LMFDB - Embedded newform 384.8.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$119.955849786$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 286x - 1680 $ x^3 - 286*x - 1680
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19.3129$ 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)

$f(q)$	$=$	$q+27.0000 q^{3} -368.444 q^{5} -1058.71 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -368.444 q^{5} -1058.71 q^{7} +729.000 q^{9} +4728.97 q^{11} -5522.34 q^{13} -9947.99 q^{15} +28487.5 q^{17} +12644.8 q^{19} -28585.1 q^{21} +16997.8 q^{23} +57626.0 q^{25} +19683.0 q^{27} +75684.6 q^{29} -69384.1 q^{31} +127682. q^{33} +390074. q^{35} +29223.4 q^{37} -149103. q^{39} +295928. q^{41} +16387.0 q^{43} -268596. q^{45} -68148.4 q^{47} +297318. q^{49} +769163. q^{51} +1.85505e6 q^{53} -1.74236e6 q^{55} +341409. q^{57} -2.75215e6 q^{59} -1.77198e6 q^{61} -771798. q^{63} +2.03467e6 q^{65} -2.76825e6 q^{67} +458941. q^{69} -4.66235e6 q^{71} -6.38405e6 q^{73} +1.55590e6 q^{75} -5.00659e6 q^{77} +7.32391e6 q^{79} +531441. q^{81} -3.05313e6 q^{83} -1.04961e7 q^{85} +2.04349e6 q^{87} -8.79581e6 q^{89} +5.84654e6 q^{91} -1.87337e6 q^{93} -4.65889e6 q^{95} +6.84797e6 q^{97} +3.44742e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 81 q^{3} - 308 q^{5} - 6 q^{7} + 2187 q^{9}+O(q^{10}) $ 3 * q + 81 * q^3 - 308 * q^5 - 6 * q^7 + 2187 * q^9	$ 3 q + 81 q^{3} - 308 q^{5} - 6 q^{7} + 2187 q^{9} + 1468 q^{11} - 3070 q^{13} - 8316 q^{15} - 5126 q^{17} + 14168 q^{19} - 162 q^{21} - 59980 q^{23} + 31529 q^{25} + 59049 q^{27} - 71984 q^{29} - 177470 q^{31} + 39636 q^{33} - 84760 q^{35} + 184014 q^{37} - 82890 q^{39} + 657250 q^{41} - 120544 q^{43} - 224532 q^{45} - 665132 q^{47} + 1204647 q^{49} - 138402 q^{51} + 111888 q^{53} - 1854224 q^{55} + 382536 q^{57} - 1033308 q^{59} - 3584306 q^{61} - 4374 q^{63} + 1230472 q^{65} - 2344900 q^{67} - 1619460 q^{69} - 6311524 q^{71} - 2112174 q^{73} + 851283 q^{75} - 6670456 q^{77} - 2018830 q^{79} + 1594323 q^{81} - 7430484 q^{83} - 9821848 q^{85} - 1943568 q^{87} - 13300410 q^{89} + 10605052 q^{91} - 4791690 q^{93} - 303008 q^{95} + 2960414 q^{97} + 1070172 q^{99}+O(q^{100}) $ 3 * q + 81 * q^3 - 308 * q^5 - 6 * q^7 + 2187 * q^9 + 1468 * q^11 - 3070 * q^13 - 8316 * q^15 - 5126 * q^17 + 14168 * q^19 - 162 * q^21 - 59980 * q^23 + 31529 * q^25 + 59049 * q^27 - 