.55940e12	−0.990527
$682$	0	0
$683$	−2.73461e12	−0.480842	−0.240421	−	0.970669i	$-0.577285\pi$
$683$	−2.73461e12	−0.480842	−0.240421	+	0.970669i	$0.577285\pi$
$684$	0	0
$685$	2.61312e11	0.0453473
$686$	0	0
$687$	−4.75867e12	−0.815043
$688$	0	0
$689$	−6.70161e11	−0.113290
$690$	0	0
$691$	−7.23661e12	−1.20749	−0.603746	−	0.797177i	$-0.706326\pi$
$691$	−7.23661e12	−1.20749	−0.603746	+	0.797177i	$0.706326\pi$
$692$	0	0
$693$	−9.22209e9	−0.00151890
$694$	0	0
$695$	−7.35199e11	−0.119529
$696$	0	0
$697$	3.55218e12	0.570095
$698$	0	0
$699$	2.97259e12	0.470964
$700$	0	0
$701$	2.35635e12	0.368561	0.184280	−	0.982874i	$-0.441005\pi$
$701$	2.35635e12	0.368561	0.184280	+	0.982874i	$0.441005\pi$
$702$	0	0
$703$	2.57827e12	0.398134
$704$	0	0
$705$	−5.86699e11	−0.0894466
$706$	0	0
$707$	7.67524e11	0.115533
$708$	0	0
$709$	1.84172e12	0.273725	0.136862	−	0.990590i	$-0.456298\pi$
$709$	1.84172e12	0.273725	0.136862	+	0.990590i	$0.456298\pi$
$710$	0	0
$711$	−8.95264e11	−0.131383
$712$	0	0
$713$	−9.65225e12	−1.39870
$714$	0	0
$715$	1.48056e10	0.00211860
$716$	0	0
$717$	−1.99274e12	−0.281588
$718$	0	0
$719$	1.00065e12	0.139637	0.0698186	−	0.997560i	$-0.477758\pi$
$719$	1.00065e12	0.139637	0.0698186	+	0.997560i	$0.477758\pi$
$720$	0	0
$721$	1.96494e12	0.270795
$722$	0	0
$723$	6.12691e12	0.833910
$724$	0	0
$725$	7.37404e12	0.991253
$726$	0	0
$727$	1.46409e13	1.94385	0.971925	−	0.235291i	$-0.0756044\pi$
$727$	1.46409e13	1.94385	0.971925	+	0.235291i	$0.0756044\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	3.17585e12	0.411369
$732$	0	0
$733$	−1.30354e12	−0.166785	−0.0833925	−	0.996517i	$-0.526576\pi$
$733$	−1.30354e12	−0.166785	−0.0833925	+	0.996517i	$0.526576\pi$
$734$	0	0
$735$	7.62453e11	0.0963652
$736$	0	0
$737$	1.22138e11	0.0152492
$738$	0	0
$739$	−5.15412e12	−0.635704	−0.317852	−	0.948140i	$-0.602961\pi$
$739$	−5.15412e12	−0.635704	−0.317852	+	0.948140i	$0.602961\pi$
$740$	0	0
$741$	−2.65806e12	−0.323879
$742$	0	0
$743$	−8.22504e12	−0.990121	−0.495060	−	0.868859i	$-0.664854\pi$
$743$	−8.22504e12	−0.990121	−0.495060	+	0.868859i	$0.664854\pi$
$744$	0	0
$745$	9.38700e11	0.111641
$746$	0	0
$747$	1.58720e11	0.0186505
$748$	0	0
$749$	−2.67740e12	−0.310846
$750$	0	0
$751$	2.09125e12	0.239898	0.119949	−	0.992780i	$-0.461727\pi$
$751$	2.09125e12	0.239898	0.119949	+	0.992780i	$0.461727\pi$
$752$	0	0
$753$	2.37035e12	0.268680
$754$	0	0
$755$	6.21110e11	0.0695677
$756$	0	0
$757$	1.40301e13	1.55285	0.776427	−	0.630208i	$-0.217030\pi$
$757$	1.40301e13	1.55285	0.776427	+	0.630208i	$0.217030\pi$
$758$	0	0
$759$	−1.66315e11	−0.0181905
$760$	0	0
$761$	4.88984e11	0.0528523	0.0264262	−	0.999651i	$-0.491587\pi$
$761$	4.88984e11	0.0528523	0.0264262	+	0.999651i	$0.491587\pi$
$762$	0	0
$763$	4.61792e11	0.0493272
