LMFDB - Embedded newform 384.8.a.q.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 620x^{2} - 700x + 83625 $ x^4 - 620x^2 - 700x + 83625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$21.8313$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} +127.307 q^{5} -547.276 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} +127.307 q^{5} -547.276 q^{7} +729.000 q^{9} +5229.39 q^{11} +11015.5 q^{13} +3437.28 q^{15} +3561.44 q^{17} +37813.3 q^{19} -14776.5 q^{21} +3189.45 q^{23} -61918.0 q^{25} +19683.0 q^{27} -3656.23 q^{29} -181441. q^{31} +141194. q^{33} -69672.0 q^{35} -133882. q^{37} +297419. q^{39} -189883. q^{41} +433286. q^{43} +92806.6 q^{45} -127956. q^{47} -524032. q^{49} +96158.9 q^{51} +1.84708e6 q^{53} +665737. q^{55} +1.02096e6 q^{57} -449110. q^{59} +357302. q^{61} -398964. q^{63} +1.40235e6 q^{65} +4.13883e6 q^{67} +86115.1 q^{69} -4.53132e6 q^{71} +5.45318e6 q^{73} -1.67179e6 q^{75} -2.86192e6 q^{77} +2.20562e6 q^{79} +531441. q^{81} +508823. q^{83} +453395. q^{85} -98718.3 q^{87} -979961. q^{89} -6.02852e6 q^{91} -4.89891e6 q^{93} +4.81389e6 q^{95} -4.99716e6 q^{97} +3.81223e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) $ 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9	$ 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9} + 3856 q^{11} - 10680 q^{13} - 9072 q^{15} + 26232 q^{17} - 15456 q^{19} + 18360 q^{21} - 11312 q^{23} + 159052 q^{25} + 78732 q^{27} + 1856 q^{29} + 71752 q^{31} + 104112 q^{33} - 179040 q^{35} + 180088 q^{37} - 288360 q^{39} + 11224 q^{41} + 66688 q^{43} - 244944 q^{45} - 1334448 q^{47} + 2401140 q^{49} + 708264 q^{51} + 864576 q^{53} - 3304896 q^{55} - 417312 q^{57} + 1878448 q^{59} + 1901176 q^{61} + 495720 q^{63} + 4366944 q^{65} + 5505488 q^{67} - 305424 q^{69} - 967696 q^{71} + 3244760 q^{73} + 4294404 q^{75} + 8979488 q^{77} + 6471816 q^{79} + 2125764 q^{81} + 17019600 q^{83} - 12122592 q^{85} + 50112 q^{87} + 13559816 q^{89} + 6692304 q^{91} + 1937304 q^{93} + 22523904 q^{95} + 2180520 q^{97} + 2811024 q^{99}+O(q^{100}) $ 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9 + 3856 * q^11 - 10680 * q^13 - 9072 * q^15 + 26232 * q^17 - 15456 * q^19 + 18360 * q^21 - 11312 * q^23 + 159052 * q^25 + 78732 * q^27 + 1856 * q^29 + 71752 * q^31 + 104112 * q^33 - 179040 * q^35 + 180088 * q^37 - 288360 * q^39 + 11224 * q^41 + 66688 * q^43 - 244944 * q^45 - 1334448 * q^47 + 2401140 * q^49 + 708264 * q^51 + 864576 * q^53 - 3304896 * q^55 - 417312 * q^57 + 1878448 * q^59 + 1901176 * q^61 + 495720 * q^63 + 4366944 * q^65 + 5505488 * q^67 - 305424 * q^69 - 967696 * q^71 + 3244760 * q^73 + 4294404 * q^75 + 8979488 * q^77 + 6471816 * q^79 + 2125764 * q^81 + 17019600 * q^83 - 12122592 * q^85 + 50112 * q^87 + 13559816 * q^89 + 6692304 * q^91 + 1937304 * q^93 + 22523904 * q^95 + 2180520 * q^97 + 2811024 * 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
$5$	127.307	0.455467	0.227733	−	0.973724i	$-0.426869\pi$
$5$	127.307	0.455467	0.227733	+	0.973724i	$0.426869\pi$
$6$	0	0
$7$	−547.276	−0.603064	−0.301532	−	0.953456i	$-0.597498\pi$
$7$	−547.276	−0.603064	−0.301532	+	0.953456i	$0.597498\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	5229.39	1.18461	0.592307	−	0.805712i	$-0.298217\pi$
$11$	5229.39	1.18461	0.592307	+	0.805712i	$0.298217\pi$
$12$	0	0
$13$	11015.5	1.39060	0.695301	−	0.718719i	$-0.255271\pi$
$13$	11015.5	1.39060	0.695301	+	0.718719i	$0.255271\pi$
$14$	0	0
$15$	3437.28	0.262964
$16$	0	0
$17$	3561.44	0.175814	0.0879072	−	0.996129i	$-0.471982\pi$
$17$	3561.44	0.175814	0.0879072	+	0.996129i	$0.471982\pi$
$18$	0	0
$19$	37813.3	1.26476	0.632378	−	0.774660i	$-0.282079\pi$
$19$	37813.3	1.26476	0.632378	+	0.774660i	$0.282079\pi$
$20$	0	0
$21$	−14776.5	−0.348179
$22$	0	0
$23$	3189.45	0.0546598	0.0273299	−	0.999626i	$-0.491300\pi$
$23$	3189.45	0.0546598	0.0273299	+	0.999626i	$0.491300\pi$
$24$	0	0
$25$	−61918.0	−0.792550
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−3656.23	−0.0278382	−0.0139191	−	0.999903i	$-0.504431\pi$
$29$	−3656.23	−0.0278382	−0.0139191	+	0.999903i	$0.504431\pi$
$30$	0	0
$31$	−181441.	−1.09388	−0.546940	−	0.837172i	$-0.684207\pi$
$31$	−181441.	−1.09388	−0.546940	+	0.837172i	$0.684207\pi$
$32$	0	0
$33$	141194.	0.683937
$34$	0	0
$35$	−69672.0	−0.274676
$36$	0	0
$37$	−133882.	−0.434527	−0.217263	−	0.976113i	$-0.569713\pi$
$37$	−133882.	−0.434527	−0.217263	+	0.976113i	$0.569713\pi$
$38$	0	0
$39$	297419.	0.802864
$40$	0	0
$41$	−189883.	−0.430272	−0.215136	−	0.976584i	$-0.569019\pi$
$41$	−189883.	−0.430272	−0.215136	+	0.976584i	$0.569019\pi$
$42$	0	0
$43$	433286.	0.831066	0.415533	−	0.909578i	$-0.363595\pi$
$43$	433286.	0.831066	0.415533	+	0.909578i	$0.363595\pi$
$44$	0	0
$45$	92806.6	0.151822
$46$	0	0
$47$	−127956.	−0.179770	−0.0898849	−	0.995952i	$-0.528650\pi$
$47$	−127956.	−0.179770	−0.0898849	+	0.995952i	$0.528650\pi$
$48$	0	0
$49$	−524032.	−0.636314
$50$	0	0
$51$	96158.9	0.101506
$52$	0	0
$53$	1.84708e6	1.70420	0.852100	−	0.523379i	$-0.175329\pi$
$53$	1.84708e6	1.70420	0.852100	+	0.523379i	$0.175329\pi$
