LMFDB - Embedded newform 384.6.d.i.193.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(193,384)]

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, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.110166016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 $ x^8 + 10x^6 + 19x^4 + 10*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.8
Root			$1.22833i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.i.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +76.5014i q^{5} +56.1972 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +76.5014i q^{5} +56.1972 q^{7} -81.0000 q^{9} +546.156i q^{11} +900.176i q^{13} -688.513 q^{15} +1120.94 q^{17} -200.589i q^{19} +505.775i q^{21} -910.627 q^{23} -2727.47 q^{25} -729.000i q^{27} +4511.97i q^{29} +5193.48 q^{31} -4915.41 q^{33} +4299.17i q^{35} +7010.01i q^{37} -8101.59 q^{39} +5505.18 q^{41} +16819.6i q^{43} -6196.62i q^{45} +27247.7 q^{47} -13648.9 q^{49} +10088.5i q^{51} -34827.1i q^{53} -41781.7 q^{55} +1805.30 q^{57} -6974.50i q^{59} -32562.1i q^{61} -4551.98 q^{63} -68864.8 q^{65} -32951.7i q^{67} -8195.64i q^{69} +69228.2 q^{71} -16144.0 q^{73} -24547.2i q^{75} +30692.5i q^{77} -57509.0 q^{79} +6561.00 q^{81} -14424.9i q^{83} +85753.5i q^{85} -40607.7 q^{87} -129260. q^{89} +50587.4i q^{91} +46741.3i q^{93} +15345.3 q^{95} -157857. q^{97} -44238.7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 648 q^{9}+O(q^{10}) $ 8 * q - 648 * q^9	$ 8 q - 648 q^{9} + 4080 q^{17} - 5528 q^{25} - 18144 q^{33} + 58704 q^{41} - 44024 q^{49} + 38880 q^{57} - 203904 q^{65} + 248816 q^{73} + 52488 q^{81} - 46800 q^{89} - 262544 q^{97}+O(q^{100}) $ 8 * q - 648 * q^9 + 4080 * q^17 - 5528 * q^25 - 18144 * q^33 + 58704 * q^41 - 44024 * q^49 + 38880 * q^57 - 203904 * q^65 + 248816 * q^73 + 52488 * q^81 - 46800 * q^89 - 262544 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	0	0
$5$	76.5014i	1.36850i	0.729248	+	0.684250i	$0.239870\pi$
$5$	76.5014i	1.36850i	−0.729248	+	0.684250i	$0.760130\pi$
$6$	0	0
$7$	56.1972	0.433481	0.216740	−	0.976229i	$-0.430457\pi$
$7$	56.1972	0.433481	0.216740	+	0.976229i	$0.430457\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	546.156i	1.36093i	0.732781	+	0.680464i	$0.238222\pi$
$11$	546.156i	1.36093i	−0.732781	+	0.680464i	$0.761778\pi$
$12$	0	0
$13$	900.176i	1.47730i	0.674088	+	0.738651i	$0.264537\pi$
$13$	900.176i	1.47730i	−0.674088	+	0.738651i	$0.735463\pi$
$14$	0	0
$15$	−688.513	−0.790103
$16$	0	0
$17$	1120.94	0.940720	0.470360	−	0.882475i	$-0.344124\pi$
$17$	1120.94	0.940720	0.470360	+	0.882475i	$0.344124\pi$
$18$	0	0
$19$	− 200.589i	− 0.127474i	−0.997967	−	0.0637371i	$-0.979698\pi$
$19$	− 200.589i	− 0.127474i	0.997967	−	0.0637371i	$-0.0203019\pi$
$20$	0	0
$21$	505.775i	0.250270i
$22$	0	0
$23$	−910.627	−0.358939	−0.179470	−	0.983764i	$-0.557438\pi$
$23$	−910.627	−0.358939	−0.179470	+	0.983764i	$0.557438\pi$
$24$	0	0
$25$	−2727.47	−0.872790
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	4511.97i	0.996256i	0.867104	+	0.498128i	$0.165979\pi$
$29$	4511.97i	0.996256i	−0.867104	+	0.498128i	$0.834021\pi$
$30$	0	0
$31$	5193.48	0.970630	0.485315	−	0.874339i	$-0.338705\pi$
$31$	5193.48	0.970630	0.485315	+	0.874339i	$0.338705\pi$
$32$	0	0
$33$	−4915.41	−0.785732
$34$	0	0
$35$	4299.17i	0.593218i
$36$	0	0
$37$	7010.01i	0.841811i	0.907105	+	0.420905i	$0.138288\pi$
$37$	7010.01i	0.841811i	−0.907105	+	0.420905i	$0.861712\pi$
$38$	0	0
$39$	−8101.59	−0.852920
$40$	0	0
$41$	5505.18	0.511460	0.255730	−	0.966748i	$-0.417684\pi$
$41$	5505.18	0.511460	0.255730	+	0.966748i	$0.417684\pi$
$42$	0	0
$43$	16819.6i	1.38722i	0.720351	+	0.693610i	$0.243981\pi$
$43$	16819.6i	1.38722i	−0.720351	+	0.693610i	$0.756019\pi$
$44$	0	0
$45$	− 6196.62i	− 0.456166i
$46$	0	0
$47$	27247.7	1.79922	0.899611	−	0.436692i	$-0.143850\pi$
$47$	27247.7	1.79922	0.899611	+	0.436692i	$0.143850\pi$
$48$	0	0
$49$	−13648.9	−0.812094
$50$	0	0
$51$	10088.5i	0.543125i
$52$	0	0
$53$	− 34827.1i	− 1.70305i	−0.524314	−	0.851525i	$-0.675678\pi$
$53$	− 34827.1i	− 1.70305i	0.524314	−	0.851525i	$-0.324322\pi$
$54$	0	0
$55$	−41781.7	−1.86243
$56$	0	0
$57$	1805.30	0.0735973
$58$	0	0
$59$	− 6974.50i	− 0.260845i	−0.991458	−	0.130423i	$-0.958367\pi$
$59$	− 6974.50i	− 0.260845i	0.991458	−	0.130423i	$-0.0416334\pi$
$60$	0	0
$61$	− 32562.1i	− 1.12044i	−0.828345	−	0.560218i	$-0.810717\pi$
$61$	− 32562.1i	− 1.12044i	0.828345	−	0.560218i	$-0.189283\pi$
$62$	0	0
$63$	−4551.98	−0.144494
$64$	0	0
$65$	−68864.8	−2.02169
$66$	0	0
$67$	− 32951.7i	− 0.896791i	−0.893835	−	0.448396i	$-0.851996\pi$
$67$	− 32951.7i	− 0.896791i	0.893835	−	0.448396i	$-0.148004\pi$
$68$	0	0
$69$	− 8195.64i	− 0.207234i
$70$	0	0
$71$	69228.2	1.62981	0.814905	−	0.579595i	$-0.196789\pi$
$71$	69228.2	1.62981	0.814905	+	0.579595i	$0.196789\pi$
$72$	0	0
$73$	−16144.0	−0.354573	−0.177286	−	0.984159i	$-0.556732\pi$
$73$	−16144.0	−0.354573	−0.177286	+	0.984159i	$0.556732\pi$
