LMFDB - Embedded newform 384.8.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,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, 8, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
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:	$119.955849786$
Analytic rank:	$1$
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} - 430x^{2} - 2448x + 12138 $ x^4 - 430x^2 - 2448x + 12138
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			1.4
Root			$-12.4323$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} +432.433 q^{5} -113.601 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} +432.433 q^{5} -113.601 q^{7} +729.000 q^{9} -4053.44 q^{11} +12738.3 q^{13} +11675.7 q^{15} -37145.9 q^{17} -43691.5 q^{19} -3067.23 q^{21} -106161. q^{23} +108873. q^{25} +19683.0 q^{27} -374.331 q^{29} -176820. q^{31} -109443. q^{33} -49124.8 q^{35} +256492. q^{37} +343935. q^{39} -853490. q^{41} +137946. q^{43} +315243. q^{45} -1.05699e6 q^{47} -810638. q^{49} -1.00294e6 q^{51} -1.02151e6 q^{53} -1.75284e6 q^{55} -1.17967e6 q^{57} -639406. q^{59} +1.60722e6 q^{61} -82815.2 q^{63} +5.50847e6 q^{65} +1.63452e6 q^{67} -2.86635e6 q^{69} +1.31315e6 q^{71} -2.28706e6 q^{73} +2.93957e6 q^{75} +460475. q^{77} +7.00761e6 q^{79} +531441. q^{81} +5.53513e6 q^{83} -1.60631e7 q^{85} -10106.9 q^{87} +2.52067e6 q^{89} -1.44709e6 q^{91} -4.77413e6 q^{93} -1.88936e7 q^{95} +4.71034e6 q^{97} -2.95496e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9}+O(q^{10}) $ 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9	$ 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9} - 4496 q^{11} + 12840 q^{13} - 5184 q^{15} - 14952 q^{17} - 21504 q^{19} - 18360 q^{21} - 54992 q^{23} + 190732 q^{25} + 78732 q^{27} - 242384 q^{29} - 151432 q^{31} - 121392 q^{33} + 273984 q^{35} - 113288 q^{37} + 346680 q^{39} - 239176 q^{41} + 1495328 q^{43} - 139968 q^{45} - 772368 q^{47} - 1577100 q^{49} - 403704 q^{51} - 2389776 q^{53} - 2590080 q^{55} - 580608 q^{57} + 141232 q^{59} - 1231304 q^{61} - 495720 q^{63} - 1041024 q^{65} - 441392 q^{67} - 1484784 q^{69} - 1507504 q^{71} - 1516840 q^{73} + 5149764 q^{75} - 12340448 q^{77} - 9540936 q^{79} + 2125764 q^{81} - 4587600 q^{83} - 6382848 q^{85} - 6544368 q^{87} + 162376 q^{89} - 4681104 q^{91} - 4088664 q^{93} - 29221248 q^{95} + 2726760 q^{97} - 3277584 q^{99}+O(q^{100}) $ 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9 - 4496 * q^11 + 12840 * q^13 - 5184 * q^15 - 14952 * q^17 - 21504 * q^19 - 18360 * q^21 - 54992 * q^23 + 190732 * q^25 + 78732 * q^27 - 242384 * q^29 - 151432 * q^31 - 121392 * q^33 + 273984 * q^35 - 113288 * q^37 + 346680 * q^39 - 239176 * q^41 + 1495328 * q^43 - 139968 * q^45 - 772368 * q^47 - 1577100 * q^49 - 403704 * q^51 - 2389776 * q^53 - 2590080 * q^55 - 580608 * q^57 + 141232 * q^59 - 1231304 * q^61 - 495720 * q^63 - 1041024 * q^65 - 441392 * q^67 - 1484784 * q^69 - 1507504 * q^71 - 1516840 * q^73 + 5149764 * q^75 - 12340448 * q^77 - 9540936 * q^79 + 2125764 * q^81 - 4587600 * q^83 - 6382848 * q^85 - 6544368 * q^87 + 162376 * q^89 - 4681104 * q^91 - 4088664 * q^93 - 29221248 * q^95 + 2726760 * q^97 - 3277584 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	432.433	1.54712	0.773559	−	0.633724i	$-0.218475\pi$
$5$	432.433	1.54712	0.773559	+	0.633724i	$0.218475\pi$
$6$	0	0
$7$	−113.601	−0.125181	−0.0625906	−	0.998039i	$-0.519936\pi$
$7$	−113.601	−0.125181	−0.0625906	+	0.998039i	$0.519936\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−4053.44	−0.918226	−0.459113	−	0.888378i	$-0.651833\pi$
$11$	−4053.44	−0.918226	−0.459113	+	0.888378i	$0.651833\pi$
$12$	0	0
$13$	12738.3	1.60809	0.804046	−	0.594567i	$-0.202676\pi$
$13$	12738.3	1.60809	0.804046	+	0.594567i	$0.202676\pi$
$14$	0	0
$15$	11675.7	0.893229
$16$	0	0
$17$	−37145.9	−1.83375	−0.916873	−	0.399179i	$-0.869295\pi$
$17$	−37145.9	−1.83375	−0.916873	+	0.399179i	$0.869295\pi$
$18$	0	0
$19$	−43691.5	−1.46137	−0.730684	−	0.682716i	$-0.760798\pi$
$19$	−43691.5	−1.46137	−0.730684	+	0.682716i	$0.760798\pi$
$20$	0	0
$21$	−3067.23	−0.0722734
$22$	0	0
$23$	−106161.	−1.81936	−0.909678	−	0.415314i	$-0.863672\pi$
$23$	−106161.	−1.81936	−0.909678	+	0.415314i	$0.863672\pi$
$24$	0	0
$25$	108873.	1.39357
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−374.331	−0.00285011	−0.00142506	−	0.999999i	$-0.500454\pi$
$29$	−374.331	−0.00285011	−0.00142506	+	0.999999i	$0.500454\pi$
$30$	0	0
$31$	−176820.	−1.06602	−0.533009	−	0.846109i	$-0.678939\pi$
$31$	−176820.	−1.06602	−0.533009	+	0.846109i	$0.678939\pi$
$32$	0	0
$33$	−109443.	−0.530138
$34$	0	0
$35$	−49124.8	−0.193670
$36$	0	0
$37$	256492.	0.832469	0.416235	−	0.909257i	$-0.363349\pi$
$37$	256492.	0.832469	0.416235	+	0.909257i	$0.363349\pi$
$38$	0	0
$39$	343935.	0.928433
$40$	0	0
$41$	−853490.	−1.93399	−0.966996	−	0.254791i	$-0.917993\pi$
$41$	−853490.	−1.93399	−0.966996	+	0.254791i	$0.917993\pi$
$42$	0	0
$43$	137946.	0.264588	0.132294	−	0.991210i	$-0.457766\pi$
$43$	137946.	0.264588	0.132294	+	0.991210i	$0.457766\pi$
$44$	0	0
$45$	315243.	0.515706
$46$	0	0
