LMFDB - Embedded newform 592.8.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,8,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, 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("592.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$592 = 2^{4} \cdot 37$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	592.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:	$184.931935087$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 10621x^{4} + 102052x^{3} + 31004503x^{2} - 305547358x - 22608804936$ x^6 - 2x^5 - 10621x^4 + 102052x^3 + 31004503x^2 - 305547358*x - 22608804936
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-26.0193$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-31.0193 q^{3} -544.510 q^{5} +1685.44 q^{7} -1224.80 q^{9} -5629.72 q^{11} +5228.81 q^{13} +16890.3 q^{15} -12575.5 q^{17} -18367.1 q^{19} -52281.0 q^{21} -40252.9 q^{23} +218366. q^{25} +105832. q^{27} +18705.8 q^{29} +169281. q^{31} +174630. q^{33} -917737. q^{35} -50653.0 q^{37} -162194. q^{39} -26437.9 q^{41} -406992. q^{43} +666918. q^{45} -175342. q^{47} +2.01715e6 q^{49} +390084. q^{51} -1.82314e6 q^{53} +3.06544e6 q^{55} +569735. q^{57} +1.24173e6 q^{59} +3.25092e6 q^{61} -2.06433e6 q^{63} -2.84714e6 q^{65} +1.66755e6 q^{67} +1.24862e6 q^{69} -262493. q^{71} +2.68612e6 q^{73} -6.77356e6 q^{75} -9.48853e6 q^{77} -363284. q^{79} -604180. q^{81} +6.71044e6 q^{83} +6.84750e6 q^{85} -580239. q^{87} +1.16083e7 q^{89} +8.81283e6 q^{91} -5.25097e6 q^{93} +1.00011e7 q^{95} +1.63488e7 q^{97} +6.89530e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 28 q^{3} - 14 q^{5} + 980 q^{7} + 8254 q^{9} - 2956 q^{11} + 2394 q^{13} + 28820 q^{15} - 45108 q^{17} - 11764 q^{19} - 135378 q^{21} - 21052 q^{23} + 194744 q^{25} - 439240 q^{27} + 288454 q^{29}+ \cdots - 25990712 q^{99}+O(q^{100})$ 6 * q - 28 * q^3 - 14 * q^5 + 980 * q^7 + 8254 * q^9 - 2956 * q^11 + 2394 * q^13 + 28820 * q^15 - 45108 * q^17 - 11764 * q^19 - 135378 * q^21 - 21052 * q^23 + 194744 * q^25 - 439240 * q^27 + 288454 * q^29 - 578868 * q^31 + 980174 * q^33 - 1243052 * q^35 - 303918 * q^37 - 1735296 * q^39 + 1176840 * q^41 - 2669236 * q^43 + 2560692 * q^45 + 131044 * q^47 + 2460856 * q^49 - 2899732 * q^51 + 983190 * q^53 + 1200168 * q^55 - 163216 * q^57 + 1215568 * q^59 + 3136358 * q^61 + 1444880 * q^63 - 1302836 * q^65 - 2179276 * q^67 - 929514 * q^69 - 325164 * q^71 + 5011444 * q^73 + 9374520 * q^75 - 26500426 * q^77 - 3173032 * q^79 - 2565226 * q^81 + 22567048 * q^83 + 1486476 * q^85 + 157228 * q^87 + 26836996 * q^89 - 17942380 * q^91 + 16734948 * q^93 + 4252048 * q^95 + 295792 * q^97 - 25990712 * q^99

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	−31.0193	−0.663296	−0.331648	−	0.943403i	$-0.607605\pi$
$3$	−31.0193	−0.663296	−0.331648	+	0.943403i	$0.607605\pi$
$4$	0	0
$5$	−544.510	−1.94810	−0.974049	−	0.226337i	$-0.927325\pi$
$5$	−544.510	−1.94810	−0.974049	+	0.226337i	$0.927325\pi$
$6$	0	0
$7$	1685.44	1.85724	0.928622	−	0.371026i	$-0.120994\pi$
$7$	1685.44	1.85724	0.928622	+	0.371026i	$0.120994\pi$
$8$	0	0
$9$	−1224.80	−0.560038
$10$	0	0
$11$	−5629.72	−1.27530	−0.637650	−	0.770327i	$-0.720093\pi$
$11$	−5629.72	−1.27530	−0.637650	+	0.770327i	$0.720093\pi$
$12$	0	0
$13$	5228.81	0.660087	0.330044	−	0.943966i	$-0.392937\pi$
$13$	5228.81	0.660087	0.330044	+	0.943966i	$0.392937\pi$
$14$	0	0
$15$	16890.3	1.29217
$16$	0	0
$17$	−12575.5	−0.620805	−0.310402	−	0.950605i	$-0.600464\pi$
$17$	−12575.5	−0.620805	−0.310402	+	0.950605i	$0.600464\pi$
$18$	0	0
$19$	−18367.1	−0.614332	−0.307166	−	0.951656i	$-0.599381\pi$
$19$	−18367.1	−0.614332	−0.307166	+	0.951656i	$0.599381\pi$
$20$	0	0
$21$	−52281.0	−1.23190
$22$	0	0
$23$	−40252.9	−0.689843	−0.344921	−	0.938632i	$-0.612094\pi$
$23$	−40252.9	−0.689843	−0.344921	+	0.938632i	$0.612094\pi$
$24$	0	0
$25$	218366.	2.79509
$26$	0	0
$27$	105832.	1.03477
$28$	0	0
$29$	18705.8	0.142424	0.0712118	−	0.997461i	$-0.477313\pi$
$29$	18705.8	0.142424	0.0712118	+	0.997461i	$0.477313\pi$
$30$	0	0
$31$	169281.	1.02057	0.510284	−	0.860006i	$-0.329540\pi$
$31$	169281.	1.02057	0.510284	+	0.860006i	$0.329540\pi$
$32$	0	0
$33$	174630.	0.845901
$34$	0	0
$35$	−917737.	−3.61809
$36$	0	0
$37$	−50653.0	−0.164399
$37$	−50653.0	−0.164399
$38$	0	0
$39$	−162194.	−0.437833
$40$	0	0
$41$	−26437.9	−0.0599078	−0.0299539	−	0.999551i	$-0.509536\pi$
$41$	−26437.9	−0.0599078	−0.0299539	+	0.999551i	$0.509536\pi$
$42$	0	0
$43$	−406992.	−0.780632	−0.390316	−	0.920681i	$-0.627634\pi$
$43$	−406992.	−0.780632	−0.390316	+	0.920681i	$0.627634\pi$
$44$	0	0
$45$	666918.	1.09101
$46$	0	0
$47$	−175342.	−0.246345	−0.123173	−	0.992385i	$-0.539307\pi$
$47$	−175342.	−0.246345	−0.123173	+	0.992385i	$0.539307\pi$
