LMFDB - Embedded newform 384.8.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.955849786$
Analytic rank:	$0$
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			1.4
Root			$-16.8500$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} +333.320 q^{5} +988.659 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} +333.320 q^{5} +988.659 q^{7} +729.000 q^{9} -4473.12 q^{11} -5599.17 q^{13} +8999.65 q^{15} +190.351 q^{17} -12312.0 q^{19} +26693.8 q^{21} +40349.6 q^{23} +32977.5 q^{25} +19683.0 q^{27} +65627.5 q^{29} +226405. q^{31} -120774. q^{33} +329540. q^{35} +608760. q^{37} -151177. q^{39} +78795.5 q^{41} +105702. q^{43} +242991. q^{45} +388155. q^{47} +153903. q^{49} +5139.48 q^{51} -53946.5 q^{53} -1.49098e6 q^{55} -332424. q^{57} +2.74850e6 q^{59} -1.42106e6 q^{61} +720732. q^{63} -1.86632e6 q^{65} -1.87548e6 q^{67} +1.08944e6 q^{69} -1.52967e6 q^{71} +4.26626e6 q^{73} +890391. q^{75} -4.42239e6 q^{77} -2.27486e6 q^{79} +531441. q^{81} +5.13213e6 q^{83} +63447.8 q^{85} +1.77194e6 q^{87} +1.00547e7 q^{89} -5.53567e6 q^{91} +6.11293e6 q^{93} -4.10384e6 q^{95} +1.40320e7 q^{97} -3.26090e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	333.320	1.19252	0.596262	−	0.802790i	$-0.296652\pi$
$5$	333.320	1.19252	0.596262	+	0.802790i	$0.296652\pi$
$6$	0	0
$7$	988.659	1.08944	0.544720	−	0.838618i	$-0.316636\pi$
$7$	988.659	1.08944	0.544720	+	0.838618i	$0.316636\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−4473.12	−1.01330	−0.506648	−	0.862153i	$-0.669115\pi$
$11$	−4473.12	−1.01330	−0.506648	+	0.862153i	$0.669115\pi$
$12$	0	0
$13$	−5599.17	−0.706841	−0.353420	−	0.935465i	$-0.614981\pi$
$13$	−5599.17	−0.706841	−0.353420	+	0.935465i	$0.614981\pi$
$14$	0	0
$15$	8999.65	0.688504
$16$	0	0
$17$	190.351	0.00939688	0.00469844	−	0.999989i	$-0.498504\pi$
$17$	190.351	0.00939688	0.00469844	+	0.999989i	$0.498504\pi$
$18$	0	0
$19$	−12312.0	−0.411804	−0.205902	−	0.978573i	$-0.566013\pi$
$19$	−12312.0	−0.411804	−0.205902	+	0.978573i	$0.566013\pi$
$20$	0	0
$21$	26693.8	0.628989
$22$	0	0
$23$	40349.6	0.691499	0.345749	−	0.938327i	$-0.387625\pi$
$23$	40349.6	0.691499	0.345749	+	0.938327i	$0.387625\pi$
$24$	0	0
$25$	32977.5	0.422111
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	65627.5	0.499681	0.249840	−	0.968287i	$-0.419622\pi$
$29$	65627.5	0.499681	0.249840	+	0.968287i	$0.419622\pi$
$30$	0	0
$31$	226405.	1.36496	0.682480	−	0.730904i	$-0.260902\pi$
$31$	226405.	1.36496	0.682480	+	0.730904i	$0.260902\pi$
$32$	0	0
$33$	−120774.	−0.585026
$34$	0	0
$35$	329540.	1.29918
$36$	0	0
$37$	608760.	1.97579	0.987893	−	0.155134i	$-0.0495808\pi$
$37$	608760.	1.97579	0.987893	+	0.155134i	$0.0495808\pi$
$38$	0	0
$39$	−151177.	−0.408095
$40$	0	0
$41$	78795.5	0.178549	0.0892746	−	0.996007i	$-0.471545\pi$
$41$	78795.5	0.178549	0.0892746	+	0.996007i	$0.471545\pi$
$42$	0	0
$43$	105702.	0.202743	0.101371	−	0.994849i	$-0.467677\pi$
$43$	105702.	0.202743	0.101371	+	0.994849i	$0.467677\pi$
$44$	0	0
$45$	242991.	0.397508
$46$	0	0
$47$	388155.	0.545335	0.272667	−	0.962108i	$-0.412094\pi$
$47$	388155.	0.545335	0.272667	+	0.962108i	$0.412094\pi$
$48$	0	0
$49$	153903.	0.186880
$50$	0	0
$51$	5139.48	0.00542529
$52$	0	0
$53$	−53946.5	−0.0497734	−0.0248867	−	0.999690i	$-0.507923\pi$
$53$	−53946.5	−0.0497734	−0.0248867	+	0.999690i	$0.507923\pi$
$54$	0	0
$55$	−1.49098e6	−1.20838
$56$	0	0
$57$	−332424.	−0.237755
$58$	0	0
$59$	2.74850e6	1.74227	0.871133	−	0.491048i	$-0.163386\pi$
$59$	2.74850e6	1.74227	0.871133	+	0.491048i	$0.163386\pi$
$60$	0	0
$61$	−1.42106e6	−0.801603	−0.400802	−	0.916165i	$-0.631268\pi$
$61$	−1.42106e6	−0.801603	−0.400802	+	0.916165i	$0.631268\pi$
$62$	0	0
$63$	720732.	0.363147
$64$	0	0
$65$	−1.86632e6	−0.842924
$66$	0	0
$67$	−1.87548e6	−0.761819	−0.380909	−	0.924612i	$-0.624389\pi$
$67$	−1.87548e6	−0.761819	−0.380909	+	0.924612i	$0.624389\pi$
$68$	0	0
$69$	1.08944e6	0.399237
$70$	0	0
$71$	−1.52967e6	−0.507218	−0.253609	−	0.967307i	$-0.581618\pi$
$71$	−1.52967e6	−0.507218	−0.253609	+	0.967307i	$0.581618\pi$
$72$	0	0
$73$	4.26626e6	1.28356	0.641782	−	0.766887i	$-0.278195\pi$
$73$	4.26626e6	1.28356	0.641782	+	0.766887i	$0.278195\pi$
$74$	0	0
$75$	890391.	0.243706
$76$	0	0
$77$	−4.42239e6	−1.10392
$78$	0	0
$79$	−2.27486e6	−0.519111	−0.259556	−	0.965728i	$-0.583576\pi$
$79$	−2.27486e6	−0.519111	−0.259556	+	0.965728i	$0.583576\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.13213e6	0.985199	0.492599	−	0.870256i	$-0.336047\pi$
$83$	5.13213e6	0.985199	0.492599	+	0.870256i	$0.336047\pi$
$84$	0	0
$85$	63447.8	0.0112060
$86$	0	0
$87$	1.77194e6	0.288491
$88$	0	0
$89$	1.00547e7	1.51183	0.755915	−	0.654669i	$-0.227192\pi$
