LMFDB - Embedded newform 592.8.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [592,8,Mod(1,592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	592.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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			1.1
Root			$-85.7890$ 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)

$f(q)$	$=$	$q-90.7890 q^{3} +133.163 q^{5} +948.062 q^{7} +6055.64 q^{9} -4655.78 q^{11} -1049.71 q^{13} -12089.8 q^{15} +37870.8 q^{17} +37462.0 q^{19} -86073.6 q^{21} -40596.4 q^{23} -60392.5 q^{25} -351229. q^{27} -66725.4 q^{29} -220090. q^{31} +422694. q^{33} +126247. q^{35} -50653.0 q^{37} +95301.7 q^{39} -101084. q^{41} -816690. q^{43} +806389. q^{45} +1.00197e6 q^{47} +75278.3 q^{49} -3.43825e6 q^{51} +1.40417e6 q^{53} -619980. q^{55} -3.40113e6 q^{57} -686480. q^{59} -2.33880e6 q^{61} +5.74112e6 q^{63} -139782. q^{65} +8039.60 q^{67} +3.68571e6 q^{69} +1.08288e6 q^{71} +142983. q^{73} +5.48297e6 q^{75} -4.41397e6 q^{77} +844178. q^{79} +1.86441e7 q^{81} +8.33149e6 q^{83} +5.04300e6 q^{85} +6.05793e6 q^{87} +8.29467e6 q^{89} -995186. q^{91} +1.99818e7 q^{93} +4.98857e6 q^{95} +4.06489e6 q^{97} -2.81937e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−90.7890	−1.94137	−0.970686	−	0.240352i	$-0.922737\pi$
$3$	−90.7890	−1.94137	−0.970686	+	0.240352i	$0.922737\pi$
$4$	0	0
$5$	133.163	0.476420	0.238210	−	0.971214i	$-0.423439\pi$
$5$	133.163	0.476420	0.238210	+	0.971214i	$0.423439\pi$
$6$	0	0
$7$	948.062	1.04470	0.522352	−	0.852730i	$-0.325055\pi$
$7$	948.062	1.04470	0.522352	+	0.852730i	$0.325055\pi$
$8$	0	0
$9$	6055.64	2.76892
$10$	0	0
$11$	−4655.78	−1.05467	−0.527337	−	0.849656i	$-0.676810\pi$
$11$	−4655.78	−1.05467	−0.527337	+	0.849656i	$0.676810\pi$
$12$	0	0
$13$	−1049.71	−0.132515	−0.0662576	−	0.997803i	$-0.521106\pi$
$13$	−1049.71	−0.132515	−0.0662576	+	0.997803i	$0.521106\pi$
$14$	0	0
$15$	−12089.8	−0.924908
$16$	0	0
$17$	37870.8	1.86953	0.934766	−	0.355264i	$-0.115609\pi$
$17$	37870.8	1.86953	0.934766	+	0.355264i	$0.115609\pi$
$18$	0	0
$19$	37462.0	1.25301	0.626503	−	0.779419i	$-0.284486\pi$
$19$	37462.0	1.25301	0.626503	+	0.779419i	$0.284486\pi$
$20$	0	0
$21$	−86073.6	−2.02816
$22$	0	0
$23$	−40596.4	−0.695729	−0.347865	−	0.937545i	$-0.613093\pi$
$23$	−40596.4	−0.695729	−0.347865	+	0.937545i	$0.613093\pi$
$24$	0	0
$25$	−60392.5	−0.773024
$26$	0	0
$27$	−351229.	−3.43414
$28$	0	0
$29$	−66725.4	−0.508041	−0.254020	−	0.967199i	$-0.581753\pi$
$29$	−66725.4	−0.508041	−0.254020	+	0.967199i	$0.581753\pi$
$30$	0	0
$31$	−220090.	−1.32689	−0.663446	−	0.748224i	$-0.730907\pi$
$31$	−220090.	−1.32689	−0.663446	+	0.748224i	$0.730907\pi$
$32$	0	0
$33$	422694.	2.04752
$34$	0	0
$35$	126247.	0.497718
$36$	0	0
$37$	−50653.0	−0.164399
$37$	−50653.0	−0.164399
$38$	0	0
$39$	95301.7	0.257261
$40$	0	0
$41$	−101084.	−0.229056	−0.114528	−	0.993420i	$-0.536536\pi$
$41$	−101084.	−0.229056	−0.114528	+	0.993420i	$0.536536\pi$
$42$	0	0
$43$	−816690.	−1.56645	−0.783227	−	0.621736i	$-0.786428\pi$
$43$	−816690.	−1.56645	−0.783227	+	0.621736i	$0.786428\pi$
$44$	0	0
$45$	806389.	1.31917
$46$	0	0
$47$	1.00197e6	1.40771	0.703857	−	0.710342i	$-0.251460\pi$
$47$	1.00197e6	1.40771	0.703857	+	0.710342i	$0.251460\pi$
$48$	0	0
$49$	75278.3	0.0914078
$50$	0	0
$51$	−3.43825e6	−3.62946
$52$	0	0
$53$	1.40417e6	1.29555	0.647776	−	0.761830i	$-0.275699\pi$
$53$	1.40417e6	1.29555	0.647776	+	0.761830i	$0.275699\pi$
$54$	0	0
$55$	−619980.	−0.502468
$56$	0	0
$57$	−3.40113e6	−2.43255
$58$	0	0
$59$	−686480.	−0.435157	−0.217578	−	0.976043i	$-0.569816\pi$
$59$	−686480.	−0.435157	−0.217578	+	0.976043i	$0.569816\pi$
$60$	0	0
$61$	−2.33880e6	−1.31929	−0.659644	−	0.751578i	$-0.729293\pi$
$61$	−2.33880e6	−1.31929	−0.659644	+	0.751578i	$0.729293\pi$
$62$	0	0
$63$	5.74112e6	2.89271
$64$	0	0
$65$	−139782.	−0.0631329
$66$	0	0
$67$	8039.60	0.00326567	0.00163284	−	0.999999i	$-0.499480\pi$
$67$	8039.60	0.00326567	0.00163284	+	0.999999i	$0.499480\pi$
$68$	0	0
$69$	3.68571e6	1.35067
$70$	0	0
$71$	1.08288e6	0.359066	0.179533	−	0.983752i	$-0.442541\pi$
$71$	1.08288e6	0.359066	0.179533	+	0.983752i	$0.442541\pi$
$72$	0	0
$73$	142983.	0.0430184	0.0215092	−	0.999769i	$-0.493153\pi$
$73$	142983.	0.0430184	0.0215092	+	0.999769i	$0.493153\pi$
$74$	0	0
$75$	5.48297e6	1.50073
$76$	0	0
$77$	−4.41397e6	−1.10182
$78$	0	0
$79$	844178.	0.192637	0.0963183	−	0.995351i	$-0.469293\pi$
$79$	844178.	0.192637	0.0963183	+	0.995351i	$0.469293\pi$
$80$	0	0
$81$	1.86441e7	3.89801
$82$	0	0
$83$	8.33149e6	1.59937	0.799685	−	0.600419i	$-0.205000\pi$
$83$	8.33149e6	1.59937	0.799685	+	0.600419i	$0.205000\pi$
$84$	0	0
$85$	5.04300e6	0.890683
$86$	0	0