$48$	0	0
$49$	2.01715e6	2.44936
$50$	0	0
$51$	390084.	0.411778
$52$	0	0
$53$	−1.82314e6	−1.68211	−0.841056	−	0.540949i	$-0.818065\pi$
$53$	−1.82314e6	−1.68211	−0.841056	+	0.540949i	$0.818065\pi$
$54$	0	0
$55$	3.06544e6	2.48441
$56$	0	0
$57$	569735.	0.407484
$58$	0	0
$59$	1.24173e6	0.787131	0.393565	−	0.919297i	$-0.371242\pi$
$59$	1.24173e6	0.787131	0.393565	+	0.919297i	$0.371242\pi$
$60$	0	0
$61$	3.25092e6	1.83380	0.916899	−	0.399120i	$-0.130684\pi$
$61$	3.25092e6	1.83380	0.916899	+	0.399120i	$0.130684\pi$
$62$	0	0
$63$	−2.06433e6	−1.04013
$64$	0	0
$65$	−2.84714e6	−1.28592
$66$	0	0
$67$	1.66755e6	0.677357	0.338678	−	0.940902i	$-0.390020\pi$
$67$	1.66755e6	0.677357	0.338678	+	0.940902i	$0.390020\pi$
$68$	0	0
$69$	1.24862e6	0.457570
$70$	0	0
$71$	−262493.	−0.0870388	−0.0435194	−	0.999053i	$-0.513857\pi$
$71$	−262493.	−0.0870388	−0.0435194	+	0.999053i	$0.513857\pi$
$72$	0	0
$73$	2.68612e6	0.808157	0.404078	−	0.914724i	$-0.367592\pi$
$73$	2.68612e6	0.808157	0.404078	+	0.914724i	$0.367592\pi$
$74$	0	0
$75$	−6.77356e6	−1.85397
$76$	0	0
$77$	−9.48853e6	−2.36854
$78$	0	0
$79$	−363284.	−0.0828993	−0.0414497	−	0.999141i	$-0.513198\pi$
$79$	−363284.	−0.0828993	−0.0414497	+	0.999141i	$0.513198\pi$
$80$	0	0
$81$	−604180.	−0.126319
$82$	0	0
$83$	6.71044e6	1.28818	0.644091	−	0.764948i	$-0.277236\pi$
$83$	6.71044e6	1.28818	0.644091	+	0.764948i	$0.277236\pi$
$84$	0	0
$85$	6.84750e6	1.20939
$86$	0	0
$87$	−580239.	−0.0944691
$88$	0	0
$89$	1.16083e7	1.74543	0.872715	−	0.488229i	$-0.162357\pi$
$89$	1.16083e7	1.74543	0.872715	+	0.488229i	$0.162357\pi$
$90$	0	0
$91$	8.81283e6	1.22594
$92$	0	0
$93$	−5.25097e6	−0.676939
$94$	0	0
$95$	1.00011e7	1.19678
$96$	0	0
$97$	1.63488e7	1.81880	0.909400	−	0.415923i	$-0.136541\pi$
$97$	1.63488e7	1.81880	0.909400	+	0.415923i	$0.136541\pi$
$98$	0	0
$99$	6.89530e6	0.714216
$100$	0	0
$101$	7.79452e6	0.752774	0.376387	−	0.926462i	$-0.377166\pi$
$101$	7.79452e6	0.752774	0.376387	+	0.926462i	$0.377166\pi$
$102$	0	0
$103$	−8.12543e6	−0.732683	−0.366341	−	0.930481i	$-0.619390\pi$
$103$	−8.12543e6	−0.732683	−0.366341	+	0.930481i	$0.619390\pi$
$104$	0	0
$105$	2.84675e7	2.39987
$106$	0	0
$107$	−1.15348e6	−0.0910264	−0.0455132	−	0.998964i	$-0.514492\pi$
$107$	−1.15348e6	−0.0910264	−0.0455132	+	0.998964i	$0.514492\pi$
$108$	0	0
$109$	−8.60358e6	−0.636336	−0.318168	−	0.948034i	$-0.603068\pi$
$109$	−8.60358e6	−0.636336	−0.318168	+	0.948034i	$0.603068\pi$
$110$	0	0
$111$	1.57122e6	0.109045
$112$	0	0
$113$	−7.76780e6	−0.506435	−0.253217	−	0.967409i	$-0.581489\pi$
$113$	−7.76780e6	−0.506435	−0.253217	+	0.967409i	$0.581489\pi$
$114$	0	0
$115$	2.19181e7	1.34388
$116$	0	0
$117$	−6.40427e6	−0.369674
$118$	0	0
$119$	−2.11953e7	−1.15299
$120$	0	0
$121$	1.22065e7	0.626388
$122$	0	0
$123$	820084.	0.0397366
$124$	0	0
$125$	−7.63627e7	−3.49700
$126$	0	0
$127$	−4.49536e6	−0.194738	−0.0973691	−	0.995248i	$-0.531043\pi$
$127$	−4.49536e6	−0.194738	−0.0973691	+	0.995248i	$0.531043\pi$
$128$	0	0
$129$	1.26246e7	0.517791
$130$	0	0
$131$	1.57640e7	0.612656	0.306328	−	0.951926i	$-0.400900\pi$
$131$	1.57640e7	0.612656	0.306328	+	0.951926i	$0.400900\pi$
$132$	0	0
$133$	−3.09566e7	−1.14097
$134$	0	0
$135$	−5.76264e7	−2.01583
$136$	0	0
$137$	−9.27787e6	−0.308267	−0.154133	−	0.988050i	$-0.549259\pi$
$137$	−9.27787e6	−0.308267	−0.154133	+	0.988050i	$0.549259\pi$
$138$	0	0
$139$	−5.30404e7	−1.67515	−0.837577	−	0.546319i	$-0.816028\pi$
$139$	−5.30404e7	−1.67515	−0.837577	+	0.546319i	$0.816028\pi$
$140$	0	0
$141$	5.43899e6	0.163400
$142$	0	0
$143$	−2.94367e7	−0.841809
$144$	0	0
$145$	−1.01855e7	−0.277455
$146$	0	0
$147$	−6.25706e7	−1.62465
$148$	0	0
$149$	2.61358e7	0.647267	0.323633	−	0.946183i	$-0.395096\pi$
$149$	2.61358e7	0.647267	0.323633	+	0.946183i	$0.395096\pi$
$150$	0	0
$151$	3.32760e7	0.786523	0.393261	−	0.919427i	$-0.371347\pi$
$151$	3.32760e7	0.786523	0.393261	+	0.919427i	$0.371347\pi$
$152$	0	0
$153$	1.54026e7	0.347674
$154$	0	0
$155$	−9.21751e7	−1.98817
$156$	0	0
$157$	−2.08133e7	−0.429232	−0.214616	−	0.976698i	$-0.568850\pi$
$157$	−2.08133e7	−0.429232	−0.214616	+	0.976698i	$0.568850\pi$
$158$	0	0
$159$	5.65525e7	1.11574
$160$	0	0
$161$	−6.78438e7	−1.28121
$162$	0	0
$163$	−4.00702e7	−0.724711	−0.362355	−	0.932040i	$-0.618027\pi$
$163$	−4.00702e7	−0.724711	−0.362355	+	0.932040i	$0.618027\pi$
$164$	0	0
$165$	−9.50877e7	−1.64790
$166$	0	0
$167$	−4.26523e7	−0.708656	−0.354328	−	0.935121i	$-0.615290\pi$
$167$	−4.26523e7	−0.708656	−0.354328	+	0.935121i	$0.615290\pi$
$168$	0	0
$169$	−3.54080e7	−0.564285
$170$	0	0
$171$	2.24961e7	0.344050
$172$	0	0
$173$	7.13528e6	0.104773	0.0523865	−	0.998627i	$-0.483317\pi$
$173$	7.13528e6	0.104773	0.0523865	+	0.998627i	$0.483317\pi$
$174$	0	0