$74$	0	0
$75$	− 24547.2i	− 0.503905i
$76$	0	0
$77$	30692.5i	0.589936i
$78$	0	0
$79$	−57509.0	−1.03674	−0.518368	−	0.855158i	$-0.673460\pi$
$79$	−57509.0	−1.03674	−0.518368	+	0.855158i	$0.673460\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	− 14424.9i	− 0.229835i	−0.993375	−	0.114918i	$-0.963340\pi$
$83$	− 14424.9i	− 0.229835i	0.993375	−	0.114918i	$-0.0366604\pi$
$84$	0	0
$85$	85753.5i	1.28737i
$86$	0	0
$87$	−40607.7	−0.575188
$88$	0	0
$89$	−129260.	−1.72977	−0.864886	−	0.501968i	$-0.832609\pi$
$89$	−129260.	−1.72977	−0.864886	+	0.501968i	$0.832609\pi$
$90$	0	0
$91$	50587.4i	0.640382i
$92$	0	0
$93$	46741.3i	0.560394i
$94$	0	0
$95$	15345.3	0.174448
$96$	0	0
$97$	−157857.	−1.70347	−0.851736	−	0.523972i	$-0.824450\pi$
$97$	−157857.	−1.70347	−0.851736	+	0.523972i	$0.824450\pi$
$98$	0	0
$99$	− 44238.7i	− 0.453643i
$100$	0	0
$101$	72079.3i	0.703084i	0.936172	+	0.351542i	$0.114342\pi$
$101$	72079.3i	0.703084i	−0.936172	+	0.351542i	$0.885658\pi$
$102$	0	0
$103$	86122.3	0.799875	0.399938	−	0.916542i	$-0.369032\pi$
$103$	86122.3	0.799875	0.399938	+	0.916542i	$0.369032\pi$
$104$	0	0
$105$	−38692.5	−0.342495
$106$	0	0
$107$	3832.40i	0.0323602i	0.999869	+	0.0161801i	$0.00515052\pi$
$107$	3832.40i	0.0323602i	−0.999869	+	0.0161801i	$0.994849\pi$
$108$	0	0
$109$	− 57259.2i	− 0.461614i	−0.973000	−	0.230807i	$-0.925863\pi$
$109$	− 57259.2i	− 0.461614i	0.973000	−	0.230807i	$-0.0741366\pi$
$110$	0	0
$111$	−63090.1	−0.486020
$112$	0	0
$113$	−101508.	−0.747833	−0.373917	−	0.927462i	$-0.621985\pi$
$113$	−101508.	−0.747833	−0.373917	+	0.927462i	$0.621985\pi$
$114$	0	0
$115$	− 69664.3i	− 0.491208i
$116$	0	0
$117$	− 72914.3i	− 0.492434i
$118$	0	0
$119$	62993.7	0.407784
$120$	0	0
$121$	−137236.	−0.852127
$122$	0	0
$123$	49546.6i	0.295292i
$124$	0	0
$125$	30411.8i	0.174087i
$126$	0	0
$127$	125288.	0.689289	0.344644	−	0.938733i	$-0.388000\pi$
$127$	125288.	0.689289	0.344644	+	0.938733i	$0.388000\pi$
$128$	0	0
$129$	−151377.	−0.800912
$130$	0	0
$131$	− 312205.i	− 1.58950i	−0.606934	−	0.794752i	$-0.707601\pi$
$131$	− 312205.i	− 1.58950i	0.606934	−	0.794752i	$-0.292399\pi$
$132$	0	0
$133$	− 11272.5i	− 0.0552577i
$134$	0	0
$135$	55769.5	0.263368
$136$	0	0
$137$	−46620.5	−0.212215	−0.106107	−	0.994355i	$-0.533839\pi$
$137$	−46620.5	−0.212215	−0.106107	+	0.994355i	$0.533839\pi$
$138$	0	0
$139$	200963.i	0.882226i	0.897452	+	0.441113i	$0.145416\pi$
$139$	200963.i	0.882226i	−0.897452	+	0.441113i	$0.854584\pi$
$140$	0	0
$141$	245229.i	1.03878i
$142$	0	0
$143$	−491637.	−2.01050
$144$	0	0
$145$	−345172.	−1.36337
$146$	0	0
$147$	− 122840.i	− 0.468863i
$148$	0	0
$149$	− 269205.i	− 0.993386i	−0.867926	−	0.496693i	$-0.834547\pi$
$149$	− 269205.i	− 0.993386i	0.867926	−	0.496693i	$-0.165453\pi$
$150$	0	0
$151$	14138.3	0.0504607	0.0252304	−	0.999682i	$-0.491968\pi$
$151$	14138.3	0.0504607	0.0252304	+	0.999682i	$0.491968\pi$
$152$	0	0
$153$	−90796.2	−0.313573
$154$	0	0
$155$	397308.i	1.32831i
$156$	0	0
$157$	− 492112.i	− 1.59336i	−0.604398	−	0.796682i	$-0.706586\pi$
$157$	− 492112.i	− 1.59336i	0.604398	−	0.796682i	$-0.293414\pi$
$158$	0	0
$159$	313444.	0.983256
$160$	0	0
$161$	−51174.7	−0.155593
$162$	0	0
$163$	542547.i	1.59944i	0.600372	+	0.799721i	$0.295019\pi$
$163$	542547.i	1.59944i	−0.600372	+	0.799721i	$0.704981\pi$
$164$	0	0
$165$	− 376036.i	− 1.07527i
$166$	0	0
$167$	−294625.	−0.817483	−0.408742	−	0.912650i	$-0.634032\pi$
$167$	−294625.	−0.817483	−0.408742	+	0.912650i	$0.634032\pi$
$168$	0	0
$169$	−439024.	−1.18242
$170$	0	0
$171$	16247.7i	0.0424914i
$172$	0	0
$173$	− 74155.8i	− 0.188378i	−0.995554	−	0.0941889i	$-0.969974\pi$
$173$	− 74155.8i	− 0.188378i	0.995554	−	0.0941889i	$-0.0300258\pi$
$174$	0	0
$175$	−153276.	−0.378338
$176$	0	0
$177$	62770.5	0.150599
$178$	0	0
$179$	− 273800.i	− 0.638707i	−0.947636	−	0.319353i	$-0.896534\pi$
$179$	− 273800.i	− 0.638707i	0.947636	−	0.319353i	$-0.103466\pi$
$180$	0	0
$181$	489240.i	1.11001i	0.831848	+	0.555003i	$0.187283\pi$
$181$	489240.i	1.11001i	−0.831848	+	0.555003i	$0.812717\pi$
$182$	0	0
$183$	293059.	0.646884
$184$	0	0
$185$	−536276.	−1.15202
$186$	0	0
$187$	612209.i	1.28025i
$188$	0	0
$189$	− 40967.8i	− 0.0834234i
$190$	0	0
$191$	655123.	1.29939	0.649695	−	0.760195i	$-0.274897\pi$
$191$	655123.	1.29939	0.649695	+	0.760195i	$0.274897\pi$
$192$	0	0
$193$	991840.	1.91667	0.958337	−	0.285640i	$-0.0922063\pi$
$193$	991840.	1.91667	0.958337	+	0.285640i	$0.0922063\pi$
$194$	0	0
$195$	− 619783.i	− 1.16722i
$196$	0	0
$197$	217157.i	0.398664i	0.979932	+	0.199332i	$0.0638773\pi$
$197$	217157.i	0.398664i	−0.979932	+	0.199332i	$0.936123\pi$
$198$	0	0
$199$	−131816.	−0.235959	−0.117979	−	0.993016i	$-0.537642\pi$
$199$	−131816.	−0.235959	−0.117979	+	0.993016i	$0.537642\pi$
$200$	0	0
$201$	296566.	0.517763
$202$	0	0
$203$	253560.i	0.431858i
$204$	0	0
$205$	421154.i	0.699933i
$206$	0	0
$207$	73760.8	0.119646