71984 * q^29 - 177470 * q^31 + 39636 * q^33 - 84760 * q^35 + 184014 * q^37 - 82890 * q^39 + 657250 * q^41 - 120544 * q^43 - 224532 * q^45 - 665132 * q^47 + 1204647 * q^49 - 138402 * q^51 + 111888 * q^53 - 1854224 * q^55 + 382536 * q^57 - 1033308 * q^59 - 3584306 * q^61 - 4374 * q^63 + 1230472 * q^65 - 2344900 * q^67 - 1619460 * q^69 - 6311524 * q^71 - 2112174 * q^73 + 851283 * q^75 - 6670456 * q^77 - 2018830 * q^79 + 1594323 * q^81 - 7430484 * q^83 - 9821848 * q^85 - 1943568 * q^87 - 13300410 * q^89 + 10605052 * q^91 - 4791690 * q^93 - 303008 * q^95 + 2960414 * q^97 + 1070172 * 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
$5$	−368.444	−1.31819	−0.659093	−	0.752062i	$-0.729059\pi$
$5$	−368.444	−1.31819	−0.659093	+	0.752062i	$0.729059\pi$
$6$	0	0
$7$	−1058.71	−1.16663	−0.583314	−	0.812246i	$-0.698244\pi$
$7$	−1058.71	−1.16663	−0.583314	+	0.812246i	$0.698244\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	4728.97	1.07125	0.535626	−	0.844455i	$-0.320076\pi$
$11$	4728.97	1.07125	0.535626	+	0.844455i	$0.320076\pi$
$12$	0	0
$13$	−5522.34	−0.697142	−0.348571	−	0.937282i	$-0.613333\pi$
$13$	−5522.34	−0.697142	−0.348571	+	0.937282i	$0.613333\pi$
$14$	0	0
$15$	−9947.99	−0.761055
$16$	0	0
$17$	28487.5	1.40632	0.703159	−	0.711033i	$-0.251772\pi$
$17$	28487.5	1.40632	0.703159	+	0.711033i	$0.251772\pi$
$18$	0	0
$19$	12644.8	0.422935	0.211467	−	0.977385i	$-0.432176\pi$
$19$	12644.8	0.422935	0.211467	+	0.977385i	$0.432176\pi$
$20$	0	0
$21$	−28585.1	−0.673554
$22$	0	0
$23$	16997.8	0.291304	0.145652	−	0.989336i	$-0.453472\pi$
$23$	16997.8	0.291304	0.145652	+	0.989336i	$0.453472\pi$
$24$	0	0
$25$	57626.0	0.737612
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	75684.6	0.576255	0.288128	−	0.957592i	$-0.406967\pi$
$29$	75684.6	0.576255	0.288128	+	0.957592i	$0.406967\pi$
$30$	0	0
$31$	−69384.1	−0.418306	−0.209153	−	0.977883i	$-0.567071\pi$
$31$	−69384.1	−0.418306	−0.209153	+	0.977883i	$0.567071\pi$
$32$	0	0
$33$	127682.	0.618488
$34$	0	0
$35$	390074.	1.53783
$36$	0	0
$37$	29223.4	0.0948472	0.0474236	−	0.998875i	$-0.484899\pi$
$37$	29223.4	0.0948472	0.0474236	+	0.998875i	$0.484899\pi$
$38$	0	0
$39$	−149103.	−0.402495
$40$	0	0
$41$	295928.	0.670568	0.335284	−	0.942117i	$-0.391168\pi$
$41$	295928.	0.670568	0.335284	+	0.942117i	$0.391168\pi$
$42$	0	0
$43$	16387.0	0.0314310	0.0157155	−	0.999877i	$-0.494997\pi$