$54$	0	0
$55$	665737.	0.539552
$56$	0	0
$57$	1.02096e6	0.730207
$58$	0	0
$59$	−449110.	−0.284689	−0.142344	−	0.989817i	$-0.545464\pi$
$59$	−449110.	−0.284689	−0.142344	+	0.989817i	$0.545464\pi$
$60$	0	0
$61$	357302.	0.201549	0.100775	−	0.994909i	$-0.467868\pi$
$61$	357302.	0.201549	0.100775	+	0.994909i	$0.467868\pi$
$62$	0	0
$63$	−398964.	−0.201021
$64$	0	0
$65$	1.40235e6	0.633372
$66$	0	0
$67$	4.13883e6	1.68119	0.840593	−	0.541667i	$-0.182207\pi$
$67$	4.13883e6	1.68119	0.840593	+	0.541667i	$0.182207\pi$
$68$	0	0
$69$	86115.1	0.0315578
$70$	0	0
$71$	−4.53132e6	−1.50252	−0.751261	−	0.660006i	$-0.770554\pi$
$71$	−4.53132e6	−1.50252	−0.751261	+	0.660006i	$0.770554\pi$
$72$	0	0
$73$	5.45318e6	1.64066	0.820332	−	0.571887i	$-0.193788\pi$
$73$	5.45318e6	1.64066	0.820332	+	0.571887i	$0.193788\pi$
$74$	0	0
$75$	−1.67179e6	−0.457579
$76$	0	0
$77$	−2.86192e6	−0.714398
$78$	0	0
$79$	2.20562e6	0.503309	0.251655	−	0.967817i	$-0.419025\pi$
$79$	2.20562e6	0.503309	0.251655	+	0.967817i	$0.419025\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	508823.	0.0976772	0.0488386	−	0.998807i	$-0.484448\pi$
$83$	508823.	0.0976772	0.0488386	+	0.998807i	$0.484448\pi$
$84$	0	0
$85$	453395.	0.0800776
$86$	0	0
$87$	−98718.3	−0.0160724
$88$	0	0
$89$	−979961.	−0.147348	−0.0736739	−	0.997282i	$-0.523472\pi$
$89$	−979961.	−0.147348	−0.0736739	+	0.997282i	$0.523472\pi$
$90$	0	0
$91$	−6.02852e6	−0.838622
$92$	0	0
$93$	−4.89891e6	−0.631552
$94$	0	0
$95$	4.81389e6	0.576054
$96$	0	0
$97$	−4.99716e6	−0.555933	−0.277966	−	0.960591i	$-0.589660\pi$
$97$	−4.99716e6	−0.555933	−0.277966	+	0.960591i	$0.589660\pi$
$98$	0	0
$99$	3.81223e6	0.394871
$100$	0	0
$101$	−1.45754e7	−1.40765	−0.703825	−	0.710373i	$-0.748526\pi$
$101$	−1.45754e7	−1.40765	−0.703825	+	0.710373i	$0.748526\pi$
$102$	0	0
$103$	4.36742e6	0.393817	0.196909	−	0.980422i	$-0.436910\pi$
$103$	4.36742e6	0.393817	0.196909	+	0.980422i	$0.436910\pi$
$104$	0	0
$105$	−1.88114e6	−0.158584
$106$	0	0
$107$	2.22120e7	1.75285	0.876424	−	0.481541i	$-0.159923\pi$
$107$	2.22120e7	1.75285	0.876424	+	0.481541i	$0.159923\pi$
$108$	0	0
$109$	1.39532e7	1.03201	0.516003	−	0.856587i	$-0.327419\pi$
$109$	1.39532e7	1.03201	0.516003	+	0.856587i	$0.327419\pi$
$110$	0	0
$111$	−3.61482e6	−0.250874
$112$	0	0
$113$	8.88362e6	0.579182	0.289591	−	0.957150i	$-0.406481\pi$
$113$	8.88362e6	0.579182	0.289591	+	0.957150i	$0.406481\pi$
$114$	0	0
$115$	406038.	0.0248957
$116$	0	0
$117$	8.03030e6	0.463534
$118$	0	0
$119$	−1.94909e6	−0.106027
$120$	0	0
$121$	7.85937e6	0.403310
$122$	0	0
$123$	−5.12685e6	−0.248418
$124$	0	0
$125$	−1.78284e7	−0.816447
$126$	0	0
$127$	1.87925e7	0.814086	0.407043	−	0.913409i	$-0.366560\pi$
$127$	1.87925e7	0.814086	0.407043	+	0.913409i	$0.366560\pi$
$128$	0	0
$129$	1.16987e7	0.479816
$130$	0	0
$131$	1.19042e7	0.462646	0.231323	−	0.972877i	$-0.425695\pi$
$131$	1.19042e7	0.462646	0.231323	+	0.972877i	$0.425695\pi$
$132$	0	0
$133$	−2.06943e7	−0.762729
$134$	0	0
$135$	2.50578e6	0.0876546
$136$	0	0
$137$	1.37043e7	0.455340	0.227670	−	0.973738i	$-0.426889\pi$
$137$	1.37043e7	0.455340	0.227670	+	0.973738i	$0.426889\pi$
$138$	0	0
$139$	7.77637e6	0.245598	0.122799	−	0.992432i	$-0.460813\pi$
$139$	7.77637e6	0.245598	0.122799	+	0.992432i	$0.460813\pi$
$140$	0	0
$141$	−3.45480e6	−0.103790
$142$	0	0
$143$	5.76044e7	1.64733
$144$	0	0
$145$	−465463.	−0.0126794
$146$	0	0
$147$	−1.41489e7	−0.367376
$148$	0	0
$149$	−3.07847e7	−0.762399	−0.381200	−	0.924493i	$-0.624489\pi$
$149$	−3.07847e7	−0.762399	−0.381200	+	0.924493i	$0.624489\pi$
$150$	0	0
$151$	3.33794e7	0.788968	0.394484	−	0.918903i	$-0.370923\pi$
$151$	3.33794e7	0.788968	0.394484	+	0.918903i	$0.370923\pi$
$152$	0	0
$153$	2.59629e6	0.0586048
$154$	0	0
$155$	−2.30987e7	−0.498226
$156$	0	0
$157$	−7.95255e7	−1.64005	−0.820026	−	0.572326i	$-0.806041\pi$
$157$	−7.95255e7	−1.64005	−0.820026	+	0.572326i	$0.806041\pi$
$158$	0	0
$159$	4.98712e7	0.983920
$160$	0	0
$161$	−1.74551e6	−0.0329634
$162$	0	0
$163$	6.39515e7	1.15663	0.578314	−	0.815814i	$-0.303711\pi$
$163$	6.39515e7	1.15663	0.578314	+	0.815814i	$0.303711\pi$
$164$	0	0
$165$	1.79749e7	0.311510
$166$	0	0
$167$	3.35708e7	0.557769	0.278885	−	0.960325i	$-0.410035\pi$
$167$	3.35708e7	0.557769	0.278885	+	0.960325i	$0.410035\pi$
$168$	0	0
$169$	5.85928e7	0.933771
$170$	0	0
$171$	2.75659e7	0.421585
$172$	0	0
$173$	5.26949e7	0.773761	0.386881	−	0.922130i	$-0.373553\pi$
$173$	5.26949e7	0.773761	0.386881	+	0.922130i	$0.373553\pi$
$174$	0	0
$175$	3.38862e7	0.477959
$176$	0	0
$177$	−1.21260e7	−0.164365
$178$	0	0
$179$	1.04238e8	1.35844	0.679222	−	0.733933i	$-0.262317\pi$
$179$	1.04238e8	1.35844	0.679222	+	0.733933i	$0.262317\pi$
$180$	0	0
$181$	−1.10385e8	−1.38367	−0.691836	−	0.722055i	$-0.743198\pi$
$181$	−1.10385e8	−1.38367	−0.691836	+	0.722055i	$0.743198\pi$
$182$	0	0
$183$	9.64717e6	0.116365
$184$	0	0
$185$	−1.70441e7	−0.197912
$186$	0	0
$187$	1.86242e7	0.208272
$188$	0	0
$189$	−1.07720e7	−0.116060
$190$	0	0
$191$	−5.89988e7	−0.612670	−0.306335	−	0.951924i	$-0.599103\pi$