$89$	1.00547e7	1.51183	0.755915	+	0.654669i	$0.227192\pi$
$90$	0	0
$91$	−5.53567e6	−0.770061
$92$	0	0
$93$	6.11293e6	0.788060
$94$	0	0
$95$	−4.10384e6	−0.491086
$96$	0	0
$97$	1.40320e7	1.56106	0.780528	−	0.625121i	$-0.214951\pi$
$97$	1.40320e7	1.56106	0.780528	+	0.625121i	$0.214951\pi$
$98$	0	0
$99$	−3.26090e6	−0.337765
$100$	0	0
$101$	2.58976e6	0.250112	0.125056	−	0.992150i	$-0.460089\pi$
$101$	2.58976e6	0.250112	0.125056	+	0.992150i	$0.460089\pi$
$102$	0	0
$103$	2.55403e6	0.230301	0.115150	−	0.993348i	$-0.463265\pi$
$103$	2.55403e6	0.230301	0.115150	+	0.993348i	$0.463265\pi$
$104$	0	0
$105$	8.89758e6	0.750083
$106$	0	0
$107$	−8.86003e6	−0.699185	−0.349592	−	0.936902i	$-0.613680\pi$
$107$	−8.86003e6	−0.699185	−0.349592	+	0.936902i	$0.613680\pi$
$108$	0	0
$109$	−1.59339e7	−1.17850	−0.589250	−	0.807951i	$-0.700577\pi$
$109$	−1.59339e7	−1.17850	−0.589250	+	0.807951i	$0.700577\pi$
$110$	0	0
$111$	1.64365e7	1.14072
$112$	0	0
$113$	−2.87858e7	−1.87674	−0.938370	−	0.345633i	$-0.887664\pi$
$113$	−2.87858e7	−1.87674	−0.938370	+	0.345633i	$0.887664\pi$
$114$	0	0
$115$	1.34493e7	0.824628
$116$	0	0
$117$	−4.08179e6	−0.235614
$118$	0	0
$119$	188192.	0.0102373
$120$	0	0
$121$	521609.	0.0267668
$122$	0	0
$123$	2.12748e6	0.103085
$124$	0	0
$125$	−1.50486e7	−0.689145
$126$	0	0
$127$	2.83742e7	1.22917	0.614583	−	0.788852i	$-0.289324\pi$
$127$	2.83742e7	1.22917	0.614583	+	0.788852i	$0.289324\pi$
$128$	0	0
$129$	2.85396e6	0.117053
$130$	0	0
$131$	4.96447e7	1.92940	0.964702	−	0.263345i	$-0.0848258\pi$
$131$	4.96447e7	1.92940	0.964702	+	0.263345i	$0.0848258\pi$
$132$	0	0
$133$	−1.21724e7	−0.448636
$134$	0	0
$135$	6.56074e6	0.229501
$136$	0	0
$137$	−2.74212e7	−0.911097	−0.455549	−	0.890211i	$-0.650557\pi$
$137$	−2.74212e7	−0.911097	−0.455549	+	0.890211i	$0.650557\pi$
$138$	0	0
$139$	−1.65774e7	−0.523557	−0.261779	−	0.965128i	$-0.584309\pi$
$139$	−1.65774e7	−0.523557	−0.261779	+	0.965128i	$0.584309\pi$
$140$	0	0
$141$	1.04802e7	0.314849
$142$	0	0
$143$	2.50457e7	0.716238
$144$	0	0
$145$	2.18750e7	0.595881
$146$	0	0
$147$	4.15539e6	0.107895
$148$	0	0
$149$	−5.98906e7	−1.48322	−0.741612	−	0.670829i	$-0.765938\pi$
$149$	−5.98906e7	−1.48322	−0.741612	+	0.670829i	$0.765938\pi$
$150$	0	0
$151$	7.23738e7	1.71065	0.855327	−	0.518089i	$-0.173356\pi$
$151$	7.23738e7	1.71065	0.855327	+	0.518089i	$0.173356\pi$
$152$	0	0
$153$	138766.	0.00313229
$154$	0	0
$155$	7.54653e7	1.62775
$156$	0	0
$157$	4.92342e7	1.01536	0.507678	−	0.861547i	$-0.330504\pi$
$157$	4.92342e7	1.01536	0.507678	+	0.861547i	$0.330504\pi$
$158$	0	0
$159$	−1.45655e6	−0.0287367
$160$	0	0
$161$	3.98920e7	0.753347
$162$	0	0
$163$	7.71331e7	1.39503	0.697516	−	0.716569i	$-0.254288\pi$
$163$	7.71331e7	1.39503	0.697516	+	0.716569i	$0.254288\pi$
$164$	0	0
$165$	−4.02565e7	−0.697657
$166$	0	0
$167$	−3.07665e7	−0.511175	−0.255588	−	0.966786i	$-0.582269\pi$
$167$	−3.07665e7	−0.511175	−0.255588	+	0.966786i	$0.582269\pi$
$168$	0	0
$169$	−3.13979e7	−0.500376
$170$	0	0
$171$	−8.97544e6	−0.137268
$172$	0	0
$173$	−9.21239e7	−1.35273	−0.676365	−	0.736567i	$-0.736446\pi$
$173$	−9.21239e7	−1.35273	−0.676365	+	0.736567i	$0.736446\pi$
$174$	0	0
$175$	3.26035e7	0.459865
$176$	0	0
$177$	7.42096e7	1.00590
$178$	0	0
$179$	−7.85504e7	−1.02368	−0.511838	−	0.859082i	$-0.671035\pi$
$179$	−7.85504e7	−1.02368	−0.511838	+	0.859082i	$0.671035\pi$
$180$	0	0
$181$	−1.03261e8	−1.29437	−0.647187	−	0.762332i	$-0.724055\pi$
$181$	−1.03261e8	−1.29437	−0.647187	+	0.762332i	$0.724055\pi$
$182$	0	0
$183$	−3.83687e7	−0.462806
$184$	0	0
$185$	2.02912e8	2.35617
$186$	0	0
$187$	−851462.	−0.00952181
$188$	0	0
$189$	1.94598e7	0.209663
$190$	0	0
$191$	1.21640e8	1.26316	0.631581	−	0.775310i	$-0.282406\pi$
$191$	1.21640e8	1.26316	0.631581	+	0.775310i	$0.282406\pi$
$192$	0	0
$193$	4.18867e7	0.419397	0.209698	−	0.977766i	$-0.432752\pi$
$193$	4.18867e7	0.419397	0.209698	+	0.977766i	$0.432752\pi$
$194$	0	0
$195$	−5.03905e7	−0.486662
$196$	0	0
$197$	−3.58032e7	−0.333649	−0.166824	−	0.985987i	$-0.553351\pi$
$197$	−3.58032e7	−0.333649	−0.166824	+	0.985987i	$0.553351\pi$
$198$	0	0
$199$	1.16124e8	1.04457	0.522284	−	0.852772i	$-0.325080\pi$
$199$	1.16124e8	1.04457	0.522284	+	0.852772i	$0.325080\pi$
$200$	0	0
$201$	−5.06381e7	−0.439836
$202$	0	0
$203$	6.48832e7	0.544372
$204$	0	0
$205$	2.62641e7	0.212924
$206$	0	0
$207$	2.94148e7	0.230500
$208$	0	0
$209$	5.50730e7	0.417279
$210$	0	0
$211$	6.32929e7	0.463838	0.231919	−	0.972735i	$-0.425500\pi$
$211$	6.32929e7	0.463838	0.231919	+	0.972735i	$0.425500\pi$
$212$	0	0
$213$	−4.13012e7	−0.292842
$214$	0	0
$215$	3.52327e7	0.241775
$216$	0	0