$87$	6.05793e6	0.986296
$88$	0	0
$89$	8.29467e6	1.24719	0.623597	−	0.781746i	$-0.285671\pi$
$89$	8.29467e6	1.24719	0.623597	+	0.781746i	$0.285671\pi$
$90$	0	0
$91$	−995186.	−0.138439
$92$	0	0
$93$	1.99818e7	2.57599
$94$	0	0
$95$	4.98857e6	0.596957
$96$	0	0
$97$	4.06489e6	0.452218	0.226109	−	0.974102i	$-0.427399\pi$
$97$	4.06489e6	0.452218	0.226109	+	0.974102i	$0.427399\pi$
$98$	0	0
$99$	−2.81937e7	−2.92031
$100$	0	0
$101$	51781.2	0.00500089	0.00250045	−	0.999997i	$-0.499204\pi$
$101$	51781.2	0.00500089	0.00250045	+	0.999997i	$0.499204\pi$
$102$	0	0
$103$	8.90574e6	0.803045	0.401522	−	0.915849i	$-0.368481\pi$
$103$	8.90574e6	0.803045	0.401522	+	0.915849i	$0.368481\pi$
$104$	0	0
$105$	−1.14619e7	−0.966256
$106$	0	0
$107$	−23216.9	−0.00183215	−0.000916075	−	1.00000i	$-0.500292\pi$
$107$	−23216.9	−0.00183215	−0.000916075		1.00000i	$0.500292\pi$
$108$	0	0
$109$	−8.81772e6	−0.652174	−0.326087	−	0.945340i	$-0.605730\pi$
$109$	−8.81772e6	−0.652174	−0.326087	+	0.945340i	$0.605730\pi$
$110$	0	0
$111$	4.59873e6	0.319160
$112$	0	0
$113$	−2.36821e7	−1.54399	−0.771997	−	0.635626i	$-0.780742\pi$
$113$	−2.36821e7	−1.54399	−0.771997	+	0.635626i	$0.780742\pi$
$114$	0	0
$115$	−5.40596e6	−0.331459
$116$	0	0
$117$	−6.35664e6	−0.366925
$118$	0	0
$119$	3.59038e7	1.95311
$120$	0	0
$121$	2.18916e6	0.112338
$122$	0	0
$123$	9.17735e6	0.444682
$124$	0	0
$125$	−1.84455e7	−0.844704
$126$	0	0
$127$	30304.1	0.00131277	0.000656384	−	1.00000i	$-0.499791\pi$
$127$	30304.1	0.00131277	0.000656384		1.00000i	$0.499791\pi$
$128$	0	0
$129$	7.41464e7	3.04107
$130$	0	0
$131$	−1.27920e7	−0.497150	−0.248575	−	0.968613i	$-0.579962\pi$
$131$	−1.27920e7	−0.497150	−0.248575	+	0.968613i	$0.579962\pi$
$132$	0	0
$133$	3.55163e7	1.30902
$134$	0	0
$135$	−4.67709e7	−1.63609
$136$	0	0
$137$	−4.45086e7	−1.47884	−0.739422	−	0.673242i	$-0.764901\pi$
$137$	−4.45086e7	−1.47884	−0.739422	+	0.673242i	$0.764901\pi$
$138$	0	0
$139$	−3.36501e6	−0.106276	−0.0531379	−	0.998587i	$-0.516922\pi$
$139$	−3.36501e6	−0.106276	−0.0531379	+	0.998587i	$0.516922\pi$
$140$	0	0
$141$	−9.09682e7	−2.73289
$142$	0	0
$143$	4.88720e6	0.139760
$144$	0	0
$145$	−8.88539e6	−0.242041
$146$	0	0
$147$	−6.83444e6	−0.177457
$148$	0	0
$149$	−3.75616e7	−0.930234	−0.465117	−	0.885249i	$-0.653988\pi$
$149$	−3.75616e7	−0.930234	−0.465117	+	0.885249i	$0.653988\pi$
$150$	0	0
$151$	7.66361e7	1.81140	0.905699	−	0.423922i	$-0.139347\pi$
$151$	7.66361e7	1.81140	0.905699	+	0.423922i	$0.139347\pi$
$152$	0	0
$153$	2.29332e8	5.17659
$154$	0	0
$155$	−2.93080e7	−0.632158
$156$	0	0
$157$	−3.83181e7	−0.790233	−0.395116	−	0.918631i	$-0.629296\pi$
$157$	−3.83181e7	−0.790233	−0.395116	+	0.918631i	$0.629296\pi$
$158$	0	0
$159$	−1.27483e8	−2.51515
$160$	0	0
$161$	−3.84879e7	−0.726831
$162$	0	0
$163$	−4.20835e7	−0.761123	−0.380562	−	0.924756i	$-0.624269\pi$
$163$	−4.20835e7	−0.761123	−0.380562	+	0.924756i	$0.624269\pi$
$164$	0	0
$165$	5.62874e7	0.975477
$166$	0	0
$167$	1.08504e8	1.80276	0.901378	−	0.433033i	$-0.142557\pi$
$167$	1.08504e8	1.80276	0.901378	+	0.433033i	$0.142557\pi$
$168$	0	0
$169$	−6.16466e7	−0.982440
$170$	0	0
$171$	2.26856e8	3.46948
$172$	0	0
$173$	−3.54173e7	−0.520060	−0.260030	−	0.965601i	$-0.583732\pi$
$173$	−3.54173e7	−0.520060	−0.260030	+	0.965601i	$0.583732\pi$
$174$	0	0
$175$	−5.72558e7	−0.807582
$176$	0	0
$177$	6.23248e7	0.844801
$178$	0	0
$179$	−4.27586e7	−0.557235	−0.278617	−	0.960402i	$-0.589876\pi$
$179$	−4.27586e7	−0.557235	−0.278617	+	0.960402i	$0.589876\pi$
$180$	0	0
$181$	−7.04378e7	−0.882940	−0.441470	−	0.897276i	$-0.645543\pi$
$181$	−7.04378e7	−0.882940	−0.441470	+	0.897276i	$0.645543\pi$
$182$	0	0
$183$	2.12338e8	2.56123
$184$	0	0
$185$	−6.74513e6	−0.0783230
$186$	0	0
$187$	−1.76318e8	−1.97175
$188$	0	0
$189$	−3.32987e8	−3.58766
$190$	0	0
$191$	−1.29769e8	−1.34758	−0.673788	−	0.738925i	$-0.735334\pi$
$191$	−1.29769e8	−1.34758	−0.673788	+	0.738925i	$0.735334\pi$
$192$	0	0
$193$	5.09514e7	0.510159	0.255079	−	0.966920i	$-0.417898\pi$
$193$	5.09514e7	0.510159	0.255079	+	0.966920i	$0.417898\pi$
$194$	0	0
$195$	1.26907e7	0.122564
$196$	0	0
$197$	−1.80715e6	−0.0168408	−0.00842039	−	0.999965i	$-0.502680\pi$
$197$	−1.80715e6	−0.0168408	−0.00842039	+	0.999965i	$0.502680\pi$
$198$	0	0
$199$	−4.75898e7	−0.428083	−0.214041	−	0.976825i	$-0.568663\pi$
$199$	−4.75898e7	−0.428083	−0.214041	+	0.976825i	$0.568663\pi$
$200$	0	0
$201$	−729907.	−0.00633989
$202$	0	0
$203$	−6.32598e7	−0.530752
$204$	0	0
$205$	−1.34608e7	−0.109127
$206$	0	0
$207$	−2.45837e8	−1.92642
$208$	0	0
$209$	−1.74415e8	−1.32151
$210$	0	0
$211$	5.04805e7	0.369943	0.184972	−	0.982744i	$-0.440781\pi$
$211$	5.04805e7	0.369943	0.184972	+	0.982744i	$0.440781\pi$