$175$	3.68042e8	5.19116
$176$	0	0
$177$	−3.85177e7	−0.522101
$178$	0	0
$179$	−1.39174e8	−1.81373	−0.906866	−	0.421419i	$-0.861532\pi$
$179$	−1.39174e8	−1.81373	−0.906866	+	0.421419i	$0.861532\pi$
$180$	0	0
$181$	−6.11153e7	−0.766082	−0.383041	−	0.923731i	$-0.625123\pi$
$181$	−6.11153e7	−0.766082	−0.383041	+	0.923731i	$0.625123\pi$
$182$	0	0
$183$	−1.00841e8	−1.21635
$184$	0	0
$185$	2.75811e7	0.320265
$186$	0	0
$187$	7.07967e7	0.791712
$188$	0	0
$189$	1.78373e8	1.92182
$190$	0	0
$191$	1.77318e8	1.84135	0.920676	−	0.390329i	$-0.127639\pi$
$191$	1.77318e8	1.84135	0.920676	+	0.390329i	$0.127639\pi$
$192$	0	0
$193$	−6.75041e7	−0.675896	−0.337948	−	0.941165i	$-0.609733\pi$
$193$	−6.75041e7	−0.675896	−0.337948	+	0.941165i	$0.609733\pi$
$194$	0	0
$195$	8.83163e7	0.852943
$196$	0	0
$197$	−1.03097e8	−0.960761	−0.480380	−	0.877060i	$-0.659501\pi$
$197$	−1.03097e8	−0.960761	−0.480380	+	0.877060i	$0.659501\pi$
$198$	0	0
$199$	−1.78780e8	−1.60817	−0.804086	−	0.594512i	$-0.797345\pi$
$199$	−1.78780e8	−1.60817	−0.804086	+	0.594512i	$0.797345\pi$
$200$	0	0
$201$	−5.17263e7	−0.449288
$202$	0	0
$203$	3.15273e7	0.264516
$204$	0	0
$205$	1.43957e7	0.116706
$206$	0	0
$207$	4.93019e7	0.386338
$208$	0	0
$209$	1.03402e8	0.783458
$210$	0	0
$211$	−2.20832e8	−1.61836	−0.809178	−	0.587564i	$-0.800087\pi$
$211$	−2.20832e8	−1.61836	−0.809178	+	0.587564i	$0.800087\pi$
$212$	0	0
$213$	8.14233e6	0.0577325
$214$	0	0
$215$	2.21611e8	1.52075
$216$	0	0
$217$	2.85312e8	1.89544
$218$	0	0
$219$	−8.33216e7	−0.536047
$220$	0	0
$221$	−6.57551e7	−0.409786
$222$	0	0
$223$	−2.27015e8	−1.37085	−0.685423	−	0.728146i	$-0.740382\pi$
$223$	−2.27015e8	−1.37085	−0.685423	+	0.728146i	$0.740382\pi$
$224$	0	0
$225$	−2.67456e8	−1.56535
$226$	0	0
$227$	2.27056e8	1.28838	0.644188	−	0.764867i	$-0.277195\pi$
$227$	2.27056e8	1.28838	0.644188	+	0.764867i	$0.277195\pi$
$228$	0	0
$229$	1.74743e8	0.961557	0.480778	−	0.876842i	$-0.340354\pi$
$229$	1.74743e8	0.961557	0.480778	+	0.876842i	$0.340354\pi$
$230$	0	0
$231$	2.94327e8	1.57105
$232$	0	0
$233$	2.66617e8	1.38083	0.690417	−	0.723411i	$-0.257427\pi$
$233$	2.66617e8	1.38083	0.690417	+	0.723411i	$0.257427\pi$
$234$	0	0
$235$	9.54755e7	0.479904
$236$	0	0
$237$	1.12688e7	0.0549868
$238$	0	0
$239$	−7.94730e7	−0.376554	−0.188277	−	0.982116i	$-0.560290\pi$
$239$	−7.94730e7	−0.376554	−0.188277	+	0.982116i	$0.560290\pi$
$240$	0	0
$241$	5.79710e7	0.266779	0.133389	−	0.991064i	$-0.457414\pi$
$241$	5.79710e7	0.266779	0.133389	+	0.991064i	$0.457414\pi$
$242$	0	0
$243$	−2.12713e8	−0.950980
$244$	0	0
$245$	−1.09836e9	−4.77159
$246$	0	0
$247$	−9.60382e7	−0.405513
$248$	0	0
$249$	−2.08153e8	−0.854447
$250$	0	0
$251$	−3.14343e8	−1.25472	−0.627358	−	0.778731i	$-0.715864\pi$
$251$	−3.14343e8	−1.25472	−0.627358	+	0.778731i	$0.715864\pi$
$252$	0	0
$253$	2.26613e8	0.879756
$254$	0	0
$255$	−2.12405e8	−0.802183
$256$	0	0
$257$	3.09191e8	1.13622	0.568108	−	0.822954i	$-0.307675\pi$
$257$	3.09191e8	1.13622	0.568108	+	0.822954i	$0.307675\pi$
$258$	0	0
$259$	−8.53724e7	−0.305329
$260$	0	0
$261$	−2.29109e7	−0.0797627
$262$	0	0
$263$	3.09539e8	1.04923	0.524614	−	0.851340i	$-0.324210\pi$
$263$	3.09539e8	1.04923	0.524614	+	0.851340i	$0.324210\pi$
$264$	0	0
$265$	9.92718e8	3.27692
$266$	0	0
$267$	−3.60080e8	−1.15774
$268$	0	0
$269$	−2.59451e8	−0.812686	−0.406343	−	0.913721i	$-0.633196\pi$
$269$	−2.59451e8	−0.812686	−0.406343	+	0.913721i	$0.633196\pi$
$270$	0	0
$271$	−7.90601e7	−0.241304	−0.120652	−	0.992695i	$-0.538499\pi$
$271$	−7.90601e7	−0.241304	−0.120652	+	0.992695i	$0.538499\pi$
$272$	0	0
$273$	−2.73368e8	−0.813164
$274$	0	0
$275$	−1.22934e9	−3.56457
$276$	0	0
$277$	2.41348e8	0.682283	0.341142	−	0.940012i	$-0.389186\pi$
$277$	2.41348e8	0.682283	0.341142	+	0.940012i	$0.389186\pi$
$278$	0	0
$279$	−2.07336e8	−0.571557
$280$	0	0
$281$	−1.24454e8	−0.334609	−0.167304	−	0.985905i	$-0.553506\pi$
$281$	−1.24454e8	−0.334609	−0.167304	+	0.985905i	$0.553506\pi$
$282$	0	0
$283$	−4.05747e8	−1.06415	−0.532074	−	0.846698i	$-0.678587\pi$
$283$	−4.05747e8	−1.06415	−0.532074	+	0.846698i	$0.678587\pi$
$284$	0	0
$285$	−3.10226e8	−0.793819
$286$	0	0
$287$	−4.45593e7	−0.111263
$288$	0	0
$289$	−2.52195e8	−0.614601
$290$	0	0
$291$	−5.07128e8	−1.20640
$292$	0	0
$293$	4.99201e8	1.15941	0.579707	−	0.814825i	$-0.303167\pi$
$293$	4.99201e8	1.15941	0.579707	+	0.814825i	$0.303167\pi$
$294$	0	0
$295$	−6.76137e8	−1.53341
$296$	0	0
$297$	−5.95803e8	−1.31964
$298$	0	0
$299$	−2.10475e8	−0.455357
$300$	0	0
$301$	−6.85959e8	−1.44983
$302$	0	0
$303$	−2.41781e8	−0.499312
$304$	0	0
$305$	−1.77016e9	−3.57242
$306$	0	0
$307$	3.21540e8	0.634235	0.317118	−	0.948386i	$-0.397285\pi$
$307$	3.21540e8	0.634235	0.317118	+	0.948386i	$0.397285\pi$
$308$	0	0