$43$	16387.0	0.0314310	0.0157155	+	0.999877i	$0.494997\pi$
$44$	0	0
$45$	−268596.	−0.439395
$46$	0	0
$47$	−68148.4	−0.0957444	−0.0478722	−	0.998853i	$-0.515244\pi$
$47$	−68148.4	−0.0957444	−0.0478722	+	0.998853i	$0.515244\pi$
$48$	0	0
$49$	297318.	0.361023
$50$	0	0
$51$	769163.	0.811938
$52$	0	0
$53$	1.85505e6	1.71155	0.855777	−	0.517346i	$-0.173080\pi$
$53$	1.85505e6	1.71155	0.855777	+	0.517346i	$0.173080\pi$
$54$	0	0
$55$	−1.74236e6	−1.41211
$56$	0	0
$57$	341409.	0.244181
$58$	0	0
$59$	−2.75215e6	−1.74458	−0.872289	−	0.488991i	$-0.837365\pi$
$59$	−2.75215e6	−1.74458	−0.872289	+	0.488991i	$0.837365\pi$
$60$	0	0
$61$	−1.77198e6	−0.999547	−0.499774	−	0.866156i	$-0.666583\pi$
$61$	−1.77198e6	−0.999547	−0.499774	+	0.866156i	$0.666583\pi$
$62$	0	0
$63$	−771798.	−0.388876
$64$	0	0
$65$	2.03467e6	0.918963
$66$	0	0
$67$	−2.76825e6	−1.12446	−0.562230	−	0.826981i	$-0.690057\pi$
$67$	−2.76825e6	−1.12446	−0.562230	+	0.826981i	$0.690057\pi$
$68$	0	0
$69$	458941.	0.168184
$70$	0	0
$71$	−4.66235e6	−1.54597	−0.772985	−	0.634425i	$-0.781237\pi$
$71$	−4.66235e6	−1.54597	−0.772985	+	0.634425i	$0.781237\pi$
$72$	0	0
$73$	−6.38405e6	−1.92073	−0.960364	−	0.278748i	$-0.910081\pi$
$73$	−6.38405e6	−1.92073	−0.960364	+	0.278748i	$0.910081\pi$
$74$	0	0
$75$	1.55590e6	0.425861
$76$	0	0
$77$	−5.00659e6	−1.24975
$78$	0	0
$79$	7.32391e6	1.67128	0.835638	−	0.549281i	$-0.185098\pi$
$79$	7.32391e6	1.67128	0.835638	+	0.549281i	$0.185098\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−3.05313e6	−0.586100	−0.293050	−	0.956097i	$-0.594670\pi$
$83$	−3.05313e6	−0.586100	−0.293050	+	0.956097i	$0.594670\pi$
$84$	0	0
$85$	−1.04961e7	−1.85379
$86$	0	0
$87$	2.04349e6	0.332701
$88$	0	0
$89$	−8.79581e6	−1.32255	−0.661273	−	0.750145i	$-0.729984\pi$
$89$	−8.79581e6	−1.32255	−0.661273	+	0.750145i	$0.729984\pi$
$90$	0	0
$91$	5.84654e6	0.813307
$92$	0	0
$93$	−1.87337e6	−0.241509
$94$	0	0
$95$	−4.65889e6	−0.557506
$96$	0	0
$97$	6.84797e6	0.761835	0.380917	−	0.924609i	$-0.375608\pi$
$97$	6.84797e6	0.761835	0.380917	+	0.924609i	$0.375608\pi$
$98$	0	0
$99$	3.44742e6	0.357084
$100$	0	0
$101$	1.01398e7	0.979274	0.489637	−	0.871926i	$-0.337129\pi$
$101$	1.01398e7	0.979274	0.489637	+	0.871926i	$0.337129\pi$
$102$	0	0
$103$	1.54107e7	1.38961	0.694803	−	0.719200i	$-0.255491\pi$
$103$	1.54107e7	1.38961	0.694803	+	0.719200i	$0.255491\pi$