$212$	0	0
$213$	−9.83131e7	−0.697080
$214$	0	0
$215$	−1.08753e8	−0.746290
$216$	0	0
$217$	−2.08659e8	−1.38621
$218$	0	0
$219$	−1.29813e7	−0.0835146
$220$	0	0
$221$	−3.97532e7	−0.247742
$222$	0	0
$223$	−1.34484e8	−0.812089	−0.406044	−	0.913853i	$-0.633092\pi$
$223$	−1.34484e8	−0.812089	−0.406044	+	0.913853i	$0.633092\pi$
$224$	0	0
$225$	−3.65715e8	−2.14044
$226$	0	0
$227$	−1.82525e8	−1.03569	−0.517847	−	0.855473i	$-0.673266\pi$
$227$	−1.82525e8	−1.03569	−0.517847	+	0.855473i	$0.673266\pi$
$228$	0	0
$229$	1.55708e8	0.856816	0.428408	−	0.903585i	$-0.359075\pi$
$229$	1.55708e8	0.856816	0.428408	+	0.903585i	$0.359075\pi$
$230$	0	0
$231$	4.00740e8	2.13905
$232$	0	0
$233$	6.82458e7	0.353452	0.176726	−	0.984260i	$-0.443449\pi$
$233$	6.82458e7	0.353452	0.176726	+	0.984260i	$0.443449\pi$
$234$	0	0
$235$	1.33426e8	0.670663
$236$	0	0
$237$	−7.66420e7	−0.373979
$238$	0	0
$239$	1.59910e8	0.757677	0.378838	−	0.925463i	$-0.376324\pi$
$239$	1.59910e8	0.757677	0.378838	+	0.925463i	$0.376324\pi$
$240$	0	0
$241$	2.46558e8	1.13465	0.567323	−	0.823496i	$-0.307979\pi$
$241$	2.46558e8	1.13465	0.567323	+	0.823496i	$0.307979\pi$
$242$	0	0
$243$	−9.24538e8	−4.13336
$244$	0	0
$245$	1.00243e7	0.0435485
$246$	0	0
$247$	−3.93241e7	−0.166042
$248$	0	0
$249$	−7.56407e8	−3.10497
$250$	0	0
$251$	3.82771e8	1.52785	0.763926	−	0.645304i	$-0.223269\pi$
$251$	3.82771e8	1.52785	0.763926	+	0.645304i	$0.223269\pi$
$252$	0	0
$253$	1.89008e8	0.733768
$254$	0	0
$255$	−4.57849e8	−1.72915
$256$	0	0
$257$	1.28126e8	0.470837	0.235419	−	0.971894i	$-0.424354\pi$
$257$	1.28126e8	0.470837	0.235419	+	0.971894i	$0.424354\pi$
$258$	0	0
$259$	−4.80222e7	−0.171748
$260$	0	0
$261$	−4.04065e8	−1.40673
$262$	0	0
$263$	−1.53745e8	−0.521142	−0.260571	−	0.965455i	$-0.583911\pi$
$263$	−1.53745e8	−0.521142	−0.260571	+	0.965455i	$0.583911\pi$
$264$	0	0
$265$	1.86985e8	0.617227
$266$	0	0
$267$	−7.53064e8	−2.42127
$268$	0	0
$269$	4.21270e8	1.31956	0.659778	−	0.751461i	$-0.270650\pi$
$269$	4.21270e8	1.31956	0.659778	+	0.751461i	$0.270650\pi$
$270$	0	0
$271$	−5.02799e8	−1.53462	−0.767312	−	0.641274i	$-0.778406\pi$
$271$	−5.02799e8	−1.53462	−0.767312	+	0.641274i	$0.778406\pi$
$272$	0	0
$273$	9.03519e7	0.268762
$274$	0	0
$275$	2.81174e8	0.815289
$276$	0	0
$277$	3.20002e8	0.904636	0.452318	−	0.891857i	$-0.350597\pi$
$277$	3.20002e8	0.904636	0.452318	+	0.891857i	$0.350597\pi$
$278$	0	0
$279$	−1.33279e9	−3.67406
$280$	0	0
$281$	−4.74241e8	−1.27505	−0.637524	−	0.770430i	$-0.720042\pi$
$281$	−4.74241e8	−1.27505	−0.637524	+	0.770430i	$0.720042\pi$
$282$	0	0
$283$	5.32308e8	1.39608	0.698040	−	0.716058i	$-0.254056\pi$
$283$	5.32308e8	1.39608	0.698040	+	0.716058i	$0.254056\pi$
$284$	0	0
$285$	−4.52907e8	−1.15892
$286$	0	0
$287$	−9.58343e7	−0.239295
$288$	0	0
$289$	1.02386e9	2.49515
$290$	0	0
$291$	−3.69047e8	−0.877923
$292$	0	0
$293$	−2.15803e8	−0.501211	−0.250606	−	0.968089i	$-0.580630\pi$
$293$	−2.15803e8	−0.501211	−0.250606	+	0.968089i	$0.580630\pi$
$294$	0	0
$295$	−9.14140e7	−0.207317
$296$	0	0
$297$	1.63525e9	3.62190
$298$	0	0
$299$	4.26143e7	0.0921947
$300$	0	0
$301$	−7.74272e8	−1.63648
$302$	0	0
$303$	−4.70116e6	−0.00970859
$304$	0	0
$305$	−3.11443e8	−0.628535
$306$	0	0
$307$	5.97092e8	1.17776	0.588881	−	0.808220i	$-0.299569\pi$
$307$	5.97092e8	1.17776	0.588881	+	0.808220i	$0.299569\pi$
$308$	0	0
$309$	−8.08543e8	−1.55901
$310$	0	0
$311$	−2.38042e8	−0.448738	−0.224369	−	0.974504i	$-0.572032\pi$
$311$	−2.38042e8	−0.448738	−0.224369	+	0.974504i	$0.572032\pi$
$312$	0	0
$313$	3.27112e8	0.602964	0.301482	−	0.953472i	$-0.402519\pi$
$313$	3.27112e8	0.602964	0.301482	+	0.953472i	$0.402519\pi$
$314$	0	0
$315$	7.64507e8	1.37814
$316$	0	0
$317$	−8.33077e8	−1.46885	−0.734426	−	0.678689i	$-0.762548\pi$
$317$	−8.33077e8	−1.46885	−0.734426	+	0.678689i	$0.762548\pi$
$318$	0	0
$319$	3.10659e8	0.535817
$320$	0	0
$321$	2.10784e6	0.00355688
$322$	0	0
$323$	1.41871e9	2.34253
$324$	0	0
$325$	6.33944e7	0.102437
$326$	0	0
$327$	8.00551e8	1.26611
$328$	0	0
$329$	9.49934e8	1.47064
$330$	0	0
$331$	4.86383e8	0.737192	0.368596	−	0.929590i	$-0.379839\pi$
$331$	4.86383e8	0.737192	0.368596	+	0.929590i	$0.379839\pi$
$332$	0	0
$333$	−3.06736e8	−0.455208
$334$	0	0
$335$	1.07058e6	0.00155583
$336$	0	0
$337$	−9.74289e8	−1.38670	−0.693351	−	0.720600i	$-0.743867\pi$
$337$	−9.74289e8	−1.38670	−0.693351	+	0.720600i	$0.743867\pi$
$338$	0	0
$339$	2.15007e9	2.99747
$340$	0	0
$341$	1.02469e9	1.39944
$342$	0	0
$343$	−7.09401e8	−0.949210
$344$	0	0
$345$	4.90801e8	0.643486
$346$	0	0
$347$	−8.61556e8	−1.10696	−0.553478	−	0.832864i	$-0.686700\pi$
$347$	−8.61556e8	−1.10696	−0.553478	+	0.832864i	$0.686700\pi$
$348$	0	0