$309$	2.52045e8	0.485986
$310$	0	0
$311$	1.03297e8	0.194726	0.0973631	−	0.995249i	$-0.468959\pi$
$311$	1.03297e8	0.194726	0.0973631	+	0.995249i	$0.468959\pi$
$312$	0	0
$313$	5.25918e8	0.969422	0.484711	−	0.874674i	$-0.338925\pi$
$313$	5.25918e8	0.969422	0.484711	+	0.874674i	$0.338925\pi$
$314$	0	0
$315$	1.12405e9	2.02627
$316$	0	0
$317$	−6.26026e7	−0.110379	−0.0551893	−	0.998476i	$-0.517576\pi$
$317$	−6.26026e7	−0.110379	−0.0551893	+	0.998476i	$0.517576\pi$
$318$	0	0
$319$	−1.05308e8	−0.181633
$320$	0	0
$321$	3.57802e7	0.0603775
$322$	0	0
$323$	2.30976e8	0.381381
$324$	0	0
$325$	1.14180e9	1.84500
$326$	0	0
$327$	2.66877e8	0.422079
$328$	0	0
$329$	−2.95528e8	−0.457523
$330$	0	0
$331$	3.10780e8	0.471037	0.235518	−	0.971870i	$-0.424321\pi$
$331$	3.10780e8	0.471037	0.235518	+	0.971870i	$0.424321\pi$
$332$	0	0
$333$	6.20400e7	0.0920697
$334$	0	0
$335$	−9.07998e8	−1.31956
$336$	0	0
$337$	−9.56000e8	−1.36067	−0.680335	−	0.732901i	$-0.738166\pi$
$337$	−9.56000e8	−1.36067	−0.680335	+	0.732901i	$0.738166\pi$
$338$	0	0
$339$	2.40952e8	0.335916
$340$	0	0
$341$	−9.53003e8	−1.30153
$342$	0	0
$343$	2.01175e9	2.69181
$344$	0	0
$345$	−6.79885e8	−0.891392
$346$	0	0
$347$	−1.37536e9	−1.76711	−0.883555	−	0.468327i	$-0.844857\pi$
$347$	−1.37536e9	−1.76711	−0.883555	+	0.468327i	$0.844857\pi$
$348$	0	0
$349$	−1.38134e9	−1.73945	−0.869725	−	0.493537i	$-0.835704\pi$
$349$	−1.38134e9	−1.73945	−0.869725	+	0.493537i	$0.835704\pi$
$350$	0	0
$351$	5.53374e8	0.683037
$352$	0	0
$353$	−6.48672e8	−0.784899	−0.392450	−	0.919774i	$-0.628372\pi$
$353$	−6.48672e8	−0.784899	−0.392450	+	0.919774i	$0.628372\pi$
$354$	0	0
$355$	1.42930e8	0.169560
$356$	0	0
$357$	6.57462e8	0.764772
$358$	0	0
$359$	−3.64068e8	−0.415291	−0.207645	−	0.978204i	$-0.566580\pi$
$359$	−3.64068e8	−0.415291	−0.207645	+	0.978204i	$0.566580\pi$
$360$	0	0
$361$	−5.56521e8	−0.622596
$362$	0	0
$363$	−3.78638e8	−0.415481
$364$	0	0
$365$	−1.46262e9	−1.57437
$366$	0	0
$367$	−3.90647e8	−0.412528	−0.206264	−	0.978496i	$-0.566131\pi$
$367$	−3.90647e8	−0.412528	−0.206264	+	0.978496i	$0.566131\pi$
$368$	0	0
$369$	3.23812e7	0.0335506
$370$	0	0
$371$	−3.07279e9	−3.12409
$372$	0	0
$373$	−8.58735e7	−0.0856797	−0.0428399	−	0.999082i	$-0.513641\pi$
$373$	−8.58735e7	−0.0856797	−0.0428399	+	0.999082i	$0.513641\pi$
$374$	0	0
$375$	2.36872e9	2.31955
$376$	0	0
$377$	9.78089e7	0.0940121
$378$	0	0
$379$	−3.53330e8	−0.333383	−0.166691	−	0.986009i	$-0.553308\pi$
$379$	−3.53330e8	−0.333383	−0.166691	+	0.986009i	$0.553308\pi$
$380$	0	0
$381$	1.39443e8	0.129169
$382$	0	0
$383$	4.90315e8	0.445943	0.222971	−	0.974825i	$-0.428424\pi$
$383$	4.90315e8	0.445943	0.222971	+	0.974825i	$0.428424\pi$
$384$	0	0
$385$	5.16660e9	4.61415
$386$	0	0
$387$	4.98486e8	0.437184
$388$	0	0
$389$	9.97474e7	0.0859168	0.0429584	−	0.999077i	$-0.486322\pi$
$389$	9.97474e7	0.0859168	0.0429584	+	0.999077i	$0.486322\pi$
$390$	0	0
$391$	5.06202e8	0.428258
$392$	0	0
$393$	−4.88988e8	−0.406372
$394$	0	0
$395$	1.97812e8	0.161496
$396$	0	0
$397$	−2.09994e8	−0.168438	−0.0842189	−	0.996447i	$-0.526839\pi$
$397$	−2.09994e8	−0.168438	−0.0842189	+	0.996447i	$0.526839\pi$
$398$	0	0
$399$	9.60252e8	0.756798
$400$	0	0
$401$	5.55414e8	0.430141	0.215071	−	0.976598i	$-0.431002\pi$
$401$	5.55414e8	0.430141	0.215071	+	0.976598i	$0.431002\pi$
$402$	0	0
$403$	8.85138e8	0.673664
$404$	0	0
$405$	3.28982e8	0.246082
$406$	0	0
$407$	2.85162e8	0.209658
$408$	0	0
$409$	1.80096e9	1.30158	0.650792	−	0.759256i	$-0.274437\pi$
$409$	1.80096e9	1.30158	0.650792	+	0.759256i	$0.274437\pi$
$410$	0	0
$411$	2.87793e8	0.204472
$412$	0	0
$413$	2.09286e9	1.46189
$414$	0	0
$415$	−3.65390e9	−2.50951
$416$	0	0
$417$	1.64527e9	1.11112
$418$	0	0
$419$	1.69929e7	0.0112854	0.00564272	−	0.999984i	$-0.498204\pi$
$419$	1.69929e7	0.0112854	0.00564272	+	0.999984i	$0.498204\pi$
$420$	0	0
$421$	2.76374e9	1.80514	0.902568	−	0.430548i	$-0.141680\pi$
$421$	2.76374e9	1.80514	0.902568	+	0.430548i	$0.141680\pi$
$422$	0	0
$423$	2.14760e8	0.137963
$424$	0	0
$425$	−2.74607e9	−1.73520
$426$	0	0
$427$	5.47921e9	3.40581
$428$	0	0
$429$	9.13107e8	0.558369
$430$	0	0
$431$	2.95133e9	1.77561	0.887804	−	0.460222i	$-0.152230\pi$
$431$	2.95133e9	1.77561	0.887804	+	0.460222i	$0.152230\pi$
$432$	0	0
$433$	−2.29366e9	−1.35776	−0.678878	−	0.734251i	$-0.737533\pi$
$433$	−2.29366e9	−1.35776	−0.678878	+	0.734251i	$0.737533\pi$
$434$	0	0
$435$	3.15946e8	0.184035
$436$	0	0
$437$	7.39331e8	0.423793
$438$	0	0
$439$	−2.21394e9	−1.24893	−0.624467	−	0.781052i	$-0.714684\pi$
$439$	−2.21394e9	−1.24893	−0.624467	+	0.781052i	$0.714684\pi$
$440$	0	0
$441$	−2.47061e9	−1.37173
$442$	0	0
$443$	2.80148e9	1.53100	0.765498	−	0.643438i	$-0.222493\pi$
$443$	2.80148e9	1.53100	0.765498	+	0.643438i	$0.222493\pi$