$104$	0	0
$105$	1.05320e7	0.887868
$106$	0	0
$107$	7.22867e6	0.570447	0.285224	−	0.958461i	$-0.407932\pi$
$107$	7.22867e6	0.570447	0.285224	+	0.958461i	$0.407932\pi$
$108$	0	0
$109$	−1.12169e7	−0.829625	−0.414812	−	0.909907i	$-0.636153\pi$
$109$	−1.12169e7	−0.829625	−0.414812	+	0.909907i	$0.636153\pi$
$110$	0	0
$111$	789031.	0.0547600
$112$	0	0
$113$	−1.49419e7	−0.974162	−0.487081	−	0.873357i	$-0.661938\pi$
$113$	−1.49419e7	−0.974162	−0.487081	+	0.873357i	$0.661938\pi$
$114$	0	0
$115$	−6.26275e6	−0.383992
$116$	0	0
$117$	−4.02579e6	−0.232381
$118$	0	0
$119$	−3.01599e7	−1.64065
$120$	0	0
$121$	2.87595e6	0.147582
$122$	0	0
$123$	7.99006e6	0.387153
$124$	0	0
$125$	7.55274e6	0.345875
$126$	0	0
$127$	−2.92110e7	−1.26542	−0.632709	−	0.774390i	$-0.718057\pi$
$127$	−2.92110e7	−1.26542	−0.632709	+	0.774390i	$0.718057\pi$
$128$	0	0
$129$	442448.	0.0181467
$130$	0	0
$131$	1.38297e7	0.537481	0.268741	−	0.963213i	$-0.413393\pi$
$131$	1.38297e7	0.537481	0.268741	+	0.963213i	$0.413393\pi$
$132$	0	0
$133$	−1.33871e7	−0.493408
$134$	0	0
$135$	−7.25208e6	−0.253685
$136$	0	0
$137$	−5.59943e7	−1.86047	−0.930233	−	0.366969i	$-0.880395\pi$
$137$	−5.59943e7	−1.86047	−0.930233	+	0.366969i	$0.880395\pi$
$138$	0	0
$139$	5.22397e7	1.64987	0.824934	−	0.565229i	$-0.191212\pi$
$139$	5.22397e7	1.64987	0.824934	+	0.565229i	$0.191212\pi$
$140$	0	0
$141$	−1.84001e6	−0.0552780
$142$	0	0
$143$	−2.61150e7	−0.746816
$144$	0	0
$145$	−2.78855e7	−0.759611
$146$	0	0
$147$	8.02759e6	0.208437
$148$	0	0
$149$	1.62977e7	0.403622	0.201811	−	0.979424i	$-0.435317\pi$
$149$	1.62977e7	0.403622	0.201811	+	0.979424i	$0.435317\pi$
$150$	0	0
$151$	−1.71334e7	−0.404971	−0.202486	−	0.979285i	$-0.564902\pi$
$151$	−1.71334e7	−0.404971	−0.202486	+	0.979285i	$0.564902\pi$
$152$	0	0
$153$	2.07674e7	0.468772
$154$	0	0
$155$	2.55641e7	0.551404
$156$	0	0
$157$	8.35265e7	1.72257	0.861283	−	0.508126i	$-0.169662\pi$
$157$	8.35265e7	1.72257	0.861283	+	0.508126i	$0.169662\pi$
$158$	0	0
$159$	5.00864e7	0.988166
$160$	0	0
$161$	−1.79957e7	−0.339843
$162$	0	0
$163$	6.70834e7	1.21327	0.606637	−	0.794979i	$-0.292518\pi$
$163$	6.70834e7	1.21327	0.606637	+	0.794979i	$0.292518\pi$
$164$	0	0
$165$	−4.70437e7	−0.815282
$166$	0	0
$167$	−5.88943e7	−0.978510	−0.489255	−	0.872141i	$-0.662731\pi$
$167$	−5.88943e7	−0.978510	−0.489255	+	0.872141i	$0.662731\pi$
$168$	0	0
$169$	−3.22523e7	−0.513992