$764$	0	0
$765$	4.63150e11	0.0488928
$766$	0	0
$767$	−1.07063e12	−0.111702
$768$	0	0
$769$	5.51278e12	0.568463	0.284232	−	0.958756i	$-0.408262\pi$
$769$	5.51278e12	0.568463	0.284232	+	0.958756i	$0.408262\pi$
$770$	0	0
$771$	2.11933e12	0.216000
$772$	0	0
$773$	7.93968e12	0.799826	0.399913	−	0.916553i	$-0.369040\pi$
$773$	7.93968e12	0.799826	0.399913	+	0.916553i	$0.369040\pi$
$774$	0	0
$775$	7.54495e12	0.751275
$776$	0	0
$777$	7.34217e11	0.0722653
$778$	0	0
$779$	5.93045e12	0.576991
$780$	0	0
$781$	−1.52975e11	−0.0147126
$782$	0	0
$783$	2.07293e12	0.197087
$784$	0	0
$785$	1.41591e12	0.133083
$786$	0	0
$787$	−2.12131e12	−0.197114	−0.0985569	−	0.995131i	$-0.531423\pi$
$787$	−2.12131e12	−0.197114	−0.0985569	+	0.995131i	$0.531423\pi$
$788$	0	0
$789$	−3.76526e11	−0.0345899
$790$	0	0
$791$	−1.29686e12	−0.117788
$792$	0	0
$793$	−2.01615e12	−0.181048
$794$	0	0
$795$	1.94952e11	0.0173092
$796$	0	0
$797$	−1.56756e13	−1.37613	−0.688067	−	0.725647i	$-0.741541\pi$
$797$	−1.56756e13	−1.37613	−0.688067	+	0.725647i	$0.741541\pi$
$798$	0	0
$799$	8.16376e12	0.708647
$800$	0	0
$801$	−5.68341e12	−0.487824
$802$	0	0
$803$	2.18286e11	0.0185271
$804$	0	0
$805$	−1.00208e12	−0.0841050
$806$	0	0
$807$	−4.17737e12	−0.346715
$808$	0	0
$809$	−1.33817e13	−1.09836	−0.549178	−	0.835705i	$-0.685059\pi$
$809$	−1.33817e13	−1.09836	−0.549178	+	0.835705i	$0.685059\pi$
$810$	0	0
$811$	−7.33572e12	−0.595455	−0.297727	−	0.954651i	$-0.596229\pi$
$811$	−7.33572e12	−0.595455	−0.297727	+	0.954651i	$0.596229\pi$
$812$	0	0
$813$	−1.16139e13	−0.932333
$814$	0	0
$815$	1.57372e12	0.124945
$816$	0	0
$817$	5.30216e12	0.416345
$818$	0	0
$819$	−7.56940e11	−0.0587874
$820$	0	0
$821$	−6.64810e12	−0.510685	−0.255343	−	0.966851i	$-0.582188\pi$
$821$	−6.64810e12	−0.510685	−0.255343	+	0.966851i	$0.582188\pi$
$822$	0	0
$823$	4.61148e12	0.350381	0.175191	−	0.984535i	$-0.443946\pi$
$823$	4.61148e12	0.350381	0.175191	+	0.984535i	$0.443946\pi$
$824$	0	0
$825$	1.30005e11	0.00977050
$826$	0	0
$827$	1.64749e13	1.22475	0.612377	−	0.790566i	$-0.290214\pi$
$827$	1.64749e13	1.22475	0.612377	+	0.790566i	$0.290214\pi$
$828$	0	0
$829$	2.33872e13	1.71982	0.859910	−	0.510446i	$-0.170520\pi$
$829$	2.33872e13	1.71982	0.859910	+	0.510446i	$0.170520\pi$
$830$	0	0
$831$	−5.87525e12	−0.427388
$832$	0	0
$833$	−1.06093e13	−0.763460
$834$	0	0
$835$	−3.59150e11	−0.0255674
$836$	0	0
$837$	2.12098e12	0.149373
$838$	0	0
$839$	3.19129e12	0.222350	0.111175	−	0.993801i	$-0.464539\pi$
$839$	3.19129e12	0.222350	0.111175	+	0.993801i	$0.464539\pi$
$840$	0	0
$841$	7.07463e11	0.0487665
$842$	0	0
$843$	1.31451e13	0.896480
$844$	0	0
$845$	−1.43868e12	−0.0970753
$846$	0	0
$847$	3.90267e12	0.260547
$848$	0	0
$849$	−1.45247e13	−0.959450
$850$	0	0
$851$	1.32412e13	0.865453