$47$	−1.05699e6	−1.48501	−0.742504	−	0.669842i	$-0.766362\pi$
$47$	−1.05699e6	−1.48501	−0.742504	+	0.669842i	$0.766362\pi$
$48$	0	0
$49$	−810638.	−0.984330
$50$	0	0
$51$	−1.00294e6	−1.05871
$52$	0	0
$53$	−1.02151e6	−0.942489	−0.471245	−	0.882003i	$-0.656195\pi$
$53$	−1.02151e6	−0.942489	−0.471245	+	0.882003i	$0.656195\pi$
$54$	0	0
$55$	−1.75284e6	−1.42060
$56$	0	0
$57$	−1.17967e6	−0.843721
$58$	0	0
$59$	−639406.	−0.405317	−0.202659	−	0.979249i	$-0.564958\pi$
$59$	−639406.	−0.405317	−0.202659	+	0.979249i	$0.564958\pi$
$60$	0	0
$61$	1.60722e6	0.906609	0.453304	−	0.891356i	$-0.350245\pi$
$61$	1.60722e6	0.906609	0.453304	+	0.891356i	$0.350245\pi$
$62$	0	0
$63$	−82815.2	−0.0417271
$64$	0	0
$65$	5.50847e6	2.48791
$66$	0	0
$67$	1.63452e6	0.663940	0.331970	−	0.943290i	$-0.392287\pi$
$67$	1.63452e6	0.663940	0.331970	+	0.943290i	$0.392287\pi$
$68$	0	0
$69$	−2.86635e6	−1.05041
$70$	0	0
$71$	1.31315e6	0.435423	0.217711	−	0.976013i	$-0.430141\pi$
$71$	1.31315e6	0.435423	0.217711	+	0.976013i	$0.430141\pi$
$72$	0	0
$73$	−2.28706e6	−0.688093	−0.344047	−	0.938953i	$-0.611798\pi$
$73$	−2.28706e6	−0.688093	−0.344047	+	0.938953i	$0.611798\pi$
$74$	0	0
$75$	2.93957e6	0.804580
$76$	0	0
$77$	460475.	0.114945
$78$	0	0
$79$	7.00761e6	1.59910	0.799549	−	0.600600i	$-0.205072\pi$
$79$	7.00761e6	1.59910	0.799549	+	0.600600i	$0.205072\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.53513e6	1.06256	0.531281	−	0.847196i	$-0.321711\pi$
$83$	5.53513e6	1.06256	0.531281	+	0.847196i	$0.321711\pi$
$84$	0	0
$85$	−1.60631e7	−2.83702
$86$	0	0
$87$	−10106.9	−0.00164551
$88$	0	0
$89$	2.52067e6	0.379010	0.189505	−	0.981880i	$-0.439312\pi$
$89$	2.52067e6	0.379010	0.189505	+	0.981880i	$0.439312\pi$
$90$	0	0
$91$	−1.44709e6	−0.201303
$92$	0	0
$93$	−4.77413e6	−0.615466
$94$	0	0
$95$	−1.88936e7	−2.26091
$96$	0	0
$97$	4.71034e6	0.524024	0.262012	−	0.965065i	$-0.415614\pi$
$97$	4.71034e6	0.524024	0.262012	+	0.965065i	$0.415614\pi$
$98$	0	0
$99$	−2.95496e6	−0.306075
$100$	0	0
$101$	1.05183e7	1.01583	0.507914	−	0.861408i	$-0.330417\pi$
$101$	1.05183e7	1.01583	0.507914	+	0.861408i	$0.330417\pi$
$102$	0	0
$103$	−433447.	−0.0390846	−0.0195423	−	0.999809i	$-0.506221\pi$
$103$	−433447.	−0.0390846	−0.0195423	+	0.999809i	$0.506221\pi$
$104$	0	0
$105$	−1.32637e6	−0.111815
$106$	0	0
$107$	1.52861e7	1.20629	0.603147	−	0.797630i	$-0.293913\pi$
$107$	1.52861e7	1.20629	0.603147	+	0.797630i	$0.293913\pi$
$108$	0	0
$109$	−7.01462e6	−0.518813	−0.259407	−	0.965768i	$-0.583527\pi$
$109$	−7.01462e6	−0.518813	−0.259407	+	0.965768i	$0.583527\pi$
$110$	0	0
$111$	6.92529e6	0.480626
$112$	0	0
$113$	1.28161e7	0.835568	0.417784	−	0.908546i	$-0.362807\pi$
$113$	1.28161e7	0.835568	0.417784	+	0.908546i	$0.362807\pi$
$114$	0	0
$115$	−4.59075e7	−2.81476
$116$	0	0
$117$	9.28625e6	0.536031
$118$	0	0
$119$	4.21981e6	0.229551
$120$	0	0
$121$	−3.05679e6	−0.156862
$122$	0	0
$123$	−2.30442e7	−1.11659
$124$	0	0
$125$	1.32964e7	0.608905
$126$	0	0
$127$	−1.64110e7	−0.710921	−0.355460	−	0.934691i	$-0.615676\pi$
$127$	−1.64110e7	−0.710921	−0.355460	+	0.934691i	$0.615676\pi$
$128$	0	0
$129$	3.72455e6	0.152760
$130$	0	0
$131$	1.27598e7	0.495901	0.247951	−	0.968773i	$-0.420243\pi$
$131$	1.27598e7	0.495901	0.247951	+	0.968773i	$0.420243\pi$
$132$	0	0
$133$	4.96340e6	0.182936
$134$	0	0
$135$	8.51157e6	0.297743
$136$	0	0
$137$	3.38703e7	1.12538	0.562688	−	0.826670i	$-0.309767\pi$
$137$	3.38703e7	1.12538	0.562688	+	0.826670i	$0.309767\pi$
$138$	0	0
$139$	−7.27928e6	−0.229899	−0.114949	−	0.993371i	$-0.536671\pi$
$139$	−7.27928e6	−0.229899	−0.114949	+	0.993371i	$0.536671\pi$
$140$	0	0
$141$	−2.85388e7	−0.857370
$142$	0	0
$143$	−5.16341e7	−1.47659
$144$	0	0
$145$	−161873.	−0.00440946
$146$	0	0
$147$	−2.18872e7	−0.568303
$148$	0	0
$149$	−3.83434e6	−0.0949596	−0.0474798	−	0.998872i	$-0.515119\pi$
$149$	−3.83434e6	−0.0949596	−0.0474798	+	0.998872i	$0.515119\pi$
$150$	0	0
$151$	−5.99823e6	−0.141776	−0.0708882	−	0.997484i	$-0.522583\pi$
$151$	−5.99823e6	−0.141776	−0.0708882	+	0.997484i	$0.522583\pi$
$152$	0	0
$153$	−2.70793e7	−0.611249
$154$	0	0
$155$	−7.64626e7	−1.64926
$156$	0	0
$157$	−5.67736e6	−0.117084	−0.0585421	−	0.998285i	$-0.518645\pi$
$157$	−5.67736e6	−0.117084	−0.0585421	+	0.998285i	$0.518645\pi$
$158$	0	0
$159$	−2.75807e7	−0.544146
$160$	0	0
$161$	1.20600e7	0.227749
$162$	0	0
$163$	−6.74490e7	−1.21989	−0.609943	−	0.792445i	$-0.708808\pi$
$163$	−6.74490e7	−1.21989	−0.609943	+	0.792445i	$0.708808\pi$
$164$	0	0
$165$	−4.73267e7	−0.820186
$166$	0	0
$167$	−9.43038e7	−1.56683	−0.783414	−	0.621500i	$-0.786524\pi$
$167$	−9.43038e7	−1.56683	−0.783414	+	0.621500i	$0.786524\pi$
$168$	0	0
$169$	9.95167e7	1.58596
$170$	0	0
$171$	−3.18511e7	−0.487122
$172$	0	0
$173$	−6.48563e7	−0.952338	−0.476169	−	0.879354i	$-0.657975\pi$
$173$	−6.48563e7	−0.952338	−0.476169	+	0.879354i	$0.657975\pi$
$174$	0	0
$175$	−1.23681e7	−0.174449
$176$	0	0
$177$	−1.72640e7	−0.234010
$178$	0	0
$179$	−9.94129e7	−1.29556	−0.647779	−	0.761828i	$-0.724302\pi$
$179$	−9.94129e7	−1.29556	−0.647779	+	0.761828i	$0.724302\pi$
$180$	0	0
$181$	3.41221e7	0.427721	0.213861	−	0.976864i	$-0.431396\pi$
