LMFDB - Embedded newform 384.8.a.h.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{366}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 366 $ x^2 - 366
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-19.1311$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} -65.0490 q^{5} -275.245 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -65.0490 q^{5} -275.245 q^{7} +729.000 q^{9} +5543.27 q^{11} +4956.94 q^{13} -1756.32 q^{15} -20343.4 q^{17} -49187.7 q^{19} -7431.62 q^{21} -55150.9 q^{23} -73893.6 q^{25} +19683.0 q^{27} +132595. q^{29} +248716. q^{31} +149668. q^{33} +17904.4 q^{35} -158346. q^{37} +133837. q^{39} +411345. q^{41} +165656. q^{43} -47420.7 q^{45} -881705. q^{47} -747783. q^{49} -549270. q^{51} +1.06905e6 q^{53} -360585. q^{55} -1.32807e6 q^{57} -1.20357e6 q^{59} +723534. q^{61} -200654. q^{63} -322444. q^{65} -1.51853e6 q^{67} -1.48907e6 q^{69} +524889. q^{71} -542977. q^{73} -1.99513e6 q^{75} -1.52576e6 q^{77} -3.24216e6 q^{79} +531441. q^{81} +2.79794e6 q^{83} +1.32331e6 q^{85} +3.58007e6 q^{87} -1.54242e6 q^{89} -1.36437e6 q^{91} +6.71534e6 q^{93} +3.19961e6 q^{95} -1.01065e7 q^{97} +4.04105e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{3} + 176 q^{5} + 980 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q + 54 * q^3 + 176 * q^5 + 980 * q^7 + 1458 * q^9	$ 2 q + 54 q^{3} + 176 q^{5} + 980 q^{7} + 1458 q^{9} + 3128 q^{11} - 8452 q^{13} + 4752 q^{15} + 5228 q^{17} - 87968 q^{19} + 26460 q^{21} - 104792 q^{23} - 93914 q^{25} + 39366 q^{27} - 98760 q^{29} + 1860 q^{31} + 84456 q^{33} + 320480 q^{35} - 455660 q^{37} - 228204 q^{39} - 205188 q^{41} - 803088 q^{43} + 128304 q^{45} - 522488 q^{47} + 4314 q^{49} + 141156 q^{51} + 2468376 q^{53} - 942784 q^{55} - 2375136 q^{57} - 989288 q^{59} + 2482292 q^{61} + 714420 q^{63} - 3554656 q^{65} - 434008 q^{67} - 2829384 q^{69} + 1554840 q^{71} + 279244 q^{73} - 2535678 q^{75} - 4557520 q^{77} + 1015396 q^{79} + 1062882 q^{81} - 2899560 q^{83} + 7487264 q^{85} - 2666520 q^{87} - 8115868 q^{89} - 18195880 q^{91} + 50220 q^{93} - 6148352 q^{95} - 2553676 q^{97} + 2280312 q^{99}+O(q^{100}) $ 2 * q + 54 * q^3 + 176 * q^5 + 980 * q^7 + 1458 * q^9 + 3128 * q^11 - 8452 * q^13 + 4752 * q^15 + 5228 * q^17 - 87968 * q^19 + 26460 * q^21 - 104792 * q^23 - 93914 * q^25 + 39366 * q^27 - 98760 * q^29 + 1860 * q^31 + 84456 * q^33 + 320480 * q^35 - 455660 * q^37 - 228204 * q^39 - 205188 * q^41 - 803088 * q^43 + 128304 * q^45 - 522488 * q^47 + 4314 * q^49 + 141156 * q^51 + 2468376 * q^53 - 942784 * q^55 - 2375136 * q^57 - 989288 * q^59 + 2482292 * q^61 + 714420 * q^63 - 3554656 * q^65 - 434008 * q^67 - 2829384 * q^69 + 1554840 * q^71 + 279244 * q^73 - 2535678 * q^75 - 4557520 * q^77 + 1015396 * q^79 + 1062882 * q^81 - 2899560 * q^83 + 7487264 * q^85 - 2666520 * q^87 - 8115868 * q^89 - 18195880 * q^91 + 50220 * q^93 - 6148352 * q^95 - 2553676 * q^97 + 2280312 * 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
$5$	−65.0490	−0.232726	−0.116363	−	0.993207i	$-0.537124\pi$
$5$	−65.0490	−0.232726	−0.116363	+	0.993207i	$0.537124\pi$
$6$	0	0
$7$	−275.245	−0.303303	−0.151651	−	0.988434i	$-0.548459\pi$
$7$	−275.245	−0.303303	−0.151651	+	0.988434i	$0.548459\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	5543.27	1.25572	0.627859	−	0.778327i	$-0.283932\pi$
$11$	5543.27	1.25572	0.627859	+	0.778327i	$0.283932\pi$
$12$	0	0
$13$	4956.94	0.625766	0.312883	−	0.949792i	$-0.398705\pi$
$13$	4956.94	0.625766	0.312883	+	0.949792i	$0.398705\pi$
$14$	0	0
$15$	−1756.32	−0.134365
$16$	0	0
$17$	−20343.4	−1.00427	−0.502136	−	0.864789i	$-0.667452\pi$
$17$	−20343.4	−1.00427	−0.502136	+	0.864789i	$0.667452\pi$
$18$	0	0
$19$	−49187.7	−1.64520	−0.822600	−	0.568621i	$-0.807477\pi$
$19$	−49187.7	−1.64520	−0.822600	+	0.568621i	$0.807477\pi$
$20$	0	0
$21$	−7431.62	−0.175112
$22$	0	0
$23$	−55150.9	−0.945159	−0.472580	−	0.881288i	$-0.656677\pi$
$23$	−55150.9	−0.945159	−0.472580	+	0.881288i	$0.656677\pi$
$24$	0	0
$25$	−73893.6	−0.945838
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	132595.	1.00957	0.504783	−	0.863246i	$-0.331572\pi$
$29$	132595.	1.00957	0.504783	+	0.863246i	$0.331572\pi$
$30$	0	0
$31$	248716.	1.49947	0.749736	−	0.661737i	$-0.230180\pi$
$31$	248716.	1.49947	0.749736	+	0.661737i	$0.230180\pi$
$32$	0	0
$33$	149668.	0.724989
$34$	0	0
$35$	17904.4	0.0705866
$36$	0	0
$37$	−158346.	−0.513926	−0.256963	−	0.966421i	$-0.582722\pi$
$37$	−158346.	−0.513926	−0.256963	+	0.966421i	$0.582722\pi$
$38$	0	0
$39$	133837.	0.361286
$40$	0	0
$41$	411345.	0.932100	0.466050	−	0.884759i	$-0.345677\pi$
$41$	411345.	0.932100	0.466050	+	0.884759i	$0.345677\pi$
$42$	0	0
$43$	165656.	0.317736	0.158868	−	0.987300i	$-0.449216\pi$
$43$	165656.	0.317736	0.158868	+	0.987300i	$0.449216\pi$
$44$	0	0
$45$	−47420.7	−0.0775755
$46$	0	0
$47$	−881705.	−1.23874	−0.619371	−	0.785099i	$-0.712612\pi$
$47$	−881705.	−1.23874	−0.619371	+	0.785099i	$0.712612\pi$
$48$	0	0
$49$	−747783.	−0.908007
$50$	0	0
$51$	−549270.	−0.579817
$52$	0	0
$53$	1.06905e6	0.986351	0.493176	−	0.869930i	$-0.335836\pi$