$349$	1.30969e9	1.64923	0.824613	−	0.565697i	$-0.191393\pi$
$349$	1.30969e9	1.64923	0.824613	+	0.565697i	$0.191393\pi$
$350$	0	0
$351$	3.68688e8	0.455076
$352$	0	0
$353$	−8.77420e8	−1.06169	−0.530843	−	0.847470i	$-0.678125\pi$
$353$	−8.77420e8	−1.06169	−0.530843	+	0.847470i	$0.678125\pi$
$354$	0	0
$355$	1.44199e8	0.171066
$356$	0	0
$357$	−3.25967e9	−3.79171
$358$	0	0
$359$	−1.49791e9	−1.70866	−0.854331	−	0.519729i	$-0.826033\pi$
$359$	−1.49791e9	−1.70866	−0.854331	+	0.519729i	$0.826033\pi$
$360$	0	0
$361$	5.09528e8	0.570023
$362$	0	0
$363$	−1.98751e8	−0.218090
$364$	0	0
$365$	1.90401e7	0.0204948
$366$	0	0
$367$	1.40720e9	1.48602	0.743008	−	0.669283i	$-0.233398\pi$
$367$	1.40720e9	1.48602	0.743008	+	0.669283i	$0.233398\pi$
$368$	0	0
$369$	−6.12131e8	−0.634237
$370$	0	0
$371$	1.33124e9	1.35347
$372$	0	0
$373$	−6.79783e8	−0.678249	−0.339125	−	0.940741i	$-0.610131\pi$
$373$	−6.79783e8	−0.678249	−0.339125	+	0.940741i	$0.610131\pi$
$374$	0	0
$375$	1.67464e9	1.63988
$376$	0	0
$377$	7.00421e7	0.0673231
$378$	0	0
$379$	−1.91260e9	−1.80462	−0.902310	−	0.431087i	$-0.858130\pi$
$379$	−1.91260e9	−1.80462	−0.902310	+	0.431087i	$0.858130\pi$
$380$	0	0
$381$	−2.75127e6	−0.00254857
$382$	0	0
$383$	−1.45283e9	−1.32135	−0.660676	−	0.750671i	$-0.729730\pi$
$383$	−1.45283e9	−1.32135	−0.660676	+	0.750671i	$0.729730\pi$
$384$	0	0
$385$	−5.87780e8	−0.524931
$386$	0	0
$387$	−4.94558e9	−4.33739
$388$	0	0
$389$	−9.78824e8	−0.843104	−0.421552	−	0.906804i	$-0.638514\pi$
$389$	−9.78824e8	−0.843104	−0.421552	+	0.906804i	$0.638514\pi$
$390$	0	0
$391$	−1.53742e9	−1.30069
$392$	0	0
$393$	1.16137e9	0.965154
$394$	0	0
$395$	1.12414e8	0.0917760
$396$	0	0
$397$	−1.37188e9	−1.10039	−0.550197	−	0.835035i	$-0.685447\pi$
$397$	−1.37188e9	−1.10039	−0.550197	+	0.835035i	$0.685447\pi$
$398$	0	0
$399$	−3.22449e9	−2.54130
$400$	0	0
$401$	3.44789e8	0.267022	0.133511	−	0.991047i	$-0.457375\pi$
$401$	3.44789e8	0.267022	0.133511	+	0.991047i	$0.457375\pi$
$402$	0	0
$403$	2.31030e8	0.175833
$404$	0	0
$405$	2.48271e9	1.85709
$406$	0	0
$407$	2.35829e8	0.173387
$408$	0	0
$409$	5.43183e8	0.392568	0.196284	−	0.980547i	$-0.437113\pi$
$409$	5.43183e8	0.392568	0.196284	+	0.980547i	$0.437113\pi$
$410$	0	0
$411$	4.04089e9	2.87099
$412$	0	0
$413$	−6.50825e8	−0.454610
$414$	0	0
$415$	1.10945e9	0.761972
$416$	0	0
$417$	3.05506e8	0.206321
$418$	0	0
$419$	−2.19669e9	−1.45888	−0.729439	−	0.684046i	$-0.760219\pi$
$419$	−2.19669e9	−1.45888	−0.729439	+	0.684046i	$0.760219\pi$
$420$	0	0
$421$	1.69532e9	1.10730	0.553650	−	0.832749i	$-0.313234\pi$
$421$	1.69532e9	1.10730	0.553650	+	0.832749i	$0.313234\pi$
$422$	0	0
$423$	6.06759e9	3.89785
$424$	0	0
$425$	−2.28711e9	−1.44519
$426$	0	0
$427$	−2.21733e9	−1.37827
$428$	0	0
$429$	−4.43704e8	−0.271327
$430$	0	0
$431$	6.19918e8	0.372962	0.186481	−	0.982459i	$-0.440292\pi$
$431$	6.19918e8	0.372962	0.186481	+	0.982459i	$0.440292\pi$
$432$	0	0
$433$	2.69875e9	1.59755	0.798777	−	0.601627i	$-0.205481\pi$
$433$	2.69875e9	1.59755	0.798777	+	0.601627i	$0.205481\pi$
$434$	0	0
$435$	8.06695e8	0.469891
$436$	0	0
$437$	−1.52082e9	−0.871752
$438$	0	0
$439$	1.55515e9	0.877297	0.438648	−	0.898659i	$-0.355457\pi$
$439$	1.55515e9	0.877297	0.438648	+	0.898659i	$0.355457\pi$
$440$	0	0
$441$	4.55858e8	0.253101
$442$	0	0
$443$	−9.06288e8	−0.495283	−0.247641	−	0.968852i	$-0.579655\pi$
$443$	−9.06288e8	−0.495283	−0.247641	+	0.968852i	$0.579655\pi$
$444$	0	0
$445$	1.10455e9	0.594188
$446$	0	0
$447$	3.41018e9	1.80593
$448$	0	0
$449$	4.68202e8	0.244102	0.122051	−	0.992524i	$-0.461053\pi$
$449$	4.68202e8	0.244102	0.122051	+	0.992524i	$0.461053\pi$
$450$	0	0
$451$	4.70627e8	0.241579
$452$	0	0
$453$	−6.95771e9	−3.51660
$454$	0	0
$455$	−1.32522e8	−0.0659553
$456$	0	0
$457$	−6.95216e8	−0.340732	−0.170366	−	0.985381i	$-0.554495\pi$
$457$	−6.95216e8	−0.340732	−0.170366	+	0.985381i	$0.554495\pi$
$458$	0	0
$459$	−1.33013e10	−6.42023
$460$	0	0
$461$	4.09996e8	0.194906	0.0974532	−	0.995240i	$-0.468930\pi$
$461$	4.09996e8	0.194906	0.0974532	+	0.995240i	$0.468930\pi$
$462$	0	0
$463$	1.35529e9	0.634600	0.317300	−	0.948325i	$-0.397224\pi$
$463$	1.35529e9	0.634600	0.317300	+	0.948325i	$0.397224\pi$
$464$	0	0
$465$	2.66084e9	1.22725
$466$	0	0
$467$	7.26978e8	0.330303	0.165151	−	0.986268i	$-0.447189\pi$
$467$	7.26978e8	0.330303	0.165151	+	0.986268i	$0.447189\pi$
$468$	0	0
$469$	7.62204e6	0.00341167
$470$	0	0
$471$	3.47886e9	1.53414
$472$	0	0
$473$	3.80233e9	1.65210
$474$	0	0
$475$	−2.26242e9	−0.968603
$476$	0	0
$477$	8.50316e9	3.58729
$478$	0	0
$479$	−2.16041e9	−0.898175	−0.449087	−	0.893488i	$-0.648251\pi$
$479$	−2.16041e9	−0.898175	−0.449087	+	0.893488i	$0.648251\pi$
$480$	0	0