$444$	0	0
$445$	−6.32082e9	−3.40027
$446$	0	0
$447$	−8.10713e8	−0.429330
$448$	0	0
$449$	−1.24071e9	−0.646856	−0.323428	−	0.946253i	$-0.604835\pi$
$449$	−1.24071e9	−0.646856	−0.323428	+	0.946253i	$0.604835\pi$
$450$	0	0
$451$	1.48838e8	0.0764003
$452$	0	0
$453$	−1.03220e9	−0.521698
$454$	0	0
$455$	−4.79867e9	−2.38826
$456$	0	0
$457$	8.16764e8	0.400304	0.200152	−	0.979765i	$-0.435856\pi$
$457$	8.16764e8	0.400304	0.200152	+	0.979765i	$0.435856\pi$
$458$	0	0
$459$	−1.33089e9	−0.642389
$460$	0	0
$461$	1.07694e9	0.511962	0.255981	−	0.966682i	$-0.417601\pi$
$461$	1.07694e9	0.511962	0.255981	+	0.966682i	$0.417601\pi$
$462$	0	0
$463$	−3.16535e9	−1.48214	−0.741068	−	0.671430i	$-0.765680\pi$
$463$	−3.16535e9	−1.48214	−0.741068	+	0.671430i	$0.765680\pi$
$464$	0	0
$465$	2.85921e9	1.31874
$466$	0	0
$467$	−2.37777e9	−1.08034	−0.540171	−	0.841555i	$-0.681640\pi$
$467$	−2.37777e9	−1.08034	−0.540171	+	0.841555i	$0.681640\pi$
$468$	0	0
$469$	2.81055e9	1.25802
$470$	0	0
$471$	6.45614e8	0.284708
$472$	0	0
$473$	2.29125e9	0.995540
$474$	0	0
$475$	−4.01076e9	−1.71711
$476$	0	0
$477$	2.23299e9	0.942047
$478$	0	0
$479$	−3.68467e9	−1.53188	−0.765939	−	0.642914i	$-0.777725\pi$
$479$	−3.68467e9	−1.53188	−0.765939	+	0.642914i	$0.777725\pi$
$480$	0	0
$481$	−2.64855e8	−0.108518
$482$	0	0
$483$	2.10447e9	0.849820
$484$	0	0
$485$	−8.90209e9	−3.54320
$486$	0	0
$487$	1.99019e9	0.780807	0.390404	−	0.920644i	$-0.372335\pi$
$487$	1.99019e9	0.780807	0.390404	+	0.920644i	$0.372335\pi$
$488$	0	0
$489$	1.24295e9	0.480698
$490$	0	0
$491$	1.03457e9	0.394433	0.197217	−	0.980360i	$-0.436810\pi$
$491$	1.03457e9	0.394433	0.197217	+	0.980360i	$0.436810\pi$
$492$	0	0
$493$	−2.35235e8	−0.0884173
$494$	0	0
$495$	−3.75456e9	−1.39136
$496$	0	0
$497$	−4.42414e8	−0.161652
$498$	0	0
$499$	1.69821e9	0.611843	0.305922	−	0.952057i	$-0.401035\pi$
$499$	1.69821e9	0.611843	0.305922	+	0.952057i	$0.401035\pi$
$500$	0	0
$501$	1.32305e9	0.470049
$502$	0	0
$503$	−9.98691e8	−0.349899	−0.174950	−	0.984577i	$-0.555976\pi$
$503$	−9.98691e8	−0.349899	−0.174950	+	0.984577i	$0.555976\pi$
$504$	0	0
$505$	−4.24420e9	−1.46648
$506$	0	0
$507$	1.09833e9	0.374288
$508$	0	0
$509$	4.86581e9	1.63547	0.817735	−	0.575594i	$-0.195229\pi$
$509$	4.86581e9	1.63547	0.817735	+	0.575594i	$0.195229\pi$
$510$	0	0
$511$	4.52729e9	1.50094
$512$	0	0
$513$	−1.94382e9	−0.635691
$514$	0	0
$515$	4.42438e9	1.42734
$516$	0	0
$517$	9.87127e8	0.314164
$518$	0	0
$519$	−2.21331e8	−0.0694956
$520$	0	0
$521$	−2.29363e9	−0.710544	−0.355272	−	0.934763i	$-0.615612\pi$
$521$	−2.29363e9	−0.710544	−0.355272	+	0.934763i	$0.615612\pi$
$522$	0	0
$523$	1.41986e8	0.0434000	0.0217000	−	0.999765i	$-0.493092\pi$
$523$	1.41986e8	0.0434000	0.0217000	+	0.999765i	$0.493092\pi$
$524$	0	0
$525$	−1.14164e10	−3.44328
$526$	0	0
$527$	−2.12880e9	−0.633574
$528$	0	0
$529$	−1.78453e9	−0.524117
$530$	0	0
$531$	−1.52088e9	−0.440823
$532$	0	0
$533$	−1.38239e8	−0.0395444
$534$	0	0
$535$	6.28082e8	0.177328
$536$	0	0
$537$	4.31709e9	1.20304
$538$	0	0
$539$	−1.13560e10	−3.12366
$540$	0	0
$541$	6.73035e9	1.82746	0.913729	−	0.406324i	$-0.133190\pi$
$541$	6.73035e9	1.82746	0.913729	+	0.406324i	$0.133190\pi$
$542$	0	0
$543$	1.89575e9	0.508139
$544$	0	0
$545$	4.68474e9	1.23964
$546$	0	0
$547$	2.10936e9	0.551056	0.275528	−	0.961293i	$-0.411147\pi$
$547$	2.10936e9	0.551056	0.275528	+	0.961293i	$0.411147\pi$
$548$	0	0
$549$	−3.98173e9	−1.02700
$550$	0	0
$551$	−3.43571e8	−0.0874955
$552$	0	0
$553$	−6.12291e8	−0.153964
$554$	0	0
$555$	−8.55545e8	−0.212431
$556$	0	0
$557$	−5.46338e9	−1.33958	−0.669790	−	0.742551i	$-0.733616\pi$
$557$	−5.46338e9	−1.33958	−0.669790	+	0.742551i	$0.733616\pi$
$558$	0	0
$559$	−2.12809e9	−0.515286
$560$	0	0
$561$	−2.19606e9	−0.525140
$562$	0	0
$563$	2.38438e9	0.563113	0.281557	−	0.959545i	$-0.409149\pi$
$563$	2.38438e9	0.563113	0.281557	+	0.959545i	$0.409149\pi$
$564$	0	0
$565$	4.22964e9	0.986584
$566$	0	0
$567$	−1.01831e9	−0.234605
$568$	0	0
$569$	4.69886e9	1.06930	0.534650	−	0.845073i	$-0.320443\pi$
$569$	4.69886e9	1.06930	0.534650	+	0.845073i	$0.320443\pi$
$570$	0	0
$571$	−2.04617e9	−0.459954	−0.229977	−	0.973196i	$-0.573865\pi$
$571$	−2.04617e9	−0.459954	−0.229977	+	0.973196i	$0.573865\pi$
$572$	0	0
$573$	−5.50029e9	−1.22136
$574$	0	0
$575$	−8.78988e9	−1.92817
$576$	0	0
$577$	−6.13969e9	−1.33055	−0.665274	−	0.746599i	$-0.731685\pi$
$577$	−6.13969e9	−1.33055	−0.665274	+	0.746599i	$0.731685\pi$
$578$	0	0
$579$	2.09393e9	0.448319
$580$	0	0
$581$	1.13100e10	2.39247
$582$	0	0
$583$	1.02638e10	2.14520
$584$	0	0
$585$	3.48719e9	0.720161
$586$	0	0
$587$	1.29772e9	0.264819	0.132410	−	0.991195i	$-0.457729\pi$
$587$	1.29772e9	0.264819	0.132410	+	0.991195i	$0.457729\pi$