$53$	1.06905e6	0.986351	0.493176	+	0.869930i	$0.335836\pi$
$54$	0	0
$55$	−360585.	−0.292239
$56$	0	0
$57$	−1.32807e6	−0.949856
$58$	0	0
$59$	−1.20357e6	−0.762936	−0.381468	−	0.924382i	$-0.624581\pi$
$59$	−1.20357e6	−0.762936	−0.381468	+	0.924382i	$0.624581\pi$
$60$	0	0
$61$	723534.	0.408136	0.204068	−	0.978957i	$-0.434584\pi$
$61$	723534.	0.408136	0.204068	+	0.978957i	$0.434584\pi$
$62$	0	0
$63$	−200654.	−0.101101
$64$	0	0
$65$	−322444.	−0.145632
$66$	0	0
$67$	−1.51853e6	−0.616826	−0.308413	−	0.951253i	$-0.599798\pi$
$67$	−1.51853e6	−0.616826	−0.308413	+	0.951253i	$0.599798\pi$
$68$	0	0
$69$	−1.48907e6	−0.545688
$70$	0	0
$71$	524889.	0.174046	0.0870229	−	0.996206i	$-0.472265\pi$
$71$	524889.	0.174046	0.0870229	+	0.996206i	$0.472265\pi$
$72$	0	0
$73$	−542977.	−0.163362	−0.0816810	−	0.996659i	$-0.526029\pi$
$73$	−542977.	−0.163362	−0.0816810	+	0.996659i	$0.526029\pi$
$74$	0	0
$75$	−1.99513e6	−0.546080
$76$	0	0
$77$	−1.52576e6	−0.380863
$78$	0	0
$79$	−3.24216e6	−0.739842	−0.369921	−	0.929063i	$-0.620615\pi$
$79$	−3.24216e6	−0.739842	−0.369921	+	0.929063i	$0.620615\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	2.79794e6	0.537113	0.268556	−	0.963264i	$-0.413453\pi$
$83$	2.79794e6	0.537113	0.268556	+	0.963264i	$0.413453\pi$
$84$	0	0
$85$	1.32331e6	0.233721
$86$	0	0
$87$	3.58007e6	0.582874
$88$	0	0
$89$	−1.54242e6	−0.231920	−0.115960	−	0.993254i	$-0.536994\pi$
$89$	−1.54242e6	−0.231920	−0.115960	+	0.993254i	$0.536994\pi$
$90$	0	0
$91$	−1.36437e6	−0.189797
$92$	0	0
$93$	6.71534e6	0.865721
$94$	0	0
$95$	3.19961e6	0.382881
$96$	0	0
$97$	−1.01065e7	−1.12435	−0.562175	−	0.827018i	$-0.690035\pi$
$97$	−1.01065e7	−1.12435	−0.562175	+	0.827018i	$0.690035\pi$
$98$	0	0
$99$	4.04105e6	0.418573
$100$	0	0
$101$	−9.26941e6	−0.895215	−0.447607	−	0.894230i	$-0.647724\pi$
$101$	−9.26941e6	−0.895215	−0.447607	+	0.894230i	$0.647724\pi$
$102$	0	0
$103$	6.69812e6	0.603980	0.301990	−	0.953311i	$-0.402349\pi$
$103$	6.69812e6	0.603980	0.301990	+	0.953311i	$0.402349\pi$
$104$	0	0
$105$	483419.	0.0407532
$106$	0	0
$107$	−1.52877e7	−1.20642	−0.603210	−	0.797583i	$-0.706112\pi$
$107$	−1.52877e7	−1.20642	−0.603210	+	0.797583i	$0.706112\pi$
$108$	0	0
$109$	−2.32632e7	−1.72058	−0.860292	−	0.509802i	$-0.829719\pi$
$109$	−2.32632e7	−1.72058	−0.860292	+	0.509802i	$0.829719\pi$
$110$	0	0
$111$	−4.27534e6	−0.296715
$112$	0	0
$113$	−8.81644e6	−0.574803	−0.287401	−	0.957810i	$-0.592791\pi$
$113$	−8.81644e6	−0.574803	−0.287401	+	0.957810i	$0.592791\pi$
$114$	0	0
$115$	3.58751e6	0.219964
$116$	0	0
$117$	3.61361e6	0.208589
$118$	0	0
$119$	5.59941e6	0.304598
$120$	0	0
$121$	1.12407e7	0.576827
$122$	0	0
$123$	1.11063e7	0.538148
$124$	0	0
$125$	9.88866e6	0.452848
$126$	0	0
$127$	1.54448e7	0.669065	0.334533	−	0.942384i	$-0.391422\pi$
$127$	1.54448e7	0.669065	0.334533	+	0.942384i	$0.391422\pi$
$128$	0	0
$129$	4.47270e6	0.183445
$130$	0	0
$131$	−2.58656e7	−1.00525	−0.502623	−	0.864506i	$-0.667632\pi$
$131$	−2.58656e7	−1.00525	−0.502623	+	0.864506i	$0.667632\pi$
$132$	0	0
$133$	1.35387e7	0.498994
$134$	0	0
$135$	−1.28036e6	−0.0447882
$136$	0	0
$137$	−2.23006e7	−0.740959	−0.370479	−	0.928841i	$-0.620807\pi$
$137$	−2.23006e7	−0.740959	−0.370479	+	0.928841i	$0.620807\pi$
$138$	0	0
$139$	−2.43314e7	−0.768451	−0.384225	−	0.923239i	$-0.625531\pi$
$139$	−2.43314e7	−0.768451	−0.384225	+	0.923239i	$0.625531\pi$
$140$	0	0
$141$	−2.38060e7	−0.715188
$142$	0	0
$143$	2.74777e7	0.785785
$144$	0	0
$145$	−8.62519e6	−0.234953
$146$	0	0
$147$	−2.01901e7	−0.524238
$148$	0	0
$149$	−7.33558e7	−1.81670	−0.908348	−	0.418215i	$-0.862656\pi$
$149$	−7.33558e7	−1.81670	−0.908348	+	0.418215i	$0.862656\pi$
$150$	0	0
$151$	4.16331e7	0.984055	0.492028	−	0.870580i	$-0.336256\pi$
$151$	4.16331e7	0.984055	0.492028	+	0.870580i	$0.336256\pi$
$152$	0	0
$153$	−1.48303e7	−0.334757
$154$	0	0
$155$	−1.61788e7	−0.348967
$156$	0	0
$157$	−3.76259e7	−0.775958	−0.387979	−	0.921668i	$-0.626827\pi$
$157$	−3.76259e7	−0.775958	−0.387979	+	0.921668i	$0.626827\pi$
$158$	0	0
$159$	2.88643e7	0.569470
$160$	0	0
$161$	1.51800e7	0.286669
$162$	0	0
$163$	3.90955e7	0.707083	0.353541	−	0.935419i	$-0.384977\pi$
$163$	3.90955e7	0.707083	0.353541	+	0.935419i	$0.384977\pi$
$164$	0	0
$165$	−9.73578e6	−0.168724
$166$	0	0
$167$	8.38058e7	1.39241	0.696204	−	0.717844i	$-0.254871\pi$
$167$	8.38058e7	1.39241	0.696204	+	0.717844i	$0.254871\pi$
$168$	0	0
$169$	−3.81773e7	−0.608417
$170$	0	0
$171$	−3.58578e7	−0.548400
$172$	0	0
$173$	−9.49967e7	−1.39491	−0.697457	−	0.716627i	$-0.745685\pi$
$173$	−9.49967e7	−1.39491	−0.697457	+	0.716627i	$0.745685\pi$
$174$	0	0
$175$	2.03389e7	0.286875
$176$	0	0
$177$	−3.24963e7	−0.440481
$178$	0	0
$179$	2.62054e7	0.341511	0.170756	−	0.985313i	$-0.445379\pi$
$179$	2.62054e7	0.341511	0.170756	+	0.985313i	$0.445379\pi$
$180$	0	0
$181$	−4.35606e7	−0.546032	−0.273016	−	0.962009i	$-0.588021\pi$
$181$	−4.35606e7	−0.546032	−0.273016	+	0.962009i	$0.588021\pi$
$182$	0	0
$183$	1.95354e7	0.235637
$184$	0	0
$185$	1.03002e7	0.119604
$186$	0	0
$187$	−1.12769e8	−1.26108
$188$	0	0
$189$	−5.41765e6	−0.0583706
$190$	0	0