$481$	5.31708e7	0.0217854
$482$	0	0
$483$	3.49428e9	1.41105
$484$	0	0
$485$	5.41295e8	0.215446
$486$	0	0
$487$	9.09190e7	0.0356700	0.0178350	−	0.999841i	$-0.494323\pi$
$487$	9.09190e7	0.0356700	0.0178350	+	0.999841i	$0.494323\pi$
$488$	0	0
$489$	3.82072e9	1.47762
$490$	0	0
$491$	4.35917e9	1.66195	0.830976	−	0.556308i	$-0.187782\pi$
$491$	4.35917e9	1.66195	0.830976	+	0.556308i	$0.187782\pi$
$492$	0	0
$493$	−2.52694e9	−0.949798
$494$	0	0
$495$	−3.75437e9	−1.39130
$496$	0	0
$497$	1.02663e9	0.375118
$498$	0	0
$499$	−3.39355e9	−1.22265	−0.611326	−	0.791379i	$-0.709364\pi$
$499$	−3.39355e9	−1.22265	−0.611326	+	0.791379i	$0.709364\pi$
$500$	0	0
$501$	−9.85094e9	−3.49982
$502$	0	0
$503$	−4.26413e9	−1.49397	−0.746986	−	0.664840i	$-0.768500\pi$
$503$	−4.26413e9	−1.49397	−0.746986	+	0.664840i	$0.768500\pi$
$504$	0	0
$505$	6.89537e6	0.00238253
$506$	0	0
$507$	5.59683e9	1.90728
$508$	0	0
$509$	−4.85962e9	−1.63339	−0.816695	−	0.577070i	$-0.804196\pi$
$509$	−4.85962e9	−1.63339	−0.816695	+	0.577070i	$0.804196\pi$
$510$	0	0
$511$	1.35557e8	0.0449415
$512$	0	0
$513$	−1.31577e10	−4.30299
$514$	0	0
$515$	1.18592e9	0.382587
$516$	0	0
$517$	−4.66498e9	−1.48468
$518$	0	0
$519$	3.21550e9	1.00963
$520$	0	0
$521$	4.97824e8	0.154221	0.0771106	−	0.997023i	$-0.475431\pi$
$521$	4.97824e8	0.154221	0.0771106	+	0.997023i	$0.475431\pi$
$522$	0	0
$523$	−3.00519e9	−0.918579	−0.459289	−	0.888287i	$-0.651896\pi$
$523$	−3.00519e9	−0.918579	−0.459289	+	0.888287i	$0.651896\pi$
$524$	0	0
$525$	5.19820e9	1.56782
$526$	0	0
$527$	−8.33500e9	−2.48067
$528$	0	0
$529$	−1.75676e9	−0.515961
$530$	0	0
$531$	−4.15707e9	−1.20492
$532$	0	0
$533$	1.06109e8	0.0303534
$534$	0	0
$535$	−3.09164e6	−0.000872873 0
$536$	0	0
$537$	3.88201e9	1.08180
$538$	0	0
$539$	−3.50479e8	−0.0964055
$540$	0	0
$541$	−2.61020e9	−0.708734	−0.354367	−	0.935106i	$-0.615304\pi$
$541$	−2.61020e9	−0.708734	−0.354367	+	0.935106i	$0.615304\pi$
$542$	0	0
$543$	6.39498e9	1.71411
$544$	0	0
$545$	−1.17420e9	−0.310709
$546$	0	0
$547$	−6.31955e8	−0.165094	−0.0825468	−	0.996587i	$-0.526305\pi$
$547$	−6.31955e8	−0.165094	−0.0825468	+	0.996587i	$0.526305\pi$
$548$	0	0
$549$	−1.41629e10	−3.65301
$550$	0	0
$551$	−2.49967e9	−0.636578
$552$	0	0
$553$	8.00333e8	0.201248
$554$	0	0
$555$	6.12383e8	0.152054
$556$	0	0
$557$	3.22387e9	0.790468	0.395234	−	0.918580i	$-0.370663\pi$
$557$	3.22387e9	0.790468	0.395234	+	0.918580i	$0.370663\pi$
$558$	0	0
$559$	8.57284e8	0.207579
$560$	0	0
$561$	1.60077e10	3.82790
$562$	0	0
$563$	−3.75859e9	−0.887658	−0.443829	−	0.896112i	$-0.646380\pi$
$563$	−3.75859e9	−0.887658	−0.443829	+	0.896112i	$0.646380\pi$
$564$	0	0
$565$	−3.15359e9	−0.735589
$566$	0	0
$567$	1.76757e10	4.07227
$568$	0	0
$569$	3.16154e8	0.0719459	0.0359730	−	0.999353i	$-0.488547\pi$
$569$	3.16154e8	0.0719459	0.0359730	+	0.999353i	$0.488547\pi$
$570$	0	0
$571$	−5.14375e9	−1.15625	−0.578127	−	0.815947i	$-0.696216\pi$
$571$	−5.14375e9	−1.15625	−0.578127	+	0.815947i	$0.696216\pi$
$572$	0	0
$573$	1.17816e10	2.61614
$574$	0	0
$575$	2.45172e9	0.537815
$576$	0	0
$577$	2.95737e8	0.0640900	0.0320450	−	0.999486i	$-0.489798\pi$
$577$	2.95737e8	0.0640900	0.0320450	+	0.999486i	$0.489798\pi$
$578$	0	0
$579$	−4.62582e9	−0.990408
$580$	0	0
$581$	7.89876e9	1.67087
$582$	0	0
$583$	−6.53753e9	−1.36639
$584$	0	0
$585$	−8.46472e8	−0.174810
$586$	0	0
$587$	−1.01541e9	−0.207209	−0.103604	−	0.994619i	$-0.533038\pi$
$587$	−1.01541e9	−0.207209	−0.103604	+	0.994619i	$0.533038\pi$
$588$	0	0
$589$	−8.24502e9	−1.66260
$590$	0	0
$591$	1.64069e8	0.0326942
$592$	0	0
$593$	−3.96291e9	−0.780409	−0.390205	−	0.920728i	$-0.627596\pi$
$593$	−3.96291e9	−0.780409	−0.390205	+	0.920728i	$0.627596\pi$
$594$	0	0
$595$	4.78108e9	0.930500
$596$	0	0
$597$	4.32063e9	0.831068
$598$	0	0
$599$	−9.42097e9	−1.79103	−0.895513	−	0.445036i	$-0.853191\pi$
$599$	−9.42097e9	−1.79103	−0.895513	+	0.445036i	$0.853191\pi$
$600$	0	0
$601$	3.58702e8	0.0674021	0.0337010	−	0.999432i	$-0.489271\pi$
$601$	3.58702e8	0.0674021	0.0337010	+	0.999432i	$0.489271\pi$
$602$	0	0
$603$	4.86849e7	0.00904240
$604$	0	0
$605$	2.91515e8	0.0535202
$606$	0	0
$607$	−6.29066e9	−1.14166	−0.570829	−	0.821069i	$-0.693378\pi$
$607$	−6.29066e9	−1.14166	−0.570829	+	0.821069i	$0.693378\pi$
$608$	0	0
$609$	5.74330e9	1.03039
$610$	0	0
$611$	−1.05178e9	−0.186544
$612$	0	0
$613$	3.78212e9	0.663168	0.331584	−	0.943426i	$-0.392417\pi$
$613$	3.78212e9	0.663168	0.331584	+	0.943426i	$0.392417\pi$
$614$	0	0
$615$	1.22209e9	0.211855
$616$	0	0
$617$	−2.05900e9	−0.352905	−0.176452	−	0.984309i	$-0.556462\pi$
$617$	−2.05900e9	−0.352905	−0.176452	+	0.984309i	$0.556462\pi$
$618$	0	0
$619$	−1.93114e9	−0.327263	−0.163632	−	0.986522i	$-0.552321\pi$