$191$	−5.89988e7	−0.612670	−0.306335	+	0.951924i	$0.599103\pi$
$192$	0	0
$193$	8.22168e7	0.823208	0.411604	−	0.911363i	$-0.364969\pi$
$193$	8.22168e7	0.823208	0.411604	+	0.911363i	$0.364969\pi$
$194$	0	0
$195$	3.78634e7	0.365678
$196$	0	0
$197$	−1.33058e8	−1.23996	−0.619982	−	0.784616i	$-0.712860\pi$
$197$	−1.33058e8	−1.23996	−0.619982	+	0.784616i	$0.712860\pi$
$198$	0	0
$199$	−1.25442e7	−0.112838	−0.0564192	−	0.998407i	$-0.517968\pi$
$199$	−1.25442e7	−0.112838	−0.0564192	+	0.998407i	$0.517968\pi$
$200$	0	0
$201$	1.11748e8	0.970633
$202$	0	0
$203$	2.00097e6	0.0167882
$204$	0	0
$205$	−2.41734e7	−0.195974
$206$	0	0
$207$	2.32511e6	0.0182199
$208$	0	0
$209$	1.97740e8	1.49825
$210$	0	0
$211$	1.98314e7	0.145333	0.0726666	−	0.997356i	$-0.476849\pi$
$211$	1.98314e7	0.145333	0.0726666	+	0.997356i	$0.476849\pi$
$212$	0	0
$213$	−1.22346e8	−0.867481
$214$	0	0
$215$	5.51603e7	0.378523
$216$	0	0
$217$	9.92984e7	0.659680
$218$	0	0
$219$	1.47236e8	0.947238
$220$	0	0
$221$	3.92310e7	0.244488
$222$	0	0
$223$	4.33375e7	0.261696	0.130848	−	0.991402i	$-0.458230\pi$
$223$	4.33375e7	0.261696	0.130848	+	0.991402i	$0.458230\pi$
$224$	0	0
$225$	−4.51382e7	−0.264183
$226$	0	0
$227$	1.00224e8	0.568699	0.284349	−	0.958721i	$-0.408222\pi$
$227$	1.00224e8	0.568699	0.284349	+	0.958721i	$0.408222\pi$
$228$	0	0
$229$	−1.73950e7	−0.0957194	−0.0478597	−	0.998854i	$-0.515240\pi$
$229$	−1.73950e7	−0.0957194	−0.0478597	+	0.998854i	$0.515240\pi$
$230$	0	0
$231$	−7.72719e7	−0.412458
$232$	0	0
$233$	2.41202e8	1.24921	0.624605	−	0.780941i	$-0.285260\pi$
$233$	2.41202e8	1.24921	0.624605	+	0.780941i	$0.285260\pi$
$234$	0	0
$235$	−1.62896e7	−0.0818792
$236$	0	0
$237$	5.95516e7	0.290586
$238$	0	0
$239$	1.05228e8	0.498584	0.249292	−	0.968428i	$-0.419802\pi$
$239$	1.05228e8	0.498584	0.249292	+	0.968428i	$0.419802\pi$
$240$	0	0
$241$	431938.	0.00198775	0.000993876	−	1.00000i	$-0.499684\pi$
$241$	431938.	0.00198775	0.000993876		1.00000i	$0.499684\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−6.67128e7	−0.289820
$246$	0	0
$247$	4.16532e8	1.75877
$248$	0	0
$249$	1.37382e7	0.0563940
$250$	0	0
$251$	4.28984e8	1.71231	0.856156	−	0.516718i	$-0.172846\pi$
$251$	4.28984e8	1.71231	0.856156	+	0.516718i	$0.172846\pi$
$252$	0	0
$253$	1.66789e7	0.0647507
$254$	0	0
$255$	1.22417e7	0.0462328
$256$	0	0
$257$	−2.64486e8	−0.971933	−0.485966	−	0.873978i	$-0.661532\pi$
$257$	−2.64486e8	−0.971933	−0.485966	+	0.873978i	$0.661532\pi$
$258$	0	0
$259$	7.32705e7	0.262047
$260$	0	0
$261$	−2.66539e6	−0.00927940
$262$	0	0
$263$	3.53075e8	1.19680	0.598400	−	0.801197i	$-0.295803\pi$
$263$	3.53075e8	1.19680	0.598400	+	0.801197i	$0.295803\pi$
$264$	0	0
$265$	2.35146e8	0.776206
$266$	0	0
$267$	−2.64589e7	−0.0850713
$268$	0	0
$269$	−9.28864e7	−0.290951	−0.145475	−	0.989362i	$-0.546471\pi$
$269$	−9.28864e7	−0.290951	−0.145475	+	0.989362i	$0.546471\pi$
$270$	0	0
$271$	−1.54543e7	−0.0471689	−0.0235845	−	0.999722i	$-0.507508\pi$
$271$	−1.54543e7	−0.0471689	−0.0235845	+	0.999722i	$0.507508\pi$
$272$	0	0
$273$	−1.62770e8	−0.484178
$274$	0	0
$275$	−3.23793e8	−0.938866
$276$	0	0
$277$	−1.41752e8	−0.400730	−0.200365	−	0.979721i	$-0.564213\pi$
$277$	−1.41752e8	−0.400730	−0.200365	+	0.979721i	$0.564213\pi$
$278$	0	0
$279$	−1.32271e8	−0.364627
$280$	0	0
$281$	7.41464e8	1.99351	0.996753	−	0.0805143i	$-0.0256563\pi$
$281$	7.41464e8	1.99351	0.996753	+	0.0805143i	$0.0256563\pi$
$282$	0	0
$283$	3.56841e8	0.935885	0.467943	−	0.883759i	$-0.344995\pi$
$283$	3.56841e8	0.935885	0.467943	+	0.883759i	$0.344995\pi$
$284$	0	0
$285$	1.29975e8	0.332585
$286$	0	0
$287$	1.03919e8	0.259482
$288$	0	0
$289$	−3.97655e8	−0.969089
$290$	0	0
$291$	−1.34923e8	−0.320968
$292$	0	0
$293$	−6.19256e7	−0.143825	−0.0719124	−	0.997411i	$-0.522910\pi$
$293$	−6.19256e7	−0.143825	−0.0719124	+	0.997411i	$0.522910\pi$
$294$	0	0
$295$	−5.71747e7	−0.129666
$296$	0	0
$297$	1.02930e8	0.227979
$298$	0	0
$299$	3.51334e7	0.0760100
$300$	0	0
$301$	−2.37127e8	−0.501186
$302$	0	0
$303$	−3.93535e8	−0.812707
$304$	0	0
$305$	4.54870e7	0.0917990
$306$	0	0
$307$	2.63097e7	0.0518957	0.0259479	−	0.999663i	$-0.491740\pi$
$307$	2.63097e7	0.0518957	0.0259479	+	0.999663i	$0.491740\pi$
$308$	0	0
$309$	1.17920e8	0.227370
$310$	0	0
$311$	−8.08590e8	−1.52429	−0.762144	−	0.647407i	$-0.775853\pi$
$311$	−8.08590e8	−1.52429	−0.762144	+	0.647407i	$0.775853\pi$
$312$	0	0
$313$	−7.35232e8	−1.35525	−0.677624	−	0.735408i	$-0.736990\pi$
$313$	−7.35232e8	−1.35525	−0.677624	+	0.735408i	$0.736990\pi$
$314$	0	0
$315$	−5.07909e7	−0.0915585
$316$	0	0
$317$	7.44841e8	1.31328	0.656638	−	0.754205i	$-0.271978\pi$
$317$	7.44841e8	1.31328	0.656638	+	0.754205i	$0.271978\pi$
$318$	0	0
$319$	−1.91199e7	−0.0329775
$320$	0	0
$321$	5.99724e8	1.01201
$322$	0	0
$323$	1.34670e8	0.222362
$324$	0	0
$325$	−6.82058e8	−1.10212
$326$	0	0
$327$	3.76738e8	0.595829
$328$	0	0
$329$	7.00271e7	0.108413
$330$	0	0
$331$	−7.46695e8	−1.13174	−0.565868	−	0.824496i	$-0.691459\pi$
$331$	−7.46695e8	−1.13174	−0.565868	+	0.824496i	$0.691459\pi$
$332$	0	0
$333$	−9.76001e7	−0.144842
$334$	0	0
$335$	5.26901e8	0.765724