$208$	0	0
$209$	109553.	0.173483
$210$	0	0
$211$	− 801861.i	− 1.23992i	−0.784634	−	0.619959i	$-0.787149\pi$
$211$	− 801861.i	− 1.23992i	0.784634	−	0.619959i	$-0.212851\pi$
$212$	0	0
$213$	623053.i	0.940971i
$214$	0	0
$215$	−1.28673e6	−1.89841
$216$	0	0
$217$	291859.	0.420750
$218$	0	0
$219$	− 145296.i	− 0.204713i
$220$	0	0
$221$	1.00904e6i	1.38973i
$222$	0	0
$223$	1.17218e6	1.57846	0.789230	−	0.614098i	$-0.210480\pi$
$223$	1.17218e6	1.57846	0.789230	+	0.614098i	$0.210480\pi$
$224$	0	0
$225$	220925.	0.290930
$226$	0	0
$227$	− 1.44311e6i	− 1.85880i	−0.369068	−	0.929402i	$-0.620323\pi$
$227$	− 1.44311e6i	− 1.85880i	0.369068	−	0.929402i	$-0.379677\pi$
$228$	0	0
$229$	659034.i	0.830461i	0.909716	+	0.415230i	$0.136299\pi$
$229$	659034.i	0.830461i	−0.909716	+	0.415230i	$0.863701\pi$
$230$	0	0
$231$	−276232.	−0.340600
$232$	0	0
$233$	125667.	0.151647	0.0758234	−	0.997121i	$-0.475841\pi$
$233$	125667.	0.151647	0.0758234	+	0.997121i	$0.475841\pi$
$234$	0	0
$235$	2.08449e6i	2.46223i
$236$	0	0
$237$	− 517581.i	− 0.598559i
$238$	0	0
$239$	−231002.	−0.261589	−0.130795	−	0.991409i	$-0.541753\pi$
$239$	−231002.	−0.261589	−0.130795	+	0.991409i	$0.541753\pi$
$240$	0	0
$241$	−453187.	−0.502614	−0.251307	−	0.967907i	$-0.580860\pi$
$241$	−453187.	−0.502614	−0.251307	+	0.967907i	$0.580860\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	− 1.04416e6i	− 1.11135i
$246$	0	0
$247$	180565.	0.188318
$248$	0	0
$249$	129824.	0.132696
$250$	0	0
$251$	− 376692.i	− 0.377400i	−0.982035	−	0.188700i	$-0.939573\pi$
$251$	− 376692.i	− 0.377400i	0.982035	−	0.188700i	$-0.0604274\pi$
$252$	0	0
$253$	− 497345.i	− 0.488491i
$254$	0	0
$255$	−771782.	−0.743266
$256$	0	0
$257$	1.82902e6	1.72737	0.863687	−	0.504029i	$-0.168150\pi$
$257$	1.82902e6	1.72737	0.863687	+	0.504029i	$0.168150\pi$
$258$	0	0
$259$	393943.i	0.364909i
$260$	0	0
$261$	− 365469.i	− 0.332085i
$262$	0	0
$263$	1.96239e6	1.74943	0.874714	−	0.484639i	$-0.161049\pi$
$263$	1.96239e6	1.74943	0.874714	+	0.484639i	$0.161049\pi$
$264$	0	0
$265$	2.66432e6	2.33062
$266$	0	0
$267$	− 1.16334e6i	− 0.998684i
$268$	0	0
$269$	− 1.18125e6i	− 0.995312i	−0.867374	−	0.497656i	$-0.834194\pi$
$269$	− 1.18125e6i	− 0.995312i	0.867374	−	0.497656i	$-0.165806\pi$
$270$	0	0
$271$	−1.75595e6	−1.45241	−0.726206	−	0.687477i	$-0.758718\pi$
$271$	−1.75595e6	−1.45241	−0.726206	+	0.687477i	$0.758718\pi$
$272$	0	0
$273$	−455287.	−0.369725
$274$	0	0
$275$	− 1.48962e6i	− 1.18780i
$276$	0	0
$277$	− 1.65314e6i	− 1.29452i	−0.762268	−	0.647262i	$-0.775914\pi$
$277$	− 1.65314e6i	− 1.29452i	0.762268	−	0.647262i	$-0.224086\pi$
$278$	0	0
$279$	−420671.	−0.323543
$280$	0	0
$281$	1.35186e6	1.02133	0.510666	−	0.859779i	$-0.329399\pi$
$281$	1.35186e6	1.02133	0.510666	+	0.859779i	$0.329399\pi$
$282$	0	0
$283$	2.58672e6i	1.91992i	0.280130	+	0.959962i	$0.409622\pi$
$283$	2.58672e6i	1.91992i	−0.280130	+	0.959962i	$0.590378\pi$
$284$	0	0
$285$	138108.i	0.100718i
$286$	0	0
$287$	309376.	0.221708
$288$	0	0
$289$	−163350.	−0.115047
$290$	0	0
$291$	− 1.42071e6i	− 0.983500i
$292$	0	0
$293$	828413.i	0.563739i	0.959453	+	0.281869i	$0.0909545\pi$
$293$	828413.i	0.563739i	−0.959453	+	0.281869i	$0.909046\pi$
$294$	0	0
$295$	533559.	0.356967
$296$	0	0
$297$	398148.	0.261911
$298$	0	0
$299$	− 819725.i	− 0.530262i
$300$	0	0
$301$	945217.i	0.601333i
$302$	0	0
$303$	−648713.	−0.405925
$304$	0	0
$305$	2.49104e6	1.53332
$306$	0	0
$307$	− 677605.i	− 0.410327i	−0.978728	−	0.205164i	$-0.934227\pi$
$307$	− 677605.i	− 0.410327i	0.978728	−	0.205164i	$-0.0657727\pi$
$308$	0	0
$309$	775100.i	0.461808i
$310$	0	0
$311$	1.95701e6	1.14734	0.573670	−	0.819086i	$-0.305519\pi$
$311$	1.95701e6	1.14734	0.573670	+	0.819086i	$0.305519\pi$
$312$	0	0
$313$	−348000.	−0.200779	−0.100390	−	0.994948i	$-0.532009\pi$
$313$	−348000.	−0.200779	−0.100390	+	0.994948i	$0.532009\pi$
$314$	0	0
$315$	− 348233.i	− 0.197739i
$316$	0	0
$317$	− 565500.i	− 0.316071i	−0.987433	−	0.158036i	$-0.949484\pi$
$317$	− 565500.i	− 0.316071i	0.987433	−	0.158036i	$-0.0505161\pi$
$318$	0	0
$319$	−2.46424e6	−1.35583
$320$	0	0
$321$	−34491.6	−0.0186832
$322$	0	0
$323$	− 224848.i	− 0.119918i
$324$	0	0
$325$	− 2.45520e6i	− 1.28937i
$326$	0	0
$327$	515333.	0.266513
$328$	0	0
$329$	1.53124e6	0.779928
$330$	0	0
$331$	2.79817e6i	1.40380i	0.712277	+	0.701898i	$0.247664\pi$
$331$	2.79817e6i	1.40380i	−0.712277	+	0.701898i	$0.752336\pi$
$332$	0	0
$333$	− 567811.i	− 0.280604i
$334$	0	0
$335$	2.52086e6	1.22726
$336$	0	0
$337$	2.31639e6	1.11106	0.555529	−	0.831497i	$-0.312516\pi$
$337$	2.31639e6	1.11106	0.555529	+	0.831497i	$0.312516\pi$
$338$	0	0
$339$	− 913574.i	− 0.431762i
$340$	0	0
$341$	2.83645e6i	1.32096i
$342$	0	0
$343$	−1.71154e6	−0.785508
$344$	0	0
$345$	626978.	0.283599
$346$	0	0
$347$	2.49118e6i	1.11066i	0.831630	+	0.555331i	$0.187408\pi$
$347$	2.49118e6i	1.11066i	−0.831630	+	0.555331i	$0.812592\pi$
$348$	0	0