$170$	0	0
$171$	9.21804e6	0.140978
$172$	0	0
$173$	−5.79040e7	−0.850250	−0.425125	−	0.905135i	$-0.639770\pi$
$173$	−5.79040e7	−0.850250	−0.425125	+	0.905135i	$0.639770\pi$
$174$	0	0
$175$	−6.10090e7	−0.860520
$176$	0	0
$177$	−7.43081e7	−1.00723
$178$	0	0
$179$	−4.73524e7	−0.617102	−0.308551	−	0.951208i	$-0.599844\pi$
$179$	−4.73524e7	−0.617102	−0.308551	+	0.951208i	$0.599844\pi$
$180$	0	0
$181$	−1.33152e8	−1.66907	−0.834534	−	0.550957i	$-0.814263\pi$
$181$	−1.33152e8	−1.66907	−0.834534	+	0.550957i	$0.814263\pi$
$182$	0	0
$183$	−4.78433e7	−0.577089
$184$	0	0
$185$	−1.07672e7	−0.125026
$186$	0	0
$187$	1.34717e8	1.50652
$188$	0	0
$189$	−2.08385e7	−0.224518
$190$	0	0
$191$	9.13177e7	0.948283	0.474142	−	0.880449i	$-0.342758\pi$
$191$	9.13177e7	0.948283	0.474142	+	0.880449i	$0.342758\pi$
$192$	0	0
$193$	−1.06396e8	−1.06530	−0.532651	−	0.846335i	$-0.678804\pi$
$193$	−1.06396e8	−1.06530	−0.532651	+	0.846335i	$0.678804\pi$
$194$	0	0
$195$	5.49362e7	0.530564
$196$	0	0
$197$	−1.91262e8	−1.78236	−0.891182	−	0.453646i	$-0.850123\pi$
$197$	−1.91262e8	−1.78236	−0.891182	+	0.453646i	$0.850123\pi$
$198$	0	0
$199$	−4.13489e7	−0.371945	−0.185972	−	0.982555i	$-0.559543\pi$
$199$	−4.13489e7	−0.371945	−0.185972	+	0.982555i	$0.559543\pi$
$200$	0	0
$201$	−7.47429e7	−0.649208
$202$	0	0
$203$	−8.01279e7	−0.672276
$204$	0	0
$205$	−1.09033e8	−0.883933
$206$	0	0
$207$	1.23914e7	0.0971012
$208$	0	0
$209$	5.97967e7	0.453070
$210$	0	0
$211$	−5.48974e7	−0.402312	−0.201156	−	0.979559i	$-0.564470\pi$
$211$	−5.48974e7	−0.402312	−0.201156	+	0.979559i	$0.564470\pi$
$212$	0	0
$213$	−1.25884e8	−0.892566
$214$	0	0
$215$	−6.03768e6	−0.0414319
$216$	0	0
$217$	7.34574e7	0.488008
$218$	0	0
$219$	−1.72369e8	−1.10893
$220$	0	0
$221$	−1.57318e8	−0.980404
$222$	0	0
$223$	−1.22016e8	−0.736799	−0.368399	−	0.929668i	$-0.620094\pi$
$223$	−1.22016e8	−0.736799	−0.368399	+	0.929668i	$0.620094\pi$
$224$	0	0
$225$	4.20093e7	0.245871
$226$	0	0
$227$	3.33351e8	1.89152	0.945762	−	0.324861i	$-0.105317\pi$
$227$	3.33351e8	1.89152	0.945762	+	0.324861i	$0.105317\pi$
$228$	0	0
$229$	−1.28881e7	−0.0709193	−0.0354597	−	0.999371i	$-0.511290\pi$
$229$	−1.28881e7	−0.0709193	−0.0354597	+	0.999371i	$0.511290\pi$
$230$	0	0
$231$	−1.35178e8	−0.721546
$232$	0	0
$233$	−3.67790e8	−1.90482	−0.952411	−	0.304817i	$-0.901405\pi$
$233$	−3.67790e8	−1.90482	−0.952411	+	0.304817i	$0.901405\pi$