$191$	−1.16280e8	−1.20750	−0.603749	−	0.797174i	$-0.706327\pi$
$191$	−1.16280e8	−1.20750	−0.603749	+	0.797174i	$0.706327\pi$
$192$	0	0
$193$	−1.09584e8	−1.09723	−0.548615	−	0.836075i	$-0.684845\pi$
$193$	−1.09584e8	−1.09723	−0.548615	+	0.836075i	$0.684845\pi$
$194$	0	0
$195$	−8.70599e6	−0.0840808
$196$	0	0
$197$	1.54530e8	1.44006	0.720029	−	0.693944i	$-0.244129\pi$
$197$	1.54530e8	1.44006	0.720029	+	0.693944i	$0.244129\pi$
$198$	0	0
$199$	1.17206e8	1.05430	0.527151	−	0.849772i	$-0.323260\pi$
$199$	1.17206e8	1.05430	0.527151	+	0.849772i	$0.323260\pi$
$200$	0	0
$201$	−4.10004e7	−0.356124
$202$	0	0
$203$	−3.64962e7	−0.306204
$204$	0	0
$205$	−2.67576e7	−0.216924
$206$	0	0
$207$	−4.02050e7	−0.315053
$208$	0	0
$209$	−2.72661e8	−2.06591
$210$	0	0
$211$	−2.34458e8	−1.71821	−0.859107	−	0.511796i	$-0.828980\pi$
$211$	−2.34458e8	−1.71821	−0.859107	+	0.511796i	$0.828980\pi$
$212$	0	0
$213$	1.41720e7	0.100485
$214$	0	0
$215$	−1.07757e7	−0.0739456
$216$	0	0
$217$	−6.84579e7	−0.454794
$218$	0	0
$219$	−1.46604e7	−0.0943171
$220$	0	0
$221$	−1.00841e8	−0.628439
$222$	0	0
$223$	2.36568e8	1.42853	0.714265	−	0.699876i	$-0.246761\pi$
$223$	2.36568e8	1.42853	0.714265	+	0.699876i	$0.246761\pi$
$224$	0	0
$225$	−5.38685e7	−0.315279
$226$	0	0
$227$	−2.20042e8	−1.24857	−0.624287	−	0.781195i	$-0.714610\pi$
$227$	−2.20042e8	−1.24857	−0.624287	+	0.781195i	$0.714610\pi$
$228$	0	0
$229$	−2.78074e8	−1.53016	−0.765079	−	0.643937i	$-0.777300\pi$
$229$	−2.78074e8	−1.53016	−0.765079	+	0.643937i	$0.777300\pi$
$230$	0	0
$231$	−4.11955e7	−0.219891
$232$	0	0
$233$	6.21028e7	0.321637	0.160818	−	0.986984i	$-0.448587\pi$
$233$	6.21028e7	0.321637	0.160818	+	0.986984i	$0.448587\pi$
$234$	0	0
$235$	5.73540e7	0.288288
$236$	0	0
$237$	−8.75382e7	−0.427148
$238$	0	0
$239$	3.49575e8	1.65633	0.828166	−	0.560483i	$-0.189385\pi$
$239$	3.49575e8	1.65633	0.828166	+	0.560483i	$0.189385\pi$
$240$	0	0
$241$	8.47411e7	0.389973	0.194987	−	0.980806i	$-0.437534\pi$
$241$	8.47411e7	0.389973	0.194987	+	0.980806i	$0.437534\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	4.86426e7	0.211317
$246$	0	0
$247$	−2.43820e8	−1.02951
$248$	0	0
$249$	7.55444e7	0.310102
$250$	0	0
$251$	−5.31962e7	−0.212335	−0.106168	−	0.994348i	$-0.533858\pi$
$251$	−5.31962e7	−0.212335	−0.106168	+	0.994348i	$0.533858\pi$
$252$	0	0
$253$	−3.05716e8	−1.18685
$254$	0	0
$255$	3.57295e7	0.134939
$256$	0	0
$257$	−4.31504e8	−1.58569	−0.792846	−	0.609422i	$-0.791401\pi$
$257$	−4.31504e8	−1.58569	−0.792846	+	0.609422i	$0.791401\pi$
$258$	0	0
$259$	4.35839e7	0.155875
$260$	0	0
$261$	9.66620e7	0.336522
$262$	0	0
$263$	5.62847e7	0.190785	0.0953927	−	0.995440i	$-0.469589\pi$
$263$	5.62847e7	0.190785	0.0953927	+	0.995440i	$0.469589\pi$
$264$	0	0
$265$	−6.95405e7	−0.229550
$266$	0	0
$267$	−4.16454e7	−0.133899
$268$	0	0
$269$	−1.57377e8	−0.492957	−0.246479	−	0.969148i	$-0.579273\pi$
$269$	−1.57377e8	−0.492957	−0.246479	+	0.969148i	$0.579273\pi$
$270$	0	0
$271$	−2.61871e7	−0.0799273	−0.0399637	−	0.999201i	$-0.512724\pi$
$271$	−2.61871e7	−0.0799273	−0.0399637	+	0.999201i	$0.512724\pi$
$272$	0	0
$273$	−3.68381e7	−0.109579
$274$	0	0
$275$	−4.09613e8	−1.18771
$276$	0	0
$277$	5.69358e8	1.60956	0.804779	−	0.593575i	$-0.202284\pi$
$277$	5.69358e8	1.60956	0.804779	+	0.593575i	$0.202284\pi$
$278$	0	0
$279$	1.81314e8	0.499824
$280$	0	0
$281$	−6.28150e7	−0.168885	−0.0844425	−	0.996428i	$-0.526911\pi$
$281$	−6.28150e7	−0.168885	−0.0844425	+	0.996428i	$0.526911\pi$
$282$	0	0
$283$	1.59079e7	0.0417216	0.0208608	−	0.999782i	$-0.493359\pi$
$283$	1.59079e7	0.0417216	0.0208608	+	0.999782i	$0.493359\pi$
$284$	0	0
$285$	8.63894e7	0.221057
$286$	0	0
$287$	−1.13221e8	−0.282708
$288$	0	0
$289$	3.51329e6	0.00856192
$290$	0	0
$291$	−2.72877e8	−0.649144
$292$	0	0
$293$	−2.96493e8	−0.688618	−0.344309	−	0.938856i	$-0.611887\pi$
$293$	−2.96493e8	−0.688618	−0.344309	+	0.938856i	$0.611887\pi$
$294$	0	0
$295$	7.82908e7	0.177555
$296$	0	0
$297$	1.09108e8	0.241663
$298$	0	0
$299$	−2.73380e8	−0.591449
$300$	0	0
$301$	−4.55959e7	−0.0963703
$302$	0	0
$303$	−2.50274e8	−0.516852
$304$	0	0
$305$	−4.70652e7	−0.0949840
$306$	0	0
$307$	4.44473e8	0.876721	0.438360	−	0.898799i	$-0.355559\pi$
$307$	4.44473e8	0.876721	0.438360	+	0.898799i	$0.355559\pi$
$308$	0	0
$309$	1.80849e8	0.348708
$310$	0	0
$311$	2.40599e8	0.453558	0.226779	−	0.973946i	$-0.427180\pi$
$311$	2.40599e8	0.453558	0.226779	+	0.973946i	$0.427180\pi$
$312$	0	0
$313$	9.75397e8	1.79794	0.898972	−	0.438005i	$-0.144315\pi$
$313$	9.75397e8	1.79794	0.898972	+	0.438005i	$0.144315\pi$
$314$	0	0
$315$	1.30523e7	0.0235289
$316$	0	0
$317$	−1.90517e8	−0.335913	−0.167956	−	0.985794i	$-0.553717\pi$
$317$	−1.90517e8	−0.335913	−0.167956	+	0.985794i	$0.553717\pi$
$318$	0	0
$319$	7.35012e8	1.26773
$320$	0	0
$321$	−4.12767e8	−0.696527
$322$	0	0
$323$	1.00064e9	1.65223
$324$	0	0
$325$	−3.66286e8	−0.591874
$326$	0	0
$327$	−6.28105e8	−0.993380
$328$	0	0
$329$	2.42685e8	0.375714
$330$	0	0
$331$	6.18272e8	0.937091	0.468546	−	0.883439i	$-0.344778\pi$
$331$	6.18272e8	0.937091	0.468546	+	0.883439i	$0.344778\pi$
$332$	0	0