$336$	0	0
$337$	−1.30266e9	−1.85408	−0.927038	−	0.374967i	$-0.877654\pi$
$337$	−1.30266e9	−1.85408	−0.927038	+	0.374967i	$0.877654\pi$
$338$	0	0
$339$	2.39858e8	0.334391
$340$	0	0
$341$	−9.48826e8	−1.29583
$342$	0	0
$343$	7.37496e8	0.986802
$344$	0	0
$345$	1.09630e7	0.0143735
$346$	0	0
$347$	−4.51272e8	−0.579809	−0.289904	−	0.957056i	$-0.593624\pi$
$347$	−4.51272e8	−0.579809	−0.289904	+	0.957056i	$0.593624\pi$
$348$	0	0
$349$	7.57452e8	0.953819	0.476910	−	0.878952i	$-0.341757\pi$
$349$	7.57452e8	0.953819	0.476910	+	0.878952i	$0.341757\pi$
$350$	0	0
$351$	2.16818e8	0.267621
$352$	0	0
$353$	−1.28645e9	−1.55661	−0.778307	−	0.627884i	$-0.783921\pi$
$353$	−1.28645e9	−1.55661	−0.778307	+	0.627884i	$0.783921\pi$
$354$	0	0
$355$	−5.76868e8	−0.684348
$356$	0	0
$357$	−5.26255e7	−0.0612149
$358$	0	0
$359$	−4.55821e8	−0.519953	−0.259976	−	0.965615i	$-0.583715\pi$
$359$	−4.55821e8	−0.519953	−0.259976	+	0.965615i	$0.583715\pi$
$360$	0	0
$361$	5.35972e8	0.599607
$362$	0	0
$363$	2.12203e8	0.232851
$364$	0	0
$365$	6.94227e8	0.747268
$366$	0	0
$367$	−1.94332e8	−0.205217	−0.102609	−	0.994722i	$-0.532719\pi$
$367$	−1.94332e8	−0.205217	−0.102609	+	0.994722i	$0.532719\pi$
$368$	0	0
$369$	−1.38425e8	−0.143424
$370$	0	0
$371$	−1.01086e9	−1.02774
$372$	0	0
$373$	−1.76082e9	−1.75684	−0.878422	−	0.477886i	$-0.841403\pi$
$373$	−1.76082e9	−1.75684	−0.878422	+	0.477886i	$0.841403\pi$
$374$	0	0
$375$	−4.81367e8	−0.471376
$376$	0	0
$377$	−4.02753e7	−0.0387118
$378$	0	0
$379$	−1.30262e9	−1.22908	−0.614539	−	0.788886i	$-0.710658\pi$
$379$	−1.30262e9	−1.22908	−0.614539	+	0.788886i	$0.710658\pi$
$380$	0	0
$381$	5.07396e8	0.470013
$382$	0	0
$383$	−1.50684e9	−1.37047	−0.685237	−	0.728321i	$-0.740301\pi$
$383$	−1.50684e9	−1.37047	−0.685237	+	0.728321i	$0.740301\pi$
$384$	0	0
$385$	−3.64342e8	−0.325384
$386$	0	0
$387$	3.15866e8	0.277022
$388$	0	0
$389$	1.87342e9	1.61366	0.806829	−	0.590785i	$-0.201182\pi$
$389$	1.87342e9	1.61366	0.806829	+	0.590785i	$0.201182\pi$
$390$	0	0
$391$	1.13590e7	0.00960997
$392$	0	0
$393$	3.21412e8	0.267109
$394$	0	0
$395$	2.80790e8	0.229241
$396$	0	0
$397$	2.33361e9	1.87181	0.935906	−	0.352251i	$-0.114584\pi$
$397$	2.33361e9	1.87181	0.935906	+	0.352251i	$0.114584\pi$
$398$	0	0
$399$	−5.58746e8	−0.440362
$400$	0	0
$401$	−3.12899e8	−0.242326	−0.121163	−	0.992633i	$-0.538662\pi$
$401$	−3.12899e8	−0.242326	−0.121163	+	0.992633i	$0.538662\pi$
$402$	0	0
$403$	−1.99866e9	−1.52115
$404$	0	0
$405$	6.76560e7	0.0506074
$406$	0	0
$407$	−7.00122e8	−0.514746
$408$	0	0
$409$	1.93419e9	1.39787	0.698936	−	0.715184i	$-0.253657\pi$
$409$	1.93419e9	1.39787	0.698936	+	0.715184i	$0.253657\pi$
$410$	0	0
$411$	3.70017e8	0.262891
$412$	0	0
$413$	2.45787e8	0.171686
$414$	0	0
$415$	6.47766e7	0.0444887
$416$	0	0
$417$	2.09962e8	0.141796
$418$	0	0
$419$	1.70074e9	1.12951	0.564754	−	0.825260i	$-0.308971\pi$
$419$	1.70074e9	1.12951	0.564754	+	0.825260i	$0.308971\pi$
$420$	0	0
$421$	1.08520e8	0.0708796	0.0354398	−	0.999372i	$-0.488717\pi$
$421$	1.08520e8	0.0708796	0.0354398	+	0.999372i	$0.488717\pi$
$422$	0	0
$423$	−9.32797e7	−0.0599233
$424$	0	0
$425$	−2.20517e8	−0.139342
$426$	0	0
$427$	−1.95543e8	−0.121547
$428$	0	0
$429$	1.55532e9	0.951084
$430$	0	0
$431$	−3.05011e9	−1.83504	−0.917519	−	0.397691i	$-0.869812\pi$
$431$	−3.05011e9	−1.83504	−0.917519	+	0.397691i	$0.869812\pi$
$432$	0	0
$433$	−2.99899e9	−1.77528	−0.887641	−	0.460537i	$-0.847657\pi$
$433$	−2.99899e9	−1.77528	−0.887641	+	0.460537i	$0.847657\pi$
$434$	0	0
$435$	−1.25675e7	−0.00732043
$436$	0	0
$437$	1.20603e8	0.0691313
$438$	0	0
$439$	−2.96994e7	−0.0167541	−0.00837706	−	0.999965i	$-0.502667\pi$
$439$	−2.96994e7	−0.0167541	−0.00837706	+	0.999965i	$0.502667\pi$
$440$	0	0
$441$	−3.82019e8	−0.212105
$442$	0	0
$443$	−1.60045e9	−0.874639	−0.437320	−	0.899306i	$-0.644072\pi$
$443$	−1.60045e9	−0.874639	−0.437320	+	0.899306i	$0.644072\pi$
$444$	0	0
$445$	−1.24756e8	−0.0671120
$446$	0	0
$447$	−8.31186e8	−0.440171
$448$	0	0
$449$	−2.82121e9	−1.47086	−0.735432	−	0.677598i	$-0.763021\pi$
$449$	−2.82121e9	−1.47086	−0.735432	+	0.677598i	$0.763021\pi$
$450$	0	0
$451$	−9.92973e8	−0.509706
$452$	0	0
$453$	9.01244e8	0.455511
$454$	0	0
$455$	−7.67472e8	−0.381964
$456$	0	0
$457$	−3.16851e9	−1.55292	−0.776459	−	0.630168i	$-0.782986\pi$
$457$	−3.16851e9	−1.55292	−0.776459	+	0.630168i	$0.782986\pi$
$458$	0	0
$459$	7.00998e7	0.0338355
$460$	0	0
$461$	3.59945e9	1.71113	0.855565	−	0.517696i	$-0.173210\pi$
$461$	3.59945e9	1.71113	0.855565	+	0.517696i	$0.173210\pi$
$462$	0	0
$463$	3.63299e9	1.70110	0.850552	−	0.525891i	$-0.176268\pi$
$463$	3.63299e9	1.70110	0.850552	+	0.525891i	$0.176268\pi$
$464$	0	0
$465$	−6.23664e8	−0.287651
$466$	0	0
$467$	3.22700e9	1.46619	0.733095	−	0.680126i	$-0.238075\pi$
$467$	3.22700e9	1.46619	0.733095	+	0.680126i	$0.238075\pi$
$468$	0	0
$469$	−2.26508e9	−1.01386
$470$	0	0
$471$	−2.14719e9	−0.946885
$472$	0	0
$473$	2.26582e9	0.984492
$474$	0	0
$475$	−2.34132e9	−1.00238
$476$	0	0
$477$	1.34652e9	0.568067
$478$	0	0
$479$	2.85867e9	1.18847	0.594236	−	0.804290i	$-0.297454\pi$