$852$	0	0
$853$	2.01905e13	1.30580	0.652899	−	0.757445i	$-0.273552\pi$
$853$	2.01905e13	1.30580	0.652899	+	0.757445i	$0.273552\pi$
$854$	0	0
$855$	7.73241e11	0.0494843
$856$	0	0
$857$	−1.40886e13	−0.892185	−0.446093	−	0.894987i	$-0.647185\pi$
$857$	−1.40886e13	−0.892185	−0.446093	+	0.894987i	$0.647185\pi$
$858$	0	0
$859$	2.11678e13	1.32650	0.663250	−	0.748398i	$-0.269177\pi$
$859$	2.11678e13	1.32650	0.663250	+	0.748398i	$0.269177\pi$
$860$	0	0
$861$	1.68882e12	0.104730
$862$	0	0
$863$	−7.28437e12	−0.447037	−0.223519	−	0.974700i	$-0.571754\pi$
$863$	−7.28437e12	−0.447037	−0.223519	+	0.974700i	$0.571754\pi$
$864$	0	0
$865$	2.65489e12	0.161240
$866$	0	0
$867$	3.16100e12	0.189993
$868$	0	0
$869$	−1.15846e11	−0.00689115
$870$	0	0
$871$	1.00250e13	0.590204
$872$	0	0
$873$	−4.09950e12	−0.238873
$874$	0	0
$875$	1.59256e12	0.0918459
$876$	0	0
$877$	2.59566e13	1.48166	0.740831	−	0.671692i	$-0.234432\pi$
$877$	2.59566e13	1.48166	0.740831	+	0.671692i	$0.234432\pi$
$878$	0	0
$879$	−1.86525e13	−1.05387
$880$	0	0
$881$	−9.85083e12	−0.550911	−0.275456	−	0.961314i	$-0.588829\pi$
$881$	−9.85083e12	−0.550911	−0.275456	+	0.961314i	$0.588829\pi$
$882$	0	0
$883$	−7.71297e11	−0.0426971	−0.0213486	−	0.999772i	$-0.506796\pi$
$883$	−7.71297e11	−0.0426971	−0.0213486	+	0.999772i	$0.506796\pi$
$884$	0	0
$885$	3.11451e11	0.0170665
$886$	0	0
$887$	9.94743e12	0.539579	0.269789	−	0.962919i	$-0.413046\pi$
$887$	9.94743e12	0.539579	0.269789	+	0.962919i	$0.413046\pi$
$888$	0	0
$889$	−3.39723e12	−0.182418
$890$	0	0
$891$	3.65460e10	0.00194263
$892$	0	0
$893$	1.36296e13	0.717219
$894$	0	0
$895$	−9.62359e10	−0.00501341
$896$	0	0
$897$	−1.36510e13	−0.704040
$898$	0	0
$899$	1.55672e13	0.794864
$900$	0	0
$901$	−2.71271e12	−0.137133
$902$	0	0
$903$	1.50990e12	0.0755708
$904$	0	0
$905$	8.88230e11	0.0440156
$906$	0	0
$907$	−1.51839e13	−0.744988	−0.372494	−	0.928035i	$-0.621497\pi$
$907$	−1.51839e13	−0.744988	−0.372494	+	0.928035i	$0.621497\pi$
$908$	0	0
$909$	−3.04160e12	−0.147763
$910$	0	0
$911$	1.30119e13	0.625903	0.312951	−	0.949769i	$-0.398682\pi$
$911$	1.30119e13	0.625903	0.312951	+	0.949769i	$0.398682\pi$
$912$	0	0
$913$	2.05382e10	0.000978235 0
$914$	0	0
$915$	5.86505e11	0.0276616
$916$	0	0
$917$	−1.83117e12	−0.0855197
$918$	0	0
$919$	2.88036e13	1.33207	0.666034	−	0.745921i	$-0.267990\pi$
$919$	2.88036e13	1.33207	0.666034	+	0.745921i	$0.267990\pi$
$920$	0	0
$921$	−2.28303e13	−1.04555
$922$	0	0
$923$	−1.25560e13	−0.569434
$924$	0	0
$925$	−1.03503e13	−0.464854
$926$	0	0
$927$	−7.78680e12	−0.346338
$928$	0	0
$929$	−2.00577e13	−0.883510	−0.441755	−	0.897136i	$-0.645644\pi$
$929$	−2.00577e13	−0.883510	−0.441755	+	0.897136i	$0.645644\pi$
$930$	0	0
$931$	−1.77126e13	−0.772695
$932$	0	0
$933$	−1.88924e13	−0.816245