$234$	0	0
$235$	2.51089e7	0.126209
$236$	0	0
$237$	1.97746e8	0.964912
$238$	0	0
$239$	−1.50083e8	−0.711113	−0.355556	−	0.934655i	$-0.615709\pi$
$239$	−1.50083e8	−0.711113	−0.355556	+	0.934655i	$0.615709\pi$
$240$	0	0
$241$	−7.57608e7	−0.348646	−0.174323	−	0.984689i	$-0.555774\pi$
$241$	−7.57608e7	−0.348646	−0.174323	+	0.984689i	$0.555774\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−1.09545e8	−0.475895
$246$	0	0
$247$	−6.98287e7	−0.294846
$248$	0	0
$249$	−8.24345e7	−0.338385
$250$	0	0
$251$	2.46051e8	0.982127	0.491063	−	0.871124i	$-0.336608\pi$
$251$	2.46051e8	0.982127	0.491063	+	0.871124i	$0.336608\pi$
$252$	0	0
$253$	8.03821e7	0.312060
$254$	0	0
$255$	−2.83393e8	−1.07028
$256$	0	0
$257$	3.21274e8	1.18062	0.590309	−	0.807177i	$-0.299006\pi$
$257$	3.21274e8	1.18062	0.590309	+	0.807177i	$0.299006\pi$
$258$	0	0
$259$	−3.09390e7	−0.110651
$260$	0	0
$261$	5.51741e7	0.192085
$262$	0	0
$263$	−1.79444e8	−0.608254	−0.304127	−	0.952632i	$-0.598365\pi$
$263$	−1.79444e8	−0.608254	−0.304127	+	0.952632i	$0.598365\pi$
$264$	0	0
$265$	−6.83483e8	−2.25614
$266$	0	0
$267$	−2.37487e8	−0.763572
$268$	0	0
$269$	7.19069e7	0.225236	0.112618	−	0.993638i	$-0.464076\pi$
$269$	7.19069e7	0.225236	0.112618	+	0.993638i	$0.464076\pi$
$270$	0	0
$271$	−5.37851e8	−1.64161	−0.820805	−	0.571209i	$-0.806475\pi$
$271$	−5.37851e8	−1.64161	−0.820805	+	0.571209i	$0.806475\pi$
$272$	0	0
$273$	1.57857e8	0.469563
$274$	0	0
$275$	2.72511e8	0.790169
$276$	0	0
$277$	2.21434e8	0.625986	0.312993	−	0.949755i	$-0.398668\pi$
$277$	2.21434e8	0.625986	0.312993	+	0.949755i	$0.398668\pi$
$278$	0	0
$279$	−5.05810e7	−0.139435
$280$	0	0
$281$	2.00421e8	0.538855	0.269428	−	0.963021i	$-0.413166\pi$
$281$	2.00421e8	0.538855	0.269428	+	0.963021i	$0.413166\pi$
$282$	0	0
$283$	5.60104e8	1.46898	0.734490	−	0.678619i	$-0.237421\pi$
$283$	5.60104e8	1.46898	0.734490	+	0.678619i	$0.237421\pi$
$284$	0	0
$285$	−1.25790e8	−0.321876
$286$	0	0
$287$	−3.13301e8	−0.782304
$288$	0	0
$289$	4.01200e8	0.977729
$290$	0	0
$291$	1.84895e8	0.439845
$292$	0	0
$293$	−4.99015e8	−1.15898	−0.579492	−	0.814978i	$-0.696749\pi$
$293$	−4.99015e8	−1.15898	−0.579492	+	0.814978i	$0.696749\pi$
$294$	0	0
$295$	1.01401e9	2.29968
$296$	0	0
$297$	9.30802e7	0.206163
$298$	0	0
$299$	−9.38678e7	−0.203080
$300$	0	0
$301$	−1.73490e7	−0.0366684
$302$	0	0