$217$	2.23837e8	1.48704
$218$	0	0
$219$	1.15189e8	0.741066
$220$	0	0
$221$	−1.06581e6	−0.00664210
$222$	0	0
$223$	−8.53064e7	−0.515127	−0.257564	−	0.966261i	$-0.582920\pi$
$223$	−8.53064e7	−0.515127	−0.257564	+	0.966261i	$0.582920\pi$
$224$	0	0
$225$	2.40406e7	0.140704
$226$	0	0
$227$	9.04507e7	0.513241	0.256621	−	0.966512i	$-0.417391\pi$
$227$	9.04507e7	0.513241	0.256621	+	0.966512i	$0.417391\pi$
$228$	0	0
$229$	−2.41674e8	−1.32986	−0.664928	−	0.746907i	$-0.731538\pi$
$229$	−2.41674e8	−1.32986	−0.664928	+	0.746907i	$0.731538\pi$
$230$	0	0
$231$	−1.19404e8	−0.637351
$232$	0	0
$233$	2.85048e8	1.47629	0.738146	−	0.674641i	$-0.235702\pi$
$233$	2.85048e8	1.47629	0.738146	+	0.674641i	$0.235702\pi$
$234$	0	0
$235$	1.29380e8	0.650324
$236$	0	0
$237$	−6.14213e7	−0.299709
$238$	0	0
$239$	1.64433e8	0.779108	0.389554	−	0.921004i	$-0.372629\pi$
$239$	1.64433e8	0.779108	0.389554	+	0.921004i	$0.372629\pi$
$240$	0	0
$241$	−1.91807e8	−0.882684	−0.441342	−	0.897339i	$-0.645497\pi$
$241$	−1.91807e8	−0.882684	−0.441342	+	0.897339i	$0.645497\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	5.12991e7	0.222858
$246$	0	0
$247$	6.89369e7	0.291080
$248$	0	0
$249$	1.38567e8	0.568805
$250$	0	0
$251$	1.95745e8	0.781328	0.390664	−	0.920533i	$-0.372245\pi$
$251$	1.95745e8	0.781328	0.390664	+	0.920533i	$0.372245\pi$
$252$	0	0
$253$	−1.80488e8	−0.700692
$254$	0	0
$255$	1.71309e6	0.00646979
$256$	0	0
$257$	2.21264e7	0.0813103	0.0406552	−	0.999173i	$-0.487055\pi$
$257$	2.21264e7	0.0813103	0.0406552	+	0.999173i	$0.487055\pi$
$258$	0	0
$259$	6.01856e8	2.15250
$260$	0	0
$261$	4.78424e7	0.166560
$262$	0	0
$263$	5.31785e7	0.180257	0.0901283	−	0.995930i	$-0.471272\pi$
$263$	5.31785e7	0.180257	0.0901283	+	0.995930i	$0.471272\pi$
$264$	0	0
$265$	−1.79815e7	−0.0593560
$266$	0	0
$267$	2.71476e8	0.872856
$268$	0	0
$269$	4.24549e8	1.32983	0.664913	−	0.746921i	$-0.268469\pi$
$269$	4.24549e8	1.32983	0.664913	+	0.746921i	$0.268469\pi$
$270$	0	0
$271$	−1.90392e8	−0.581108	−0.290554	−	0.956859i	$-0.593840\pi$
$271$	−1.90392e8	−0.581108	−0.290554	+	0.956859i	$0.593840\pi$
$272$	0	0
$273$	−1.49463e8	−0.444595
$274$	0	0
$275$	−1.47512e8	−0.427723
$276$	0	0
$277$	−3.99208e8	−1.12855	−0.564274	−	0.825587i	$-0.690844\pi$
$277$	−3.99208e8	−1.12855	−0.564274	+	0.825587i	$0.690844\pi$
$278$	0	0
$279$	1.65049e8	0.454987
$280$	0	0
$281$	7.27895e7	0.195703	0.0978513	−	0.995201i	$-0.468803\pi$
$281$	7.27895e7	0.195703	0.0978513	+	0.995201i	$0.468803\pi$
$282$	0	0
$283$	−2.89021e8	−0.758012	−0.379006	−	0.925394i	$-0.623734\pi$
$283$	−2.89021e8	−0.758012	−0.379006	+	0.925394i	$0.623734\pi$
$284$	0	0
$285$	−1.10804e8	−0.283528
$286$	0	0
$287$	7.79019e7	0.194519
$288$	0	0
$289$	−4.10302e8	−0.999912
$290$	0	0
$291$	3.78864e8	0.901276
$292$	0	0
$293$	2.83123e8	0.657565	0.328783	−	0.944406i	$-0.393362\pi$
$293$	2.83123e8	0.657565	0.328783	+	0.944406i	$0.393362\pi$
$294$	0	0
$295$	9.16132e8	2.07769
$296$	0	0
$297$	−8.80444e7	−0.195009
$298$	0	0
$299$	−2.25924e8	−0.488780
$300$	0	0
$301$	1.04504e8	0.220876
$302$	0	0
$303$	6.99235e7	0.144402
$304$	0	0
$305$	−4.73670e8	−0.955930
$306$	0	0
$307$	−9.92928e8	−1.95854	−0.979272	−	0.202548i	$-0.935078\pi$
$307$	−9.92928e8	−1.95854	−0.979272	+	0.202548i	$0.935078\pi$
$308$	0	0
$309$	6.89588e7	0.132964
$310$	0	0
$311$	6.26280e8	1.18061	0.590306	−	0.807180i	$-0.299007\pi$
$311$	6.26280e8	1.18061	0.590306	+	0.807180i	$0.299007\pi$
$312$	0	0
$313$	9.79093e8	1.80476	0.902378	−	0.430945i	$-0.141820\pi$
$313$	9.79093e8	1.80476	0.902378	+	0.430945i	$0.141820\pi$
$314$	0	0
$315$	2.40235e8	0.433061
$316$	0	0
$317$	8.09806e7	0.142782	0.0713910	−	0.997448i	$-0.477256\pi$
$317$	8.09806e7	0.142782	0.0713910	+	0.997448i	$0.477256\pi$
$318$	0	0
$319$	−2.93559e8	−0.506324
$320$	0	0
$321$	−2.39221e8	−0.403674
$322$	0	0
$323$	−2.34360e6	−0.00386967
$324$	0	0
$325$	−1.84646e8	−0.298366
$326$	0	0
$327$	−4.30215e8	−0.680407
$328$	0	0
$329$	3.83753e8	0.594109
$330$	0	0
$331$	−8.06576e8	−1.22250	−0.611248	−	0.791439i	$-0.709332\pi$
$331$	−8.06576e8	−1.22250	−0.611248	+	0.791439i	$0.709332\pi$
$332$	0	0
$333$	4.43786e8	0.658596
$334$	0	0
$335$	−6.25137e8	−0.908487
$336$	0	0
$337$	−4.60177e8	−0.654969	−0.327484	−	0.944857i	$-0.606201\pi$
$337$	−4.60177e8	−0.654969	−0.327484	+	0.944857i	$0.606201\pi$
$338$	0	0
$339$	−7.77217e8	−1.08354
$340$	0	0
$341$	−1.01274e9	−1.38311
$342$	0	0
$343$	−6.62045e8	−0.885846
$344$	0	0
$345$	3.63132e8	0.476099
$346$	0	0
$347$	4.70884e8	0.605008	0.302504	−	0.953148i	$-0.402177\pi$
$347$	4.70884e8	0.605008	0.302504	+	0.953148i	$0.402177\pi$
$348$	0	0
$349$	−4.67595e8	−0.588818	−0.294409	−	0.955680i	$-0.595123\pi$