$619$	−1.93114e9	−0.327263	−0.163632	+	0.986522i	$0.552321\pi$
$620$	0	0
$621$	1.42587e10	2.38923
$622$	0	0
$623$	7.86386e9	1.30295
$624$	0	0
$625$	2.26190e9	0.370590
$626$	0	0
$627$	1.58349e10	2.56555
$628$	0	0
$629$	−1.91827e9	−0.307349
$630$	0	0
$631$	−1.64426e9	−0.260537	−0.130268	−	0.991479i	$-0.541584\pi$
$631$	−1.64426e9	−0.260537	−0.130268	+	0.991479i	$0.541584\pi$
$632$	0	0
$633$	−4.58307e9	−0.718197
$634$	0	0
$635$	4.03539e6	0.000625429 0
$636$	0	0
$637$	−7.90201e7	−0.0121129
$638$	0	0
$639$	6.55750e9	0.994226
$640$	0	0
$641$	−1.79601e9	−0.269344	−0.134672	−	0.990890i	$-0.542998\pi$
$641$	−1.79601e9	−0.269344	−0.134672	+	0.990890i	$0.542998\pi$
$642$	0	0
$643$	−6.06484e9	−0.899666	−0.449833	−	0.893113i	$-0.648516\pi$
$643$	−6.06484e9	−0.899666	−0.449833	+	0.893113i	$0.648516\pi$
$644$	0	0
$645$	9.87359e9	1.44883
$646$	0	0
$647$	8.83053e9	1.28180	0.640902	−	0.767622i	$-0.278560\pi$
$647$	8.83053e9	1.28180	0.640902	+	0.767622i	$0.278560\pi$
$648$	0	0
$649$	3.19610e9	0.458949
$650$	0	0
$651$	1.89440e10	2.69115
$652$	0	0
$653$	1.00013e10	1.40560	0.702800	−	0.711388i	$-0.251933\pi$
$653$	1.00013e10	1.40560	0.702800	+	0.711388i	$0.251933\pi$
$654$	0	0
$655$	−1.70342e9	−0.236852
$656$	0	0
$657$	8.65852e8	0.119115
$658$	0	0
$659$	4.00313e9	0.544880	0.272440	−	0.962173i	$-0.412169\pi$
$659$	4.00313e9	0.544880	0.272440	+	0.962173i	$0.412169\pi$
$660$	0	0
$661$	1.12552e10	1.51582	0.757910	−	0.652359i	$-0.226221\pi$
$661$	1.12552e10	1.51582	0.757910	+	0.652359i	$0.226221\pi$
$662$	0	0
$663$	3.60915e9	0.480958
$664$	0	0
$665$	4.72947e9	0.623644
$666$	0	0
$667$	2.70881e9	0.353459
$668$	0	0
$669$	1.22097e10	1.57657
$670$	0	0
$671$	1.08890e10	1.39142
$672$	0	0
$673$	−1.10405e10	−1.39616	−0.698080	−	0.716020i	$-0.745962\pi$
$673$	−1.10405e10	−1.39616	−0.698080	+	0.716020i	$0.745962\pi$
$674$	0	0
$675$	2.12116e10	2.65467
$676$	0	0
$677$	−5.08333e9	−0.629634	−0.314817	−	0.949152i	$-0.601943\pi$
$677$	−5.08333e9	−0.629634	−0.314817	+	0.949152i	$0.601943\pi$
$678$	0	0
$679$	3.85377e9	0.472434
$680$	0	0
$681$	1.65712e10	2.01067
$682$	0	0
$683$	4.62507e9	0.555451	0.277725	−	0.960661i	$-0.410419\pi$
$683$	4.62507e9	0.555451	0.277725	+	0.960661i	$0.410419\pi$
$684$	0	0
$685$	−5.92692e9	−0.704551
$686$	0	0
$687$	−1.41366e10	−1.66340
$688$	0	0
$689$	−1.47397e9	−0.171681
$690$	0	0
$691$	1.38457e10	1.59640	0.798200	−	0.602393i	$-0.205786\pi$
$691$	1.38457e10	1.59640	0.798200	+	0.602393i	$0.205786\pi$
$692$	0	0
$693$	−2.67294e10	−3.05086
$694$	0	0
$695$	−4.48096e8	−0.0506319
$696$	0	0
$697$	−3.82815e9	−0.428227
$698$	0	0
$699$	−6.19597e9	−0.686181
$700$	0	0
$701$	1.30053e10	1.42596	0.712980	−	0.701184i	$-0.247345\pi$
$701$	1.30053e10	1.42596	0.712980	+	0.701184i	$0.247345\pi$
$702$	0	0
$703$	−1.89756e9	−0.205993
$704$	0	0
$705$	−1.21136e10	−1.30201
$706$	0	0
$707$	4.90918e7	0.00522446
$708$	0	0
$709$	−1.83890e10	−1.93774	−0.968872	−	0.247564i	$-0.920370\pi$
$709$	−1.83890e10	−1.93774	−0.968872	+	0.247564i	$0.920370\pi$
$710$	0	0
$711$	5.11203e9	0.533396
$712$	0	0
$713$	8.93488e9	0.923157
$714$	0	0
$715$	6.50797e8	0.0665847
$716$	0	0
$717$	−1.45181e10	−1.47093
$718$	0	0
$719$	−1.56639e9	−0.157162	−0.0785811	−	0.996908i	$-0.525039\pi$
$719$	−1.56639e9	−0.157162	−0.0785811	+	0.996908i	$0.525039\pi$
$720$	0	0
$721$	8.44319e9	0.838945
$722$	0	0
$723$	−2.23848e10	−2.20277
$724$	0	0
$725$	4.02972e9	0.392728
$726$	0	0
$727$	1.22967e10	1.18691	0.593456	−	0.804867i	$-0.297763\pi$
$727$	1.22967e10	1.18691	0.593456	+	0.804867i	$0.297763\pi$
$728$	0	0
$729$	4.31633e10	4.12637
$730$	0	0
$731$	−3.09287e10	−2.92854
$732$	0	0
$733$	−8.41188e9	−0.788913	−0.394456	−	0.918915i	$-0.629067\pi$
$733$	−8.41188e9	−0.788913	−0.394456	+	0.918915i	$0.629067\pi$
$734$	0	0
$735$	−9.10097e8	−0.0845439
$736$	0	0
$737$	−3.74307e7	−0.00344422
$738$	0	0
$739$	−4.83992e9	−0.441147	−0.220573	−	0.975370i	$-0.570793\pi$
$739$	−4.83992e9	−0.441147	−0.220573	+	0.975370i	$0.570793\pi$
$740$	0	0
$741$	3.57019e9	0.322350
$742$	0	0
$743$	−5.87021e8	−0.0525040	−0.0262520	−	0.999655i	$-0.508357\pi$
$743$	−5.87021e8	−0.0525040	−0.0262520	+	0.999655i	$0.508357\pi$
$744$	0	0
$745$	−5.00184e9	−0.443182
$746$	0	0
$747$	5.04524e10	4.42854
$748$	0	0
$749$	−2.20111e7	−0.00191406
$750$	0	0
$751$	2.24897e9	0.193751	0.0968753	−	0.995297i	$-0.469115\pi$
$751$	2.24897e9	0.193751	0.0968753	+	0.995297i	$0.469115\pi$
$752$	0	0
$753$	−3.47514e10	−2.96613
$754$	0	0
$755$	1.02051e10	0.862986
$756$	0	0
$757$	−1.99556e10	−1.67197	−0.835985	−	0.548752i	$-0.815103\pi$
$757$	−1.99556e10	−1.67197	−0.835985	+	0.548752i	$0.815103\pi$
$758$	0	0
$759$	−1.71598e10	−1.42452