$934$	0	0
$935$	5.99309e10	0.00256448
$936$	0	0
$937$	3.18883e13	1.35146	0.675731	−	0.737149i	$-0.263828\pi$
$937$	3.18883e13	1.35146	0.675731	+	0.737149i	$0.263828\pi$
$938$	0	0
$939$	2.66605e13	1.11911
$940$	0	0
$941$	3.39219e12	0.141035	0.0705175	−	0.997511i	$-0.477535\pi$
$941$	3.39219e12	0.141035	0.0705175	+	0.997511i	$0.477535\pi$
$942$	0	0
$943$	3.04569e13	1.25425
$944$	0	0
$945$	2.20197e11	0.00898189
$946$	0	0
$947$	3.55975e13	1.43828	0.719142	−	0.694863i	$-0.244535\pi$
$947$	3.55975e13	1.43828	0.719142	+	0.694863i	$0.244535\pi$
$948$	0	0
$949$	1.79167e13	0.717068
$950$	0	0
$951$	7.34007e12	0.290997
$952$	0	0
$953$	−2.43100e13	−0.954702	−0.477351	−	0.878713i	$-0.658403\pi$
$953$	−2.43100e13	−0.954702	−0.477351	+	0.878713i	$0.658403\pi$
$954$	0	0
$955$	7.01061e12	0.272735
$956$	0	0
$957$	2.68235e11	0.0103374
$958$	0	0
$959$	−1.72872e12	−0.0659994
$960$	0	0
$961$	−1.05116e13	−0.397569
$962$	0	0
$963$	1.06102e13	0.397562
$964$	0	0
$965$	2.54698e12	0.0945482
$966$	0	0
$967$	1.53573e13	0.564801	0.282401	−	0.959297i	$-0.408869\pi$
$967$	1.53573e13	0.564801	0.282401	+	0.959297i	$0.408869\pi$
$968$	0	0
$969$	−1.07594e13	−0.392042
$970$	0	0
$971$	3.16083e13	1.14108	0.570539	−	0.821271i	$-0.306734\pi$
$971$	3.16083e13	1.14108	0.570539	+	0.821271i	$0.306734\pi$
$972$	0	0
$973$	4.86373e12	0.173965
$974$	0	0
$975$	1.06707e13	0.378156
$976$	0	0
$977$	−1.05660e13	−0.371008	−0.185504	−	0.982643i	$-0.559392\pi$
$977$	−1.05660e13	−0.371008	−0.185504	+	0.982643i	$0.559392\pi$
$978$	0	0
$979$	−7.35425e11	−0.0255868
$980$	0	0
$981$	−1.83002e12	−0.0630879
$982$	0	0
$983$	−4.36692e13	−1.49171	−0.745856	−	0.666108i	$-0.767959\pi$
$983$	−4.36692e13	−1.49171	−0.745856	+	0.666108i	$0.767959\pi$
$984$	0	0
$985$	1.85260e12	0.0627074
$986$	0	0
$987$	3.88132e12	0.130182
$988$	0	0
$989$	2.72302e13	0.905040
$990$	0	0
$991$	3.20991e13	1.05721	0.528605	−	0.848868i	$-0.322715\pi$
$991$	3.20991e13	1.05721	0.528605	+	0.848868i	$0.322715\pi$
$992$	0	0
$993$	2.48868e13	0.812266
$994$	0	0
$995$	2.85627e12	0.0923837
$996$	0	0
$997$	3.79644e13	1.21688	0.608441	−	0.793599i	$-0.291795\pi$
$997$	3.79644e13	1.21688	0.608441	+	0.793599i	$0.291795\pi$
$998$	0	0
$999$	−2.90961e12	−0.0924250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.c.1.3	yes	4
4.3	odd	2		384.10.a.g.1.3	yes	4
8.3	odd	2		384.10.a.b.1.2	✓	4
8.5	even	2		384.10.a.f.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.b.1.2	✓	4	8.3	odd	2
384.10.a.c.1.3	yes	4	1.1	even	1	trivial
384.10.a.f.1.2	yes	4	8.5	even	2