$349$	−4.67595e8	−0.588818	−0.294409	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	−1.10208e8	−0.136032
$352$	0	0
$353$	2.70784e8	0.327651	0.163826	−	0.986489i	$-0.447617\pi$
$353$	2.70784e8	0.327651	0.163826	+	0.986489i	$0.447617\pi$
$354$	0	0
$355$	−5.09871e8	−0.604869
$356$	0	0
$357$	5.08119e6	0.00591053
$358$	0	0
$359$	−9.52540e8	−1.08656	−0.543279	−	0.839552i	$-0.682817\pi$
$359$	−9.52540e8	−1.08656	−0.543279	+	0.839552i	$0.682817\pi$
$360$	0	0
$361$	−7.42287e8	−0.830417
$362$	0	0
$363$	1.40835e7	0.0154538
$364$	0	0
$365$	1.42203e9	1.53068
$366$	0	0
$367$	4.33407e8	0.457683	0.228842	−	0.973464i	$-0.426506\pi$
$367$	4.33407e8	0.457683	0.228842	+	0.973464i	$0.426506\pi$
$368$	0	0
$369$	5.74419e7	0.0595164
$370$	0	0
$371$	−5.33347e7	−0.0542252
$372$	0	0
$373$	−5.62154e8	−0.560885	−0.280443	−	0.959871i	$-0.590481\pi$
$373$	−5.62154e8	−0.560885	−0.280443	+	0.959871i	$0.590481\pi$
$374$	0	0
$375$	−4.06312e8	−0.397878
$376$	0	0
$377$	−3.67459e8	−0.353195
$378$	0	0
$379$	1.15646e8	0.109117	0.0545585	−	0.998511i	$-0.482625\pi$
$379$	1.15646e8	0.109117	0.0545585	+	0.998511i	$0.482625\pi$
$380$	0	0
$381$	7.66104e8	0.709660
$382$	0	0
$383$	−4.05029e8	−0.368375	−0.184188	−	0.982891i	$-0.558965\pi$
$383$	−4.05029e8	−0.368375	−0.184188	+	0.982891i	$0.558965\pi$
$384$	0	0
$385$	−1.47407e9	−1.31646
$386$	0	0
$387$	7.70570e7	0.0675808
$388$	0	0
$389$	−2.19744e9	−1.89275	−0.946377	−	0.323064i	$-0.895287\pi$
$389$	−2.19744e9	−1.89275	−0.946377	+	0.323064i	$0.895287\pi$
$390$	0	0
$391$	7.68058e6	0.00649793
$392$	0	0
$393$	1.34041e9	1.11394
$394$	0	0
$395$	−7.58258e8	−0.619052
$396$	0	0
$397$	1.11407e9	0.893602	0.446801	−	0.894633i	$-0.352563\pi$
$397$	1.11407e9	0.893602	0.446801	+	0.894633i	$0.352563\pi$
$398$	0	0
$399$	−3.28654e8	−0.259020
$400$	0	0
$401$	−1.42853e9	−1.10633	−0.553164	−	0.833072i	$-0.686580\pi$
$401$	−1.42853e9	−1.10633	−0.553164	+	0.833072i	$0.686580\pi$
$402$	0	0
$403$	−1.26768e9	−0.964809
$404$	0	0
$405$	1.77140e8	0.132503
$406$	0	0
$407$	−2.72306e9	−2.00206
$408$	0	0
$409$	−2.37234e9	−1.71453	−0.857264	−	0.514877i	$-0.827838\pi$
$409$	−2.37234e9	−1.71453	−0.857264	+	0.514877i	$0.827838\pi$
$410$	0	0
$411$	−7.40373e8	−0.526022
$412$	0	0
$413$	2.71733e9	1.89809
$414$	0	0
$415$	1.71064e9	1.17487
$416$	0	0
$417$	−4.47589e8	−0.302276
$418$	0	0
$419$	1.38513e9	0.919904	0.459952	−	0.887944i	$-0.347867\pi$
$419$	1.38513e9	0.919904	0.459952	+	0.887944i	$0.347867\pi$
$420$	0	0
$421$	7.39264e8	0.482850	0.241425	−	0.970419i	$-0.422385\pi$
$421$	7.39264e8	0.482850	0.241425	+	0.970419i	$0.422385\pi$
$422$	0	0
$423$	2.82965e8	0.181778
$424$	0	0
$425$	6.27729e6	0.00396653
$426$	0	0
$427$	−1.40495e9	−0.873299
$428$	0	0
$429$	6.76235e8	0.413520
$430$	0	0
$431$	2.16527e8	0.130269	0.0651345	−	0.997876i	$-0.479252\pi$
$431$	2.16527e8	0.130269	0.0651345	+	0.997876i	$0.479252\pi$
$432$	0	0
$433$	−1.82129e9	−1.07813	−0.539066	−	0.842263i	$-0.681223\pi$
$433$	−1.82129e9	−1.07813	−0.539066	+	0.842263i	$0.681223\pi$
$434$	0	0
$435$	5.90624e8	0.344032
$436$	0	0
$437$	−4.96783e8	−0.284762
$438$	0	0
$439$	−1.17781e8	−0.0664428	−0.0332214	−	0.999448i	$-0.510577\pi$
$439$	−1.17781e8	−0.0664428	−0.0332214	+	0.999448i	$0.510577\pi$
$440$	0	0
$441$	1.12196e8	0.0622932
$442$	0	0
$443$	8.01738e8	0.438146	0.219073	−	0.975708i	$-0.429697\pi$
$443$	8.01738e8	0.438146	0.219073	+	0.975708i	$0.429697\pi$
$444$	0	0
$445$	3.35143e9	1.80289
$446$	0	0
$447$	−1.61705e9	−0.856339
$448$	0	0
$449$	−2.22662e8	−0.116087	−0.0580436	−	0.998314i	$-0.518486\pi$
$449$	−2.22662e8	−0.116087	−0.0580436	+	0.998314i	$0.518486\pi$
$450$	0	0
$451$	−3.52462e8	−0.180923
$452$	0	0
$453$	1.95409e9	0.987646
$454$	0	0
$455$	−1.84515e9	−0.918315
$456$	0	0
$457$	1.44172e9	0.706603	0.353301	−	0.935510i	$-0.385059\pi$
$457$	1.44172e9	0.706603	0.353301	+	0.935510i	$0.385059\pi$
$458$	0	0
$459$	3.74668e6	0.00180843
$460$	0	0
$461$	1.98623e9	0.944228	0.472114	−	0.881537i	$-0.343491\pi$
$461$	1.98623e9	0.944228	0.472114	+	0.881537i	$0.343491\pi$
$462$	0	0
$463$	−5.07392e7	−0.0237580	−0.0118790	−	0.999929i	$-0.503781\pi$
$463$	−5.07392e7	−0.0237580	−0.0118790	+	0.999929i	$0.503781\pi$
$464$	0	0
$465$	2.03756e9	0.939779
$466$	0	0
$467$	1.73869e8	0.0789976	0.0394988	−	0.999220i	$-0.487424\pi$
$467$	1.73869e8	0.0789976	0.0394988	+	0.999220i	$0.487424\pi$
$468$	0	0
$469$	−1.85421e9	−0.829956
$470$	0	0
$471$	1.32932e9	0.586216
$472$	0	0
$473$	−4.72819e8	−0.205438
$474$	0	0
$475$	−4.06018e8	−0.173827
$476$	0	0
$477$	−3.93270e7	−0.0165911
$478$	0	0
$479$	−4.39851e9	−1.82865	−0.914326	−	0.404980i	$-0.867279\pi$
$479$	−4.39851e9	−1.82865	−0.914326	+	0.404980i	$0.867279\pi$