$479$	2.85867e9	1.18847	0.594236	+	0.804290i	$0.297454\pi$
$480$	0	0
$481$	−1.47478e9	−0.604253
$482$	0	0
$483$	−4.71287e7	−0.0190314
$484$	0	0
$485$	−6.36173e8	−0.253209
$486$	0	0
$487$	−4.44089e9	−1.74228	−0.871141	−	0.491033i	$-0.836620\pi$
$487$	−4.44089e9	−1.74228	−0.871141	+	0.491033i	$0.836620\pi$
$488$	0	0
$489$	1.72669e9	0.667780
$490$	0	0
$491$	2.93593e8	0.111933	0.0559667	−	0.998433i	$-0.482176\pi$
$491$	2.93593e8	0.111933	0.0559667	+	0.998433i	$0.482176\pi$
$492$	0	0
$493$	−1.30215e7	−0.00489435
$494$	0	0
$495$	4.85322e8	0.179851
$496$	0	0
$497$	2.47988e9	0.906117
$498$	0	0
$499$	−2.28237e9	−0.822308	−0.411154	−	0.911566i	$-0.634874\pi$
$499$	−2.28237e9	−0.822308	−0.411154	+	0.911566i	$0.634874\pi$
$500$	0	0
$501$	9.06412e8	0.322028
$502$	0	0
$503$	−2.74070e9	−0.960226	−0.480113	−	0.877207i	$-0.659404\pi$
$503$	−2.74070e9	−0.960226	−0.480113	+	0.877207i	$0.659404\pi$
$504$	0	0
$505$	−1.85554e9	−0.641137
$506$	0	0
$507$	1.58200e9	0.539113
$508$	0	0
$509$	4.79364e8	0.161122	0.0805608	−	0.996750i	$-0.474329\pi$
$509$	4.79364e8	0.161122	0.0805608	+	0.996750i	$0.474329\pi$
$510$	0	0
$511$	−2.98440e9	−0.989426
$512$	0	0
$513$	7.44279e8	0.243402
$514$	0	0
$515$	5.56002e8	0.179371
$516$	0	0
$517$	−6.69130e8	−0.212958
$518$	0	0
$519$	1.42276e9	0.446731
$520$	0	0
$521$	1.89326e9	0.586515	0.293258	−	0.956033i	$-0.405261\pi$
$521$	1.89326e9	0.586515	0.293258	+	0.956033i	$0.405261\pi$
$522$	0	0
$523$	−2.02598e9	−0.619269	−0.309635	−	0.950856i	$-0.600207\pi$
$523$	−2.02598e9	−0.619269	−0.309635	+	0.950856i	$0.600207\pi$
$524$	0	0
$525$	9.14929e8	0.275950
$526$	0	0
$527$	−6.46191e8	−0.192320
$528$	0	0
$529$	−3.39465e9	−0.997012
$530$	0	0
$531$	−3.27401e8	−0.0948963
$532$	0	0
$533$	−2.09166e9	−0.598337
$534$	0	0
$535$	2.82774e9	0.798363
$536$	0	0
$537$	2.81443e9	0.784298
$538$	0	0
$539$	−2.74037e9	−0.753786
$540$	0	0
$541$	−3.90343e9	−1.05988	−0.529939	−	0.848036i	$-0.677785\pi$
$541$	−3.90343e9	−1.05988	−0.529939	+	0.848036i	$0.677785\pi$
$542$	0	0
$543$	−2.98038e9	−0.798863
$544$	0	0
$545$	1.77634e9	0.470044
$546$	0	0
$547$	−3.78626e9	−0.989133	−0.494567	−	0.869140i	$-0.664673\pi$
$547$	−3.78626e9	−0.989133	−0.494567	+	0.869140i	$0.664673\pi$
$548$	0	0
$549$	2.60473e8	0.0671831
$550$	0	0
$551$	−1.38254e8	−0.0352085
$552$	0	0
$553$	−1.20708e9	−0.303528
$554$	0	0
$555$	−4.60191e8	−0.114265
$556$	0	0
$557$	6.00470e9	1.47231	0.736153	−	0.676815i	$-0.236640\pi$
$557$	6.00470e9	1.47231	0.736153	+	0.676815i	$0.236640\pi$
$558$	0	0
$559$	4.77287e9	1.15568
$560$	0	0
$561$	5.02852e8	0.120246
$562$	0	0
$563$	−1.31073e9	−0.309553	−0.154776	−	0.987950i	$-0.549466\pi$
$563$	−1.31073e9	−0.309553	−0.154776	+	0.987950i	$0.549466\pi$
$564$	0	0
$565$	1.13094e9	0.263798
$566$	0	0
$567$	−2.90845e8	−0.0670071
$568$	0	0
$569$	3.32825e9	0.757395	0.378698	−	0.925520i	$-0.376372\pi$
$569$	3.32825e9	0.757395	0.378698	+	0.925520i	$0.376372\pi$
$570$	0	0
$571$	6.19528e9	1.39263	0.696313	−	0.717739i	$-0.254823\pi$
$571$	6.19528e9	1.39263	0.696313	+	0.717739i	$0.254823\pi$
$572$	0	0
$573$	−1.59297e9	−0.353725
$574$	0	0
$575$	−1.97484e8	−0.0433206
$576$	0	0
$577$	−3.76996e9	−0.816999	−0.408500	−	0.912759i	$-0.633948\pi$
$577$	−3.76996e9	−0.816999	−0.408500	+	0.912759i	$0.633948\pi$
$578$	0	0
$579$	2.21985e9	0.475280
$580$	0	0
$581$	−2.78467e8	−0.0589056
$582$	0	0
$583$	9.65911e9	2.01882
$584$	0	0
$585$	1.02231e9	0.211124
$586$	0	0
$587$	1.10174e9	0.224825	0.112412	−	0.993662i	$-0.464142\pi$
$587$	1.10174e9	0.224825	0.112412	+	0.993662i	$0.464142\pi$
$588$	0	0
$589$	−6.86088e9	−1.38349
$590$	0	0
$591$	−3.59257e9	−0.715894
$592$	0	0
$593$	−6.94155e9	−1.36699	−0.683494	−	0.729956i	$-0.739541\pi$
$593$	−6.94155e9	−1.36699	−0.683494	+	0.729956i	$0.739541\pi$
$594$	0	0
$595$	−2.48133e8	−0.0482919
$596$	0	0
$597$	−3.38693e8	−0.0651473
$598$	0	0
$599$	−1.59345e9	−0.302932	−0.151466	−	0.988462i	$-0.548399\pi$
$599$	−1.59345e9	−0.302932	−0.151466	+	0.988462i	$0.548399\pi$
$600$	0	0
$601$	−7.81030e8	−0.146760	−0.0733799	−	0.997304i	$-0.523379\pi$
$601$	−7.81030e8	−0.146760	−0.0733799	+	0.997304i	$0.523379\pi$
$602$	0	0
$603$	3.01721e9	0.560396
$604$	0	0
$605$	1.00055e9	0.183694
$606$	0	0
$607$	5.89120e9	1.06916	0.534581	−	0.845117i	$-0.320469\pi$
$607$	5.89120e9	1.06916	0.534581	+	0.845117i	$0.320469\pi$
$608$	0	0
$609$	5.40262e7	0.00969268
$610$	0	0
$611$	−1.40950e9	−0.249988
$612$	0	0
$613$	−4.35937e9	−0.764385	−0.382192	−	0.924083i	$-0.624831\pi$
$613$	−4.35937e9	−0.764385	−0.382192	+	0.924083i	$0.624831\pi$
$614$	0	0
$615$	−6.52682e8	−0.113146
$616$	0	0
$617$	2.37160e9	0.406483	0.203242	−	0.979129i	$-0.434852\pi$
$617$	2.37160e9	0.406483	0.203242	+	0.979129i	$0.434852\pi$
$618$	0	0
$619$	−6.81078e9	−1.15420	−0.577098	−	0.816675i	$-0.695815\pi$
$619$	−6.81078e9	−1.15420	−0.577098	+	0.816675i	$0.695815\pi$
$620$	0	0
$621$	6.27779e7	0.0105193
$622$	0	0
$623$	5.36309e8	0.0888602
$624$	0	0
$625$	2.56766e9	0.420686
$626$	0	0
$627$	5.33899e9	0.865013
$628$	0	0
$629$	−4.76813e8	−0.0763960
$630$	0	0