$333$	−1.15434e8	−0.171309
$334$	0	0
$335$	9.87791e7	0.143552
$336$	0	0
$337$	−1.17191e9	−1.66798	−0.833990	−	0.551779i	$-0.813949\pi$
$337$	−1.17191e9	−1.66798	−0.833990	+	0.551779i	$0.813949\pi$
$338$	0	0
$339$	−2.38044e8	−0.331862
$340$	0	0
$341$	1.37870e9	1.88291
$342$	0	0
$343$	4.32500e8	0.578704
$344$	0	0
$345$	9.68628e7	0.126996
$346$	0	0
$347$	4.14800e8	0.532949	0.266474	−	0.963842i	$-0.414141\pi$
$347$	4.14800e8	0.532949	0.266474	+	0.963842i	$0.414141\pi$
$348$	0	0
$349$	−1.48010e9	−1.86381	−0.931905	−	0.362703i	$-0.881854\pi$
$349$	−1.48010e9	−1.86381	−0.931905	+	0.362703i	$0.881854\pi$
$350$	0	0
$351$	9.75675e7	0.120429
$352$	0	0
$353$	1.02147e9	1.23599	0.617993	−	0.786183i	$-0.287946\pi$
$353$	1.02147e9	1.23599	0.617993	+	0.786183i	$0.287946\pi$
$354$	0	0
$355$	−3.41435e7	−0.0405050
$356$	0	0
$357$	1.51184e8	0.175860
$358$	0	0
$359$	−1.26250e9	−1.44012	−0.720062	−	0.693910i	$-0.755887\pi$
$359$	−1.26250e9	−1.44012	−0.720062	+	0.693910i	$0.755887\pi$
$360$	0	0
$361$	1.52555e9	1.70668
$362$	0	0
$363$	3.03499e8	0.333031
$364$	0	0
$365$	3.53201e7	0.0380187
$366$	0	0
$367$	−8.24186e8	−0.870350	−0.435175	−	0.900346i	$-0.643313\pi$
$367$	−8.24186e8	−0.870350	−0.435175	+	0.900346i	$0.643313\pi$
$368$	0	0
$369$	2.99870e8	0.310700
$370$	0	0
$371$	−2.94250e8	−0.299163
$372$	0	0
$373$	3.68578e8	0.367746	0.183873	−	0.982950i	$-0.441136\pi$
$373$	3.68578e8	0.367746	0.183873	+	0.982950i	$0.441136\pi$
$374$	0	0
$375$	2.66994e8	0.261452
$376$	0	0
$377$	6.57267e8	0.631753
$378$	0	0
$379$	8.53429e8	0.805248	0.402624	−	0.915365i	$-0.368098\pi$
$379$	8.53429e8	0.805248	0.402624	+	0.915365i	$0.368098\pi$
$380$	0	0
$381$	4.17009e8	0.386285
$382$	0	0
$383$	−1.18529e9	−1.07803	−0.539013	−	0.842298i	$-0.681203\pi$
$383$	−1.18529e9	−1.07803	−0.539013	+	0.842298i	$0.681203\pi$
$384$	0	0
$385$	9.92491e7	0.0886368
$386$	0	0
$387$	1.20763e8	0.105912
$388$	0	0
$389$	1.11972e9	0.964461	0.482231	−	0.876044i	$-0.339827\pi$
$389$	1.11972e9	0.964461	0.482231	+	0.876044i	$0.339827\pi$
$390$	0	0
$391$	1.12195e9	0.949197
$392$	0	0
$393$	−6.98370e8	−0.580379
$394$	0	0
$395$	2.10899e8	0.172181
$396$	0	0
$397$	−6.33522e8	−0.508153	−0.254077	−	0.967184i	$-0.581772\pi$
$397$	−6.33522e8	−0.508153	−0.254077	+	0.967184i	$0.581772\pi$
$398$	0	0
$399$	3.65544e8	0.288094
$400$	0	0
$401$	−1.45494e9	−1.12679	−0.563393	−	0.826189i	$-0.690504\pi$
$401$	−1.45494e9	−1.12679	−0.563393	+	0.826189i	$0.690504\pi$
$402$	0	0
$403$	1.23287e9	0.938319
$404$	0	0
$405$	−3.45697e7	−0.0258585
$406$	0	0
$407$	−8.77754e8	−0.645346
$408$	0	0
$409$	1.51094e9	1.09198	0.545990	−	0.837792i	$-0.316154\pi$
$409$	1.51094e9	1.09198	0.545990	+	0.837792i	$0.316154\pi$
$410$	0	0
$411$	−6.02115e8	−0.427793
$412$	0	0
$413$	3.31276e8	0.231401
$414$	0	0
$415$	−1.82003e8	−0.125000
$416$	0	0
$417$	−6.56949e8	−0.443665
$418$	0	0
$419$	−5.43994e8	−0.361281	−0.180641	−	0.983549i	$-0.557817\pi$
$419$	−5.43994e8	−0.361281	−0.180641	+	0.983549i	$0.557817\pi$
$420$	0	0
$421$	3.01260e9	1.96768	0.983839	−	0.179055i	$-0.0573039\pi$
$421$	3.01260e9	1.96768	0.983839	+	0.179055i	$0.0573039\pi$
$422$	0	0
$423$	−6.42763e8	−0.412914
$424$	0	0
$425$	1.50324e9	0.949879
$426$	0	0
$427$	−1.99149e8	−0.123789
$428$	0	0
$429$	7.41897e8	0.453673
$430$	0	0
$431$	−2.03054e9	−1.22164	−0.610818	−	0.791771i	$-0.709159\pi$
$431$	−2.03054e9	−1.22164	−0.610818	+	0.791771i	$0.709159\pi$
$432$	0	0
$433$	2.89154e9	1.71168	0.855838	−	0.517244i	$-0.173042\pi$
$433$	2.89154e9	1.71168	0.855838	+	0.517244i	$0.173042\pi$
$434$	0	0
$435$	−2.32880e8	−0.135650
$436$	0	0
$437$	2.71274e9	1.55498
$438$	0	0
$439$	−2.66416e9	−1.50291	−0.751457	−	0.659782i	$-0.770648\pi$
$439$	−2.66416e9	−1.50291	−0.751457	+	0.659782i	$0.770648\pi$
$440$	0	0
$441$	−5.45134e8	−0.302669
$442$	0	0
$443$	−875964.	−0.000478711 0	−0.000239356	−	1.00000i	$-0.500076\pi$
$443$	−875964.	−0.000478711 0	−0.000239356		1.00000i	$0.500076\pi$
$444$	0	0
$445$	1.00333e8	0.0539739
$446$	0	0
$447$	−1.98061e9	−1.04887
$448$	0	0
$449$	−2.61727e9	−1.36454	−0.682271	−	0.731099i	$-0.739008\pi$
$449$	−2.61727e9	−1.36454	−0.682271	+	0.731099i	$0.739008\pi$
$450$	0	0
$451$	2.28020e9	1.17045
$452$	0	0
$453$	1.12409e9	0.568145
$454$	0	0
$455$	8.87511e7	0.0441707
$456$	0	0
$457$	2.90146e9	1.42203	0.711016	−	0.703176i	$-0.248235\pi$
$457$	2.90146e9	1.42203	0.711016	+	0.703176i	$0.248235\pi$
$458$	0	0
$459$	−4.00418e8	−0.193272
$460$	0	0
$461$	−2.10787e9	−1.00205	−0.501027	−	0.865432i	$-0.667044\pi$
$461$	−2.10787e9	−1.00205	−0.501027	+	0.865432i	$0.667044\pi$
$462$	0	0
$463$	1.78584e9	0.836200	0.418100	−	0.908401i	$-0.362696\pi$
$463$	1.78584e9	0.836200	0.418100	+	0.908401i	$0.362696\pi$
$464$	0	0
$465$	−4.36826e8	−0.201476
$466$	0	0
$467$	1.88405e9	0.856021	0.428011	−	0.903774i	$-0.359215\pi$
$467$	1.88405e9	0.856021	0.428011	+	0.903774i	$0.359215\pi$
$468$	0	0
$469$	4.17969e8	0.187085
$470$	0	0
$471$	−1.01590e9	−0.448000
$472$	0	0
$473$	9.18275e8	0.398987
$474$	0	0
$475$	3.63466e9	1.55609
$476$	0	0
$477$	7.79336e8	0.328784
$478$	0	0
$479$	−9.53861e8	−0.396562	−0.198281	−	0.980145i	$-0.563536\pi$