$349$	2.35137e6i	1.03337i	0.856175	+	0.516686i	$0.172834\pi$
$349$	2.35137e6i	1.03337i	−0.856175	+	0.516686i	$0.827166\pi$
$350$	0	0
$351$	656228.	0.284307
$352$	0	0
$353$	−696267.	−0.297399	−0.148699	−	0.988882i	$-0.547509\pi$
$353$	−696267.	−0.297399	−0.148699	+	0.988882i	$0.547509\pi$
$354$	0	0
$355$	5.29605e6i	2.23039i
$356$	0	0
$357$	566944.i	0.235434i
$358$	0	0
$359$	−275050.	−0.112636	−0.0563178	−	0.998413i	$-0.517936\pi$
$359$	−275050.	−0.112636	−0.0563178	+	0.998413i	$0.517936\pi$
$360$	0	0
$361$	2.43586e6	0.983750
$362$	0	0
$363$	− 1.23512e6i	− 0.491976i
$364$	0	0
$365$	− 1.23504e6i	− 0.485232i
$366$	0	0
$367$	2.95659e6	1.14584	0.572922	−	0.819610i	$-0.305810\pi$
$367$	2.95659e6	1.14584	0.572922	+	0.819610i	$0.305810\pi$
$368$	0	0
$369$	−445920.	−0.170487
$370$	0	0
$371$	− 1.95719e6i	− 0.738240i
$372$	0	0
$373$	− 298474.i	− 0.111080i	−0.998456	−	0.0555398i	$-0.982312\pi$
$373$	− 298474.i	− 0.111080i	0.998456	−	0.0555398i	$-0.0176880\pi$
$374$	0	0
$375$	−273706.	−0.100509
$376$	0	0
$377$	−4.06156e6	−1.47177
$378$	0	0
$379$	2.93586e6i	1.04987i	0.851141	+	0.524936i	$0.175911\pi$
$379$	2.93586e6i	1.04987i	−0.851141	+	0.524936i	$0.824089\pi$
$380$	0	0
$381$	1.12759e6i	0.397961i
$382$	0	0
$383$	−2.95674e6	−1.02995	−0.514976	−	0.857205i	$-0.672199\pi$
$383$	−2.95674e6	−1.02995	−0.514976	+	0.857205i	$0.672199\pi$
$384$	0	0
$385$	−2.34802e6	−0.807327
$386$	0	0
$387$	− 1.36239e6i	− 0.462407i
$388$	0	0
$389$	1.40012e6i	0.469127i	0.972101	+	0.234564i	$0.0753661\pi$
$389$	1.40012e6i	0.469127i	−0.972101	+	0.234564i	$0.924634\pi$
$390$	0	0
$391$	−1.02076e6	−0.337661
$392$	0	0
$393$	2.80984e6	0.917700
$394$	0	0
$395$	− 4.39952e6i	− 1.41877i
$396$	0	0
$397$	5.27254e6i	1.67897i	0.543380	+	0.839487i	$0.317144\pi$
$397$	5.27254e6i	1.67897i	−0.543380	+	0.839487i	$0.682856\pi$
$398$	0	0
$399$	101453.	0.0319030
$400$	0	0
$401$	−1.77125e6	−0.550072	−0.275036	−	0.961434i	$-0.588690\pi$
$401$	−1.77125e6	−0.550072	−0.275036	+	0.961434i	$0.588690\pi$
$402$	0	0
$403$	4.67504e6i	1.43391i
$404$	0	0
$405$	501926.i	0.152055i
$406$	0	0
$407$	−3.82856e6	−1.14564
$408$	0	0
$409$	401562.	0.118698	0.0593491	−	0.998237i	$-0.481097\pi$
$409$	401562.	0.118698	0.0593491	+	0.998237i	$0.481097\pi$
$410$	0	0
$411$	− 419585.i	− 0.122522i
$412$	0	0
$413$	− 391948.i	− 0.113071i
$414$	0	0
$415$	1.10352e6	0.314530
$416$	0	0
$417$	−1.80867e6	−0.509353
$418$	0	0
$419$	− 4.22653e6i	− 1.17611i	−0.808820	−	0.588056i	$-0.799894\pi$
$419$	− 4.22653e6i	− 1.17611i	0.808820	−	0.588056i	$-0.200106\pi$
$420$	0	0
$421$	− 467543.i	− 0.128563i	−0.997932	−	0.0642816i	$-0.979524\pi$
$421$	− 467543.i	− 0.128563i	0.997932	−	0.0642816i	$-0.0204756\pi$
$422$	0	0
$423$	−2.20706e6	−0.599741
$424$	0	0
$425$	−3.05733e6	−0.821050
$426$	0	0
$427$	− 1.82990e6i	− 0.485688i
$428$	0	0
$429$	− 4.42473e6i	− 1.16076i
$430$	0	0
$431$	−632828.	−0.164094	−0.0820469	−	0.996628i	$-0.526146\pi$
$431$	−632828.	−0.164094	−0.0820469	+	0.996628i	$0.526146\pi$
$432$	0	0
$433$	−3.32144e6	−0.851347	−0.425674	−	0.904877i	$-0.639963\pi$
$433$	−3.32144e6	−0.851347	−0.425674	+	0.904877i	$0.639963\pi$
$434$	0	0
$435$	− 3.10655e6i	− 0.787145i
$436$	0	0
$437$	182662.i	0.0457555i
$438$	0	0
$439$	7.34440e6	1.81884	0.909422	−	0.415876i	$-0.136525\pi$
$439$	7.34440e6	1.81884	0.909422	+	0.415876i	$0.136525\pi$
$440$	0	0
$441$	1.10556e6	0.270698
$442$	0	0
$443$	− 1.04224e6i	− 0.252323i	−0.992010	−	0.126162i	$-0.959734\pi$
$443$	− 1.04224e6i	− 0.252323i	0.992010	−	0.126162i	$-0.0402658\pi$
$444$	0	0
$445$	− 9.88857e6i	− 2.36719i
$446$	0	0
$447$	2.42285e6	0.573532
$448$	0	0
$449$	5.05190e6	1.18260	0.591302	−	0.806450i	$-0.298614\pi$
$449$	5.05190e6	1.18260	0.591302	+	0.806450i	$0.298614\pi$
$450$	0	0
$451$	3.00669e6i	0.696061i
$452$	0	0
$453$	127244.i	0.0291335i
$454$	0	0
$455$	−3.87001e6	−0.876362
$456$	0	0
$457$	−2.44643e6	−0.547953	−0.273976	−	0.961736i	$-0.588339\pi$
$457$	−2.44643e6	−0.547953	−0.273976	+	0.961736i	$0.588339\pi$
$458$	0	0
$459$	− 817165.i	− 0.181042i
$460$	0	0
$461$	− 3.78789e6i	− 0.830129i	−0.909792	−	0.415064i	$-0.863759\pi$
$461$	− 3.78789e6i	− 0.830129i	0.909792	−	0.415064i	$-0.136241\pi$
$462$	0	0
$463$	6.82027e6	1.47859	0.739297	−	0.673379i	$-0.235158\pi$
$463$	6.82027e6	1.47859	0.739297	+	0.673379i	$0.235158\pi$
$464$	0	0
$465$	−3.57577e6	−0.766898
$466$	0	0
$467$	3.24065e6i	0.687606i	0.939042	+	0.343803i	$0.111715\pi$
$467$	3.24065e6i	0.687606i	−0.939042	+	0.343803i	$0.888285\pi$
$468$	0	0
$469$	− 1.85180e6i	− 0.388742i
$470$	0	0
$471$	4.42901e6	0.919929
$472$	0	0
$473$	−9.18615e6	−1.88791
$474$	0	0
$475$	547099.i	0.111258i
$476$	0	0
$477$	2.82099e6i	0.567683i
$478$	0	0
$479$	−7.21068e6	−1.43594	−0.717972	−	0.696072i	$-0.754929\pi$
$479$	−7.21068e6	−1.43594	−0.717972	+	0.696072i	$0.754929\pi$
$480$	0	0
$481$	−6.31025e6	−1.24361
$482$	0	0
$483$	− 460573.i	− 0.0898318i
$484$	0	0
$485$	− 1.20763e7i	− 2.33120i
$486$	0	0