$181$	3.41221e7	0.427721	0.213861	+	0.976864i	$0.431396\pi$
$182$	0	0
$183$	4.33948e7	0.523431
$184$	0	0
$185$	1.10916e8	1.28793
$186$	0	0
$187$	1.50569e8	1.68379
$188$	0	0
$189$	−2.23601e6	−0.0240911
$190$	0	0
$191$	6.84169e7	0.710471	0.355236	−	0.934777i	$-0.384401\pi$
$191$	6.84169e7	0.710471	0.355236	+	0.934777i	$0.384401\pi$
$192$	0	0
$193$	6.69131e7	0.669978	0.334989	−	0.942222i	$-0.391267\pi$
$193$	6.69131e7	0.669978	0.334989	+	0.942222i	$0.391267\pi$
$194$	0	0
$195$	1.48729e8	1.43639
$196$	0	0
$197$	3.44534e7	0.321070	0.160535	−	0.987030i	$-0.448678\pi$
$197$	3.44534e7	0.321070	0.160535	+	0.987030i	$0.448678\pi$
$198$	0	0
$199$	−8.85955e7	−0.796941	−0.398470	−	0.917181i	$-0.630459\pi$
$199$	−8.85955e7	−0.796941	−0.398470	+	0.917181i	$0.630459\pi$
$200$	0	0
$201$	4.41321e7	0.383326
$202$	0	0
$203$	42524.3	0.000356781 0
$204$	0	0
$205$	−3.69077e8	−2.99211
$206$	0	0
$207$	−7.73914e7	−0.606452
$208$	0	0
$209$	1.77101e8	1.34186
$210$	0	0
$211$	−1.06986e8	−0.784037	−0.392019	−	0.919957i	$-0.628223\pi$
$211$	−1.06986e8	−0.784037	−0.392019	+	0.919957i	$0.628223\pi$
$212$	0	0
$213$	3.54551e7	0.251391
$214$	0	0
$215$	5.96525e7	0.409349
$216$	0	0
$217$	2.00869e7	0.133445
$218$	0	0
$219$	−6.17506e7	−0.397271
$220$	0	0
$221$	−4.73176e8	−2.94883
$222$	0	0
$223$	9.90851e7	0.598331	0.299165	−	0.954201i	$-0.403292\pi$
$223$	9.90851e7	0.598331	0.299165	+	0.954201i	$0.403292\pi$
$224$	0	0
$225$	7.93684e7	0.464525
$226$	0	0
$227$	4.42843e7	0.251281	0.125640	−	0.992076i	$-0.459901\pi$
$227$	4.42843e7	0.251281	0.125640	+	0.992076i	$0.459901\pi$
$228$	0	0
$229$	−3.12338e7	−0.171870	−0.0859350	−	0.996301i	$-0.527388\pi$
$229$	−3.12338e7	−0.171870	−0.0859350	+	0.996301i	$0.527388\pi$
$230$	0	0
$231$	1.24328e7	0.0663633
$232$	0	0
$233$	−2.59987e8	−1.34650	−0.673249	−	0.739416i	$-0.735102\pi$
$233$	−2.59987e8	−1.34650	−0.673249	+	0.739416i	$0.735102\pi$
$234$	0	0
$235$	−4.57077e8	−2.29748
$236$	0	0
$237$	1.89206e8	0.923240
$238$	0	0
$239$	−4.10330e8	−1.94420	−0.972099	−	0.234570i	$-0.924632\pi$
$239$	−4.10330e8	−1.94420	−0.972099	+	0.234570i	$0.924632\pi$
$240$	0	0
$241$	3.23104e8	1.48690	0.743451	−	0.668790i	$-0.233188\pi$
$241$	3.23104e8	1.48690	0.743451	+	0.668790i	$0.233188\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−3.50546e8	−1.52287
$246$	0	0
$247$	−5.56557e8	−2.35001
$248$	0	0
$249$	1.49448e8	0.613470
$250$	0	0
$251$	1.99561e7	0.0796560	0.0398280	−	0.999207i	$-0.487319\pi$
$251$	1.99561e7	0.0796560	0.0398280	+	0.999207i	$0.487319\pi$
$252$	0	0
$253$	4.30318e8	1.67058
$254$	0	0
$255$	−4.33703e8	−1.63796
$256$	0	0
$257$	−2.79734e8	−1.02797	−0.513984	−	0.857800i	$-0.671831\pi$
$257$	−2.79734e8	−1.02797	−0.513984	+	0.857800i	$0.671831\pi$
$258$	0	0
$259$	−2.91378e7	−0.104210
$260$	0	0
$261$	−272887.	−0.000950038 0
$262$	0	0
$263$	5.96019e7	0.202030	0.101015	−	0.994885i	$-0.467791\pi$
$263$	5.96019e7	0.202030	0.101015	+	0.994885i	$0.467791\pi$
$264$	0	0
$265$	−4.41734e8	−1.45814
$266$	0	0
$267$	6.80580e7	0.218822
$268$	0	0
$269$	3.68116e8	1.15306	0.576530	−	0.817076i	$-0.304406\pi$
$269$	3.68116e8	1.15306	0.576530	+	0.817076i	$0.304406\pi$
$270$	0	0
$271$	3.94782e8	1.20494	0.602470	−	0.798142i	$-0.294183\pi$
$271$	3.94782e8	1.20494	0.602470	+	0.798142i	$0.294183\pi$
$272$	0	0
$273$	−3.90714e7	−0.116222
$274$	0	0
$275$	−4.41310e8	−1.27962
$276$	0	0
$277$	5.76998e6	0.0163116	0.00815578	−	0.999967i	$-0.497404\pi$
$277$	5.76998e6	0.0163116	0.00815578	+	0.999967i	$0.497404\pi$
$278$	0	0
$279$	−1.28902e8	−0.355339
$280$	0	0
$281$	−6.16673e8	−1.65799	−0.828996	−	0.559254i	$-0.811088\pi$
$281$	−6.16673e8	−1.65799	−0.828996	+	0.559254i	$0.811088\pi$
$282$	0	0
$283$	5.71460e8	1.49877	0.749383	−	0.662137i	$-0.230350\pi$
$283$	5.71460e8	1.49877	0.749383	+	0.662137i	$0.230350\pi$
$284$	0	0
$285$	−5.10128e8	−1.30534
$286$	0	0
$287$	9.69573e7	0.242100
$288$	0	0
$289$	9.69476e8	2.36263
$290$	0	0
$291$	1.27179e8	0.302546
$292$	0	0
$293$	1.62023e7	0.0376304	0.0188152	−	0.999823i	$-0.494011\pi$
$293$	1.62023e7	0.0376304	0.0188152	+	0.999823i	$0.494011\pi$
$294$	0	0
$295$	−2.76500e8	−0.627073
$296$	0	0
$297$	−7.97839e7	−0.176713
$298$	0	0
$299$	−1.35232e9	−2.92569
$300$	0	0
$301$	−1.56709e7	−0.0331215
$302$	0	0
$303$	2.83994e8	0.586488
$304$	0	0
$305$	6.95013e8	1.40263
$306$	0	0
$307$	−7.76916e8	−1.53246	−0.766232	−	0.642565i	$-0.777870\pi$
$307$	−7.76916e8	−1.53246	−0.766232	+	0.642565i	$0.777870\pi$
$308$	0	0
$309$	−1.17031e7	−0.0225655
$310$	0	0
$311$	5.53634e8	1.04367	0.521833	−	0.853047i	$-0.325248\pi$
$311$	5.53634e8	1.04367	0.521833	+	0.853047i	$0.325248\pi$
$312$	0	0
$313$	4.23238e7	0.0780153	0.0390077	−	0.999239i	$-0.487580\pi$
$313$	4.23238e7	0.0780153	0.0390077	+	0.999239i	$0.487580\pi$
$314$	0	0
$315$	−3.58120e7	−0.0645567
$316$	0	0
$317$	−4.97720e8	−0.877561	−0.438781	−	0.898594i	$-0.644589\pi$
$317$	−4.97720e8	−0.877561	−0.438781	+	0.898594i	$0.644589\pi$
$318$	0	0
$319$	1.51733e6	0.00261705
$320$	0	0
$321$	4.12724e8	0.696454
$322$	0	0
$323$	1.62296e9	2.67978
$324$	0	0
$325$	1.38686e9	2.24100
$326$	0	0
$327$	−1.89395e8	−0.299537
$328$	0	0