$760$	0	0
$761$	−5.23703e9	−0.430764	−0.215382	−	0.976530i	$-0.569100\pi$
$761$	−5.23703e9	−0.430764	−0.215382	+	0.976530i	$0.569100\pi$
$762$	0	0
$763$	−8.35974e9	−0.681329
$764$	0	0
$765$	3.05386e10	2.46623
$766$	0	0
$767$	7.20602e8	0.0576649
$768$	0	0
$769$	−7.11248e9	−0.564000	−0.282000	−	0.959414i	$-0.590998\pi$
$769$	−7.11248e9	−0.564000	−0.282000	+	0.959414i	$0.590998\pi$
$770$	0	0
$771$	−1.16324e10	−0.914070
$772$	0	0
$773$	5.47777e9	0.426556	0.213278	−	0.976992i	$-0.431586\pi$
$773$	5.47777e9	0.426556	0.213278	+	0.976992i	$0.431586\pi$
$774$	0	0
$775$	1.32918e10	1.02572
$776$	0	0
$777$	4.35988e9	0.333427
$778$	0	0
$779$	−3.78682e9	−0.287008
$780$	0	0
$781$	−5.04163e9	−0.378698
$782$	0	0
$783$	2.34359e10	1.74468
$784$	0	0
$785$	−5.10257e9	−0.376483
$786$	0	0
$787$	1.97867e10	1.44698	0.723488	−	0.690337i	$-0.242538\pi$
$787$	1.97867e10	1.44698	0.723488	+	0.690337i	$0.242538\pi$
$788$	0	0
$789$	1.39584e10	1.01173
$790$	0	0
$791$	−2.24521e10	−1.61302
$792$	0	0
$793$	2.45506e9	0.174826
$794$	0	0
$795$	−1.69761e10	−1.19827
$796$	0	0
$797$	−1.91992e10	−1.34332	−0.671661	−	0.740859i	$-0.734419\pi$
$797$	−1.91992e10	−1.34332	−0.671661	+	0.740859i	$0.734419\pi$
$798$	0	0
$799$	3.79456e10	2.63177
$800$	0	0
$801$	5.02295e10	3.45338
$802$	0	0
$803$	−6.65697e8	−0.0453704
$804$	0	0
$805$	−5.12518e9	−0.346277
$806$	0	0
$807$	−3.82467e10	−2.56175
$808$	0	0
$809$	9.47189e8	0.0628951	0.0314476	−	0.999505i	$-0.489988\pi$
$809$	9.47189e8	0.0628951	0.0314476	+	0.999505i	$0.489988\pi$
$810$	0	0
$811$	1.41110e9	0.0928931	0.0464466	−	0.998921i	$-0.485210\pi$
$811$	1.41110e9	0.0928931	0.0464466	+	0.998921i	$0.485210\pi$
$812$	0	0
$813$	4.56486e10	2.97928
$814$	0	0
$815$	−5.60398e9	−0.362614
$816$	0	0
$817$	−3.05948e10	−1.96278
$818$	0	0
$819$	−6.02649e9	−0.383328
$820$	0	0
$821$	2.01870e7	0.00127313	0.000636563	−	1.00000i	$-0.499797\pi$
$821$	2.01870e7	0.00127313	0.000636563		1.00000i	$0.499797\pi$
$822$	0	0
$823$	2.14990e9	0.134437	0.0672185	−	0.997738i	$-0.478588\pi$
$823$	2.14990e9	0.134437	0.0672185	+	0.997738i	$0.478588\pi$
$824$	0	0
$825$	−2.55275e10	−1.58278
$826$	0	0
$827$	−1.71093e10	−1.05187	−0.525937	−	0.850524i	$-0.676285\pi$
$827$	−1.71093e10	−1.05187	−0.525937	+	0.850524i	$0.676285\pi$
$828$	0	0
$829$	−7.25127e8	−0.0442052	−0.0221026	−	0.999756i	$-0.507036\pi$
$829$	−7.25127e8	−0.0442052	−0.0221026	+	0.999756i	$0.507036\pi$
$830$	0	0
$831$	−2.90527e10	−1.75623
$832$	0	0
$833$	2.85085e9	0.170890
$834$	0	0
$835$	1.44487e10	0.858869
$836$	0	0
$837$	7.73023e10	4.55673
$838$	0	0
$839$	−1.09113e9	−0.0637839	−0.0318920	−	0.999491i	$-0.510153\pi$
$839$	−1.09113e9	−0.0637839	−0.0318920	+	0.999491i	$0.510153\pi$
$840$	0	0
$841$	−1.27976e10	−0.741895
$842$	0	0
$843$	4.30558e10	2.47534
$844$	0	0
$845$	−8.20908e9	−0.468054
$846$	0	0
$847$	2.07545e9	0.117360
$848$	0	0
$849$	−4.83277e10	−2.71031
$850$	0	0
$851$	2.05633e9	0.114377
$852$	0	0
$853$	1.53367e10	0.846076	0.423038	−	0.906112i	$-0.360964\pi$
$853$	1.53367e10	0.846076	0.423038	+	0.906112i	$0.360964\pi$
$854$	0	0
$855$	3.02089e10	1.65293
$856$	0	0
$857$	−1.85211e10	−1.00516	−0.502578	−	0.864532i	$-0.667615\pi$
$857$	−1.85211e10	−1.00516	−0.502578	+	0.864532i	$0.667615\pi$
$858$	0	0
$859$	8.76910e6	0.000472040 0	0.000236020	−	1.00000i	$-0.499925\pi$
$859$	8.76910e6	0.000472040 0	0.000236020		1.00000i	$0.499925\pi$
$860$	0	0
$861$	8.70070e9	0.464561
$862$	0	0
$863$	1.82668e10	0.967443	0.483721	−	0.875222i	$-0.339285\pi$
$863$	1.82668e10	0.967443	0.483721	+	0.875222i	$0.339285\pi$
$864$	0	0
$865$	−4.71628e9	−0.247767
$866$	0	0
$867$	−9.29549e10	−4.84402
$868$	0	0
$869$	−3.93031e9	−0.203169
$870$	0	0
$871$	−8.43922e6	−0.000432752 0
$872$	0	0
$873$	2.46155e10	1.25216
$874$	0	0
$875$	−1.74874e10	−0.882466
$876$	0	0
$877$	−2.99806e10	−1.50086	−0.750432	−	0.660948i	$-0.770154\pi$
$877$	−2.99806e10	−1.50086	−0.750432	+	0.660948i	$0.770154\pi$
$878$	0	0
$879$	1.95925e10	0.973037
$880$	0	0
$881$	3.82446e8	0.0188432	0.00942160	−	0.999956i	$-0.497001\pi$
$881$	3.82446e8	0.0188432	0.00942160	+	0.999956i	$0.497001\pi$
$882$	0	0
$883$	5.12232e9	0.250382	0.125191	−	0.992133i	$-0.460046\pi$
$883$	5.12232e9	0.250382	0.125191	+	0.992133i	$0.460046\pi$
$884$	0	0
$885$	8.29939e9	0.402480
$886$	0	0
$887$	−2.34581e10	−1.12865	−0.564326	−	0.825552i	$-0.690864\pi$
$887$	−2.34581e10	−1.12865	−0.564326	+	0.825552i	$0.690864\pi$
$888$	0	0
$889$	2.87301e7	0.00137145
$890$	0	0
$891$	−8.68028e10	−4.11114
$892$	0	0
$893$	3.75360e10	1.76387
$894$	0	0
$895$	−5.69389e9	−0.265478
$896$	0	0
$897$	−3.86891e9	−0.178984
$898$	0	0
$899$	1.46856e10	0.674115
$900$	0	0
$901$	5.31771e10	2.42208
$902$	0	0