$303$	2.73774e8	0.565384
$304$	0	0
$305$	6.52874e8	1.31759
$306$	0	0
$307$	−8.21105e8	−1.61963	−0.809813	−	0.586689i	$-0.800431\pi$
$307$	−8.21105e8	−1.61963	−0.809813	+	0.586689i	$0.800431\pi$
$308$	0	0
$309$	4.16089e8	0.802290
$310$	0	0
$311$	5.43818e7	0.102516	0.0512581	−	0.998685i	$-0.483677\pi$
$311$	5.43818e7	0.102516	0.0512581	+	0.998685i	$0.483677\pi$
$312$	0	0
$313$	−6.71990e8	−1.23868	−0.619338	−	0.785124i	$-0.712599\pi$
$313$	−6.71990e8	−1.23868	−0.619338	+	0.785124i	$0.712599\pi$
$314$	0	0
$315$	2.84364e8	0.512611
$316$	0	0
$317$	−1.49707e8	−0.263958	−0.131979	−	0.991253i	$-0.542133\pi$
$317$	−1.49707e8	−0.263958	−0.131979	+	0.991253i	$0.542133\pi$
$318$	0	0
$319$	3.57910e8	0.617315
$320$	0	0
$321$	1.95174e8	0.329348
$322$	0	0
$323$	3.60218e8	0.594780
$324$	0	0
$325$	−3.18230e8	−0.514221
$326$	0	0
$327$	−3.02857e8	−0.478984
$328$	0	0
$329$	7.21492e7	0.111698
$330$	0	0
$331$	−2.29194e8	−0.347380	−0.173690	−	0.984800i	$-0.555569\pi$
$331$	−2.29194e8	−0.347380	−0.173690	+	0.984800i	$0.555569\pi$
$332$	0	0
$333$	2.13038e7	0.0316157
$334$	0	0
$335$	1.01995e9	1.48225
$336$	0	0
$337$	−5.68268e8	−0.808814	−0.404407	−	0.914579i	$-0.632522\pi$
$337$	−5.68268e8	−0.808814	−0.404407	+	0.914579i	$0.632522\pi$
$338$	0	0
$339$	−4.03431e8	−0.562433
$340$	0	0
$341$	−3.28115e8	−0.448111
$342$	0	0
$343$	5.57118e8	0.745449
$344$	0	0
$345$	−1.69094e8	−0.221698
$346$	0	0
$347$	−2.99189e8	−0.384408	−0.192204	−	0.981355i	$-0.561564\pi$
$347$	−2.99189e8	−0.384408	−0.192204	+	0.981355i	$0.561564\pi$
$348$	0	0
$349$	1.48423e9	1.86901	0.934503	−	0.355954i	$-0.115844\pi$
$349$	1.48423e9	1.86901	0.934503	+	0.355954i	$0.115844\pi$
$350$	0	0
$351$	−1.08696e8	−0.134165
$352$	0	0
$353$	−6.05417e8	−0.732560	−0.366280	−	0.930505i	$-0.619369\pi$
$353$	−6.05417e8	−0.732560	−0.366280	+	0.930505i	$0.619369\pi$
$354$	0	0
$355$	1.71782e9	2.03787
$356$	0	0
$357$	−8.14318e8	−0.947230
$358$	0	0
$359$	−2.74734e8	−0.313388	−0.156694	−	0.987647i	$-0.550084\pi$
$359$	−2.74734e8	−0.313388	−0.156694	+	0.987647i	$0.550084\pi$
$360$	0	0
$361$	−7.33982e8	−0.821126
$362$	0	0
$363$	7.76507e7	0.0852064
$364$	0	0
$365$	2.35216e9	2.53188
$366$	0	0
$367$	8.66951e8	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$367$	8.66951e8	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$368$	0	0
$369$	2.15732e8	0.223523
$370$	0	0
$371$	−1.96396e9	−1.99675