$329$	1.20075e8	0.185895
$330$	0	0
$331$	3.63918e8	0.551576	0.275788	−	0.961218i	$-0.411061\pi$
$331$	3.63918e8	0.551576	0.275788	+	0.961218i	$0.411061\pi$
$332$	0	0
$333$	1.86983e8	0.277490
$334$	0	0
$335$	7.06820e8	1.02719
$336$	0	0
$337$	−3.40846e8	−0.485124	−0.242562	−	0.970136i	$-0.577988\pi$
$337$	−3.40846e8	−0.485124	−0.242562	+	0.970136i	$0.577988\pi$
$338$	0	0
$339$	3.46035e8	0.482415
$340$	0	0
$341$	7.16728e8	0.978845
$342$	0	0
$343$	1.85645e8	0.248401
$344$	0	0
$345$	−1.23950e9	−1.62510
$346$	0	0
$347$	5.28966e8	0.679634	0.339817	−	0.940492i	$-0.389635\pi$
$347$	5.28966e8	0.679634	0.339817	+	0.940492i	$0.389635\pi$
$348$	0	0
$349$	−8.70598e8	−1.09630	−0.548149	−	0.836380i	$-0.684667\pi$
$349$	−8.70598e8	−1.09630	−0.548149	+	0.836380i	$0.684667\pi$
$350$	0	0
$351$	2.50729e8	0.309478
$352$	0	0
$353$	−2.81501e8	−0.340618	−0.170309	−	0.985391i	$-0.554477\pi$
$353$	−2.81501e8	−0.340618	−0.170309	+	0.985391i	$0.554477\pi$
$354$	0	0
$355$	5.67850e8	0.673650
$356$	0	0
$357$	1.13935e8	0.132531
$358$	0	0
$359$	−2.81915e8	−0.321580	−0.160790	−	0.986989i	$-0.551404\pi$
$359$	−2.81915e8	−0.321580	−0.160790	+	0.986989i	$0.551404\pi$
$360$	0	0
$361$	1.01508e9	1.13559
$362$	0	0
$363$	−8.25333e7	−0.0905641
$364$	0	0
$365$	−9.88999e8	−1.06456
$366$	0	0
$367$	9.40385e8	0.993058	0.496529	−	0.868020i	$-0.334608\pi$
$367$	9.40385e8	0.993058	0.496529	+	0.868020i	$0.334608\pi$
$368$	0	0
$369$	−6.22194e8	−0.644664
$370$	0	0
$371$	1.16044e8	0.117982
$372$	0	0
$373$	4.21483e8	0.420532	0.210266	−	0.977644i	$-0.432567\pi$
$373$	4.21483e8	0.420532	0.210266	+	0.977644i	$0.432567\pi$
$374$	0	0
$375$	3.59003e8	0.351552
$376$	0	0
$377$	−4.76835e6	−0.00458325
$378$	0	0
$379$	−9.02613e8	−0.851656	−0.425828	−	0.904804i	$-0.640017\pi$
$379$	−9.02613e8	−0.851656	−0.425828	+	0.904804i	$0.640017\pi$
$380$	0	0
$381$	−4.43096e8	−0.410450
$382$	0	0
$383$	−1.31601e9	−1.19691	−0.598456	−	0.801156i	$-0.704219\pi$
$383$	−1.31601e9	−1.19691	−0.598456	+	0.801156i	$0.704219\pi$
$384$	0	0
$385$	1.99124e8	0.177833
$386$	0	0
$387$	1.00563e8	0.0881961
$388$	0	0
$389$	8.43615e8	0.726642	0.363321	−	0.931664i	$-0.381643\pi$
$389$	8.43615e8	0.726642	0.363321	+	0.931664i	$0.381643\pi$
$390$	0	0
$391$	3.94344e9	3.33624
$392$	0	0
$393$	3.44515e8	0.286309
$394$	0	0
$395$	3.03032e9	2.47399
$396$	0	0
$397$	3.65135e8	0.292878	0.146439	−	0.989220i	$-0.453219\pi$
$397$	3.65135e8	0.292878	0.146439	+	0.989220i	$0.453219\pi$
$398$	0	0
$399$	1.34012e8	0.105618
$400$	0	0
$401$	1.58858e9	1.23028	0.615139	−	0.788419i	$-0.289100\pi$
$401$	1.58858e9	1.23028	0.615139	+	0.788419i	$0.289100\pi$
$402$	0	0
$403$	−2.25239e9	−1.71426
$404$	0	0
$405$	2.29812e8	0.171902
$406$	0	0
$407$	−1.03968e9	−0.764395
$408$	0	0
$409$	−1.58379e9	−1.14463	−0.572315	−	0.820034i	$-0.693954\pi$
$409$	−1.58379e9	−1.14463	−0.572315	+	0.820034i	$0.693954\pi$
$410$	0	0
$411$	9.14499e8	0.649736
$412$	0	0
$413$	7.26372e7	0.0507381
$414$	0	0
$415$	2.39357e9	1.64391
$416$	0	0
$417$	−1.96541e8	−0.132732
$418$	0	0
$419$	2.44460e8	0.162353	0.0811763	−	0.996700i	$-0.474132\pi$
$419$	2.44460e8	0.162353	0.0811763	+	0.996700i	$0.474132\pi$
$420$	0	0
$421$	2.73275e9	1.78489	0.892446	−	0.451154i	$-0.148987\pi$
$421$	2.73275e9	1.78489	0.892446	+	0.451154i	$0.148987\pi$
$422$	0	0
$423$	−7.70546e8	−0.495003
$424$	0	0
$425$	−4.04418e9	−2.55546
$426$	0	0
$427$	−1.82581e8	−0.113490
$428$	0	0
$429$	−1.39412e9	−0.852511
$430$	0	0
$431$	−2.09606e9	−1.26105	−0.630527	−	0.776167i	$-0.717161\pi$
$431$	−2.09606e9	−1.26105	−0.630527	+	0.776167i	$0.717161\pi$
$432$	0	0
$433$	−1.61587e9	−0.956533	−0.478266	−	0.878215i	$-0.658735\pi$
$433$	−1.61587e9	−0.956533	−0.478266	+	0.878215i	$0.658735\pi$
$434$	0	0
$435$	−4.37056e6	−0.00254580
$436$	0	0
$437$	4.63834e9	2.65875
$438$	0	0
$439$	4.01819e7	0.0226676	0.0113338	−	0.999936i	$-0.496392\pi$
$439$	4.01819e7	0.0226676	0.0113338	+	0.999936i	$0.496392\pi$
$440$	0	0
$441$	−5.90955e8	−0.328110
$442$	0	0
$443$	2.59996e9	1.42087	0.710434	−	0.703764i	$-0.248499\pi$
$443$	2.59996e9	1.42087	0.710434	+	0.703764i	$0.248499\pi$
$444$	0	0
$445$	1.09002e9	0.586373
$446$	0	0
$447$	−1.03527e8	−0.0548250
$448$	0	0
$449$	−1.16295e9	−0.606317	−0.303158	−	0.952940i	$-0.598041\pi$
$449$	−1.16295e9	−0.606317	−0.303158	+	0.952940i	$0.598041\pi$
$450$	0	0
$451$	3.45957e9	1.77584
$452$	0	0
$453$	−1.61952e8	−0.0818546
$454$	0	0
$455$	−6.25768e8	−0.311439
$456$	0	0
$457$	−3.77081e9	−1.84811	−0.924056	−	0.382257i	$-0.875147\pi$
$457$	−3.77081e9	−1.84811	−0.924056	+	0.382257i	$0.875147\pi$
$458$	0	0
$459$	−7.31142e8	−0.352905
$460$	0	0
$461$	−4.47538e8	−0.212753	−0.106377	−	0.994326i	$-0.533925\pi$
$461$	−4.47538e8	−0.212753	−0.106377	+	0.994326i	$0.533925\pi$
$462$	0	0
$463$	−2.40368e9	−1.12549	−0.562746	−	0.826630i	$-0.690255\pi$
$463$	−2.40368e9	−1.12549	−0.562746	+	0.826630i	$0.690255\pi$
$464$	0	0
$465$	−2.06449e9	−0.952199
$466$	0	0
$467$	−1.53548e9	−0.697646	−0.348823	−	0.937189i	$-0.613419\pi$
$467$	−1.53548e9	−0.697646	−0.348823	+	0.937189i	$0.613419\pi$
$468$	0	0
$469$	−1.85683e8	−0.0831128
$470$	0	0
$471$	−1.53289e8	−0.0675986