$487$	−9.86714e6	−1.88525	−0.942625	−	0.333854i	$-0.891651\pi$
$487$	−9.86714e6	−1.88525	−0.942625	+	0.333854i	$0.891651\pi$
$488$	0	0
$489$	−4.88292e6	−0.923438
$490$	0	0
$491$	− 5.06928e6i	− 0.948949i	−0.880269	−	0.474474i	$-0.842638\pi$
$491$	− 5.06928e6i	− 0.948949i	0.880269	−	0.474474i	$-0.157362\pi$
$492$	0	0
$493$	5.05764e6i	0.937197i
$494$	0	0
$495$	3.38432e6	0.620810
$496$	0	0
$497$	3.89043e6	0.706491
$498$	0	0
$499$	5.15490e6i	0.926764i	0.886159	+	0.463382i	$0.153364\pi$
$499$	5.15490e6i	0.926764i	−0.886159	+	0.463382i	$0.846636\pi$
$500$	0	0
$501$	− 2.65163e6i	− 0.471974i
$502$	0	0
$503$	3.07607e6	0.542096	0.271048	−	0.962566i	$-0.412630\pi$
$503$	3.07607e6	0.542096	0.271048	+	0.962566i	$0.412630\pi$
$504$	0	0
$505$	−5.51417e6	−0.962169
$506$	0	0
$507$	− 3.95122e6i	− 0.682670i
$508$	0	0
$509$	1.12309e7i	1.92140i	0.277585	+	0.960701i	$0.410466\pi$
$509$	1.12309e7i	1.92140i	−0.277585	+	0.960701i	$0.589534\pi$
$510$	0	0
$511$	−907251.	−0.153700
$512$	0	0
$513$	−146229.	−0.0245324
$514$	0	0
$515$	6.58848e6i	1.09463i
$516$	0	0
$517$	1.48815e7i	2.44861i
$518$	0	0
$519$	667402.	0.108760
$520$	0	0
$521$	−9.05083e6	−1.46081	−0.730406	−	0.683013i	$-0.760669\pi$
$521$	−9.05083e6	−1.46081	−0.730406	+	0.683013i	$0.760669\pi$
$522$	0	0
$523$	− 8.34283e6i	− 1.33370i	−0.745190	−	0.666852i	$-0.767641\pi$
$523$	− 8.34283e6i	− 1.33370i	0.745190	−	0.666852i	$-0.232359\pi$
$524$	0	0
$525$	− 1.37949e6i	− 0.218433i
$526$	0	0
$527$	5.82158e6	0.913091
$528$	0	0
$529$	−5.60710e6	−0.871163
$530$	0	0
$531$	564935.i	0.0869485i
$532$	0	0
$533$	4.95563e6i	0.755581i
$534$	0	0
$535$	−293184.	−0.0442850
$536$	0	0
$537$	2.46420e6	0.368758
$538$	0	0
$539$	− 7.45442e6i	− 1.10520i
$540$	0	0
$541$	− 5.48101e6i	− 0.805133i	−0.915391	−	0.402566i	$-0.868118\pi$
$541$	− 5.48101e6i	− 0.805133i	0.915391	−	0.402566i	$-0.131882\pi$
$542$	0	0
$543$	−4.40316e6	−0.640862
$544$	0	0
$545$	4.38041e6	0.631718
$546$	0	0
$547$	2.04402e6i	0.292090i	0.989278	+	0.146045i	$0.0466545\pi$
$547$	2.04402e6i	0.292090i	−0.989278	+	0.146045i	$0.953346\pi$
$548$	0	0
$549$	2.63753e6i	0.373479i
$550$	0	0
$551$	905050.	0.126997
$552$	0	0
$553$	−3.23185e6	−0.449405
$554$	0	0
$555$	− 4.82648e6i	− 0.665118i
$556$	0	0
$557$	7.39999e6i	1.01063i	0.862934	+	0.505316i	$0.168624\pi$
$557$	7.39999e6i	1.01063i	−0.862934	+	0.505316i	$0.831376\pi$
$558$	0	0
$559$	−1.51406e7	−2.04934
$560$	0	0
$561$	−5.50988e6	−0.739154
$562$	0	0
$563$	− 5.30287e6i	− 0.705082i	−0.935796	−	0.352541i	$-0.885318\pi$
$563$	− 5.30287e6i	− 0.705082i	0.935796	−	0.352541i	$-0.114682\pi$
$564$	0	0
$565$	− 7.76552e6i	− 1.02341i
$566$	0	0
$567$	368710.	0.0481645
$568$	0	0
$569$	−5.96823e6	−0.772796	−0.386398	−	0.922332i	$-0.626281\pi$
$569$	−5.96823e6	−0.772796	−0.386398	+	0.922332i	$0.626281\pi$
$570$	0	0
$571$	1.27901e7i	1.64166i	0.571172	+	0.820830i	$0.306489\pi$
$571$	1.27901e7i	1.64166i	−0.571172	+	0.820830i	$0.693511\pi$
$572$	0	0
$573$	5.89611e6i	0.750203i
$574$	0	0
$575$	2.48371e6	0.313278
$576$	0	0
$577$	650009.	0.0812793	0.0406396	−	0.999174i	$-0.487060\pi$
$577$	650009.	0.0812793	0.0406396	+	0.999174i	$0.487060\pi$
$578$	0	0
$579$	8.92656e6i	1.10659i
$580$	0	0
$581$	− 810638.i	− 0.0996292i
$582$	0	0
$583$	1.90210e7	2.31773
$584$	0	0
$585$	5.57805e6	0.673895
$586$	0	0
$587$	− 6.73154e6i	− 0.806342i	−0.915125	−	0.403171i	$-0.867908\pi$
$587$	− 6.73154e6i	− 0.806342i	0.915125	−	0.403171i	$-0.132092\pi$
$588$	0	0
$589$	− 1.04175e6i	− 0.123730i
$590$	0	0
$591$	−1.95441e6	−0.230169
$592$	0	0
$593$	−1.69469e6	−0.197904	−0.0989518	−	0.995092i	$-0.531549\pi$
$593$	−1.69469e6	−0.197904	−0.0989518	+	0.995092i	$0.531549\pi$
$594$	0	0
$595$	4.81911e6i	0.558052i
$596$	0	0
$597$	− 1.18635e6i	− 0.136231i
$598$	0	0
$599$	5.40888e6	0.615943	0.307971	−	0.951396i	$-0.400350\pi$
$599$	5.40888e6	0.615943	0.307971	+	0.951396i	$0.400350\pi$
$600$	0	0
$601$	1.53834e7	1.73726	0.868632	−	0.495458i	$-0.165000\pi$
$601$	1.53834e7	1.73726	0.868632	+	0.495458i	$0.165000\pi$
$602$	0	0
$603$	2.66909e6i	0.298930i
$604$	0	0
$605$	− 1.04987e7i	− 1.16613i
$606$	0	0
$607$	−8.53390e6	−0.940104	−0.470052	−	0.882639i	$-0.655765\pi$
$607$	−8.53390e6	−0.940104	−0.470052	+	0.882639i	$0.655765\pi$
$608$	0	0
$609$	−2.28204e6	−0.249333
$610$	0	0
$611$	2.45277e7i	2.65799i
$612$	0	0
$613$	7.89807e6i	0.848926i	0.905445	+	0.424463i	$0.139537\pi$
$613$	7.89807e6i	0.848926i	−0.905445	+	0.424463i	$0.860463\pi$
$614$	0	0
$615$	−3.79039e6	−0.404106
$616$	0	0
$617$	−1.04516e7	−1.10528	−0.552638	−	0.833421i	$-0.686379\pi$
$617$	−1.04516e7	−1.10528	−0.552638	+	0.833421i	$0.686379\pi$
$618$	0	0
$619$	− 9.95278e6i	− 1.04404i	−0.852933	−	0.522021i	$-0.825178\pi$
$619$	− 9.95278e6i	− 1.04404i	0.852933	−	0.522021i	$-0.174822\pi$
$620$	0	0
$621$	663847.i	0.0690779i
$622$	0	0
$623$	−7.26405e6	−0.749823
$624$	0	0
$625$	−1.08499e7	−1.11103
$626$	0	0
$627$	985975.i	0.100161i
$628$	0	0