$588$	0	0
$589$	−3.10920e9	−0.626968
$590$	0	0
$591$	3.19800e9	0.637269
$592$	0	0
$593$	−1.01765e9	−0.200404	−0.100202	−	0.994967i	$-0.531949\pi$
$593$	−1.01765e9	−0.200404	−0.100202	+	0.994967i	$0.531949\pi$
$594$	0	0
$595$	1.15410e10	2.24613
$596$	0	0
$597$	5.54562e9	1.06670
$598$	0	0
$599$	−1.36024e9	−0.258595	−0.129298	−	0.991606i	$-0.541272\pi$
$599$	−1.36024e9	−0.258595	−0.129298	+	0.991606i	$0.541272\pi$
$600$	0	0
$601$	5.83436e9	1.09631	0.548154	−	0.836378i	$-0.315331\pi$
$601$	5.83436e9	1.09631	0.548154	+	0.836378i	$0.315331\pi$
$602$	0	0
$603$	−2.04242e9	−0.379346
$604$	0	0
$605$	−6.64658e9	−1.22027
$606$	0	0
$607$	−4.09613e9	−0.743384	−0.371692	−	0.928356i	$-0.621222\pi$
$607$	−4.09613e9	−0.743384	−0.371692	+	0.928356i	$0.621222\pi$
$608$	0	0
$609$	−9.77956e8	−0.175452
$610$	0	0
$611$	−9.16831e8	−0.162609
$612$	0	0
$613$	−5.20044e8	−0.0911861	−0.0455930	−	0.998960i	$-0.514518\pi$
$613$	−5.20044e8	−0.0911861	−0.0455930	+	0.998960i	$0.514518\pi$
$614$	0	0
$615$	−4.46544e8	−0.0774108
$616$	0	0
$617$	3.87499e9	0.664160	0.332080	−	0.943251i	$-0.392250\pi$
$617$	3.87499e9	0.664160	0.332080	+	0.943251i	$0.392250\pi$
$618$	0	0
$619$	6.93110e9	1.17459	0.587293	−	0.809375i	$-0.300194\pi$
$619$	6.93110e9	1.17459	0.587293	+	0.809375i	$0.300194\pi$
$620$	0	0
$621$	−4.26004e9	−0.713827
$622$	0	0
$623$	1.95650e10	3.24169
$624$	0	0
$625$	2.45204e10	4.01742
$626$	0	0
$627$	−3.20745e9	−0.519665
$628$	0	0
$629$	6.36988e8	0.102060
$630$	0	0
$631$	−1.27564e9	−0.202128	−0.101064	−	0.994880i	$-0.532225\pi$
$631$	−1.27564e9	−0.202128	−0.101064	+	0.994880i	$0.532225\pi$
$632$	0	0
$633$	6.85006e9	1.07345
$634$	0	0
$635$	2.44777e9	0.379369
$636$	0	0
$637$	1.05473e10	1.61679
$638$	0	0
$639$	3.21502e8	0.0487450
$640$	0	0
$641$	8.45969e9	1.26868	0.634339	−	0.773055i	$-0.281272\pi$
$641$	8.45969e9	1.26868	0.634339	+	0.773055i	$0.281272\pi$
$642$	0	0
$643$	2.53383e9	0.375871	0.187935	−	0.982181i	$-0.439820\pi$
$643$	2.53383e9	0.375871	0.187935	+	0.982181i	$0.439820\pi$
$644$	0	0
$645$	−6.87423e9	−1.00871
$646$	0	0
$647$	9.30854e8	0.135119	0.0675595	−	0.997715i	$-0.478479\pi$
$647$	9.30854e8	0.135119	0.0675595	+	0.997715i	$0.478479\pi$
$648$	0	0
$649$	−6.99061e9	−1.00383
$650$	0	0
$651$	−8.85018e9	−1.25724
$652$	0	0
$653$	4.54702e9	0.639044	0.319522	−	0.947579i	$-0.396478\pi$
$653$	4.54702e9	0.639044	0.319522	+	0.947579i	$0.396478\pi$
$654$	0	0
$655$	−8.58365e9	−1.19351
$656$	0	0
$657$	−3.28997e9	−0.452599
$658$	0	0
$659$	−5.26043e9	−0.716015	−0.358007	−	0.933719i	$-0.616544\pi$
$659$	−5.26043e9	−0.716015	−0.358007	+	0.933719i	$0.616544\pi$
$660$	0	0
$661$	−9.97829e8	−0.134385	−0.0671925	−	0.997740i	$-0.521404\pi$
$661$	−9.97829e8	−0.134385	−0.0671925	+	0.997740i	$0.521404\pi$
$662$	0	0
$663$	2.03968e9	0.271809
$664$	0	0
$665$	1.68562e10	2.22271
$666$	0	0
$667$	−7.52962e8	−0.0982500
$668$	0	0
$669$	7.04186e9	0.909276
$670$	0	0
$671$	−1.83017e10	−2.33864
$672$	0	0
$673$	−2.48723e9	−0.314530	−0.157265	−	0.987556i	$-0.550268\pi$
$673$	−2.48723e9	−0.314530	−0.157265	+	0.987556i	$0.550268\pi$
$674$	0	0
$675$	2.31101e10	2.89226
$676$	0	0
$677$	−6.47115e9	−0.801532	−0.400766	−	0.916180i	$-0.631256\pi$
$677$	−6.47115e9	−0.801532	−0.400766	+	0.916180i	$0.631256\pi$
$678$	0	0
$679$	2.75549e10	3.37796
$680$	0	0
$681$	−7.04312e9	−0.854575
$682$	0	0
$683$	4.86398e9	0.584143	0.292071	−	0.956397i	$-0.405655\pi$
$683$	4.86398e9	0.584143	0.292071	+	0.956397i	$0.405655\pi$
$684$	0	0
$685$	5.05189e9	0.600533
$686$	0	0
$687$	−5.42040e9	−0.637797
$688$	0	0
$689$	−9.53287e9	−1.11034
$690$	0	0
$691$	−7.51546e9	−0.866527	−0.433264	−	0.901267i	$-0.642638\pi$
$691$	−7.51546e9	−0.866527	−0.433264	+	0.901267i	$0.642638\pi$
$692$	0	0
$693$	1.16216e10	1.32647
$694$	0	0
$695$	2.88810e10	3.26336
$696$	0	0
$697$	3.32470e8	0.0371910
$698$	0	0
$699$	−8.27026e9	−0.915902
$700$	0	0
$701$	−1.02509e10	−1.12396	−0.561978	−	0.827152i	$-0.689960\pi$
$701$	−1.02509e10	−1.12396	−0.561978	+	0.827152i	$0.689960\pi$
$702$	0	0
$703$	9.30350e8	0.100996
$704$	0	0
$705$	−2.96158e9	−0.318319
$706$	0	0
$707$	1.31372e10	1.39809
$708$	0	0
$709$	5.01078e9	0.528012	0.264006	−	0.964521i	$-0.414956\pi$
$709$	5.01078e9	0.528012	0.264006	+	0.964521i	$0.414956\pi$
$710$	0	0
$711$	4.44951e8	0.0464268
$712$	0	0
$713$	−6.81405e9	−0.704032
$714$	0	0
$715$	1.60286e10	1.63993
$716$	0	0
$717$	2.46520e9	0.249767
$718$	0	0
$719$	−1.55089e10	−1.55607	−0.778035	−	0.628221i	$-0.783783\pi$
$719$	−1.55089e10	−1.55607	−0.778035	+	0.628221i	$0.783783\pi$
$720$	0	0
$721$	−1.36949e10	−1.36077
$722$	0	0
$723$	−1.79822e9	−0.176953
$724$	0	0
$725$	4.08470e9	0.398086
$726$	0	0
$727$	9.09964e8	0.0878322	0.0439161	−	0.999035i	$-0.486017\pi$