$631$	1.19562e10	1.89449	0.947244	−	0.320513i	$-0.103855\pi$
$631$	1.19562e10	1.89449	0.947244	+	0.320513i	$0.103855\pi$
$632$	0	0
$633$	5.35448e8	0.0839082
$634$	0	0
$635$	2.39241e9	0.370789
$636$	0	0
$637$	−5.77247e9	−0.884858
$638$	0	0
$639$	−3.30333e9	−0.500840
$640$	0	0
$641$	6.16170e9	0.924054	0.462027	−	0.886866i	$-0.347122\pi$
$641$	6.16170e9	0.924054	0.462027	+	0.886866i	$0.347122\pi$
$642$	0	0
$643$	−3.93538e9	−0.583779	−0.291890	−	0.956452i	$-0.594284\pi$
$643$	−3.93538e9	−0.583779	−0.291890	+	0.956452i	$0.594284\pi$
$644$	0	0
$645$	1.48933e9	0.218540
$646$	0	0
$647$	−2.51681e9	−0.365330	−0.182665	−	0.983175i	$-0.558472\pi$
$647$	−2.51681e9	−0.365330	−0.182665	+	0.983175i	$0.558472\pi$
$648$	0	0
$649$	−2.34857e9	−0.337246
$650$	0	0
$651$	2.68106e9	0.380866
$652$	0	0
$653$	−8.76277e9	−1.23153	−0.615765	−	0.787930i	$-0.711153\pi$
$653$	−8.76277e9	−1.23153	−0.615765	+	0.787930i	$0.711153\pi$
$654$	0	0
$655$	1.51548e9	0.210720
$656$	0	0
$657$	3.97537e9	0.546888
$658$	0	0
$659$	1.02903e10	1.40064	0.700321	−	0.713828i	$-0.253040\pi$
$659$	1.02903e10	1.40064	0.700321	+	0.713828i	$0.253040\pi$
$660$	0	0
$661$	−1.35666e10	−1.82711	−0.913557	−	0.406710i	$-0.866676\pi$
$661$	−1.35666e10	−1.82711	−0.913557	+	0.406710i	$0.866676\pi$
$662$	0	0
$663$	1.05924e9	0.141155
$664$	0	0
$665$	−2.63453e9	−0.347397
$666$	0	0
$667$	−1.16614e7	−0.00152163
$668$	0	0
$669$	1.17011e9	0.151090
$670$	0	0
$671$	1.86847e9	0.238758
$672$	0	0
$673$	1.18271e9	0.149563	0.0747816	−	0.997200i	$-0.476174\pi$
$673$	1.18271e9	0.149563	0.0747816	+	0.997200i	$0.476174\pi$
$674$	0	0
$675$	−1.21873e9	−0.152526
$676$	0	0
$677$	5.21396e9	0.645814	0.322907	−	0.946431i	$-0.395340\pi$
$677$	5.21396e9	0.645814	0.322907	+	0.946431i	$0.395340\pi$
$678$	0	0
$679$	2.73483e9	0.335263
$680$	0	0
$681$	2.70606e9	0.328339
$682$	0	0
$683$	1.13287e10	1.36053	0.680266	−	0.732966i	$-0.261864\pi$
$683$	1.13287e10	1.36053	0.680266	+	0.732966i	$0.261864\pi$
$684$	0	0
$685$	1.74465e9	0.207392
$686$	0	0
$687$	−4.69665e8	−0.0552636
$688$	0	0
$689$	2.03465e10	2.36986
$690$	0	0
$691$	6.33579e9	0.730513	0.365256	−	0.930907i	$-0.380981\pi$
$691$	6.33579e9	0.730513	0.365256	+	0.930907i	$0.380981\pi$
$692$	0	0
$693$	−2.08634e9	−0.238133
$694$	0	0
$695$	9.89984e8	0.111862
$696$	0	0
$697$	−6.76257e8	−0.0756480
$698$	0	0
$699$	6.51246e9	0.721232
$700$	0	0
$701$	5.13852e9	0.563411	0.281705	−	0.959501i	$-0.409100\pi$
$701$	5.13852e9	0.563411	0.281705	+	0.959501i	$0.409100\pi$
$702$	0	0
$703$	−5.06252e9	−0.549570
$704$	0	0
$705$	−4.39820e8	−0.0472730
$706$	0	0
$707$	7.97675e9	0.848903
$708$	0	0
$709$	6.15300e9	0.648374	0.324187	−	0.945993i	$-0.394909\pi$
$709$	6.15300e9	0.648374	0.324187	+	0.945993i	$0.394909\pi$
$710$	0	0
$711$	1.60789e9	0.167770
$712$	0	0
$713$	−5.78697e8	−0.0597913
$714$	0	0
$715$	7.33343e9	0.750302
$716$	0	0
$717$	2.84115e9	0.287858
$718$	0	0
$719$	1.61162e10	1.61700	0.808500	−	0.588496i	$-0.200280\pi$
$719$	1.61162e10	1.61700	0.808500	+	0.588496i	$0.200280\pi$
$720$	0	0
$721$	−2.39019e9	−0.237497
$722$	0	0
$723$	1.16623e7	0.00114763
$724$	0	0
$725$	2.26387e8	0.0220632
$726$	0	0
$727$	−1.62382e8	−0.0156736	−0.00783679	−	0.999969i	$-0.502495\pi$
$727$	−1.62382e8	−0.0156736	−0.00783679	+	0.999969i	$0.502495\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.54312e9	0.146113
$732$	0	0
$733$	−1.59813e9	−0.149882	−0.0749408	−	0.997188i	$-0.523877\pi$
$733$	−1.59813e9	−0.149882	−0.0749408	+	0.997188i	$0.523877\pi$
$734$	0	0
$735$	−1.80125e9	−0.167327
$736$	0	0
$737$	2.16436e10	1.99156
$738$	0	0
$739$	−1.00921e10	−0.919872	−0.459936	−	0.887952i	$-0.652128\pi$
$739$	−1.00921e10	−0.919872	−0.459936	+	0.887952i	$0.652128\pi$
$740$	0	0
$741$	1.12464e10	1.01543
$742$	0	0
$743$	−1.70676e9	−0.152655	−0.0763277	−	0.997083i	$-0.524320\pi$
$743$	−1.70676e9	−0.152655	−0.0763277	+	0.997083i	$0.524320\pi$
$744$	0	0
$745$	−3.91910e9	−0.347247
$746$	0	0
$747$	3.70932e8	0.0325591
$748$	0	0
$749$	−1.21561e10	−1.05708
$750$	0	0
$751$	5.56063e9	0.479054	0.239527	−	0.970890i	$-0.423008\pi$
$751$	5.56063e9	0.479054	0.239527	+	0.970890i	$0.423008\pi$
$752$	0	0
$753$	1.15826e10	0.988603
$754$	0	0
$755$	4.24943e9	0.359349
$756$	0	0
$757$	−1.48328e10	−1.24276	−0.621379	−	0.783510i	$-0.713427\pi$
$757$	−1.48328e10	−1.24276	−0.621379	+	0.783510i	$0.713427\pi$
$758$	0	0
$759$	4.50329e8	0.0373839
$760$	0	0
$761$	−7.66624e9	−0.630574	−0.315287	−	0.948996i	$-0.602101\pi$
$761$	−7.66624e9	−0.630574	−0.315287	+	0.948996i	$0.602101\pi$
$762$	0	0
$763$	−7.63628e9	−0.622366
$764$	0	0
$765$	3.30525e8	0.0266925
$766$	0	0
$767$	−4.94717e9	−0.395888
$768$	0	0
$769$	−1.51989e9	−0.120523	−0.0602614	−	0.998183i	$-0.519193\pi$
$769$	−1.51989e9	−0.120523	−0.0602614	+	0.998183i	$0.519193\pi$
$770$	0	0
$771$	−7.14111e9	−0.561146
$772$	0	0
$773$	5.63668e9	0.438930	0.219465	−	0.975620i	$-0.429569\pi$
$773$	5.63668e9	0.438930	0.219465	+	0.975620i	$0.429569\pi$
$774$	0	0
$775$	1.12345e10	0.866955
$776$	0	0
$777$	1.97830e9	0.151293
$778$	0	0
$779$	−7.18010e9	−0.544189
$780$	0	0
$781$	−2.36961e10	−1.77991
$782$	0	0