$629$	7.85781e6i	0.791908i
$630$	0	0
$631$	−1.06747e7	−1.06728	−0.533642	−	0.845710i	$-0.679177\pi$
$631$	−1.06747e7	−1.06728	−0.533642	+	0.845710i	$0.679177\pi$
$632$	0	0
$633$	7.21675e6	0.715867
$634$	0	0
$635$	9.58473e6i	0.943291i
$636$	0	0
$637$	− 1.22864e7i	− 1.19971i
$638$	0	0
$639$	−5.60748e6	−0.543270
$640$	0	0
$641$	−1.12741e6	−0.108377	−0.0541883	−	0.998531i	$-0.517257\pi$
$641$	−1.12741e6	−0.108377	−0.0541883	+	0.998531i	$0.517257\pi$
$642$	0	0
$643$	− 7.22697e6i	− 0.689332i	−0.938725	−	0.344666i	$-0.887992\pi$
$643$	− 7.22697e6i	− 0.689332i	0.938725	−	0.344666i	$-0.112008\pi$
$644$	0	0
$645$	− 1.15805e7i	− 1.09605i
$646$	0	0
$647$	1.47804e7	1.38811	0.694055	−	0.719922i	$-0.255822\pi$
$647$	1.47804e7	1.38811	0.694055	+	0.719922i	$0.255822\pi$
$648$	0	0
$649$	3.80917e6	0.354992
$650$	0	0
$651$	2.62673e6i	0.242920i
$652$	0	0
$653$	− 4.89041e6i	− 0.448810i	−0.974496	−	0.224405i	$-0.927956\pi$
$653$	− 4.89041e6i	− 0.448810i	0.974496	−	0.224405i	$-0.0720438\pi$
$654$	0	0
$655$	2.38841e7	2.17523
$656$	0	0
$657$	1.30767e6	0.118191
$658$	0	0
$659$	− 1.09879e6i	− 0.0985602i	−0.998785	−	0.0492801i	$-0.984307\pi$
$659$	− 1.09879e6i	− 0.0985602i	0.998785	−	0.0492801i	$-0.0156927\pi$
$660$	0	0
$661$	− 1.78915e6i	− 0.159273i	−0.996824	−	0.0796366i	$-0.974624\pi$
$661$	− 1.78915e6i	− 0.159273i	0.996824	−	0.0796366i	$-0.0253760\pi$
$662$	0	0
$663$	−9.08139e6	−0.802359
$664$	0	0
$665$	862365.	0.0756200
$666$	0	0
$667$	− 4.10872e6i	− 0.357595i
$668$	0	0
$669$	1.05497e7i	0.911324i
$670$	0	0
$671$	1.77840e7	1.52483
$672$	0	0
$673$	1.22144e7	1.03952	0.519761	−	0.854312i	$-0.326021\pi$
$673$	1.22144e7	1.03952	0.519761	+	0.854312i	$0.326021\pi$
$674$	0	0
$675$	1.98832e6i	0.167968i
$676$	0	0
$677$	− 6.19988e6i	− 0.519890i	−0.965623	−	0.259945i	$-0.916296\pi$
$677$	− 6.19988e6i	− 0.519890i	0.965623	−	0.259945i	$-0.0837044\pi$
$678$	0	0
$679$	−8.87113e6	−0.738422
$680$	0	0
$681$	1.29880e7	1.07318
$682$	0	0
$683$	− 1.49551e7i	− 1.22670i	−0.789812	−	0.613349i	$-0.789822\pi$
$683$	− 1.49551e7i	− 1.22670i	0.789812	−	0.613349i	$-0.210178\pi$
$684$	0	0
$685$	− 3.56654e6i	− 0.290416i
$686$	0	0
$687$	−5.93131e6	−0.479467
$688$	0	0
$689$	3.13505e7	2.51592
$690$	0	0
$691$	− 566183.i	− 0.0451088i	−0.999746	−	0.0225544i	$-0.992820\pi$
$691$	− 566183.i	− 0.0451088i	0.999746	−	0.0225544i	$-0.00717990\pi$
$692$	0	0
$693$	− 2.48609e6i	− 0.196645i
$694$	0	0
$695$	−1.53740e7	−1.20733
$696$	0	0
$697$	6.17098e6	0.481141
$698$	0	0
$699$	1.13101e6i	0.0875533i
$700$	0	0
$701$	− 316263.i	− 0.0243082i	−0.999926	−	0.0121541i	$-0.996131\pi$
$701$	− 316263.i	− 0.0243082i	0.999926	−	0.0121541i	$-0.00386887\pi$
$702$	0	0
$703$	1.40613e6	0.107309
$704$	0	0
$705$	−1.87604e7	−1.42157
$706$	0	0
$707$	4.05066e6i	0.304773i
$708$	0	0
$709$	− 971009.i	− 0.0725451i	−0.999342	−	0.0362725i	$-0.988452\pi$
$709$	− 971009.i	− 0.0725451i	0.999342	−	0.0362725i	$-0.0115484\pi$
$710$	0	0
$711$	4.65823e6	0.345578
$712$	0	0
$713$	−4.72932e6	−0.348397
$714$	0	0
$715$	− 3.76109e7i	− 2.75137i
$716$	0	0
$717$	− 2.07901e6i	− 0.151029i
$718$	0	0
$719$	1.60574e7	1.15839	0.579194	−	0.815190i	$-0.303367\pi$
$719$	1.60574e7	1.15839	0.579194	+	0.815190i	$0.303367\pi$
$720$	0	0
$721$	4.83983e6	0.346731
$722$	0	0
$723$	− 4.07868e6i	− 0.290184i
$724$	0	0
$725$	− 1.23062e7i	− 0.869522i
$726$	0	0
$727$	−3.75285e6	−0.263345	−0.131672	−	0.991293i	$-0.542035\pi$
$727$	−3.75285e6	−0.263345	−0.131672	+	0.991293i	$0.542035\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.88538e7i	1.30498i
$732$	0	0
$733$	− 4.16860e6i	− 0.286570i	−0.989681	−	0.143285i	$-0.954233\pi$
$733$	− 4.16860e6i	− 0.286570i	0.989681	−	0.143285i	$-0.0457665\pi$
$734$	0	0
$735$	9.39742e6	0.641638
$736$	0	0
$737$	1.79968e7	1.22047
$738$	0	0
$739$	4.06467e6i	0.273788i	0.990586	+	0.136894i	$0.0437120\pi$
$739$	4.06467e6i	0.273788i	−0.990586	+	0.136894i	$0.956288\pi$
$740$	0	0
$741$	1.62509e6i	0.108725i
$742$	0	0
$743$	−1.29765e7	−0.862354	−0.431177	−	0.902267i	$-0.641902\pi$
$743$	−1.29765e7	−0.862354	−0.431177	+	0.902267i	$0.641902\pi$
$744$	0	0
$745$	2.05946e7	1.35945
$746$	0	0
$747$	1.16842e6i	0.0766118i
$748$	0	0
$749$	215370.i	0.0140275i
$750$	0	0
$751$	−2.99173e7	−1.93563	−0.967816	−	0.251661i	$-0.919023\pi$
$751$	−2.99173e7	−1.93563	−0.967816	+	0.251661i	$0.919023\pi$
$752$	0	0
$753$	3.39023e6	0.217892
$754$	0	0
$755$	1.08160e6i	0.0690554i
$756$	0	0
$757$	8.50221e6i	0.539253i	0.962965	+	0.269626i	$0.0869002\pi$
$757$	8.50221e6i	0.539253i	−0.962965	+	0.269626i	$0.913100\pi$
$758$	0	0
$759$	4.47610e6	0.282030
$760$	0	0
$761$	1.45847e7	0.912928	0.456464	−	0.889742i	$-0.349116\pi$
$761$	1.45847e7	0.912928	0.456464	+	0.889742i	$0.349116\pi$
$762$	0	0
$763$	− 3.21781e6i	− 0.200101i
$764$	0	0
$765$	− 6.94604e6i	− 0.429125i
$766$	0	0
$767$	6.27828e6	0.385347
$768$	0	0
$769$	−2.23049e7	−1.36014	−0.680071	−	0.733146i	$-0.738051\pi$