$480$	0	0
$481$	−3.40855e9	−1.39657
$482$	0	0
$483$	1.07708e9	0.434945
$484$	0	0
$485$	4.67715e9	1.86159
$486$	0	0
$487$	2.47748e9	0.971985	0.485993	−	0.873963i	$-0.338458\pi$
$487$	2.47748e9	0.971985	0.485993	+	0.873963i	$0.338458\pi$
$488$	0	0
$489$	2.08259e9	0.805423
$490$	0	0
$491$	−2.94868e9	−1.12420	−0.562099	−	0.827070i	$-0.690006\pi$
$491$	−2.94868e9	−1.12420	−0.562099	+	0.827070i	$0.690006\pi$
$492$	0	0
$493$	1.24923e7	0.00469544
$494$	0	0
$495$	−1.08693e9	−0.402793
$496$	0	0
$497$	−1.51232e9	−0.552583
$498$	0	0
$499$	−2.77793e9	−1.00085	−0.500425	−	0.865780i	$-0.666823\pi$
$499$	−2.77793e9	−1.00085	−0.500425	+	0.865780i	$0.666823\pi$
$500$	0	0
$501$	−8.30694e8	−0.295127
$502$	0	0
$503$	3.53359e7	0.0123802	0.00619011	−	0.999981i	$-0.498030\pi$
$503$	3.53359e7	0.0123802	0.00619011	+	0.999981i	$0.498030\pi$
$504$	0	0
$505$	8.63220e8	0.298265
$506$	0	0
$507$	−8.47742e8	−0.288892
$508$	0	0
$509$	2.25851e8	0.0759118	0.0379559	−	0.999279i	$-0.487915\pi$
$509$	2.25851e8	0.0759118	0.0379559	+	0.999279i	$0.487915\pi$
$510$	0	0
$511$	4.21788e9	1.39837
$512$	0	0
$513$	−2.42337e8	−0.0792517
$514$	0	0
$515$	8.51310e8	0.274639
$516$	0	0
$517$	−1.73626e9	−0.552585
$518$	0	0
$519$	−2.48735e9	−0.780999
$520$	0	0
$521$	5.35395e9	1.65860	0.829302	−	0.558801i	$-0.188738\pi$
$521$	5.35395e9	1.65860	0.829302	+	0.558801i	$0.188738\pi$
$522$	0	0
$523$	1.52388e8	0.0465796	0.0232898	−	0.999729i	$-0.492586\pi$
$523$	1.52388e8	0.0465796	0.0232898	+	0.999729i	$0.492586\pi$
$524$	0	0
$525$	8.80293e8	0.265503
$526$	0	0
$527$	4.30964e7	0.0128264
$528$	0	0
$529$	−1.77674e9	−0.521829
$530$	0	0
$531$	2.00366e9	0.580755
$532$	0	0
$533$	−4.41189e8	−0.126206
$534$	0	0
$535$	−2.95323e9	−0.833794
$536$	0	0
$537$	−2.12086e9	−0.591020
$538$	0	0
$539$	−6.88428e8	−0.189364
$540$	0	0
$541$	−6.17832e9	−1.67757	−0.838784	−	0.544465i	$-0.816733\pi$
$541$	−6.17832e9	−1.67757	−0.838784	+	0.544465i	$0.816733\pi$
$542$	0	0
$543$	−2.78804e9	−0.747307
$544$	0	0
$545$	−5.31109e9	−1.40539
$546$	0	0
$547$	−7.45053e9	−1.94640	−0.973198	−	0.229968i	$-0.926138\pi$
$547$	−7.45053e9	−1.94640	−0.973198	+	0.229968i	$0.926138\pi$
$548$	0	0
$549$	−1.03596e9	−0.267201
$550$	0	0
$551$	−8.08005e8	−0.205771
$552$	0	0
$553$	−2.24906e9	−0.565541
$554$	0	0
$555$	5.47863e9	1.36034
$556$	0	0
$557$	−4.12387e9	−1.01114	−0.505570	−	0.862785i	$-0.668718\pi$
$557$	−4.12387e9	−1.01114	−0.505570	+	0.862785i	$0.668718\pi$
$558$	0	0
$559$	−5.91845e8	−0.143307
$560$	0	0
$561$	−2.29895e7	−0.00549742
$562$	0	0
$563$	6.22608e9	1.47040	0.735199	−	0.677851i	$-0.237088\pi$
$563$	6.22608e9	1.47040	0.735199	+	0.677851i	$0.237088\pi$
$564$	0	0
$565$	−9.59490e9	−2.23805
$566$	0	0
$567$	5.25414e8	0.121049
$568$	0	0
$569$	3.66211e9	0.833372	0.416686	−	0.909050i	$-0.363191\pi$
$569$	3.66211e9	0.833372	0.416686	+	0.909050i	$0.363191\pi$
$570$	0	0
$571$	−7.17442e9	−1.61273	−0.806363	−	0.591421i	$-0.798567\pi$
$571$	−7.17442e9	−1.61273	−0.806363	+	0.591421i	$0.798567\pi$
$572$	0	0
$573$	3.28428e9	0.729287
$574$	0	0
$575$	1.33063e9	0.291890
$576$	0	0
$577$	−3.58603e9	−0.777138	−0.388569	−	0.921420i	$-0.627031\pi$
$577$	−3.58603e9	−0.777138	−0.388569	+	0.921420i	$0.627031\pi$
$578$	0	0
$579$	1.13094e9	0.242139
$580$	0	0
$581$	5.07392e9	1.07332
$582$	0	0
$583$	2.41309e8	0.0504352
$584$	0	0
$585$	−1.36054e9	−0.280975
$586$	0	0
$587$	−3.64466e9	−0.743744	−0.371872	−	0.928284i	$-0.621284\pi$
$587$	−3.64466e9	−0.743744	−0.371872	+	0.928284i	$0.621284\pi$
$588$	0	0
$589$	−2.78749e9	−0.562096
$590$	0	0
$591$	−9.66686e8	−0.192632
$592$	0	0
$593$	9.35245e9	1.84176	0.920882	−	0.389841i	$-0.127470\pi$
$593$	9.35245e9	1.84176	0.920882	+	0.389841i	$0.127470\pi$
$594$	0	0
$595$	6.27283e7	0.0122083
$596$	0	0
$597$	3.13535e9	0.603081
$598$	0	0
$599$	4.87909e9	0.927566	0.463783	−	0.885949i	$-0.346492\pi$
$599$	4.87909e9	0.927566	0.463783	+	0.885949i	$0.346492\pi$
$600$	0	0
$601$	2.85323e9	0.536138	0.268069	−	0.963400i	$-0.413614\pi$
$601$	2.85323e9	0.536138	0.268069	+	0.963400i	$0.413614\pi$
$602$	0	0
$603$	−1.36723e9	−0.253940
$604$	0	0
$605$	1.73863e8	0.0319200
$606$	0	0
$607$	−7.08828e8	−0.128641	−0.0643206	−	0.997929i	$-0.520488\pi$
$607$	−7.08828e8	−0.128641	−0.0643206	+	0.997929i	$0.520488\pi$
$608$	0	0
$609$	1.75185e9	0.314294
$610$	0	0
$611$	−2.17335e9	−0.385465
$612$	0	0
$613$	−4.15658e9	−0.728827	−0.364413	−	0.931237i	$-0.618730\pi$
$613$	−4.15658e9	−0.728827	−0.364413	+	0.931237i	$0.618730\pi$
$614$	0	0
$615$	7.09132e8	0.122932
$616$	0	0
$617$	6.45359e9	1.10612	0.553061	−	0.833141i	$-0.313460\pi$
$617$	6.45359e9	1.10612	0.553061	+	0.833141i	$0.313460\pi$
$618$	0	0