$479$	−9.53861e8	−0.396562	−0.198281	+	0.980145i	$0.563536\pi$
$480$	0	0
$481$	−7.84910e8	−0.321597
$482$	0	0
$483$	4.09860e8	0.165509
$484$	0	0
$485$	6.57421e8	0.261666
$486$	0	0
$487$	2.59346e9	1.01748	0.508742	−	0.860919i	$-0.330111\pi$
$487$	2.59346e9	1.01748	0.508742	+	0.860919i	$0.330111\pi$
$488$	0	0
$489$	1.05558e9	0.408234
$490$	0	0
$491$	−2.95226e9	−1.12556	−0.562780	−	0.826606i	$-0.690268\pi$
$491$	−2.95226e9	−1.12556	−0.562780	+	0.826606i	$0.690268\pi$
$492$	0	0
$493$	−2.69743e9	−1.01388
$494$	0	0
$495$	−2.62866e8	−0.0974129
$496$	0	0
$497$	−1.44473e8	−0.0527886
$498$	0	0
$499$	4.13817e8	0.149093	0.0745463	−	0.997218i	$-0.476249\pi$
$499$	4.13817e8	0.149093	0.0745463	+	0.997218i	$0.476249\pi$
$500$	0	0
$501$	2.26276e9	0.803907
$502$	0	0
$503$	3.99193e9	1.39860	0.699302	−	0.714827i	$-0.253494\pi$
$503$	3.99193e9	1.39860	0.699302	+	0.714827i	$0.253494\pi$
$504$	0	0
$505$	6.02966e8	0.208340
$506$	0	0
$507$	−1.03079e9	−0.351270
$508$	0	0
$509$	3.11683e9	1.04761	0.523807	−	0.851837i	$-0.324511\pi$
$509$	3.11683e9	1.04761	0.523807	+	0.851837i	$0.324511\pi$
$510$	0	0
$511$	1.49452e8	0.0495482
$512$	0	0
$513$	−9.68161e8	−0.316619
$514$	0	0
$515$	−4.35706e8	−0.140562
$516$	0	0
$517$	−4.88753e9	−1.55551
$518$	0	0
$519$	−2.56491e9	−0.805354
$520$	0	0
$521$	4.65025e8	0.144060	0.0720301	−	0.997402i	$-0.477052\pi$
$521$	4.65025e8	0.144060	0.0720301	+	0.997402i	$0.477052\pi$
$522$	0	0
$523$	1.35138e9	0.413070	0.206535	−	0.978439i	$-0.433781\pi$
$523$	1.35138e9	0.413070	0.206535	+	0.978439i	$0.433781\pi$
$524$	0	0
$525$	5.49149e8	0.165628
$526$	0	0
$527$	−5.05972e9	−1.50588
$528$	0	0
$529$	−3.63206e8	−0.106674
$530$	0	0
$531$	−8.77400e8	−0.254312
$532$	0	0
$533$	2.03901e9	0.583276
$534$	0	0
$535$	9.94448e8	0.280766
$536$	0	0
$537$	7.07546e8	0.197172
$538$	0	0
$539$	−4.14517e9	−1.14020
$540$	0	0
$541$	2.89823e9	0.786941	0.393470	−	0.919337i	$-0.371274\pi$
$541$	2.89823e9	0.786941	0.393470	+	0.919337i	$0.371274\pi$
$542$	0	0
$543$	−1.17614e9	−0.315252
$544$	0	0
$545$	1.51325e9	0.400425
$546$	0	0
$547$	2.18287e9	0.570259	0.285129	−	0.958489i	$-0.407963\pi$
$547$	2.18287e9	0.570259	0.285129	+	0.958489i	$0.407963\pi$
$548$	0	0
$549$	5.27456e8	0.136045
$550$	0	0
$551$	−6.52205e9	−1.66094
$552$	0	0
$553$	8.92387e8	0.224396
$554$	0	0
$555$	2.78106e8	0.0690535
$556$	0	0
$557$	−9.04284e8	−0.221723	−0.110862	−	0.993836i	$-0.535361\pi$
$557$	−9.04284e8	−0.221723	−0.110862	+	0.993836i	$0.535361\pi$
$558$	0	0
$559$	8.21145e8	0.198828
$560$	0	0
$561$	−3.04476e9	−0.728086
$562$	0	0
$563$	3.09873e9	0.731820	0.365910	−	0.930650i	$-0.380758\pi$
$563$	3.09873e9	0.731820	0.365910	+	0.930650i	$0.380758\pi$
$564$	0	0
$565$	5.73501e8	0.133772
$566$	0	0
$567$	−1.46277e8	−0.0337003
$568$	0	0
$569$	−3.33477e9	−0.758880	−0.379440	−	0.925216i	$-0.623883\pi$
$569$	−3.33477e9	−0.758880	−0.379440	+	0.925216i	$0.623883\pi$
$570$	0	0
$571$	1.76279e9	0.396255	0.198128	−	0.980176i	$-0.436514\pi$
$571$	1.76279e9	0.396255	0.198128	+	0.980176i	$0.436514\pi$
$572$	0	0
$573$	−3.13955e9	−0.697149
$574$	0	0
$575$	4.07530e9	0.893968
$576$	0	0
$577$	4.31922e9	0.936030	0.468015	−	0.883720i	$-0.344969\pi$
$577$	4.31922e9	0.936030	0.468015	+	0.883720i	$0.344969\pi$
$578$	0	0
$579$	−2.95878e9	−0.633486
$580$	0	0
$581$	−7.70120e8	−0.162908
$582$	0	0
$583$	5.92603e9	1.23858
$584$	0	0
$585$	−2.35062e8	−0.0485441
$586$	0	0
$587$	3.62609e9	0.739955	0.369978	−	0.929041i	$-0.379365\pi$
$587$	3.62609e9	0.739955	0.369978	+	0.929041i	$0.379365\pi$
$588$	0	0
$589$	−1.22338e10	−2.46693
$590$	0	0
$591$	4.17230e9	0.831418
$592$	0	0
$593$	−6.25587e9	−1.23196	−0.615980	−	0.787762i	$-0.711240\pi$
$593$	−6.25587e9	−1.23196	−0.615980	+	0.787762i	$0.711240\pi$
$594$	0	0
$595$	−3.64236e8	−0.0708881
$596$	0	0
$597$	3.16457e9	0.608701
$598$	0	0
$599$	1.87712e8	0.0356860	0.0178430	−	0.999841i	$-0.494320\pi$
$599$	1.87712e8	0.0356860	0.0178430	+	0.999841i	$0.494320\pi$
$600$	0	0
$601$	8.67744e9	1.63054	0.815269	−	0.579082i	$-0.196589\pi$
$601$	8.67744e9	1.63054	0.815269	+	0.579082i	$0.196589\pi$
$602$	0	0
$603$	−1.10701e9	−0.205609
$604$	0	0
$605$	−7.31198e8	−0.134243
$606$	0	0
$607$	9.47758e9	1.72003	0.860017	−	0.510265i	$-0.170453\pi$
$607$	9.47758e9	1.72003	0.860017	+	0.510265i	$0.170453\pi$
$608$	0	0
$609$	−9.85397e8	−0.176787
$610$	0	0
$611$	−4.37056e9	−0.775162
$612$	0	0
$613$	−9.39813e9	−1.64790	−0.823948	−	0.566665i	$-0.808233\pi$
$613$	−9.39813e9	−1.64790	−0.823948	+	0.566665i	$0.808233\pi$
$614$	0	0
$615$	−7.22454e8	−0.125241
$616$	0	0
$617$	9.02430e8	0.154673	0.0773367	−	0.997005i	$-0.475358\pi$
$617$	9.02430e8	0.154673	0.0773367	+	0.997005i	$0.475358\pi$
$618$	0	0
$619$	9.94894e9	1.68601	0.843004	−	0.537908i	$-0.180785\pi$
$619$	9.94894e9	1.68601	0.843004	+	0.537908i	$0.180785\pi$
$620$	0	0
$621$	−1.08553e9	−0.181896
$622$	0	0
$623$	4.24544e8	0.0703419
$624$	0	0
$625$	5.12969e9	0.840449
$626$	0	0
$627$	−7.36184e9	−1.19275
$628$	0	0
$629$	3.22128e9	0.516121
$630$	0	0
$631$	5.16471e8	0.0818358	0.0409179	−	0.999163i	$-0.486972\pi$