$769$	−2.23049e7	−1.36014	−0.680071	+	0.733146i	$0.738051\pi$
$770$	0	0
$771$	1.64612e7i	0.997299i
$772$	0	0
$773$	1.89823e7i	1.14262i	0.820736	+	0.571308i	$0.193564\pi$
$773$	1.89823e7i	1.14262i	−0.820736	+	0.571308i	$0.806436\pi$
$774$	0	0
$775$	−1.41650e7	−0.847156
$776$	0	0
$777$	−3.54549e6	−0.210680
$778$	0	0
$779$	− 1.10428e6i	− 0.0651980i
$780$	0	0
$781$	3.78094e7i	2.21805i
$782$	0	0
$783$	3.28922e6	0.191729
$784$	0	0
$785$	3.76473e7	2.18052
$786$	0	0
$787$	− 2.43444e7i	− 1.40108i	−0.713615	−	0.700538i	$-0.752944\pi$
$787$	− 2.43444e7i	− 1.40108i	0.713615	−	0.700538i	$-0.247056\pi$
$788$	0	0
$789$	1.76615e7i	1.01003i
$790$	0	0
$791$	−5.70448e6	−0.324171
$792$	0	0
$793$	2.93116e7	1.65522
$794$	0	0
$795$	2.39789e7i	1.34559i
$796$	0	0
$797$	3.23426e7i	1.80356i	0.432199	+	0.901778i	$0.357738\pi$
$797$	3.23426e7i	1.80356i	−0.432199	+	0.901778i	$0.642262\pi$
$798$	0	0
$799$	3.05430e7	1.69256
$800$	0	0
$801$	1.04701e7	0.576591
$802$	0	0
$803$	− 8.81717e6i	− 0.482548i
$804$	0	0
$805$	− 3.91494e6i	− 0.212929i
$806$	0	0
$807$	1.06312e7	0.574644
$808$	0	0
$809$	6.88480e6	0.369845	0.184923	−	0.982753i	$-0.440797\pi$
$809$	6.88480e6	0.369845	0.184923	+	0.982753i	$0.440797\pi$
$810$	0	0
$811$	− 1.66897e7i	− 0.891040i	−0.895272	−	0.445520i	$-0.853019\pi$
$811$	− 1.66897e7i	− 0.891040i	0.895272	−	0.445520i	$-0.146981\pi$
$812$	0	0
$813$	− 1.58036e7i	− 0.838551i
$814$	0	0
$815$	−4.15056e7	−2.18883
$816$	0	0
$817$	3.37383e6	0.176835
$818$	0	0
$819$	− 4.09758e6i	− 0.213461i
$820$	0	0
$821$	8.32338e6i	0.430965i	0.976508	+	0.215483i	$0.0691324\pi$
$821$	8.32338e6i	0.430965i	−0.976508	+	0.215483i	$0.930868\pi$
$822$	0	0
$823$	2.20928e7	1.13698	0.568488	−	0.822692i	$-0.307529\pi$
$823$	2.20928e7	1.13698	0.568488	+	0.822692i	$0.307529\pi$
$824$	0	0
$825$	1.34066e7	0.685779
$826$	0	0
$827$	2.07662e7i	1.05583i	0.849298	+	0.527914i	$0.177026\pi$
$827$	2.07662e7i	1.05583i	−0.849298	+	0.527914i	$0.822974\pi$
$828$	0	0
$829$	3.60390e6i	0.182132i	0.995845	+	0.0910660i	$0.0290274\pi$
$829$	3.60390e6i	0.182132i	−0.995845	+	0.0910660i	$0.970973\pi$
$830$	0	0
$831$	1.48783e7	0.747394
$832$	0	0
$833$	−1.52996e7	−0.763953
$834$	0	0
$835$	− 2.25392e7i	− 1.11872i
$836$	0	0
$837$	− 3.78604e6i	− 0.186798i
$838$	0	0
$839$	1.06438e7	0.522024	0.261012	−	0.965336i	$-0.415944\pi$
$839$	1.06438e7	0.522024	0.261012	+	0.965336i	$0.415944\pi$
$840$	0	0
$841$	153313.	0.00747463
$842$	0	0
$843$	1.21668e7i	0.589667i
$844$	0	0
$845$	− 3.35860e7i	− 1.61814i
$846$	0	0
$847$	−7.71228e6	−0.369381
$848$	0	0
$849$	−2.32805e7	−1.10847
$850$	0	0
$851$	− 6.38351e6i	− 0.302159i
$852$	0	0
$853$	− 1.55497e6i	− 0.0731729i	−0.999330	−	0.0365864i	$-0.988352\pi$
$853$	− 1.55497e6i	− 0.0731729i	0.999330	−	0.0365864i	$-0.0116484\pi$
$854$	0	0
$855$	−1.24297e6	−0.0581495
$856$	0	0
$857$	−2.24570e7	−1.04448	−0.522239	−	0.852799i	$-0.674903\pi$
$857$	−2.24570e7	−1.04448	−0.522239	+	0.852799i	$0.674903\pi$
$858$	0	0
$859$	− 950968.i	− 0.0439727i	−0.999758	−	0.0219863i	$-0.993001\pi$
$859$	− 950968.i	− 0.0439727i	0.999758	−	0.0219863i	$-0.00699903\pi$
$860$	0	0
$861$	2.78438e6i	0.128003i
$862$	0	0
$863$	−2.33496e7	−1.06721	−0.533607	−	0.845732i	$-0.679164\pi$
$863$	−2.33496e7	−1.06721	−0.533607	+	0.845732i	$0.679164\pi$
$864$	0	0
$865$	5.67302e6	0.257795
$866$	0	0
$867$	− 1.47015e6i	− 0.0664223i
$868$	0	0
$869$	− 3.14089e7i	− 1.41092i
$870$	0	0
$871$	2.96624e7	1.32483
$872$	0	0
$873$	1.27864e7	0.567824
$874$	0	0
$875$	1.70906e6i	0.0754635i
$876$	0	0
$877$	− 4.66760e6i	− 0.204925i	−0.994737	−	0.102462i	$-0.967328\pi$
$877$	− 4.66760e6i	− 0.204925i	0.994737	−	0.102462i	$-0.0326722\pi$
$878$	0	0
$879$	−7.45572e6	−0.325475
$880$	0	0
$881$	9.13912e6	0.396702	0.198351	−	0.980131i	$-0.436441\pi$
$881$	9.13912e6	0.396702	0.198351	+	0.980131i	$0.436441\pi$
$882$	0	0
$883$	− 2.88960e7i	− 1.24720i	−0.781744	−	0.623599i	$-0.785670\pi$
$883$	− 2.88960e7i	− 1.24720i	0.781744	−	0.623599i	$-0.214330\pi$
$884$	0	0
$885$	4.80203e6i	0.206095i
$886$	0	0
$887$	1.14694e7	0.489476	0.244738	−	0.969589i	$-0.421298\pi$
$887$	1.14694e7	0.489476	0.244738	+	0.969589i	$0.421298\pi$
$888$	0	0
$889$	7.04085e6	0.298793
$890$	0	0
$891$	3.58333e6i	0.151214i
$892$	0	0
$893$	− 5.46558e6i	− 0.229354i
$894$	0	0
$895$	2.09461e7	0.874070
$896$	0	0
$897$	7.37752e6	0.306147
$898$	0	0
$899$	2.34328e7i	0.966996i
$900$	0	0
$901$	− 3.90391e7i	− 1.60209i
$902$	0	0
$903$	−8.50695e6	−0.347180
$904$	0	0
$905$	−3.74275e7	−1.51904
$906$	0	0
$907$	− 1.60176e7i	− 0.646516i	−0.946311	−	0.323258i	$-0.895222\pi$
$907$	− 1.60176e7i	− 0.646516i	0.946311	−	0.323258i	$-0.104778\pi$
$908$	0	0
$909$	− 5.83842e6i	− 0.234361i
$910$	0	0
$911$	−9.49641e6	−0.379108	−0.189554	−	0.981870i	$-0.560704\pi$
$911$	−9.49641e6	−0.379108	−0.189554	+	0.981870i	$0.560704\pi$
$912$	0	0
$913$	7.87824e6	0.312790
$914$	0	0
$915$	2.24194e7i	0.885261i
$916$	0	0
$917$	− 1.75451e7i	− 0.689019i