$619$	9.63841e9	1.63338	0.816692	−	0.577074i	$-0.195806\pi$
$619$	9.63841e9	1.63338	0.816692	+	0.577074i	$0.195806\pi$
$620$	0	0
$621$	7.94201e8	0.133079
$622$	0	0
$623$	9.94065e9	1.64705
$624$	0	0
$625$	−7.59237e9	−1.24393
$626$	0	0
$627$	1.48697e9	0.240916
$628$	0	0
$629$	1.15878e8	0.0185662
$630$	0	0
$631$	4.63824e9	0.734937	0.367468	−	0.930036i	$-0.380225\pi$
$631$	4.63824e9	0.734937	0.367468	+	0.930036i	$0.380225\pi$
$632$	0	0
$633$	1.70891e9	0.267797
$634$	0	0
$635$	9.45770e9	1.46581
$636$	0	0
$637$	−8.61731e8	−0.132094
$638$	0	0
$639$	−1.11513e9	−0.169073
$640$	0	0
$641$	5.64527e9	0.846607	0.423304	−	0.905988i	$-0.360870\pi$
$641$	5.64527e9	0.846607	0.423304	+	0.905988i	$0.360870\pi$
$642$	0	0
$643$	−4.82868e9	−0.716292	−0.358146	−	0.933666i	$-0.616591\pi$
$643$	−4.82868e9	−0.716292	−0.358146	+	0.933666i	$0.616591\pi$
$644$	0	0
$645$	9.51284e8	0.139589
$646$	0	0
$647$	2.29901e9	0.333715	0.166858	−	0.985981i	$-0.446638\pi$
$647$	2.29901e9	0.333715	0.166858	+	0.985981i	$0.446638\pi$
$648$	0	0
$649$	−1.22944e10	−1.76543
$650$	0	0
$651$	6.04360e9	0.858544
$652$	0	0
$653$	1.17082e10	1.64548	0.822742	−	0.568416i	$-0.192443\pi$
$653$	1.17082e10	1.64548	0.822742	+	0.568416i	$0.192443\pi$
$654$	0	0
$655$	1.65476e10	2.30086
$656$	0	0
$657$	3.11010e9	0.427855
$658$	0	0
$659$	6.96735e9	0.948350	0.474175	−	0.880431i	$-0.342746\pi$
$659$	6.96735e9	0.948350	0.474175	+	0.880431i	$0.342746\pi$
$660$	0	0
$661$	−2.92769e9	−0.394294	−0.197147	−	0.980374i	$-0.563168\pi$
$661$	−2.92769e9	−0.394294	−0.197147	+	0.980374i	$0.563168\pi$
$662$	0	0
$663$	−2.87768e7	−0.00383482
$664$	0	0
$665$	−4.05729e9	−0.535008
$666$	0	0
$667$	2.64804e9	0.345529
$668$	0	0
$669$	−2.30327e9	−0.297409
$670$	0	0
$671$	6.35659e9	0.812260
$672$	0	0
$673$	1.40230e10	1.77332	0.886662	−	0.462417i	$-0.153018\pi$
$673$	1.40230e10	1.77332	0.886662	+	0.462417i	$0.153018\pi$
$674$	0	0
$675$	6.49095e8	0.0812354
$676$	0	0
$677$	2.19699e8	0.0272124	0.0136062	−	0.999907i	$-0.495669\pi$
$677$	2.19699e8	0.0272124	0.0136062	+	0.999907i	$0.495669\pi$
$678$	0	0
$679$	1.38729e10	1.70068
$680$	0	0
$681$	2.44217e9	0.296320
$682$	0	0
$683$	2.71753e8	0.0326364	0.0163182	−	0.999867i	$-0.494806\pi$
$683$	2.71753e8	0.0326364	0.0163182	+	0.999867i	$0.494806\pi$
$684$	0	0
$685$	−9.14005e9	−1.08650
$686$	0	0
$687$	−6.52519e9	−0.767793
$688$	0	0
$689$	3.02055e8	0.0351819
$690$	0	0
$691$	−5.99526e9	−0.691250	−0.345625	−	0.938373i	$-0.612333\pi$
$691$	−5.99526e9	−0.691250	−0.345625	+	0.938373i	$0.612333\pi$
$692$	0	0
$693$	−3.22392e9	−0.367975
$694$	0	0
$695$	−5.52558e9	−0.624354
$696$	0	0
$697$	1.49988e7	0.00167781
$698$	0	0
$699$	7.69629e9	0.852337
$700$	0	0
$701$	−1.22308e10	−1.34104	−0.670520	−	0.741891i	$-0.733929\pi$
$701$	−1.22308e10	−1.34104	−0.670520	+	0.741891i	$0.733929\pi$
$702$	0	0
$703$	−7.49505e9	−0.813637
$704$	0	0
$705$	3.49326e9	0.375465
$706$	0	0
$707$	2.56039e9	0.272482
$708$	0	0
$709$	−3.58261e9	−0.377518	−0.188759	−	0.982023i	$-0.560447\pi$
$709$	−3.58261e9	−0.377518	−0.188759	+	0.982023i	$0.560447\pi$
$710$	0	0
$711$	−1.65838e9	−0.173037
$712$	0	0
$713$	9.13534e9	0.943868
$714$	0	0
$715$	8.34825e9	0.854131
$716$	0	0
$717$	4.43970e9	0.449818
$718$	0	0
$719$	−1.18977e10	−1.19375	−0.596874	−	0.802335i	$-0.703591\pi$
$719$	−1.18977e10	−1.19375	−0.596874	+	0.802335i	$0.703591\pi$
$720$	0	0
$721$	2.52506e9	0.250899
$722$	0	0
$723$	−5.17880e9	−0.509618
$724$	0	0
$725$	2.16423e9	0.210921
$726$	0	0
$727$	1.49319e10	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$727$	1.49319e10	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.01205e7	0.00190515
$732$	0	0
$733$	−1.37369e10	−1.28832	−0.644161	−	0.764890i	$-0.722793\pi$
$733$	−1.37369e10	−1.28832	−0.644161	+	0.764890i	$0.722793\pi$
$734$	0	0
$735$	1.38508e9	0.128667
$736$	0	0
$737$	8.38926e9	0.771947
$738$	0	0
$739$	6.78972e9	0.618865	0.309433	−	0.950921i	$-0.399861\pi$
$739$	6.78972e9	0.618865	0.309433	+	0.950921i	$0.399861\pi$
$740$	0	0
$741$	1.86130e9	0.168055
$742$	0	0
$743$	−4.45145e9	−0.398144	−0.199072	−	0.979985i	$-0.563793\pi$
$743$	−4.45145e9	−0.398144	−0.199072	+	0.979985i	$0.563793\pi$
$744$	0	0
$745$	−1.99627e10	−1.76878
$746$	0	0
$747$	3.74132e9	0.328400
$748$	0	0
$749$	−8.75955e9	−0.761720
$750$	0	0
$751$	−1.03314e9	−0.0890058	−0.0445029	−	0.999009i	$-0.514170\pi$
$751$	−1.03314e9	−0.0890058	−0.0445029	+	0.999009i	$0.514170\pi$
$752$	0	0
$753$	5.28512e9	0.451100
$754$	0	0
$755$	2.41237e10	2.03999
$756$	0	0
$757$	1.56909e9	0.131466	0.0657328	−	0.997837i	$-0.479061\pi$
$757$	1.56909e9	0.131466	0.0657328	+	0.997837i	$0.479061\pi$
$758$	0	0
$759$	−4.87319e9	−0.404545
$760$	0	0