$727$	9.09964e8	0.0878322	0.0439161	+	0.999035i	$0.486017\pi$
$728$	0	0
$729$	7.91954e9	0.757101
$730$	0	0
$731$	5.11814e9	0.484620
$732$	0	0
$733$	−6.80412e9	−0.638128	−0.319064	−	0.947733i	$-0.603369\pi$
$733$	−6.80412e9	−0.638128	−0.319064	+	0.947733i	$0.603369\pi$
$734$	0	0
$735$	3.40703e10	3.16498
$736$	0	0
$737$	−9.38784e9	−0.863832
$738$	0	0
$739$	−9.88530e9	−0.901020	−0.450510	−	0.892771i	$-0.648758\pi$
$739$	−9.88530e9	−0.901020	−0.450510	+	0.892771i	$0.648758\pi$
$740$	0	0
$741$	2.97904e9	0.268975
$742$	0	0
$743$	1.40609e10	1.25763	0.628814	−	0.777556i	$-0.283541\pi$
$743$	1.40609e10	1.25763	0.628814	+	0.777556i	$0.283541\pi$
$744$	0	0
$745$	−1.42312e10	−1.26094
$746$	0	0
$747$	−8.21897e9	−0.721432
$748$	0	0
$749$	−1.94412e9	−0.169058
$750$	0	0
$751$	−1.89107e10	−1.62918	−0.814590	−	0.580038i	$-0.803038\pi$
$751$	−1.89107e10	−1.62918	−0.814590	+	0.580038i	$0.803038\pi$
$752$	0	0
$753$	9.75069e9	0.832248
$754$	0	0
$755$	−1.81191e10	−1.53222
$756$	0	0
$757$	−1.68703e9	−0.141348	−0.0706738	−	0.997499i	$-0.522515\pi$
$757$	−1.68703e9	−0.141348	−0.0706738	+	0.997499i	$0.522515\pi$
$758$	0	0
$759$	−7.02936e9	−0.583539
$760$	0	0
$761$	1.35093e10	1.11118	0.555591	−	0.831456i	$-0.312492\pi$
$761$	1.35093e10	1.11118	0.555591	+	0.831456i	$0.312492\pi$
$762$	0	0
$763$	−1.45008e10	−1.18183
$764$	0	0
$765$	−8.38685e9	−0.677304
$766$	0	0
$767$	6.49280e9	0.519575
$768$	0	0
$769$	−2.04375e10	−1.62064	−0.810318	−	0.585990i	$-0.800706\pi$
$769$	−2.04375e10	−1.62064	−0.810318	+	0.585990i	$0.800706\pi$
$770$	0	0
$771$	−9.59089e9	−0.753648
$772$	0	0
$773$	−3.22437e9	−0.251083	−0.125541	−	0.992088i	$-0.540067\pi$
$773$	−3.22437e9	−0.251083	−0.125541	+	0.992088i	$0.540067\pi$
$774$	0	0
$775$	3.69652e10	2.85258
$776$	0	0
$777$	2.64819e9	0.202524
$778$	0	0
$779$	4.85588e8	0.0368033
$780$	0	0
$781$	1.47776e9	0.111000
$782$	0	0
$783$	1.97966e9	0.147375
$784$	0	0
$785$	1.13331e10	0.836187
$786$	0	0
$787$	1.34167e10	0.981147	0.490574	−	0.871400i	$-0.336787\pi$
$787$	1.34167e10	0.981147	0.490574	+	0.871400i	$0.336787\pi$
$788$	0	0
$789$	−9.60167e9	−0.695949
$790$	0	0
$791$	−1.30921e10	−0.940573
$792$	0	0
$793$	1.69984e10	1.21047
$794$	0	0
$795$	−3.07934e10	−2.17357
$796$	0	0
$797$	1.04838e10	0.733528	0.366764	−	0.930314i	$-0.380466\pi$
$797$	1.04838e10	0.733528	0.366764	+	0.930314i	$0.380466\pi$
$798$	0	0
$799$	2.20502e9	0.152932
$800$	0	0
$801$	−1.42179e10	−0.977508
$802$	0	0
$803$	−1.51221e10	−1.03064
$804$	0	0
$805$	3.69416e10	2.49592
$806$	0	0
$807$	8.04800e9	0.539052
$808$	0	0
$809$	8.66031e9	0.575060	0.287530	−	0.957772i	$-0.407166\pi$
$809$	8.66031e9	0.575060	0.287530	+	0.957772i	$0.407166\pi$
$810$	0	0
$811$	−7.78352e9	−0.512393	−0.256196	−	0.966625i	$-0.582469\pi$
$811$	−7.78352e9	−0.512393	−0.256196	+	0.966625i	$0.582469\pi$
$812$	0	0
$813$	2.45239e9	0.160056
$814$	0	0
$815$	2.18186e10	1.41181
$816$	0	0
$817$	7.47528e9	0.479568
$818$	0	0
$819$	−1.07940e10	−0.686575
$820$	0	0
$821$	7.91243e9	0.499009	0.249505	−	0.968374i	$-0.419732\pi$
$821$	7.91243e9	0.499009	0.249505	+	0.968374i	$0.419732\pi$
$822$	0	0
$823$	4.48091e9	0.280199	0.140100	−	0.990137i	$-0.455258\pi$
$823$	4.48091e9	0.280199	0.140100	+	0.990137i	$0.455258\pi$
$824$	0	0
$825$	3.81332e10	2.36437
$826$	0	0
$827$	7.15443e8	0.0439851	0.0219926	−	0.999758i	$-0.492999\pi$
$827$	7.15443e8	0.0439851	0.0219926	+	0.999758i	$0.492999\pi$
$828$	0	0
$829$	−2.36866e10	−1.44398	−0.721990	−	0.691903i	$-0.756773\pi$
$829$	−2.36866e10	−1.44398	−0.721990	+	0.691903i	$0.756773\pi$
$830$	0	0
$831$	−7.48645e9	−0.452556
$832$	0	0
$833$	−2.53668e10	−1.52057
$834$	0	0
$835$	2.32246e10	1.38053
$836$	0	0
$837$	1.79153e10	1.05605
$838$	0	0
$839$	−2.37629e10	−1.38909	−0.694547	−	0.719447i	$-0.744395\pi$
$839$	−2.37629e10	−1.38909	−0.694547	+	0.719447i	$0.744395\pi$
$840$	0	0
$841$	−1.69000e10	−0.979715
$842$	0	0
$843$	3.86048e9	0.221945
$844$	0	0
$845$	1.92800e10	1.09928
$846$	0	0
$847$	2.05733e10	1.16336
$848$	0	0
$849$	1.25860e10	0.705846
$850$	0	0
$851$	2.03893e9	0.113409
$852$	0	0
$853$	−3.28503e10	−1.81225	−0.906126	−	0.423009i	$-0.860974\pi$
$853$	−3.28503e10	−1.81225	−0.906126	+	0.423009i	$0.860974\pi$
$854$	0	0
$855$	−1.22494e10	−0.670242
$856$	0	0
$857$	−5.67832e9	−0.308168	−0.154084	−	0.988058i	$-0.549243\pi$
$857$	−5.67832e9	−0.308168	−0.154084	+	0.988058i	$0.549243\pi$
$858$	0	0
$859$	1.22964e10	0.661914	0.330957	−	0.943646i	$-0.392628\pi$
$859$	1.22964e10	0.661914	0.330957	+	0.943646i	$0.392628\pi$
$860$	0	0
$861$	1.38220e9	0.0738006
$862$	0	0
$863$	−1.75550e10	−0.929743	−0.464871	−	0.885378i	$-0.653899\pi$
$863$	−1.75550e10	−0.929743	−0.464871	+	0.885378i	$0.653899\pi$
$864$	0	0
$865$	−3.88523e9	−0.204108
$866$	0	0
$867$	7.82290e9	0.407663