$783$	−7.19657e7	−0.00535746
$784$	0	0
$785$	−1.01241e10	−0.746989
$786$	0	0
$787$	−1.58634e10	−1.16007	−0.580035	−	0.814591i	$-0.696961\pi$
$787$	−1.58634e10	−1.16007	−0.580035	+	0.814591i	$0.696961\pi$
$788$	0	0
$789$	9.53302e9	0.690973
$790$	0	0
$791$	−4.86179e9	−0.349284
$792$	0	0
$793$	3.93587e9	0.280275
$794$	0	0
$795$	6.34894e9	0.448143
$796$	0	0
$797$	−1.81464e10	−1.26965	−0.634827	−	0.772654i	$-0.718929\pi$
$797$	−1.81464e10	−1.26965	−0.634827	+	0.772654i	$0.718929\pi$
$798$	0	0
$799$	−4.55706e8	−0.0316061
$800$	0	0
$801$	−7.14391e8	−0.0491159
$802$	0	0
$803$	2.85168e10	1.94355
$804$	0	0
$805$	−2.22215e8	−0.0150137
$806$	0	0
$807$	−2.50793e9	−0.167980
$808$	0	0
$809$	2.65839e10	1.76522	0.882611	−	0.470103i	$-0.155783\pi$
$809$	2.65839e10	1.76522	0.882611	+	0.470103i	$0.155783\pi$
$810$	0	0
$811$	1.05526e10	0.694680	0.347340	−	0.937739i	$-0.387085\pi$
$811$	1.05526e10	0.694680	0.347340	+	0.937739i	$0.387085\pi$
$812$	0	0
$813$	−4.17265e8	−0.0272330
$814$	0	0
$815$	8.14145e9	0.526806
$816$	0	0
$817$	1.63840e10	1.05110
$818$	0	0
$819$	−4.39479e9	−0.279541
$820$	0	0
$821$	1.56032e10	0.984041	0.492021	−	0.870584i	$-0.336258\pi$
$821$	1.56032e10	0.984041	0.492021	+	0.870584i	$0.336258\pi$
$822$	0	0
$823$	−1.10871e10	−0.693298	−0.346649	−	0.937995i	$-0.612681\pi$
$823$	−1.10871e10	−0.693298	−0.346649	+	0.937995i	$0.612681\pi$
$824$	0	0
$825$	−8.74242e9	−0.542054
$826$	0	0
$827$	−1.69158e10	−1.03998	−0.519989	−	0.854173i	$-0.674064\pi$
$827$	−1.69158e10	−1.03998	−0.519989	+	0.854173i	$0.674064\pi$
$828$	0	0
$829$	−1.30871e10	−0.797816	−0.398908	−	0.916991i	$-0.630611\pi$
$829$	−1.30871e10	−0.797816	−0.398908	+	0.916991i	$0.630611\pi$
$830$	0	0
$831$	−3.82732e9	−0.231361
$832$	0	0
$833$	−1.86631e9	−0.111873
$834$	0	0
$835$	4.27379e9	0.254045
$836$	0	0
$837$	−3.57130e9	−0.210517
$838$	0	0
$839$	−1.49451e10	−0.873637	−0.436819	−	0.899550i	$-0.643895\pi$
$839$	−1.49451e10	−0.873637	−0.436819	+	0.899550i	$0.643895\pi$
$840$	0	0
$841$	−1.72365e10	−0.999225
$842$	0	0
$843$	2.00195e10	1.15095
$844$	0	0
$845$	7.45926e9	0.425302
$846$	0	0
$847$	−4.30124e9	−0.243222
$848$	0	0
$849$	9.63472e9	0.540334
$850$	0	0
$851$	−4.27010e8	−0.0237511
$852$	0	0
$853$	1.99352e10	1.09976	0.549880	−	0.835244i	$-0.314673\pi$
$853$	1.99352e10	1.09976	0.549880	+	0.835244i	$0.314673\pi$
$854$	0	0
$855$	3.50932e9	0.192018
$856$	0	0
$857$	−3.20597e10	−1.73991	−0.869955	−	0.493131i	$-0.835852\pi$
$857$	−3.20597e10	−1.73991	−0.869955	+	0.493131i	$0.835852\pi$
$858$	0	0
$859$	−6.38167e7	−0.00343525	−0.00171762	−	0.999999i	$-0.500547\pi$
$859$	−6.38167e7	−0.00343525	−0.00171762	+	0.999999i	$0.500547\pi$
$860$	0	0
$861$	2.80580e9	0.149812
$862$	0	0
$863$	−1.71545e10	−0.908532	−0.454266	−	0.890866i	$-0.650098\pi$
$863$	−1.71545e10	−0.908532	−0.454266	+	0.890866i	$0.650098\pi$
$864$	0	0
$865$	6.70841e9	0.352422
$866$	0	0
$867$	−1.07367e10	−0.559504
$868$	0	0
$869$	1.15340e10	0.596227
$870$	0	0
$871$	4.55913e10	2.33786
$872$	0	0
$873$	−3.64293e9	−0.185311
$874$	0	0
$875$	9.75707e9	0.492370
$876$	0	0
$877$	7.31771e9	0.366333	0.183167	−	0.983082i	$-0.441365\pi$
$877$	7.31771e9	0.366333	0.183167	+	0.983082i	$0.441365\pi$
$878$	0	0
$879$	−1.67199e9	−0.0830373
$880$	0	0
$881$	−2.58423e10	−1.27326	−0.636628	−	0.771171i	$-0.719671\pi$
$881$	−2.58423e10	−1.27326	−0.636628	+	0.771171i	$0.719671\pi$
$882$	0	0
$883$	6.05970e9	0.296203	0.148101	−	0.988972i	$-0.452684\pi$
$883$	6.05970e9	0.296203	0.148101	+	0.988972i	$0.452684\pi$
$884$	0	0
$885$	−1.54372e9	−0.0748628
$886$	0	0
$887$	−2.61018e10	−1.25585	−0.627926	−	0.778273i	$-0.716096\pi$
$887$	−2.61018e10	−1.25585	−0.627926	+	0.778273i	$0.716096\pi$
$888$	0	0
$889$	−1.02847e10	−0.490946
$890$	0	0
$891$	2.77911e9	0.131624
$892$	0	0
$893$	−4.83842e9	−0.227365
$894$	0	0
$895$	1.32702e10	0.618726
$896$	0	0
$897$	9.48601e8	0.0438844
$898$	0	0
$899$	6.63391e8	0.0304516
$900$	0	0
$901$	6.57827e9	0.299623
$902$	0	0
$903$	−6.40244e9	−0.289360
$904$	0	0
$905$	−1.40527e10	−0.630216
$906$	0	0
$907$	−2.85690e10	−1.27136	−0.635681	−	0.771951i	$-0.719281\pi$
$907$	−2.85690e10	−1.27136	−0.635681	+	0.771951i	$0.719281\pi$
$908$	0	0
$909$	−1.06254e10	−0.469217
$910$	0	0
$911$	2.80813e10	1.23056	0.615281	−	0.788308i	$-0.289043\pi$
$911$	2.80813e10	1.23056	0.615281	+	0.788308i	$0.289043\pi$
$912$	0	0
$913$	2.66083e9	0.115710
$914$	0	0
$915$	1.22815e9	0.0530002
$916$	0	0
$917$	−6.51486e9	−0.279005
$918$	0	0
$919$	−1.38008e10	−0.586544	−0.293272	−	0.956029i	$-0.594744\pi$
$919$	−1.38008e10	−0.586544	−0.293272	+	0.956029i	$0.594744\pi$
$920$	0	0
$921$	7.10362e8	0.0299620
$922$	0	0
$923$	−4.99148e10	−2.08941
$924$	0	0
$925$	8.28971e9	0.344384
$926$	0	0
$927$	3.18385e9	0.131272
$928$	0	0
$929$	−8.28889e9	−0.339189	−0.169594	−	0.985514i	$-0.554246\pi$
$929$	−8.28889e9	−0.339189	−0.169594	+	0.985514i	$0.554246\pi$
$930$	0	0
$931$	−1.98154e10	−0.804781
$932$	0	0
$933$	−2.18319e10	−0.880048
$934$	0	0
$935$	2.37098e9	0.0948610
$936$	0	0
$937$	−4.79384e10	−1.90368	−0.951842	−	0.306589i	$-0.900812\pi$