$472$	0	0
$473$	−5.59158e8	−0.242952
$474$	0	0
$475$	−4.75682e9	−2.03652
$476$	0	0
$477$	−7.44680e8	−0.314163
$478$	0	0
$479$	2.27435e9	0.945545	0.472773	−	0.881184i	$-0.343253\pi$
$479$	2.27435e9	0.945545	0.472773	+	0.881184i	$0.343253\pi$
$480$	0	0
$481$	3.26728e9	1.33869
$482$	0	0
$483$	3.25620e8	0.131491
$484$	0	0
$485$	2.03691e9	0.810728
$486$	0	0
$487$	9.46655e8	0.371399	0.185699	−	0.982607i	$-0.440545\pi$
$487$	9.46655e8	0.371399	0.185699	+	0.982607i	$0.440545\pi$
$488$	0	0
$489$	−1.82112e9	−0.704301
$490$	0	0
$491$	3.30221e9	1.25898	0.629491	−	0.777008i	$-0.283263\pi$
$491$	3.30221e9	1.25898	0.629491	+	0.777008i	$0.283263\pi$
$492$	0	0
$493$	1.39048e7	0.00522639
$494$	0	0
$495$	−1.27782e9	−0.473534
$496$	0	0
$497$	−1.49176e8	−0.0545067
$498$	0	0
$499$	3.60745e9	1.29972	0.649858	−	0.760055i	$-0.274828\pi$
$499$	3.60745e9	1.29972	0.649858	+	0.760055i	$0.274828\pi$
$500$	0	0
$501$	−2.54620e9	−0.904609
$502$	0	0
$503$	1.59124e9	0.557503	0.278751	−	0.960363i	$-0.410079\pi$
$503$	1.59124e9	0.557503	0.278751	+	0.960363i	$0.410079\pi$
$504$	0	0
$505$	4.54845e9	1.57160
$506$	0	0
$507$	2.68695e9	0.915655
$508$	0	0
$509$	−1.02545e9	−0.344668	−0.172334	−	0.985039i	$-0.555131\pi$
$509$	−1.02545e9	−0.344668	−0.172334	+	0.985039i	$0.555131\pi$
$510$	0	0
$511$	2.59812e8	0.0861363
$512$	0	0
$513$	−8.59980e8	−0.281240
$514$	0	0
$515$	−1.87437e8	−0.0604685
$516$	0	0
$517$	4.28445e9	1.36357
$518$	0	0
$519$	−1.75112e9	−0.549832
$520$	0	0
$521$	8.27831e8	0.256454	0.128227	−	0.991745i	$-0.459071\pi$
$521$	8.27831e8	0.256454	0.128227	+	0.991745i	$0.459071\pi$
$522$	0	0
$523$	−1.35030e9	−0.412737	−0.206368	−	0.978474i	$-0.566165\pi$
$523$	−1.35030e9	−0.412737	−0.206368	+	0.978474i	$0.566165\pi$
$524$	0	0
$525$	−3.33938e8	−0.100718
$526$	0	0
$527$	6.56812e9	1.95481
$528$	0	0
$529$	7.86534e9	2.31006
$530$	0	0
$531$	−4.66127e8	−0.135106
$532$	0	0
$533$	−1.08720e10	−3.11004
$534$	0	0
$535$	6.61020e9	1.86628
$536$	0	0
$537$	−2.68415e9	−0.747991
$538$	0	0
$539$	3.28587e9	0.903837
$540$	0	0
$541$	8.86725e8	0.240768	0.120384	−	0.992727i	$-0.461587\pi$
$541$	8.86725e8	0.240768	0.120384	+	0.992727i	$0.461587\pi$
$542$	0	0
$543$	9.21297e8	0.246945
$544$	0	0
$545$	−3.03335e9	−0.802666
$546$	0	0
$547$	−5.01911e9	−1.31121	−0.655603	−	0.755106i	$-0.727585\pi$
$547$	−5.01911e9	−1.31121	−0.655603	+	0.755106i	$0.727585\pi$
$548$	0	0
$549$	1.17166e9	0.302203
$550$	0	0
$551$	1.63551e7	0.00416506
$552$	0	0
$553$	−7.96072e8	−0.200177
$554$	0	0
$555$	2.99472e9	0.743586
$556$	0	0
$557$	−1.03603e9	−0.254026	−0.127013	−	0.991901i	$-0.540539\pi$
$557$	−1.03603e9	−0.254026	−0.127013	+	0.991901i	$0.540539\pi$
$558$	0	0
$559$	1.75721e9	0.425483
$560$	0	0
$561$	4.06535e9	0.972138
$562$	0	0
$563$	3.16795e9	0.748168	0.374084	−	0.927395i	$-0.377957\pi$
$563$	3.16795e9	0.748168	0.374084	+	0.927395i	$0.377957\pi$
$564$	0	0
$565$	5.54210e9	1.29272
$566$	0	0
$567$	−6.03723e7	−0.0139090
$568$	0	0
$569$	−7.99353e9	−1.81905	−0.909527	−	0.415645i	$-0.863556\pi$
$569$	−7.99353e9	−1.81905	−0.909527	+	0.415645i	$0.863556\pi$
$570$	0	0
$571$	−2.13771e9	−0.480532	−0.240266	−	0.970707i	$-0.577235\pi$
$571$	−2.13771e9	−0.480532	−0.240266	+	0.970707i	$0.577235\pi$
$572$	0	0
$573$	1.84726e9	0.410191
$574$	0	0
$575$	−1.15581e10	−2.53541
$576$	0	0
$577$	7.47985e9	1.62098	0.810490	−	0.585753i	$-0.199201\pi$
$577$	7.47985e9	1.62098	0.810490	+	0.585753i	$0.199201\pi$
$578$	0	0
$579$	1.80665e9	0.386812
$580$	0	0
$581$	−6.28796e8	−0.133013
$582$	0	0
$583$	4.14062e9	0.865418
$584$	0	0
$585$	4.01568e9	0.829303
$586$	0	0
$587$	−5.69132e9	−1.16139	−0.580697	−	0.814120i	$-0.697220\pi$
$587$	−5.69132e9	−1.16139	−0.580697	+	0.814120i	$0.697220\pi$
$588$	0	0
$589$	7.72552e9	1.55784
$590$	0	0
$591$	9.30241e8	0.185370
$592$	0	0
$593$	2.47655e9	0.487703	0.243852	−	0.969813i	$-0.421589\pi$
$593$	2.47655e9	0.487703	0.243852	+	0.969813i	$0.421589\pi$
$594$	0	0
$595$	1.82478e9	0.355142
$596$	0	0
$597$	−2.39208e9	−0.460114
$598$	0	0
$599$	−2.30462e9	−0.438133	−0.219066	−	0.975710i	$-0.570301\pi$
$599$	−2.30462e9	−0.438133	−0.219066	+	0.975710i	$0.570301\pi$
$600$	0	0
$601$	1.23573e9	0.232201	0.116101	−	0.993237i	$-0.462960\pi$
$601$	1.23573e9	0.232201	0.116101	+	0.993237i	$0.462960\pi$
$602$	0	0
$603$	1.19157e9	0.221313
$604$	0	0
$605$	−1.32186e9	−0.242684
$606$	0	0
$607$	2.14170e9	0.388686	0.194343	−	0.980934i	$-0.437743\pi$
$607$	2.14170e9	0.388686	0.194343	+	0.980934i	$0.437743\pi$
$608$	0	0
$609$	1.14816e6	0.000205987 0
$610$	0	0
$611$	−1.34643e10	−2.38803
$612$	0	0
$613$	7.90385e7	0.0138588	0.00692942	−	0.999976i	$-0.497794\pi$
$613$	7.90385e7	0.0138588	0.00692942	+	0.999976i	$0.497794\pi$
$614$	0	0
$615$	−9.96507e9	−1.72750
$616$	0	0
$617$	1.19571e9	0.204940	0.102470	−	0.994736i	$-0.467325\pi$
$617$	1.19571e9	0.204940	0.102470	+	0.994736i	$0.467325\pi$
$618$	0	0
$619$	−6.46093e9	−1.09491	−0.547455	−	0.836835i	$-0.684403\pi$
$619$	−6.46093e9	−1.09491	−0.547455	+	0.836835i	$0.684403\pi$
$620$	0	0
$621$	−2.08957e9	−0.350135
$622$	0	0
$623$	−2.86351e8	−0.0474449
$624$	0	0
$625$	−2.75590e9	−0.451526
$626$	0	0