$903$	7.02954e10	3.17702
$904$	0	0
$905$	−9.37975e9	−0.420650
$906$	0	0
$907$	−2.54092e10	−1.13075	−0.565373	−	0.824835i	$-0.691268\pi$
$907$	−2.54092e10	−1.13075	−0.565373	+	0.824835i	$0.691268\pi$
$908$	0	0
$909$	3.13568e8	0.0138471
$910$	0	0
$911$	−7.96177e9	−0.348895	−0.174448	−	0.984666i	$-0.555814\pi$
$911$	−7.96177e9	−0.348895	−0.174448	+	0.984666i	$0.555814\pi$
$912$	0	0
$913$	−3.87896e10	−1.68682
$914$	0	0
$915$	2.82756e10	1.22022
$916$	0	0
$917$	−1.21276e10	−0.519375
$918$	0	0
$919$	−2.30253e10	−0.978588	−0.489294	−	0.872119i	$-0.662746\pi$
$919$	−2.30253e10	−0.978588	−0.489294	+	0.872119i	$0.662746\pi$
$920$	0	0
$921$	−5.42094e10	−2.28647
$922$	0	0
$923$	−1.13670e9	−0.0475817
$924$	0	0
$925$	3.05906e9	0.127084
$926$	0	0
$927$	5.39299e10	2.22357
$928$	0	0
$929$	3.32878e10	1.36217	0.681083	−	0.732206i	$-0.261509\pi$
$929$	3.32878e10	1.36217	0.681083	+	0.732206i	$0.261509\pi$
$930$	0	0
$931$	2.82007e9	0.114535
$932$	0	0
$933$	2.16116e10	0.871167
$934$	0	0
$935$	−2.34791e10	−0.939380
$936$	0	0
$937$	1.77970e10	0.706737	0.353369	−	0.935484i	$-0.385036\pi$
$937$	1.77970e10	0.706737	0.353369	+	0.935484i	$0.385036\pi$
$938$	0	0
$939$	−2.96982e10	−1.17058
$940$	0	0
$941$	−3.32182e10	−1.29961	−0.649803	−	0.760102i	$-0.725149\pi$
$941$	−3.32182e10	−1.29961	−0.649803	+	0.760102i	$0.725149\pi$
$942$	0	0
$943$	4.10367e9	0.159361
$944$	0	0
$945$	−4.43417e10	−1.70923
$946$	0	0
$947$	−1.50372e10	−0.575364	−0.287682	−	0.957726i	$-0.592885\pi$
$947$	−1.50372e10	−0.575364	−0.287682	+	0.957726i	$0.592885\pi$
$948$	0	0
$949$	−1.50090e8	−0.00570059
$950$	0	0
$951$	7.56342e10	2.85159
$952$	0	0
$953$	−1.27447e10	−0.476983	−0.238492	−	0.971145i	$-0.576653\pi$
$953$	−1.27447e10	−0.476983	−0.238492	+	0.971145i	$0.576653\pi$
$954$	0	0
$955$	−1.72804e10	−0.642012
$956$	0	0
$957$	−2.82044e10	−1.04022
$958$	0	0
$959$	−4.21969e10	−1.54496
$960$	0	0
$961$	2.09272e10	0.760640
$962$	0	0
$963$	−1.40593e8	−0.00507308
$964$	0	0
$965$	6.78486e9	0.243050
$966$	0	0
$967$	−2.50300e10	−0.890161	−0.445080	−	0.895491i	$-0.646825\pi$
$967$	−2.50300e10	−0.890161	−0.445080	+	0.895491i	$0.646825\pi$
$968$	0	0
$969$	−1.28804e11	−4.54773
$970$	0	0
$971$	−1.01634e10	−0.356265	−0.178132	−	0.984007i	$-0.557006\pi$
$971$	−1.01634e10	−0.356265	−0.178132	+	0.984007i	$0.557006\pi$
$972$	0	0
$973$	−3.19024e9	−0.111027
$974$	0	0
$975$	−5.75551e9	−0.198869
$976$	0	0
$977$	3.59483e10	1.23324	0.616620	−	0.787261i	$-0.288502\pi$
$977$	3.59483e10	1.23324	0.616620	+	0.787261i	$0.288502\pi$
$978$	0	0
$979$	−3.86182e10	−1.31538
$980$	0	0
$981$	−5.33969e10	−1.80582
$982$	0	0
$983$	−2.69073e10	−0.903510	−0.451755	−	0.892142i	$-0.649202\pi$
$983$	−2.69073e10	−0.903510	−0.451755	+	0.892142i	$0.649202\pi$
$984$	0	0
$985$	−2.40646e8	−0.00802328
$986$	0	0
$987$	−8.62435e10	−2.85507
$988$	0	0
$989$	3.31547e10	1.08983
$990$	0	0
$991$	1.91531e10	0.625147	0.312573	−	0.949894i	$-0.398809\pi$
$991$	1.91531e10	0.625147	0.312573	+	0.949894i	$0.398809\pi$
$992$	0	0
$993$	−4.41582e10	−1.43116
$994$	0	0
$995$	−6.33722e9	−0.203947
$996$	0	0
$997$	−2.26750e10	−0.724625	−0.362313	−	0.932057i	$-0.618013\pi$
$997$	−2.26750e10	−0.724625	−0.362313	+	0.932057i	$0.618013\pi$
$998$	0	0
$999$	1.77908e10	0.564569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.8.a.c.1.1		6
4.3	odd	2		74.8.a.c.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.8.a.c.1.6	✓	6	4.3	odd	2
592.8.a.c.1.1		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-90.7890 q^{3} +133.163 q^{5} +948.062 q^{7} +6055.64 q^{9} -4655.78 q^{11} -1049.71 q^{13} -12089.8 q^{15} +37870.8 q^{17} +37462.0 q^{19} -86073.6 q^{21} -40596.4 q^{23} -60392.5 q^{25} -351229. q^{27} -66725.4 q^{29} -220090. q^{31} +422694. q^{33} +126247. q^{35} -50653.0 q^{37} +95301.7 q^{39} -101084. q^{41} -816690. q^{43} +806389. q^{45} +1.00197e6 q^{47} +75278.3 q^{49} -3.43825e6 q^{51} +1.40417e6 q^{53} -619980. q^{55} -3.40113e6 q^{57} -686480. q^{59} -2.33880e6 q^{61} +5.74112e6 q^{63} -139782. q^{65} +8039.60 q^{67} +3.68571e6 q^{69} +1.08288e6 q^{71} +142983. q^{73} +5.48297e6 q^{75} -4.41397e6 q^{77} +844178. q^{79} +1.86441e7 q^{81} +8.33149e6 q^{83} +5.04300e6 q^{85} +6.05793e6 q^{87} +8.29467e6 q^{89} -995186. q^{91} +1.99818e7 q^{93} +4.98857e6 q^{95} +4.06489e6 q^{97} -2.81937e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 28 q^{3} - 14 q^{5} + 980 q^{7} + 8254 q^{9} - 2956 q^{11} + 2394 q^{13} + 28820 q^{15} - 45108 q^{17} - 11764 q^{19} - 135378 q^{21} - 21052 q^{23} + 194744 q^{25} - 439240 q^{27} + 288454 q^{29}+ \cdots - 25990712 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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