$868$	0	0
$869$	2.04518e9	0.105721
$870$	0	0
$871$	8.71931e9	0.447115
$872$	0	0
$873$	−2.00241e10	−1.01860
$874$	0	0
$875$	−1.28704e11	−6.49479
$876$	0	0
$877$	−9.75144e7	−0.00488169	−0.00244084	−	0.999997i	$-0.500777\pi$
$877$	−9.75144e7	−0.00488169	−0.00244084	+	0.999997i	$0.500777\pi$
$878$	0	0
$879$	−1.54848e10	−0.769035
$880$	0	0
$881$	−1.68618e10	−0.830786	−0.415393	−	0.909642i	$-0.636356\pi$
$881$	−1.68618e10	−0.830786	−0.415393	+	0.909642i	$0.636356\pi$
$882$	0	0
$883$	−2.81287e10	−1.37495	−0.687476	−	0.726207i	$-0.741281\pi$
$883$	−2.81287e10	−1.37495	−0.687476	+	0.726207i	$0.741281\pi$
$884$	0	0
$885$	2.09733e10	1.01710
$886$	0	0
$887$	−1.49073e10	−0.717241	−0.358621	−	0.933483i	$-0.616753\pi$
$887$	−1.49073e10	−0.717241	−0.358621	+	0.933483i	$0.616753\pi$
$888$	0	0
$889$	−7.57664e9	−0.361676
$890$	0	0
$891$	3.40136e9	0.161095
$892$	0	0
$893$	3.22053e9	0.151338
$894$	0	0
$895$	7.57817e10	3.53333
$896$	0	0
$897$	6.52879e9	0.302036
$898$	0	0
$899$	3.16653e9	0.145353
$900$	0	0
$901$	2.29270e10	1.04426
$902$	0	0
$903$	2.12780e10	0.961664
$904$	0	0
$905$	3.32779e10	1.49240
$906$	0	0
$907$	−2.00171e10	−0.890790	−0.445395	−	0.895334i	$-0.646937\pi$
$907$	−2.00171e10	−0.890790	−0.445395	+	0.895334i	$0.646937\pi$
$908$	0	0
$909$	−9.54676e9	−0.421582
$910$	0	0
$911$	1.55429e10	0.681113	0.340556	−	0.940224i	$-0.389385\pi$
$911$	1.55429e10	0.681113	0.340556	+	0.940224i	$0.389385\pi$
$912$	0	0
$913$	−3.77779e10	−1.64282
$914$	0	0
$915$	5.49090e10	2.36957
$916$	0	0
$917$	2.65692e10	1.13785
$918$	0	0
$919$	7.07850e9	0.300841	0.150420	−	0.988622i	$-0.451937\pi$
$919$	7.07850e9	0.300841	0.150420	+	0.988622i	$0.451937\pi$
$920$	0	0
$921$	−9.97394e9	−0.420686
$922$	0	0
$923$	−1.37252e9	−0.0574532
$924$	0	0
$925$	−1.10609e10	−0.459509
$926$	0	0
$927$	9.95205e9	0.410330
$928$	0	0
$929$	−9.26783e9	−0.379248	−0.189624	−	0.981857i	$-0.560727\pi$
$929$	−9.26783e9	−0.379248	−0.189624	+	0.981857i	$0.560727\pi$
$930$	0	0
$931$	−3.70493e10	−1.50472
$932$	0	0
$933$	−3.20419e9	−0.129161
$934$	0	0
$935$	−3.85495e10	−1.54233
$936$	0	0
$937$	−2.95508e10	−1.17349	−0.586746	−	0.809771i	$-0.699591\pi$
$937$	−2.95508e10	−1.17349	−0.586746	+	0.809771i	$0.699591\pi$
$938$	0	0
$939$	−1.63136e10	−0.643014
$940$	0	0
$941$	1.39298e10	0.544979	0.272489	−	0.962159i	$-0.412153\pi$
$941$	1.39298e10	0.544979	0.272489	+	0.962159i	$0.412153\pi$
$942$	0	0
$943$	1.06420e9	0.0413269
$944$	0	0
$945$	−9.71257e10	−3.74389
$946$	0	0
$947$	2.48316e10	0.950122	0.475061	−	0.879953i	$-0.342426\pi$
$947$	2.48316e10	0.950122	0.475061	+	0.879953i	$0.342426\pi$
$948$	0	0
$949$	1.40452e10	0.533454
$950$	0	0
$951$	1.94189e9	0.0732137
$952$	0	0
$953$	−3.01791e10	−1.12949	−0.564744	−	0.825266i	$-0.691025\pi$
$953$	−3.01791e10	−1.12949	−0.564744	+	0.825266i	$0.691025\pi$
$954$	0	0
$955$	−9.65515e10	−3.58713
$956$	0	0
$957$	3.26658e9	0.120476
$958$	0	0
$959$	−1.56373e10	−0.572526
$960$	0	0
$961$	1.14339e9	0.0415589
$962$	0	0
$963$	1.41279e9	0.0509783
$964$	0	0
$965$	3.67567e10	1.31671
$966$	0	0
$967$	2.93661e10	1.04437	0.522184	−	0.852833i	$-0.325118\pi$
$967$	2.93661e10	1.04437	0.522184	+	0.852833i	$0.325118\pi$
$968$	0	0
$969$	−7.16472e9	−0.252968
$970$	0	0
$971$	2.39542e10	0.839683	0.419841	−	0.907598i	$-0.362086\pi$
$971$	2.39542e10	0.839683	0.419841	+	0.907598i	$0.362086\pi$
$972$	0	0
$973$	−8.93961e10	−3.11117
$974$	0	0
$975$	−3.54177e10	−1.22378
$976$	0	0
$977$	2.91663e10	1.00058	0.500289	−	0.865859i	$-0.333227\pi$
$977$	2.91663e10	1.00058	0.500289	+	0.865859i	$0.333227\pi$
$978$	0	0
$979$	−6.53513e10	−2.22595
$980$	0	0
$981$	1.05377e10	0.356372
$982$	0	0
$983$	−4.44637e9	−0.149303	−0.0746514	−	0.997210i	$-0.523784\pi$
$983$	−4.44637e9	−0.149303	−0.0746514	+	0.997210i	$0.523784\pi$
$984$	0	0
$985$	5.61375e10	1.87166
$986$	0	0
$987$	9.16707e9	0.303473
$988$	0	0
$989$	1.63826e10	0.538514
$990$	0	0
$991$	−3.36917e10	−1.09968	−0.549838	−	0.835271i	$-0.685311\pi$
$991$	−3.36917e10	−1.09968	−0.549838	+	0.835271i	$0.685311\pi$
$992$	0	0
$993$	−9.64017e9	−0.312437
$994$	0	0
$995$	9.73474e10	3.13288
$996$	0	0
$997$	−7.98360e9	−0.255132	−0.127566	−	0.991830i	$-0.540717\pi$
$997$	−7.98360e9	−0.255132	−0.127566	+	0.991830i	$0.540717\pi$
$998$	0	0
$999$	−5.36069e9	−0.170115

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.8.a.c.1.3		6
4.3	odd	2		74.8.a.c.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.8.a.c.1.4	✓	6	4.3	odd	2
592.8.a.c.1.3		6	1.1	even	1	trivial

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Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	592.8.a.c.1.3

Level	$592$
Weight	$8$
Character	592.1
Self dual	yes
Analytic conductor	$184.932$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$1$