$627$	4.78172e9	0.774726
$628$	0	0
$629$	−9.52763e9	−1.52654
$630$	0	0
$631$	−6.83737e9	−1.08339	−0.541697	−	0.840574i	$-0.682218\pi$
$631$	−6.83737e9	−1.08339	−0.541697	+	0.840574i	$0.682218\pi$
$632$	0	0
$633$	−2.88861e9	−0.452664
$634$	0	0
$635$	−7.09664e9	−1.09988
$636$	0	0
$637$	−1.03262e10	−1.58289
$638$	0	0
$639$	9.57288e8	0.145141
$640$	0	0
$641$	3.47453e9	0.521067	0.260533	−	0.965465i	$-0.416102\pi$
$641$	3.47453e9	0.521067	0.260533	+	0.965465i	$0.416102\pi$
$642$	0	0
$643$	−1.11949e10	−1.66066	−0.830329	−	0.557274i	$-0.811847\pi$
$643$	−1.11949e10	−1.66066	−0.830329	+	0.557274i	$0.811847\pi$
$644$	0	0
$645$	1.61062e9	0.236338
$646$	0	0
$647$	2.53754e9	0.368339	0.184170	−	0.982894i	$-0.441040\pi$
$647$	2.53754e9	0.368339	0.184170	+	0.982894i	$0.441040\pi$
$648$	0	0
$649$	2.59180e9	0.372173
$650$	0	0
$651$	5.42346e8	0.0770448
$652$	0	0
$653$	−9.03386e9	−1.26963	−0.634815	−	0.772664i	$-0.718924\pi$
$653$	−9.03386e9	−1.26963	−0.634815	+	0.772664i	$0.718924\pi$
$654$	0	0
$655$	5.51777e9	0.767218
$656$	0	0
$657$	−1.66727e9	−0.229364
$658$	0	0
$659$	7.45165e9	1.01427	0.507135	−	0.861867i	$-0.330705\pi$
$659$	7.45165e9	1.01427	0.507135	+	0.861867i	$0.330705\pi$
$660$	0	0
$661$	6.46685e9	0.870939	0.435470	−	0.900203i	$-0.356582\pi$
$661$	6.46685e9	0.870939	0.435470	+	0.900203i	$0.356582\pi$
$662$	0	0
$663$	−1.27758e10	−1.70251
$664$	0	0
$665$	2.14634e9	0.283023
$666$	0	0
$667$	3.97393e7	0.00518537
$668$	0	0
$669$	2.67530e9	0.345446
$670$	0	0
$671$	−6.51476e9	−0.832472
$672$	0	0
$673$	−2.88478e9	−0.364804	−0.182402	−	0.983224i	$-0.558387\pi$
$673$	−2.88478e9	−0.364804	−0.182402	+	0.983224i	$0.558387\pi$
$674$	0	0
$675$	2.14295e9	0.268193
$676$	0	0
$677$	1.13785e10	1.40937	0.704686	−	0.709519i	$-0.251088\pi$
$677$	1.13785e10	1.40937	0.704686	+	0.709519i	$0.251088\pi$
$678$	0	0
$679$	−5.35100e8	−0.0655980
$680$	0	0
$681$	1.19568e9	0.145077
$682$	0	0
$683$	4.64556e9	0.557912	0.278956	−	0.960304i	$-0.410012\pi$
$683$	4.64556e9	0.557912	0.278956	+	0.960304i	$0.410012\pi$
$684$	0	0
$685$	1.46466e10	1.74109
$686$	0	0
$687$	−8.43312e8	−0.0992292
$688$	0	0
$689$	−1.30123e10	−1.51561
$690$	0	0
$691$	−6.10377e9	−0.703760	−0.351880	−	0.936045i	$-0.614458\pi$
$691$	−6.10377e9	−0.703760	−0.351880	+	0.936045i	$0.614458\pi$
$692$	0	0
$693$	3.35686e8	0.0383149
$694$	0	0
$695$	−3.14780e9	−0.355681
$696$	0	0
$697$	3.17036e10	3.54645
$698$	0	0
$699$	−7.01965e9	−0.777401
$700$	0	0
$701$	−7.94877e9	−0.871539	−0.435770	−	0.900058i	$-0.643524\pi$
$701$	−7.94877e9	−0.871539	−0.435770	+	0.900058i	$0.643524\pi$
$702$	0	0
$703$	−1.12065e10	−1.21654
$704$	0	0
$705$	−1.23411e10	−1.32645
$706$	0	0
$707$	−1.19489e9	−0.127163
$708$	0	0
$709$	−1.41207e10	−1.48797	−0.743986	−	0.668196i	$-0.767067\pi$
$709$	−1.41207e10	−1.48797	−0.743986	+	0.668196i	$0.767067\pi$
$710$	0	0
$711$	5.10855e9	0.533033
$712$	0	0
$713$	1.87714e10	1.93947
$714$	0	0
$715$	−2.23283e10	−2.28446
$716$	0	0
$717$	−1.10789e10	−1.12248
$718$	0	0
$719$	−6.37203e9	−0.639333	−0.319666	−	0.947530i	$-0.603571\pi$
$719$	−6.37203e9	−0.639333	−0.319666	+	0.947530i	$0.603571\pi$
$720$	0	0
$721$	4.92401e7	0.00489266
$722$	0	0
$723$	8.72380e9	0.858463
$724$	0	0
$725$	−4.07545e7	−0.00397184
$726$	0	0
$727$	−9.33015e9	−0.900572	−0.450286	−	0.892884i	$-0.648678\pi$
$727$	−9.33015e9	−0.900572	−0.450286	+	0.892884i	$0.648678\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−5.12414e9	−0.485188
$732$	0	0
$733$	3.25394e9	0.305173	0.152587	−	0.988290i	$-0.451240\pi$
$733$	3.25394e9	0.305173	0.152587	+	0.988290i	$0.451240\pi$
$734$	0	0
$735$	−9.46475e9	−0.879232
$736$	0	0
$737$	−6.62543e9	−0.609646
$738$	0	0
$739$	−2.37031e9	−0.216047	−0.108024	−	0.994148i	$-0.534452\pi$
$739$	−2.37031e9	−0.216047	−0.108024	+	0.994148i	$0.534452\pi$
$740$	0	0
$741$	−1.50270e10	−1.35678
$742$	0	0
$743$	1.66748e9	0.149142	0.0745711	−	0.997216i	$-0.476241\pi$
$743$	1.66748e9	0.149142	0.0745711	+	0.997216i	$0.476241\pi$
$744$	0	0
$745$	−1.65809e9	−0.146914
$746$	0	0
$747$	4.03511e9	0.354187
$748$	0	0
$749$	−1.73652e9	−0.151005
$750$	0	0
$751$	1.84380e10	1.58845	0.794224	−	0.607625i	$-0.207877\pi$
$751$	1.84380e10	1.58845	0.794224	+	0.607625i	$0.207877\pi$
$752$	0	0
$753$	5.38816e8	0.0459894
$754$	0	0
$755$	−2.59383e9	−0.219345
$756$	0	0
$757$	−1.43696e10	−1.20395	−0.601974	−	0.798515i	$-0.705619\pi$
$757$	−1.43696e10	−1.20395	−0.601974	+	0.798515i	$0.705619\pi$
$758$	0	0
$759$	1.16186e10	0.964510
$760$	0	0
$761$	−1.38939e10	−1.14282	−0.571409	−	0.820665i	$-0.693603\pi$
$761$	−1.38939e10	−1.14282	−0.571409	+	0.820665i	$0.693603\pi$
$762$	0	0
$763$	7.96868e8	0.0649457
$764$	0	0
$765$	−1.17100e10	−0.945674
$766$	0	0
$767$	−8.14497e9	−0.651787
$768$	0	0
$769$	1.34109e10	1.06345	0.531723	−	0.846918i	$-0.321545\pi$
$769$	1.34109e10	1.06345	0.531723	+	0.846918i	$0.321545\pi$
$770$	0	0
$771$	−7.55282e9	−0.593497
$772$	0	0
$773$	−2.16390e10	−1.68503	−0.842516	−	0.538672i	$-0.818926\pi$
$773$	−2.16390e10	−1.68503	−0.842516	+	0.538672i	$0.818926\pi$
$774$	0	0
$775$	−1.92509e10	−1.48558
$776$	0	0
$777$	−7.86720e8	−0.0601654
$778$	0	0