$631$	5.16471e8	0.0818358	0.0409179	+	0.999163i	$0.486972\pi$
$632$	0	0
$633$	−6.33038e9	−0.992011
$634$	0	0
$635$	−1.00467e9	−0.155709
$636$	0	0
$637$	−3.70672e9	−0.568200
$638$	0	0
$639$	3.82644e8	0.0580152
$640$	0	0
$641$	−6.32326e9	−0.948283	−0.474141	−	0.880449i	$-0.657241\pi$
$641$	−6.32326e9	−0.948283	−0.474141	+	0.880449i	$0.657241\pi$
$642$	0	0
$643$	−1.34254e9	−0.199153	−0.0995766	−	0.995030i	$-0.531749\pi$
$643$	−1.34254e9	−0.199153	−0.0995766	+	0.995030i	$0.531749\pi$
$644$	0	0
$645$	−2.90945e8	−0.0426925
$646$	0	0
$647$	6.15547e9	0.893504	0.446752	−	0.894658i	$-0.352581\pi$
$647$	6.15547e9	0.893504	0.446752	+	0.894658i	$0.352581\pi$
$648$	0	0
$649$	−6.67170e9	−0.958032
$650$	0	0
$651$	−1.84836e9	−0.262575
$652$	0	0
$653$	9.94403e9	1.39755	0.698773	−	0.715343i	$-0.253730\pi$
$653$	9.94403e9	1.39755	0.698773	+	0.715343i	$0.253730\pi$
$654$	0	0
$655$	1.68253e9	0.233947
$656$	0	0
$657$	−3.95830e8	−0.0544540
$658$	0	0
$659$	−5.58150e9	−0.759718	−0.379859	−	0.925044i	$-0.624027\pi$
$659$	−5.58150e9	−0.759718	−0.379859	+	0.925044i	$0.624027\pi$
$660$	0	0
$661$	−4.82898e9	−0.650354	−0.325177	−	0.945653i	$-0.605424\pi$
$661$	−4.82898e9	−0.650354	−0.325177	+	0.945653i	$0.605424\pi$
$662$	0	0
$663$	−2.72270e9	−0.362830
$664$	0	0
$665$	−8.80677e8	−0.116129
$666$	0	0
$667$	−7.31275e9	−0.954201
$668$	0	0
$669$	6.38734e9	0.824762
$670$	0	0
$671$	4.01075e9	0.512503
$672$	0	0
$673$	−3.20000e9	−0.404666	−0.202333	−	0.979317i	$-0.564852\pi$
$673$	−3.20000e9	−0.404666	−0.202333	+	0.979317i	$0.564852\pi$
$674$	0	0
$675$	−1.45445e9	−0.182027
$676$	0	0
$677$	−4.97836e9	−0.616632	−0.308316	−	0.951284i	$-0.599765\pi$
$677$	−4.97836e9	−0.616632	−0.308316	+	0.951284i	$0.599765\pi$
$678$	0	0
$679$	2.78178e9	0.341018
$680$	0	0
$681$	−5.94112e9	−0.720865
$682$	0	0
$683$	−1.56314e10	−1.87726	−0.938630	−	0.344926i	$-0.887904\pi$
$683$	−1.56314e10	−1.87726	−0.938630	+	0.344926i	$0.887904\pi$
$684$	0	0
$685$	1.45063e9	0.172441
$686$	0	0
$687$	−7.50800e9	−0.883437
$688$	0	0
$689$	5.29921e9	0.617225
$690$	0	0
$691$	−7.62162e8	−0.0878768	−0.0439384	−	0.999034i	$-0.513991\pi$
$691$	−7.62162e8	−0.0878768	−0.0439384	+	0.999034i	$0.513991\pi$
$692$	0	0
$693$	−1.11228e9	−0.126954
$694$	0	0
$695$	1.58274e9	0.178839
$696$	0	0
$697$	−8.36813e9	−0.936081
$698$	0	0
$699$	1.67678e9	0.185697
$700$	0	0
$701$	−1.20366e10	−1.31974	−0.659872	−	0.751378i	$-0.729390\pi$
$701$	−1.20366e10	−1.31974	−0.659872	+	0.751378i	$0.729390\pi$
$702$	0	0
$703$	7.78866e9	0.845510
$704$	0	0
$705$	1.54856e9	0.166443
$706$	0	0
$707$	2.55136e9	0.271521
$708$	0	0
$709$	−2.95168e9	−0.311034	−0.155517	−	0.987833i	$-0.549704\pi$
$709$	−2.95168e9	−0.311034	−0.155517	+	0.987833i	$0.549704\pi$
$710$	0	0
$711$	−2.36353e9	−0.246614
$712$	0	0
$713$	−1.37169e10	−1.41724
$714$	0	0
$715$	−1.78740e9	−0.182873
$716$	0	0
$717$	9.43852e9	0.956283
$718$	0	0
$719$	−1.15653e10	−1.16040	−0.580198	−	0.814476i	$-0.697025\pi$
$719$	−1.15653e10	−1.16040	−0.580198	+	0.814476i	$0.697025\pi$
$720$	0	0
$721$	−1.84362e9	−0.183189
$722$	0	0
$723$	2.28801e9	0.225151
$724$	0	0
$725$	−9.79795e9	−0.954887
$726$	0	0
$727$	1.35771e10	1.31050	0.655249	−	0.755413i	$-0.272564\pi$
$727$	1.35771e10	1.31050	0.655249	+	0.755413i	$0.272564\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−3.36999e9	−0.319093
$732$	0	0
$733$	−7.68831e9	−0.721053	−0.360527	−	0.932749i	$-0.617403\pi$
$733$	−7.68831e9	−0.721053	−0.360527	+	0.932749i	$0.617403\pi$
$734$	0	0
$735$	1.31335e9	0.122004
$736$	0	0
$737$	−8.41764e9	−0.774559
$738$	0	0
$739$	−6.20836e9	−0.565876	−0.282938	−	0.959138i	$-0.591309\pi$
$739$	−6.20836e9	−0.565876	−0.282938	+	0.959138i	$0.591309\pi$
$740$	0	0
$741$	−6.58315e9	−0.594388
$742$	0	0
$743$	4.70406e9	0.420738	0.210369	−	0.977622i	$-0.432533\pi$
$743$	4.70406e9	0.420738	0.210369	+	0.977622i	$0.432533\pi$
$744$	0	0
$745$	4.77172e9	0.422793
$746$	0	0
$747$	2.03970e9	0.179038
$748$	0	0
$749$	4.20786e9	0.365910
$750$	0	0
$751$	1.36220e10	1.17355	0.586774	−	0.809750i	$-0.300398\pi$
$751$	1.36220e10	1.17355	0.586774	+	0.809750i	$0.300398\pi$
$752$	0	0
$753$	−1.43630e9	−0.122592
$754$	0	0
$755$	−2.70819e9	−0.229016
$756$	0	0
$757$	−6.49409e9	−0.544105	−0.272052	−	0.962282i	$-0.587702\pi$
$757$	−6.49409e9	−0.544105	−0.272052	+	0.962282i	$0.587702\pi$
$758$	0	0
$759$	−8.25434e9	−0.685230
$760$	0	0
$761$	−1.76658e10	−1.45307	−0.726537	−	0.687128i	$-0.758871\pi$
$761$	−1.76658e10	−1.45307	−0.726537	+	0.687128i	$0.758871\pi$
$762$	0	0
$763$	6.40307e9	0.521858
$764$	0	0
$765$	9.64697e8	0.0779069
$766$	0	0
$767$	−5.96601e9	−0.477420
$768$	0	0
$769$	1.98045e10	1.57044	0.785219	−	0.619218i	$-0.212550\pi$
$769$	1.98045e10	1.57044	0.785219	+	0.619218i	$0.212550\pi$
$770$	0	0
$771$	−1.16506e10	−0.915500
$772$	0	0
$773$	2.80123e8	0.0218132	0.0109066	−	0.999941i	$-0.496528\pi$
$773$	2.80123e8	0.0218132	0.0109066	+	0.999941i	$0.496528\pi$
$774$	0	0
$775$	−1.83786e10	−1.41826
$776$	0	0
$777$	1.17676e9	0.0899945
$778$	0	0
$779$	−2.02331e10	−1.53349
$780$	0	0
$781$	2.90960e9	0.218552
$782$	0	0
$783$	2.60987e9	0.194291
$784$	0	0
$785$	2.44753e9	0.180586