$918$	0	0
$919$	−1.33408e7	−0.521066	−0.260533	−	0.965465i	$-0.583898\pi$
$919$	−1.33408e7	−0.521066	−0.260533	+	0.965465i	$0.583898\pi$
$920$	0	0
$921$	6.09844e6	0.236903
$922$	0	0
$923$	6.23175e7i	2.40772i
$924$	0	0
$925$	− 1.91196e7i	− 0.734724i
$926$	0	0
$927$	−6.97590e6	−0.266625
$928$	0	0
$929$	3.20359e7	1.21786	0.608931	−	0.793223i	$-0.291599\pi$
$929$	3.20359e7	1.21786	0.608931	+	0.793223i	$0.291599\pi$
$930$	0	0
$931$	2.73781e6i	0.103521i
$932$	0	0
$933$	1.76131e7i	0.662417i
$934$	0	0
$935$	−4.68348e7	−1.75202
$936$	0	0
$937$	−1.51237e7	−0.562740	−0.281370	−	0.959599i	$-0.590789\pi$
$937$	−1.51237e7	−0.562740	−0.281370	+	0.959599i	$0.590789\pi$
$938$	0	0
$939$	− 3.13200e6i	− 0.115920i
$940$	0	0
$941$	1.11906e7i	0.411985i	0.978554	+	0.205992i	$0.0660422\pi$
$941$	1.11906e7i	0.411985i	−0.978554	+	0.205992i	$0.933958\pi$
$942$	0	0
$943$	−5.01317e6	−0.183583
$944$	0	0
$945$	3.13409e6	0.114165
$946$	0	0
$947$	− 1.37177e6i	− 0.0497058i	−0.999691	−	0.0248529i	$-0.992088\pi$
$947$	− 1.37177e6i	− 0.0497058i	0.999691	−	0.0248529i	$-0.00791174\pi$
$948$	0	0
$949$	− 1.45325e7i	− 0.523811i
$950$	0	0
$951$	5.08950e6	0.182484
$952$	0	0
$953$	−4.65945e7	−1.66189	−0.830946	−	0.556354i	$-0.812200\pi$
$953$	−4.65945e7	−1.66189	−0.830946	+	0.556354i	$0.812200\pi$
$954$	0	0
$955$	5.01179e7i	1.77821i
$956$	0	0
$957$	− 2.21782e7i	− 0.782790i
$958$	0	0
$959$	−2.61994e6	−0.0919910
$960$	0	0
$961$	−1.65697e6	−0.0578769
$962$	0	0
$963$	− 310425.i	− 0.0107867i
$964$	0	0
$965$	7.58771e7i	2.62297i
$966$	0	0
$967$	2.36734e7	0.814132	0.407066	−	0.913399i	$-0.366552\pi$
$967$	2.36734e7	0.814132	0.407066	+	0.913399i	$0.366552\pi$
$968$	0	0
$969$	2.02363e6	0.0692344
$970$	0	0
$971$	1.77699e7i	0.604833i	0.953176	+	0.302417i	$0.0977934\pi$
$971$	1.77699e7i	0.604833i	−0.953176	+	0.302417i	$0.902207\pi$
$972$	0	0
$973$	1.12936e7i	0.382428i
$974$	0	0
$975$	2.20968e7	0.744420
$976$	0	0
$977$	−3.53620e6	−0.118522	−0.0592611	−	0.998243i	$-0.518874\pi$
$977$	−3.53620e6	−0.118522	−0.0592611	+	0.998243i	$0.518874\pi$
$978$	0	0
$979$	− 7.05961e7i	− 2.35410i
$980$	0	0
$981$	4.63799e6i	0.153871i
$982$	0	0
$983$	3.78022e7	1.24777	0.623884	−	0.781517i	$-0.285554\pi$
$983$	3.78022e7	1.24777	0.623884	+	0.781517i	$0.285554\pi$
$984$	0	0
$985$	−1.66128e7	−0.545572
$986$	0	0
$987$	1.37812e7i	0.450292i
$988$	0	0
$989$	− 1.53164e7i	− 0.497928i
$990$	0	0
$991$	−2.16530e7	−0.700381	−0.350190	−	0.936679i	$-0.613883\pi$
$991$	−2.16530e7	−0.700381	−0.350190	+	0.936679i	$0.613883\pi$
$992$	0	0
$993$	−2.51835e7	−0.810482
$994$	0	0
$995$	− 1.00841e7i	− 0.322910i
$996$	0	0
$997$	8.04784e6i	0.256414i	0.991747	+	0.128207i	$0.0409221\pi$
$997$	8.04784e6i	0.256414i	−0.991747	+	0.128207i	$0.959078\pi$
$998$	0	0
$999$	5.11030e6	0.162007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.i.193.8	yes	8
4.3	odd	2	inner	384.6.d.i.193.4	yes	8
8.3	odd	2	inner	384.6.d.i.193.5	yes	8
8.5	even	2	inner	384.6.d.i.193.1	✓	8
16.3	odd	4		768.6.a.y.1.4		4
16.5	even	4		768.6.a.y.1.1		4
16.11	odd	4		768.6.a.x.1.1		4
16.13	even	4		768.6.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.i.193.1	✓	8	8.5	even	2	inner
384.6.d.i.193.4	yes	8	4.3	odd	2	inner
384.6.d.i.193.5	yes	8	8.3	odd	2	inner
384.6.d.i.193.8	yes	8	1.1	even	1	trivial
768.6.a.x.1.1		4	16.11	odd	4
768.6.a.x.1.4		4	16.13	even	4
768.6.a.y.1.1		4	16.5	even	4
768.6.a.y.1.4		4	16.3	odd	4

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +76.5014i q^{5} +56.1972 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +76.5014i q^{5} +56.1972 q^{7} -81.0000 q^{9} +546.156i q^{11} +900.176i q^{13} -688.513 q^{15} +1120.94 q^{17} -200.589i q^{19} +505.775i q^{21} -910.627 q^{23} -2727.47 q^{25} -729.000i q^{27} +4511.97i q^{29} +5193.48 q^{31} -4915.41 q^{33} +4299.17i q^{35} +7010.01i q^{37} -8101.59 q^{39} +5505.18 q^{41} +16819.6i q^{43} -6196.62i q^{45} +27247.7 q^{47} -13648.9 q^{49} +10088.5i q^{51} -34827.1i q^{53} -41781.7 q^{55} +1805.30 q^{57} -6974.50i q^{59} -32562.1i q^{61} -4551.98 q^{63} -68864.8 q^{65} -32951.7i q^{67} -8195.64i q^{69} +69228.2 q^{71} -16144.0 q^{73} -24547.2i q^{75} +30692.5i q^{77} -57509.0 q^{79} +6561.00 q^{81} -14424.9i q^{83} +85753.5i q^{85} -40607.7 q^{87} -129260. q^{89} +50587.4i q^{91} +46741.3i q^{93} +15345.3 q^{95} -157857. q^{97} -44238.7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 648 q^{9}+O(q^{10}) \) 8 * q - 648 * q^9	\( 8 q - 648 q^{9} + 4080 q^{17} - 5528 q^{25} - 18144 q^{33} + 58704 q^{41} - 44024 q^{49} + 38880 q^{57} - 203904 q^{65} + 248816 q^{73} + 52488 q^{81} - 46800 q^{89} - 262544 q^{97}+O(q^{100}) \) 8 * q - 648 * q^9 + 4080 * q^17 - 5528 * q^25 - 18144 * q^33 + 58704 * q^41 - 44024 * q^49 + 38880 * q^57 - 203904 * q^65 + 248816 * q^73 + 52488 * q^81 - 46800 * q^89 - 262544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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