384.10.a.g.1.3	yes	4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} +250.262 q^{5} -1655.62 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} +250.262 q^{5} -1655.62 q^{7} +6561.00 q^{9} +848.983 q^{11} +69683.7 q^{13} -20271.3 q^{15} +282069. q^{17} +470922. q^{19} +134105. q^{21} +2.41851e6 q^{23} -1.89049e6 q^{25} -531441. q^{27} -3.90059e6 q^{29} -3.99099e6 q^{31} -68767.7 q^{33} -414339. q^{35} +5.47494e6 q^{37} -5.64438e6 q^{39} +1.25933e7 q^{41} +1.12591e7 q^{43} +1.64197e6 q^{45} +2.89424e7 q^{47} -3.76125e7 q^{49} -2.28476e7 q^{51} -9.61718e6 q^{53} +212469. q^{55} -3.81447e7 q^{57} -1.53642e7 q^{59} -2.89328e7 q^{61} -1.08625e7 q^{63} +1.74392e7 q^{65} +1.43864e8 q^{67} -1.95899e8 q^{69} -1.80186e8 q^{71} +2.57115e8 q^{73} +1.53130e8 q^{75} -1.40559e6 q^{77} -1.36452e8 q^{79} +4.30467e7 q^{81} +2.41915e7 q^{83} +7.05914e7 q^{85} +3.15948e8 q^{87} -8.66242e8 q^{89} -1.15370e8 q^{91} +3.23271e8 q^{93} +1.17854e8 q^{95} -6.24828e8 q^{97} +5.57018e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{3} + 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q - 324 * q^3 + 240 * q^5 + 4840 * q^7 + 26244 * q^9	\( 4 q - 324 q^{3} + 240 q^{5} + 4840 q^{7} + 26244 q^{9} + 99664 q^{11} + 60840 q^{13} - 19440 q^{15} - 434952 q^{17} - 631776 q^{19} - 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} - 2125764 q^{27} + 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} + 25567008 q^{35} + 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} - 14177920 q^{43} + 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} + 35231112 q^{51} - 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} + 115028080 q^{59} - 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} + 231785104 q^{67} - 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} - 485314092 q^{75} + 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} + 607613328 q^{83} - 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} + 847629840 q^{91} - 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} + 653895504 q^{99}+O(q^{100}) \) 4 * q - 324 * q^3 + 240 * q^5 + 4840 * q^7 + 26244 * q^9 + 99664 * q^11 + 60840 * q^13 - 19440 * q^15 - 434952 * q^17 - 631776 * q^19 - 392040 * q^21 + 749392 * q^23 + 5991532 * q^25 - 2125764 * q^27 + 7908544 * q^29 + 11351240 * q^31 - 8072784 * q^33 + 25567008 * q^35 + 13592920 * q^37 - 4928040 * q^39 - 18838888 * q^41 - 14177920 * q^43 + 1574640 * q^45 - 37779120 * q^47 + 9409332 * q^49 + 35231112 * q^51 - 115336512 * q^53 - 184580544 * q^55 + 51173856 * q^57 + 115028080 * q^59 - 173228648 * q^61 + 31755240 * q^63 - 328077984 * q^65 + 231785104 * q^67 - 60700752 * q^69 + 197476208 * q^71 + 44629400 * q^73 - 485314092 * q^75 + 308117920 * q^77 - 355774584 * q^79 + 172186884 * q^81 + 607613328 * q^83 - 1087351392 * q^85 - 640592064 * q^87 - 1157146424 * q^89 + 847629840 * q^91 - 919450440 * q^93 - 329699328 * q^95 - 1599536472 * q^97 + 653895504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more