$786$	0	0
$787$	6.73837e9	0.492769	0.246384	−	0.969172i	$-0.420757\pi$
$787$	6.73837e9	0.492769	0.246384	+	0.969172i	$0.420757\pi$
$788$	0	0
$789$	1.51969e9	0.110150
$790$	0	0
$791$	2.42668e9	0.174339
$792$	0	0
$793$	3.58652e9	0.255398
$794$	0	0
$795$	−1.87759e9	−0.132531
$796$	0	0
$797$	1.01524e10	0.710339	0.355169	−	0.934802i	$-0.384423\pi$
$797$	1.01524e10	0.710339	0.355169	+	0.934802i	$0.384423\pi$
$798$	0	0
$799$	1.79368e10	1.24403
$800$	0	0
$801$	−1.12442e9	−0.0773066
$802$	0	0
$803$	−3.00987e9	−0.205137
$804$	0	0
$805$	−9.87445e8	−0.0667156
$806$	0	0
$807$	−4.24919e9	−0.284609
$808$	0	0
$809$	−1.39524e10	−0.926462	−0.463231	−	0.886237i	$-0.653310\pi$
$809$	−1.39524e10	−0.926462	−0.463231	+	0.886237i	$0.653310\pi$
$810$	0	0
$811$	2.65022e10	1.74466	0.872328	−	0.488921i	$-0.162610\pi$
$811$	2.65022e10	1.74466	0.872328	+	0.488921i	$0.162610\pi$
$812$	0	0
$813$	−7.07053e8	−0.0461461
$814$	0	0
$815$	−2.54312e9	−0.164557
$816$	0	0
$817$	−8.14821e9	−0.522739
$818$	0	0
$819$	−9.94628e8	−0.0632655
$820$	0	0
$821$	−4.48371e9	−0.282772	−0.141386	−	0.989955i	$-0.545156\pi$
$821$	−4.48371e9	−0.282772	−0.141386	+	0.989955i	$0.545156\pi$
$822$	0	0
$823$	1.00006e8	0.00625354	0.00312677	−	0.999995i	$-0.499005\pi$
$823$	1.00006e8	0.00625354	0.00312677	+	0.999995i	$0.499005\pi$
$824$	0	0
$825$	−1.10595e10	−0.685722
$826$	0	0
$827$	−9.16338e8	−0.0563361	−0.0281680	−	0.999603i	$-0.508967\pi$
$827$	−9.16338e8	−0.0563361	−0.0281680	+	0.999603i	$0.508967\pi$
$828$	0	0
$829$	−3.85040e8	−0.0234728	−0.0117364	−	0.999931i	$-0.503736\pi$
$829$	−3.85040e8	−0.0234728	−0.0117364	+	0.999931i	$0.503736\pi$
$830$	0	0
$831$	1.53727e10	0.929278
$832$	0	0
$833$	1.52124e10	0.911886
$834$	0	0
$835$	−5.45148e9	−0.324050
$836$	0	0
$837$	4.89548e9	0.288574
$838$	0	0
$839$	1.54534e10	0.903350	0.451675	−	0.892183i	$-0.350827\pi$
$839$	1.54534e10	0.903350	0.451675	+	0.892183i	$0.350827\pi$
$840$	0	0
$841$	3.31631e8	0.0192251
$842$	0	0
$843$	−1.69601e9	−0.0975058
$844$	0	0
$845$	2.48339e9	0.141595
$846$	0	0
$847$	−3.09395e9	−0.174953
$848$	0	0
$849$	4.29514e8	0.0240880
$850$	0	0
$851$	8.73291e9	0.485742
$852$	0	0
$853$	−1.52946e10	−0.843753	−0.421876	−	0.906653i	$-0.638628\pi$
$853$	−1.52946e10	−0.843753	−0.421876	+	0.906653i	$0.638628\pi$
$854$	0	0
$855$	2.33252e9	0.127627
$856$	0	0
$857$	2.34165e9	0.127084	0.0635419	−	0.997979i	$-0.479760\pi$
$857$	2.34165e9	0.127084	0.0635419	+	0.997979i	$0.479760\pi$
$858$	0	0
$859$	2.11238e9	0.113709	0.0568546	−	0.998382i	$-0.481893\pi$
$859$	2.11238e9	0.113709	0.0568546	+	0.998382i	$0.481893\pi$
$860$	0	0
$861$	−3.05696e9	−0.163222
$862$	0	0
$863$	−3.54620e10	−1.87813	−0.939065	−	0.343740i	$-0.888306\pi$
$863$	−3.54620e10	−1.87813	−0.939065	+	0.343740i	$0.888306\pi$
$864$	0	0
$865$	6.17944e9	0.324633
$866$	0	0
$867$	9.48588e7	0.00494323
$868$	0	0
$869$	−1.79722e10	−0.929033
$870$	0	0
$871$	−7.52728e9	−0.385988
$872$	0	0
$873$	−7.36767e9	−0.374783
$874$	0	0
$875$	−2.72181e9	−0.137350
$876$	0	0
$877$	2.20646e10	1.10458	0.552289	−	0.833652i	$-0.313754\pi$
$877$	2.20646e10	1.10458	0.552289	+	0.833652i	$0.313754\pi$
$878$	0	0
$879$	−8.00532e9	−0.397574
$880$	0	0
$881$	2.74750e10	1.35370	0.676850	−	0.736121i	$-0.263345\pi$
$881$	2.74750e10	1.35370	0.676850	+	0.736121i	$0.263345\pi$
$882$	0	0
$883$	−2.14560e9	−0.104878	−0.0524392	−	0.998624i	$-0.516700\pi$
$883$	−2.14560e9	−0.104878	−0.0524392	+	0.998624i	$0.516700\pi$
$884$	0	0
$885$	2.11385e9	0.102512
$886$	0	0
$887$	1.01285e10	0.487317	0.243658	−	0.969861i	$-0.421652\pi$
$887$	1.01285e10	0.487317	0.243658	+	0.969861i	$0.421652\pi$
$888$	0	0
$889$	−4.25110e9	−0.202929
$890$	0	0
$891$	2.94592e9	0.139524
$892$	0	0
$893$	4.33690e10	2.03798
$894$	0	0
$895$	−1.70464e9	−0.0794787
$896$	0	0
$897$	−7.38125e9	−0.341473
$898$	0	0
$899$	3.29786e10	1.51382
$900$	0	0
$901$	−2.17480e10	−0.990565
$902$	0	0
$903$	−1.23109e9	−0.0556394
$904$	0	0
$905$	2.83357e9	0.127076
$906$	0	0
$907$	−3.37970e10	−1.50402	−0.752008	−	0.659154i	$-0.770914\pi$
$907$	−3.37970e10	−1.50402	−0.752008	+	0.659154i	$0.770914\pi$
$908$	0	0
$909$	−6.75740e9	−0.298405
$910$	0	0
$911$	−1.02764e10	−0.450325	−0.225163	−	0.974321i	$-0.572291\pi$
$911$	−1.02764e10	−0.450325	−0.225163	+	0.974321i	$0.572291\pi$
$912$	0	0
$913$	1.55098e10	0.674462
$914$	0	0
$915$	−1.27076e9	−0.0548390
$916$	0	0
$917$	7.11937e9	0.304894
$918$	0	0
$919$	2.82318e10	1.19987	0.599935	−	0.800049i	$-0.295193\pi$
$919$	2.82318e10	1.19987	0.599935	+	0.800049i	$0.295193\pi$
$920$	0	0
$921$	1.20008e10	0.506175
$922$	0	0
$923$	2.60184e9	0.108912
$924$	0	0
$925$	1.17007e10	0.486091
$926$	0	0
$927$	4.88293e9	0.201327
$928$	0	0
$929$	1.51907e10	0.621617	0.310808	−	0.950473i	$-0.399400\pi$
$929$	1.51907e10	0.621617	0.310808	+	0.950473i	$0.399400\pi$
$930$	0	0
$931$	3.67817e10	1.49385
$932$	0	0
$933$	6.49619e9	0.261862
$934$	0	0
$935$	7.33550e9	0.293487
$936$	0	0
$937$	−9.14565e9	−0.363183	−0.181592	−	0.983374i	$-0.558125\pi$
$937$	−9.14565e9	−0.363183	−0.181592	+	0.983374i	$0.558125\pi$
$938$	0	0
$939$	2.63357e10	1.03804
$940$	0	0