$937$	−4.79384e10	−1.90368	−0.951842	+	0.306589i	$0.900812\pi$
$938$	0	0
$939$	−1.98513e10	−0.782453
$940$	0	0
$941$	−6.21433e9	−0.243126	−0.121563	−	0.992584i	$-0.538791\pi$
$941$	−6.21433e9	−0.243126	−0.121563	+	0.992584i	$0.538791\pi$
$942$	0	0
$943$	−6.05622e8	−0.0235186
$944$	0	0
$945$	−1.37135e9	−0.0528613
$946$	0	0
$947$	7.84554e8	0.0300191	0.0150096	−	0.999887i	$-0.495222\pi$
$947$	7.84554e8	0.0300191	0.0150096	+	0.999887i	$0.495222\pi$
$948$	0	0
$949$	6.00695e10	2.28151
$950$	0	0
$951$	2.01107e10	0.758221
$952$	0	0
$953$	7.93826e9	0.297098	0.148549	−	0.988905i	$-0.452540\pi$
$953$	7.93826e9	0.297098	0.148549	+	0.988905i	$0.452540\pi$
$954$	0	0
$955$	−7.51095e9	−0.279051
$956$	0	0
$957$	−5.16237e8	−0.0190396
$958$	0	0
$959$	−7.50005e9	−0.274599
$960$	0	0
$961$	5.40825e9	0.196574
$962$	0	0
$963$	1.61925e10	0.584283
$964$	0	0
$965$	1.04667e10	0.374944
$966$	0	0
$967$	−2.52503e10	−0.897996	−0.448998	−	0.893533i	$-0.648219\pi$
$967$	−2.52503e10	−0.897996	−0.448998	+	0.893533i	$0.648219\pi$
$968$	0	0
$969$	3.63608e9	0.128381
$970$	0	0
$971$	−3.91233e10	−1.37141	−0.685707	−	0.727878i	$-0.740507\pi$
$971$	−3.91233e10	−1.37141	−0.685707	+	0.727878i	$0.740507\pi$
$972$	0	0
$973$	−4.25582e9	−0.148111
$974$	0	0
$975$	−1.84156e10	−0.636310
$976$	0	0
$977$	3.42856e10	1.17620	0.588099	−	0.808789i	$-0.299877\pi$
$977$	3.42856e10	1.17620	0.588099	+	0.808789i	$0.299877\pi$
$978$	0	0
$979$	−5.12460e9	−0.174550
$980$	0	0
$981$	1.01719e10	0.344002
$982$	0	0
$983$	−4.97643e10	−1.67101	−0.835507	−	0.549480i	$-0.814826\pi$
$983$	−4.97643e10	−1.67101	−0.835507	+	0.549480i	$0.814826\pi$
$984$	0	0
$985$	−1.69392e10	−0.564762
$986$	0	0
$987$	1.89073e9	0.0625921
$988$	0	0
$989$	1.38194e9	0.0454259
$990$	0	0
$991$	−5.64824e10	−1.84355	−0.921775	−	0.387725i	$-0.873261\pi$
$991$	−5.64824e10	−1.84355	−0.921775	+	0.387725i	$0.873261\pi$
$992$	0	0
$993$	−2.01608e10	−0.653409
$994$	0	0
$995$	−1.59696e9	−0.0513941
$996$	0	0
$997$	−1.37061e10	−0.438006	−0.219003	−	0.975724i	$-0.570280\pi$
$997$	−1.37061e10	−0.438006	−0.219003	+	0.975724i	$0.570280\pi$
$998$	0	0
$999$	−2.63520e9	−0.0836247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.q.1.3	yes	4
4.3	odd	2		384.8.a.m.1.3	✓	4
8.3	odd	2		384.8.a.t.1.2	yes	4
8.5	even	2		384.8.a.p.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.m.1.3	✓	4	4.3	odd	2
384.8.a.p.1.2	yes	4	8.5	even	2
384.8.a.q.1.3	yes	4	1.1	even	1	trivial
384.8.a.t.1.2	yes	4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} +127.307 q^{5} -547.276 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} +127.307 q^{5} -547.276 q^{7} +729.000 q^{9} +5229.39 q^{11} +11015.5 q^{13} +3437.28 q^{15} +3561.44 q^{17} +37813.3 q^{19} -14776.5 q^{21} +3189.45 q^{23} -61918.0 q^{25} +19683.0 q^{27} -3656.23 q^{29} -181441. q^{31} +141194. q^{33} -69672.0 q^{35} -133882. q^{37} +297419. q^{39} -189883. q^{41} +433286. q^{43} +92806.6 q^{45} -127956. q^{47} -524032. q^{49} +96158.9 q^{51} +1.84708e6 q^{53} +665737. q^{55} +1.02096e6 q^{57} -449110. q^{59} +357302. q^{61} -398964. q^{63} +1.40235e6 q^{65} +4.13883e6 q^{67} +86115.1 q^{69} -4.53132e6 q^{71} +5.45318e6 q^{73} -1.67179e6 q^{75} -2.86192e6 q^{77} +2.20562e6 q^{79} +531441. q^{81} +508823. q^{83} +453395. q^{85} -98718.3 q^{87} -979961. q^{89} -6.02852e6 q^{91} -4.89891e6 q^{93} +4.81389e6 q^{95} -4.99716e6 q^{97} +3.81223e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9} + 3856 q^{11} - 10680 q^{13} - 9072 q^{15} + 26232 q^{17} - 15456 q^{19} + 18360 q^{21} - 11312 q^{23} + 159052 q^{25} + 78732 q^{27} + 1856 q^{29} + 71752 q^{31} + 104112 q^{33} - 179040 q^{35} + 180088 q^{37} - 288360 q^{39} + 11224 q^{41} + 66688 q^{43} - 244944 q^{45} - 1334448 q^{47} + 2401140 q^{49} + 708264 q^{51} + 864576 q^{53} - 3304896 q^{55} - 417312 q^{57} + 1878448 q^{59} + 1901176 q^{61} + 495720 q^{63} + 4366944 q^{65} + 5505488 q^{67} - 305424 q^{69} - 967696 q^{71} + 3244760 q^{73} + 4294404 q^{75} + 8979488 q^{77} + 6471816 q^{79} + 2125764 q^{81} + 17019600 q^{83} - 12122592 q^{85} + 50112 q^{87} + 13559816 q^{89} + 6692304 q^{91} + 1937304 q^{93} + 22523904 q^{95} + 2180520 q^{97} + 2811024 q^{99}+O(q^{100}) \) 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9 + 3856 * q^11 - 10680 * q^13 - 9072 * q^15 + 26232 * q^17 - 15456 * q^19 + 18360 * q^21 - 11312 * q^23 + 159052 * q^25 + 78732 * q^27 + 1856 * q^29 + 71752 * q^31 + 104112 * q^33 - 179040 * q^35 + 180088 * q^37 - 288360 * q^39 + 11224 * q^41 + 66688 * q^43 - 244944 * q^45 - 1334448 * q^47 + 2401140 * q^49 + 708264 * q^51 + 864576 * q^53 - 3304896 * q^55 - 417312 * q^57 + 1878448 * q^59 + 1901176 * q^61 + 495720 * q^63 + 4366944 * q^65 + 5505488 * q^67 - 305424 * q^69 - 967696 * q^71 + 3244760 * q^73 + 4294404 * q^75 + 8979488 * q^77 + 6471816 * q^79 + 2125764 * q^81 + 17019600 * q^83 - 12122592 * q^85 + 50112 * q^87 + 13559816 * q^89 + 6692304 * q^91 + 1937304 * q^93 + 22523904 * q^95 + 2180520 * q^97 + 2811024 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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