$779$	3.72902e10	2.82627
$780$	0	0
$781$	−5.32279e9	−0.399816
$782$	0	0
$783$	−7.36795e6	−0.000548505 0
$784$	0	0
$785$	−2.45508e9	−0.181143
$786$	0	0
$787$	2.64777e10	1.93629	0.968143	−	0.250399i	$-0.0805618\pi$
$787$	2.64777e10	1.93629	0.968143	+	0.250399i	$0.0805618\pi$
$788$	0	0
$789$	1.60925e9	0.116642
$790$	0	0
$791$	−1.45592e9	−0.104597
$792$	0	0
$793$	2.04733e10	1.45791
$794$	0	0
$795$	−1.19268e10	−0.841859
$796$	0	0
$797$	−2.13742e9	−0.149550	−0.0747749	−	0.997200i	$-0.523824\pi$
$797$	−2.13742e9	−0.149550	−0.0747749	+	0.997200i	$0.523824\pi$
$798$	0	0
$799$	3.92628e10	2.72313
$800$	0	0
$801$	1.83757e9	0.126337
$802$	0	0
$803$	9.27046e9	0.631825
$804$	0	0
$805$	5.21514e9	0.352355
$806$	0	0
$807$	9.93914e9	0.665719
$808$	0	0
$809$	1.20167e10	0.797931	0.398966	−	0.916966i	$-0.369369\pi$
$809$	1.20167e10	0.797931	0.398966	+	0.916966i	$0.369369\pi$
$810$	0	0
$811$	2.11984e10	1.39550	0.697751	−	0.716340i	$-0.254184\pi$
$811$	2.11984e10	1.39550	0.697751	+	0.716340i	$0.254184\pi$
$812$	0	0
$813$	1.06591e10	0.695672
$814$	0	0
$815$	−2.91672e10	−1.88731
$816$	0	0
$817$	−6.02708e9	−0.386661
$818$	0	0
$819$	−1.05493e9	−0.0671010
$820$	0	0
$821$	2.03473e9	0.128323	0.0641617	−	0.997940i	$-0.479563\pi$
$821$	2.03473e9	0.128323	0.0641617	+	0.997940i	$0.479563\pi$
$822$	0	0
$823$	1.57345e10	0.983907	0.491953	−	0.870622i	$-0.336283\pi$
$823$	1.57345e10	0.983907	0.491953	+	0.870622i	$0.336283\pi$
$824$	0	0
$825$	−1.19154e10	−0.738786
$826$	0	0
$827$	8.93057e9	0.549048	0.274524	−	0.961580i	$-0.411480\pi$
$827$	8.93057e9	0.549048	0.274524	+	0.961580i	$0.411480\pi$
$828$	0	0
$829$	1.93007e10	1.17661	0.588304	−	0.808640i	$-0.299796\pi$
$829$	1.93007e10	1.17661	0.588304	+	0.808640i	$0.299796\pi$
$830$	0	0
$831$	1.55790e8	0.00941748
$832$	0	0
$833$	3.01118e10	1.80501
$834$	0	0
$835$	−4.07800e10	−2.42407
$836$	0	0
$837$	−3.48034e9	−0.205155
$838$	0	0
$839$	−1.14938e10	−0.671886	−0.335943	−	0.941882i	$-0.609055\pi$
$839$	−1.14938e10	−0.671886	−0.335943	+	0.941882i	$0.609055\pi$
$840$	0	0
$841$	−1.72497e10	−0.999992
$842$	0	0
$843$	−1.66502e10	−0.957243
$844$	0	0
$845$	4.30343e10	2.45367
$846$	0	0
$847$	3.47255e8	0.0196361
$848$	0	0
$849$	1.54294e10	0.865313
$850$	0	0
$851$	−2.72295e10	−1.51456
$852$	0	0
$853$	−1.27124e10	−0.701303	−0.350652	−	0.936506i	$-0.614040\pi$
$853$	−1.27124e10	−0.701303	−0.350652	+	0.936506i	$0.614040\pi$
$854$	0	0
$855$	−1.37735e10	−0.753636
$856$	0	0
$857$	1.47260e10	0.799196	0.399598	−	0.916691i	$-0.369150\pi$
$857$	1.47260e10	0.799196	0.399598	+	0.916691i	$0.369150\pi$
$858$	0	0
$859$	−3.15430e10	−1.69796	−0.848980	−	0.528425i	$-0.822783\pi$
$859$	−3.15430e10	−1.69796	−0.848980	+	0.528425i	$0.822783\pi$
$860$	0	0
$861$	2.61785e9	0.139776
$862$	0	0
$863$	−5.41290e9	−0.286677	−0.143338	−	0.989674i	$-0.545784\pi$
$863$	−5.41290e9	−0.286677	−0.143338	+	0.989674i	$0.545784\pi$
$864$	0	0
$865$	−2.80460e10	−1.47338
$866$	0	0
$867$	2.61759e10	1.36406
$868$	0	0
$869$	−2.84049e10	−1.46833
$870$	0	0
$871$	2.08211e10	1.06768
$872$	0	0
$873$	3.43384e9	0.174675
$874$	0	0
$875$	−1.51049e9	−0.0762235
$876$	0	0
$877$	−2.27278e9	−0.113778	−0.0568890	−	0.998381i	$-0.518118\pi$
$877$	−2.27278e9	−0.113778	−0.0568890	+	0.998381i	$0.518118\pi$
$878$	0	0
$879$	4.37461e8	0.0217259
$880$	0	0
$881$	−2.10440e10	−1.03684	−0.518422	−	0.855125i	$-0.673480\pi$
$881$	−2.10440e10	−1.03684	−0.518422	+	0.855125i	$0.673480\pi$
$882$	0	0
$883$	−3.13811e10	−1.53393	−0.766966	−	0.641688i	$-0.778234\pi$
$883$	−3.13811e10	−1.53393	−0.766966	+	0.641688i	$0.778234\pi$
$884$	0	0
$885$	−7.46550e9	−0.362041
$886$	0	0
$887$	2.34607e10	1.12878	0.564389	−	0.825509i	$-0.309112\pi$
$887$	2.34607e10	1.12878	0.564389	+	0.825509i	$0.309112\pi$
$888$	0	0
$889$	1.86430e9	0.0889939
$890$	0	0
$891$	−2.15416e9	−0.102025
$892$	0	0
$893$	4.61815e10	2.17014
$894$	0	0
$895$	−4.29894e10	−2.00438
$896$	0	0
$897$	−3.65125e10	−1.68915
$898$	0	0
$899$	6.61890e7	0.00303827
$900$	0	0
$901$	3.79448e10	1.72829
$902$	0	0
$903$	−4.23113e8	−0.0191227
$904$	0	0
$905$	1.47555e10	0.661736
$906$	0	0
$907$	7.17694e9	0.319384	0.159692	−	0.987167i	$-0.448950\pi$
$907$	7.17694e9	0.319384	0.159692	+	0.987167i	$0.448950\pi$
$908$	0	0
$909$	7.66783e9	0.338609
$910$	0	0
$911$	−2.07817e10	−0.910684	−0.455342	−	0.890317i	$-0.650483\pi$
$911$	−2.07817e10	−0.910684	−0.455342	+	0.890317i	$0.650483\pi$
$912$	0	0
$913$	−2.24363e10	−0.975672
$914$	0	0
$915$	1.87653e10	0.809809
$916$	0	0
$917$	−1.44953e9	−0.0620775
$918$	0	0
$919$	−1.48634e10	−0.631705	−0.315852	−	0.948808i	$-0.602290\pi$
$919$	−1.48634e10	−0.631705	−0.315852	+	0.948808i	$0.602290\pi$
$920$	0	0
$921$	−2.09767e10	−0.884768
$922$	0	0
$923$	1.67274e10	0.700200
$924$	0	0
$925$	2.79251e10	1.16011
$926$	0	0
$927$	−3.15983e8	−0.0130282
$928$	0	0
$929$	−1.87907e10	−0.768934	−0.384467	−	0.923139i	$-0.625615\pi$
$929$	−1.87907e10	−0.768934	−0.384467	+	0.923139i	$0.625615\pi$
$930$	0	0
$931$	3.54180e10	1.43847
$932$	0	0
$933$	1.49481e10	0.602561
$934$	0	0
$935$	6.51107e10	2.60503
$936$	0	0
$937$	−1.88100e10	−0.746966	−0.373483	−	0.927637i	$-0.621837\pi$