$761$	−1.35877e9	−0.111764	−0.0558818	−	0.998437i	$-0.517797\pi$
$761$	−1.35877e9	−0.111764	−0.0558818	+	0.998437i	$0.517797\pi$
$762$	0	0
$763$	−1.57532e10	−1.28390
$764$	0	0
$765$	4.62535e7	0.00373533
$766$	0	0
$767$	−1.53893e10	−1.23150
$768$	0	0
$769$	2.01788e10	1.60012	0.800061	−	0.599918i	$-0.204800\pi$
$769$	2.01788e10	1.60012	0.800061	+	0.599918i	$0.204800\pi$
$770$	0	0
$771$	5.97414e8	0.0469445
$772$	0	0
$773$	−4.53602e9	−0.353221	−0.176610	−	0.984281i	$-0.556513\pi$
$773$	−4.53602e9	−0.353221	−0.176610	+	0.984281i	$0.556513\pi$
$774$	0	0
$775$	7.46626e9	0.576165
$776$	0	0
$777$	1.62501e10	1.24275
$778$	0	0
$779$	−9.70129e8	−0.0735273
$780$	0	0
$781$	6.84241e9	0.513961
$782$	0	0
$783$	1.29175e9	0.0961636
$784$	0	0
$785$	1.64108e10	1.21084
$786$	0	0
$787$	1.78287e10	1.30379	0.651896	−	0.758308i	$-0.273974\pi$
$787$	1.78287e10	1.30379	0.651896	+	0.758308i	$0.273974\pi$
$788$	0	0
$789$	1.43582e9	0.104071
$790$	0	0
$791$	−2.84593e10	−2.04459
$792$	0	0
$793$	7.95678e9	0.566606
$794$	0	0
$795$	−4.85499e8	−0.0342692
$796$	0	0
$797$	4.20827e9	0.294442	0.147221	−	0.989104i	$-0.452967\pi$
$797$	4.20827e9	0.294442	0.147221	+	0.989104i	$0.452967\pi$
$798$	0	0
$799$	7.38857e7	0.00512445
$800$	0	0
$801$	7.32986e9	0.503944
$802$	0	0
$803$	−1.90835e10	−1.30063
$804$	0	0
$805$	1.32968e10	0.898383
$806$	0	0
$807$	1.14628e10	0.767775
$808$	0	0
$809$	−2.62494e10	−1.74301	−0.871505	−	0.490387i	$-0.836855\pi$
$809$	−2.62494e10	−1.74301	−0.871505	+	0.490387i	$0.836855\pi$
$810$	0	0
$811$	−1.23174e10	−0.810864	−0.405432	−	0.914125i	$-0.632879\pi$
$811$	−1.23174e10	−0.810864	−0.405432	+	0.914125i	$0.632879\pi$
$812$	0	0
$813$	−5.14059e9	−0.335503
$814$	0	0
$815$	2.57100e10	1.66361
$816$	0	0
$817$	−1.30141e9	−0.0834902
$818$	0	0
$819$	−4.03550e9	−0.256687
$820$	0	0
$821$	−1.47049e10	−0.927386	−0.463693	−	0.885996i	$-0.653476\pi$
$821$	−1.47049e10	−0.927386	−0.463693	+	0.885996i	$0.653476\pi$
$822$	0	0
$823$	2.76241e10	1.72738	0.863691	−	0.504021i	$-0.168147\pi$
$823$	2.76241e10	1.72738	0.863691	+	0.504021i	$0.168147\pi$
$824$	0	0
$825$	−3.98282e9	−0.246946
$826$	0	0
$827$	7.85420e9	0.482873	0.241436	−	0.970417i	$-0.422381\pi$
$827$	7.85420e9	0.482873	0.241436	+	0.970417i	$0.422381\pi$
$828$	0	0
$829$	−2.34405e10	−1.42898	−0.714489	−	0.699647i	$-0.753341\pi$
$829$	−2.34405e10	−1.42898	−0.714489	+	0.699647i	$0.753341\pi$
$830$	0	0
$831$	−1.07786e10	−0.651568
$832$	0	0
$833$	2.92957e7	0.00175609
$834$	0	0
$835$	−1.02551e10	−0.609588
$836$	0	0
$837$	4.45633e9	0.262687
$838$	0	0
$839$	−1.79573e10	−1.04972	−0.524862	−	0.851187i	$-0.675883\pi$
$839$	−1.79573e10	−1.04972	−0.524862	+	0.851187i	$0.675883\pi$
$840$	0	0
$841$	−1.29429e10	−0.750319
$842$	0	0
$843$	1.96532e9	0.112989
$844$	0	0
$845$	−1.04655e10	−0.596710
$846$	0	0
$847$	5.15694e8	0.0291608
$848$	0	0
$849$	−7.80355e9	−0.437639
$850$	0	0
$851$	2.45632e10	1.36625
$852$	0	0
$853$	3.22147e10	1.77719	0.888593	−	0.458696i	$-0.151683\pi$
$853$	3.22147e10	1.77719	0.888593	+	0.458696i	$0.151683\pi$
$854$	0	0
$855$	−2.99170e9	−0.163695
$856$	0	0
$857$	4.32148e9	0.234531	0.117265	−	0.993101i	$-0.462587\pi$
$857$	4.32148e9	0.234531	0.117265	+	0.993101i	$0.462587\pi$
$858$	0	0
$859$	−1.65926e10	−0.893179	−0.446590	−	0.894739i	$-0.647362\pi$
$859$	−1.65926e10	−0.893179	−0.446590	+	0.894739i	$0.647362\pi$
$860$	0	0
$861$	2.10335e9	0.112305
$862$	0	0
$863$	3.34157e10	1.76975	0.884876	−	0.465827i	$-0.154243\pi$
$863$	3.34157e10	1.76975	0.884876	+	0.465827i	$0.154243\pi$
$864$	0	0
$865$	−3.07068e10	−1.61316
$866$	0	0
$867$	−1.10782e10	−0.577299
$868$	0	0
$869$	1.01757e10	0.526013
$870$	0	0
$871$	1.05012e10	0.538485
$872$	0	0
$873$	1.02293e10	0.520352
$874$	0	0
$875$	−1.48779e10	−0.750783
$876$	0	0
$877$	−2.35786e10	−1.18038	−0.590188	−	0.807266i	$-0.700946\pi$
$877$	−2.35786e10	−1.18038	−0.590188	+	0.807266i	$0.700946\pi$
$878$	0	0
$879$	7.64433e9	0.379646
$880$	0	0
$881$	1.32265e10	0.651673	0.325836	−	0.945426i	$-0.394354\pi$
$881$	1.32265e10	0.651673	0.325836	+	0.945426i	$0.394354\pi$
$882$	0	0
$883$	−1.61758e10	−0.790685	−0.395343	−	0.918534i	$-0.629374\pi$
$883$	−1.61758e10	−0.790685	−0.395343	+	0.918534i	$0.629374\pi$
$884$	0	0
$885$	2.47356e10	1.19956
$886$	0	0
$887$	−5.57010e9	−0.267997	−0.133999	−	0.990982i	$-0.542782\pi$
$887$	−5.57010e9	−0.267997	−0.133999	+	0.990982i	$0.542782\pi$
$888$	0	0
$889$	2.80524e10	1.33910
$890$	0	0
$891$	−2.37720e9	−0.112588
$892$	0	0
$893$	−4.77896e9	−0.224571
$894$	0	0
$895$	−2.61824e10	−1.22076
$896$	0	0
$897$	−6.09995e9	−0.282197
$898$	0	0
$899$	1.48584e10	0.682044
$900$	0	0
$901$	−1.02688e7	−0.000467715 0
$902$	0	0