$941$	−4.70181e9	−0.183951	−0.0919754	−	0.995761i	$-0.529318\pi$
$941$	−4.70181e9	−0.183951	−0.0919754	+	0.995761i	$0.529318\pi$
$942$	0	0
$943$	−2.26860e10	−0.880983
$944$	0	0
$945$	3.52413e8	0.0135844
$946$	0	0
$947$	4.28407e10	1.63920	0.819600	−	0.572936i	$-0.194196\pi$
$947$	4.28407e10	1.63920	0.819600	+	0.572936i	$0.194196\pi$
$948$	0	0
$949$	−2.69150e9	−0.102226
$950$	0	0
$951$	−5.14396e9	−0.193939
$952$	0	0
$953$	4.14045e10	1.54961	0.774804	−	0.632201i	$-0.217848\pi$
$953$	4.14045e10	1.54961	0.774804	+	0.632201i	$0.217848\pi$
$954$	0	0
$955$	7.56387e9	0.281017
$956$	0	0
$957$	1.98453e10	0.731925
$958$	0	0
$959$	6.13812e9	0.224735
$960$	0	0
$961$	3.43472e10	1.24842
$962$	0	0
$963$	−1.11447e10	−0.402140
$964$	0	0
$965$	7.12835e9	0.255354
$966$	0	0
$967$	−4.22566e10	−1.50280	−0.751402	−	0.659845i	$-0.770622\pi$
$967$	−4.22566e10	−1.50280	−0.751402	+	0.659845i	$0.770622\pi$
$968$	0	0
$969$	2.70173e10	0.953914
$970$	0	0
$971$	2.50043e10	0.876491	0.438245	−	0.898855i	$-0.355600\pi$
$971$	2.50043e10	0.876491	0.438245	+	0.898855i	$0.355600\pi$
$972$	0	0
$973$	6.69711e9	0.233073
$974$	0	0
$975$	−9.88973e9	−0.341718
$976$	0	0
$977$	1.67463e10	0.574498	0.287249	−	0.957856i	$-0.407259\pi$
$977$	1.67463e10	0.574498	0.287249	+	0.957856i	$0.407259\pi$
$978$	0	0
$979$	−8.55006e9	−0.291226
$980$	0	0
$981$	−1.69588e10	−0.573528
$982$	0	0
$983$	3.79707e9	0.127500	0.0637502	−	0.997966i	$-0.479694\pi$
$983$	3.79707e9	0.127500	0.0637502	+	0.997966i	$0.479694\pi$
$984$	0	0
$985$	−1.00520e10	−0.335139
$986$	0	0
$987$	6.55249e9	0.216918
$988$	0	0
$989$	−9.13605e9	−0.300311
$990$	0	0
$991$	7.52745e9	0.245692	0.122846	−	0.992426i	$-0.460798\pi$
$991$	7.52745e9	0.245692	0.122846	+	0.992426i	$0.460798\pi$
$992$	0	0
$993$	1.66934e10	0.541030
$994$	0	0
$995$	−7.62415e9	−0.245364
$996$	0	0
$997$	1.29435e9	0.0413637	0.0206819	−	0.999786i	$-0.493416\pi$
$997$	1.29435e9	0.0413637	0.0206819	+	0.999786i	$0.493416\pi$
$998$	0	0
$999$	−3.11672e9	−0.0989051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.h.1.1	yes	2
4.3	odd	2		384.8.a.f.1.1	yes	2
8.3	odd	2		384.8.a.g.1.2	yes	2
8.5	even	2		384.8.a.e.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.e.1.2	✓	2	8.5	even	2
384.8.a.f.1.1	yes	2	4.3	odd	2
384.8.a.g.1.2	yes	2	8.3	odd	2
384.8.a.h.1.1	yes	2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -65.0490 q^{5} -275.245 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -65.0490 q^{5} -275.245 q^{7} +729.000 q^{9} +5543.27 q^{11} +4956.94 q^{13} -1756.32 q^{15} -20343.4 q^{17} -49187.7 q^{19} -7431.62 q^{21} -55150.9 q^{23} -73893.6 q^{25} +19683.0 q^{27} +132595. q^{29} +248716. q^{31} +149668. q^{33} +17904.4 q^{35} -158346. q^{37} +133837. q^{39} +411345. q^{41} +165656. q^{43} -47420.7 q^{45} -881705. q^{47} -747783. q^{49} -549270. q^{51} +1.06905e6 q^{53} -360585. q^{55} -1.32807e6 q^{57} -1.20357e6 q^{59} +723534. q^{61} -200654. q^{63} -322444. q^{65} -1.51853e6 q^{67} -1.48907e6 q^{69} +524889. q^{71} -542977. q^{73} -1.99513e6 q^{75} -1.52576e6 q^{77} -3.24216e6 q^{79} +531441. q^{81} +2.79794e6 q^{83} +1.32331e6 q^{85} +3.58007e6 q^{87} -1.54242e6 q^{89} -1.36437e6 q^{91} +6.71534e6 q^{93} +3.19961e6 q^{95} -1.01065e7 q^{97} +4.04105e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{3} + 176 q^{5} + 980 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q + 54 * q^3 + 176 * q^5 + 980 * q^7 + 1458 * q^9	\( 2 q + 54 q^{3} + 176 q^{5} + 980 q^{7} + 1458 q^{9} + 3128 q^{11} - 8452 q^{13} + 4752 q^{15} + 5228 q^{17} - 87968 q^{19} + 26460 q^{21} - 104792 q^{23} - 93914 q^{25} + 39366 q^{27} - 98760 q^{29} + 1860 q^{31} + 84456 q^{33} + 320480 q^{35} - 455660 q^{37} - 228204 q^{39} - 205188 q^{41} - 803088 q^{43} + 128304 q^{45} - 522488 q^{47} + 4314 q^{49} + 141156 q^{51} + 2468376 q^{53} - 942784 q^{55} - 2375136 q^{57} - 989288 q^{59} + 2482292 q^{61} + 714420 q^{63} - 3554656 q^{65} - 434008 q^{67} - 2829384 q^{69} + 1554840 q^{71} + 279244 q^{73} - 2535678 q^{75} - 4557520 q^{77} + 1015396 q^{79} + 1062882 q^{81} - 2899560 q^{83} + 7487264 q^{85} - 2666520 q^{87} - 8115868 q^{89} - 18195880 q^{91} + 50220 q^{93} - 6148352 q^{95} - 2553676 q^{97} + 2280312 q^{99}+O(q^{100}) \) 2 * q + 54 * q^3 + 176 * q^5 + 980 * q^7 + 1458 * q^9 + 3128 * q^11 - 8452 * q^13 + 4752 * q^15 + 5228 * q^17 - 87968 * q^19 + 26460 * q^21 - 104792 * q^23 - 93914 * q^25 + 39366 * q^27 - 98760 * q^29 + 1860 * q^31 + 84456 * q^33 + 320480 * q^35 - 455660 * q^37 - 228204 * q^39 - 205188 * q^41 - 803088 * q^43 + 128304 * q^45 - 522488 * q^47 + 4314 * q^49 + 141156 * q^51 + 2468376 * q^53 - 942784 * q^55 - 2375136 * q^57 - 989288 * q^59 + 2482292 * q^61 + 714420 * q^63 - 3554656 * q^65 - 434008 * q^67 - 2829384 * q^69 + 1554840 * q^71 + 279244 * q^73 - 2535678 * q^75 - 4557520 * q^77 + 1015396 * q^79 + 1062882 * q^81 - 2899560 * q^83 + 7487264 * q^85 - 2666520 * q^87 - 8115868 * q^89 - 18195880 * q^91 + 50220 * q^93 - 6148352 * q^95 - 2553676 * q^97 + 2280312 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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