$937$	−1.88100e10	−0.746966	−0.373483	+	0.927637i	$0.621837\pi$
$938$	0	0
$939$	1.14274e9	0.0450422
$940$	0	0
$941$	9.09611e9	0.355871	0.177935	−	0.984042i	$-0.443058\pi$
$941$	9.09611e9	0.355871	0.177935	+	0.984042i	$0.443058\pi$
$942$	0	0
$943$	9.06074e10	3.51862
$944$	0	0
$945$	−9.66923e8	−0.0372718
$946$	0	0
$947$	8.08994e9	0.309543	0.154771	−	0.987950i	$-0.450536\pi$
$947$	8.08994e9	0.309543	0.154771	+	0.987950i	$0.450536\pi$
$948$	0	0
$949$	−2.91333e10	−1.10652
$950$	0	0
$951$	−1.34384e10	−0.506660
$952$	0	0
$953$	−2.38739e10	−0.893506	−0.446753	−	0.894657i	$-0.647420\pi$
$953$	−2.38739e10	−0.893506	−0.446753	+	0.894657i	$0.647420\pi$
$954$	0	0
$955$	2.95857e10	1.09918
$956$	0	0
$957$	4.09678e7	0.00151095
$958$	0	0
$959$	−3.84771e9	−0.140876
$960$	0	0
$961$	3.75259e9	0.136395
$962$	0	0
$963$	1.11436e10	0.402098
$964$	0	0
$965$	2.89354e10	1.03653
$966$	0	0
$967$	−3.58904e10	−1.27639	−0.638197	−	0.769873i	$-0.720320\pi$
$967$	−3.58904e10	−1.27639	−0.638197	+	0.769873i	$0.720320\pi$
$968$	0	0
$969$	4.38199e10	1.54717
$970$	0	0
$971$	−1.06301e10	−0.372624	−0.186312	−	0.982491i	$-0.559654\pi$
$971$	−1.06301e10	−0.372624	−0.186312	+	0.982491i	$0.559654\pi$
$972$	0	0
$973$	8.26934e8	0.0287790
$974$	0	0
$975$	3.74452e10	1.29384
$976$	0	0
$977$	6.00337e9	0.205951	0.102976	−	0.994684i	$-0.467164\pi$
$977$	6.00337e9	0.205951	0.102976	+	0.994684i	$0.467164\pi$
$978$	0	0
$979$	−1.02174e10	−0.348017
$980$	0	0
$981$	−5.11366e9	−0.172938
$982$	0	0
$983$	−5.26370e9	−0.176748	−0.0883739	−	0.996087i	$-0.528167\pi$
$983$	−5.26370e9	−0.176748	−0.0883739	+	0.996087i	$0.528167\pi$
$984$	0	0
$985$	1.48988e10	0.496733
$986$	0	0
$987$	3.24203e9	0.107327
$988$	0	0
$989$	−1.46445e10	−0.481381
$990$	0	0
$991$	−1.42058e10	−0.463669	−0.231835	−	0.972755i	$-0.574473\pi$
$991$	−1.42058e10	−0.463669	−0.231835	+	0.972755i	$0.574473\pi$
$992$	0	0
$993$	9.82578e9	0.318453
$994$	0	0
$995$	−3.83116e10	−1.23296
$996$	0	0
$997$	−4.78802e10	−1.53011	−0.765055	−	0.643965i	$-0.777288\pi$
$997$	−4.78802e10	−1.53011	−0.765055	+	0.643965i	$0.777288\pi$
$998$	0	0
$999$	5.04854e9	0.160209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.r.1.4	yes	4
4.3	odd	2		384.8.a.n.1.4	✓	4
8.3	odd	2		384.8.a.s.1.1	yes	4
8.5	even	2		384.8.a.o.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.n.1.4	✓	4	4.3	odd	2
384.8.a.o.1.1	yes	4	8.5	even	2
384.8.a.r.1.4	yes	4	1.1	even	1	trivial
384.8.a.s.1.1	yes	4	8.3	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} +432.433 q^{5} -113.601 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} +432.433 q^{5} -113.601 q^{7} +729.000 q^{9} -4053.44 q^{11} +12738.3 q^{13} +11675.7 q^{15} -37145.9 q^{17} -43691.5 q^{19} -3067.23 q^{21} -106161. q^{23} +108873. q^{25} +19683.0 q^{27} -374.331 q^{29} -176820. q^{31} -109443. q^{33} -49124.8 q^{35} +256492. q^{37} +343935. q^{39} -853490. q^{41} +137946. q^{43} +315243. q^{45} -1.05699e6 q^{47} -810638. q^{49} -1.00294e6 q^{51} -1.02151e6 q^{53} -1.75284e6 q^{55} -1.17967e6 q^{57} -639406. q^{59} +1.60722e6 q^{61} -82815.2 q^{63} +5.50847e6 q^{65} +1.63452e6 q^{67} -2.86635e6 q^{69} +1.31315e6 q^{71} -2.28706e6 q^{73} +2.93957e6 q^{75} +460475. q^{77} +7.00761e6 q^{79} +531441. q^{81} +5.53513e6 q^{83} -1.60631e7 q^{85} -10106.9 q^{87} +2.52067e6 q^{89} -1.44709e6 q^{91} -4.77413e6 q^{93} -1.88936e7 q^{95} +4.71034e6 q^{97} -2.95496e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9	\( 4 q + 108 q^{3} - 192 q^{5} - 680 q^{7} + 2916 q^{9} - 4496 q^{11} + 12840 q^{13} - 5184 q^{15} - 14952 q^{17} - 21504 q^{19} - 18360 q^{21} - 54992 q^{23} + 190732 q^{25} + 78732 q^{27} - 242384 q^{29} - 151432 q^{31} - 121392 q^{33} + 273984 q^{35} - 113288 q^{37} + 346680 q^{39} - 239176 q^{41} + 1495328 q^{43} - 139968 q^{45} - 772368 q^{47} - 1577100 q^{49} - 403704 q^{51} - 2389776 q^{53} - 2590080 q^{55} - 580608 q^{57} + 141232 q^{59} - 1231304 q^{61} - 495720 q^{63} - 1041024 q^{65} - 441392 q^{67} - 1484784 q^{69} - 1507504 q^{71} - 1516840 q^{73} + 5149764 q^{75} - 12340448 q^{77} - 9540936 q^{79} + 2125764 q^{81} - 4587600 q^{83} - 6382848 q^{85} - 6544368 q^{87} + 162376 q^{89} - 4681104 q^{91} - 4088664 q^{93} - 29221248 q^{95} + 2726760 q^{97} - 3277584 q^{99}+O(q^{100}) \) 4 * q + 108 * q^3 - 192 * q^5 - 680 * q^7 + 2916 * q^9 - 4496 * q^11 + 12840 * q^13 - 5184 * q^15 - 14952 * q^17 - 21504 * q^19 - 18360 * q^21 - 54992 * q^23 + 190732 * q^25 + 78732 * q^27 - 242384 * q^29 - 151432 * q^31 - 121392 * q^33 + 273984 * q^35 - 113288 * q^37 + 346680 * q^39 - 239176 * q^41 + 1495328 * q^43 - 139968 * q^45 - 772368 * q^47 - 1577100 * q^49 - 403704 * q^51 - 2389776 * q^53 - 2590080 * q^55 - 580608 * q^57 + 141232 * q^59 - 1231304 * q^61 - 495720 * q^63 - 1041024 * q^65 - 441392 * q^67 - 1484784 * q^69 - 1507504 * q^71 - 1516840 * q^73 + 5149764 * q^75 - 12340448 * q^77 - 9540936 * q^79 + 2125764 * q^81 - 4587600 * q^83 - 6382848 * q^85 - 6544368 * q^87 + 162376 * q^89 - 4681104 * q^91 - 4088664 * q^93 - 29221248 * q^95 + 2726760 * q^97 - 3277584 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more