$903$	2.82159e9	0.127523
$904$	0	0
$905$	−3.44188e10	−1.54357
$906$	0	0
$907$	−9.37708e9	−0.417294	−0.208647	−	0.977991i	$-0.566906\pi$
$907$	−9.37708e9	−0.417294	−0.208647	+	0.977991i	$0.566906\pi$
$908$	0	0
$909$	1.88794e9	0.0833707
$910$	0	0
$911$	2.79943e10	1.22675	0.613374	−	0.789793i	$-0.289812\pi$
$911$	2.79943e10	1.22675	0.613374	+	0.789793i	$0.289812\pi$
$912$	0	0
$913$	−2.29566e10	−0.998297
$914$	0	0
$915$	−1.27891e10	−0.551907
$916$	0	0
$917$	4.90816e10	2.10197
$918$	0	0
$919$	−3.88071e10	−1.64933	−0.824665	−	0.565622i	$-0.808636\pi$
$919$	−3.88071e10	−1.64933	−0.824665	+	0.565622i	$0.808636\pi$
$920$	0	0
$921$	−2.68091e10	−1.13077
$922$	0	0
$923$	8.56489e9	0.358522
$924$	0	0
$925$	2.00754e10	0.834002
$926$	0	0
$927$	1.86189e9	0.0767670
$928$	0	0
$929$	−7.98348e9	−0.326691	−0.163346	−	0.986569i	$-0.552229\pi$
$929$	−7.98348e9	−0.326691	−0.163346	+	0.986569i	$0.552229\pi$
$930$	0	0
$931$	−1.89486e9	−0.0769578
$932$	0	0
$933$	1.69096e10	0.681627
$934$	0	0
$935$	−2.83810e8	−0.0113550
$936$	0	0
$937$	−2.97710e10	−1.18224	−0.591119	−	0.806584i	$-0.701314\pi$
$937$	−2.97710e10	−1.18224	−0.591119	+	0.806584i	$0.701314\pi$
$938$	0	0
$939$	2.64355e10	1.04198
$940$	0	0
$941$	−5.39177e9	−0.210944	−0.105472	−	0.994422i	$-0.533635\pi$
$941$	−5.39177e9	−0.210944	−0.105472	+	0.994422i	$0.533635\pi$
$942$	0	0
$943$	3.17936e9	0.123467
$944$	0	0
$945$	6.48634e9	0.250028
$946$	0	0
$947$	−1.59034e10	−0.608507	−0.304254	−	0.952591i	$-0.598407\pi$
$947$	−1.59034e10	−0.608507	−0.304254	+	0.952591i	$0.598407\pi$
$948$	0	0
$949$	−2.38875e10	−0.907275
$950$	0	0
$951$	2.18648e9	0.0824352
$952$	0	0
$953$	−3.42029e10	−1.28008	−0.640041	−	0.768341i	$-0.721082\pi$
$953$	−3.42029e10	−1.28008	−0.640041	+	0.768341i	$0.721082\pi$
$954$	0	0
$955$	4.05450e10	1.50635
$956$	0	0
$957$	−7.92611e9	−0.292326
$958$	0	0
$959$	−2.71102e10	−0.992586
$960$	0	0
$961$	2.37465e10	0.863115
$962$	0	0
$963$	−6.45896e9	−0.233062
$964$	0	0
$965$	1.39617e10	0.500141
$966$	0	0
$967$	−1.86938e10	−0.664821	−0.332411	−	0.943135i	$-0.607862\pi$
$967$	−1.86938e10	−0.664821	−0.332411	+	0.943135i	$0.607862\pi$
$968$	0	0
$969$	−6.32772e7	−0.00223416
$970$	0	0
$971$	−5.04205e10	−1.76742	−0.883711	−	0.468033i	$-0.844963\pi$
$971$	−5.04205e10	−1.76742	−0.883711	+	0.468033i	$0.844963\pi$
$972$	0	0
$973$	−1.63894e10	−0.570384
$974$	0	0
$975$	−4.98545e9	−0.172261
$976$	0	0
$977$	−1.64060e10	−0.562822	−0.281411	−	0.959587i	$-0.590802\pi$
$977$	−1.64060e10	−0.562822	−0.281411	+	0.959587i	$0.590802\pi$
$978$	0	0
$979$	−4.49758e10	−1.53193
$980$	0	0
$981$	−1.16158e10	−0.392833
$982$	0	0
$983$	−5.03584e10	−1.69096	−0.845482	−	0.534004i	$-0.820687\pi$
$983$	−5.03584e10	−1.69096	−0.845482	+	0.534004i	$0.820687\pi$
$984$	0	0
$985$	−1.19339e10	−0.397884
$986$	0	0
$987$	1.03613e10	0.343009
$988$	0	0
$989$	4.26504e9	0.140196
$990$	0	0
$991$	−3.66269e10	−1.19548	−0.597739	−	0.801691i	$-0.703934\pi$
$991$	−3.66269e10	−1.19548	−0.597739	+	0.801691i	$0.703934\pi$
$992$	0	0
$993$	−2.17775e10	−0.705808
$994$	0	0
$995$	3.87065e10	1.24567
$996$	0	0
$997$	3.37142e10	1.07741	0.538704	−	0.842495i	$-0.318914\pi$
$997$	3.37142e10	1.07741	0.538704	+	0.842495i	$0.318914\pi$
$998$	0	0
$999$	1.19822e10	0.380240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.q.1.4	yes	4
4.3	odd	2		384.8.a.m.1.4	✓	4
8.3	odd	2		384.8.a.t.1.1	yes	4
8.5	even	2		384.8.a.p.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.m.1.4	✓	4	4.3	odd	2
384.8.a.p.1.1	yes	4	8.5	even	2
384.8.a.q.1.4	yes	4	1.1	even	1	trivial
384.8.a.t.1.1	yes	4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} +333.320 q^{5} +988.659 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} +333.320 q^{5} +988.659 q^{7} +729.000 q^{9} -4473.12 q^{11} -5599.17 q^{13} +8999.65 q^{15} +190.351 q^{17} -12312.0 q^{19} +26693.8 q^{21} +40349.6 q^{23} +32977.5 q^{25} +19683.0 q^{27} +65627.5 q^{29} +226405. q^{31} -120774. q^{33} +329540. q^{35} +608760. q^{37} -151177. q^{39} +78795.5 q^{41} +105702. q^{43} +242991. q^{45} +388155. q^{47} +153903. q^{49} +5139.48 q^{51} -53946.5 q^{53} -1.49098e6 q^{55} -332424. q^{57} +2.74850e6 q^{59} -1.42106e6 q^{61} +720732. q^{63} -1.86632e6 q^{65} -1.87548e6 q^{67} +1.08944e6 q^{69} -1.52967e6 q^{71} +4.26626e6 q^{73} +890391. q^{75} -4.42239e6 q^{77} -2.27486e6 q^{79} +531441. q^{81} +5.13213e6 q^{83} +63447.8 q^{85} +1.77194e6 q^{87} +1.00547e7 q^{89} -5.53567e6 q^{91} +6.11293e6 q^{93} -4.10384e6 q^{95} +1.40320e7 q^{97} -3.26090e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 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

Properties

Related objects

Downloads

Learn more