$372$	0	0
$373$	−5.09365e8	−0.508216	−0.254108	−	0.967176i	$-0.581782\pi$
$373$	−5.09365e8	−0.508216	−0.254108	+	0.967176i	$0.581782\pi$
$374$	0	0
$375$	2.03924e8	0.199691
$376$	0	0
$377$	−4.17956e8	−0.401732
$378$	0	0
$379$	−9.85372e8	−0.929743	−0.464871	−	0.885378i	$-0.653899\pi$
$379$	−9.85372e8	−0.929743	−0.464871	+	0.885378i	$0.653899\pi$
$380$	0	0
$381$	−7.88698e8	−0.730590
$382$	0	0
$383$	3.47060e8	0.315652	0.157826	−	0.987467i	$-0.449551\pi$
$383$	3.47060e8	0.315652	0.157826	+	0.987467i	$0.449551\pi$
$384$	0	0
$385$	1.84465e9	1.64741
$386$	0	0
$387$	1.19461e7	0.0104770
$388$	0	0
$389$	−8.19819e8	−0.706146	−0.353073	−	0.935596i	$-0.614863\pi$
$389$	−8.19819e8	−0.706146	−0.353073	+	0.935596i	$0.614863\pi$
$390$	0	0
$391$	4.84226e8	0.409665
$392$	0	0
$393$	3.73402e8	0.310315
$394$	0	0
$395$	−2.69845e9	−2.20305
$396$	0	0
$397$	1.57699e9	1.26492	0.632460	−	0.774593i	$-0.282045\pi$
$397$	1.57699e9	1.26492	0.632460	+	0.774593i	$0.282045\pi$
$398$	0	0
$399$	−3.61452e8	−0.284869
$400$	0	0
$401$	5.99350e8	0.464168	0.232084	−	0.972696i	$-0.425446\pi$
$401$	5.99350e8	0.464168	0.232084	+	0.972696i	$0.425446\pi$
$402$	0	0
$403$	3.83162e8	0.291619
$404$	0	0
$405$	−1.95806e8	−0.146465
$406$	0	0
$407$	1.38196e8	0.101605
$408$	0	0
$409$	−4.08431e8	−0.295180	−0.147590	−	0.989049i	$-0.547152\pi$
$409$	−4.08431e8	−0.295180	−0.147590	+	0.989049i	$0.547152\pi$
$410$	0	0
$411$	−1.51185e9	−1.07414
$412$	0	0
$413$	2.91372e9	2.03527
$414$	0	0
$415$	1.12491e9	0.772589
$416$	0	0
$417$	1.41047e9	0.952552
$418$	0	0
$419$	−1.46270e9	−0.971417	−0.485708	−	0.874121i	$-0.661438\pi$
$419$	−1.46270e9	−0.971417	−0.485708	+	0.874121i	$0.661438\pi$
$420$	0	0
$421$	−1.76741e8	−0.115438	−0.0577191	−	0.998333i	$-0.518383\pi$
$421$	−1.76741e8	−0.115438	−0.0577191	+	0.998333i	$0.518383\pi$
$422$	0	0
$423$	−4.96802e7	−0.0319148
$424$	0	0
$425$	1.64162e9	1.03732
$426$	0	0
$427$	1.87600e9	1.16610
$428$	0	0
$429$	−7.05104e8	−0.431174
$430$	0	0
$431$	−1.56757e9	−0.943099	−0.471549	−	0.881840i	$-0.656305\pi$
$431$	−1.56757e9	−0.943099	−0.471549	+	0.881840i	$0.656305\pi$
$432$	0	0
$433$	2.75419e9	1.63037	0.815185	−	0.579201i	$-0.196635\pi$
$433$	2.75419e9	1.63037	0.815185	+	0.579201i	$0.196635\pi$
$434$	0	0
$435$	−7.52910e8	−0.438562
$436$	0	0
$437$	2.14934e8	0.123202
$438$	0	0
$439$	−5.51524e8	−0.311128	−0.155564	−	0.987826i	$-0.549719\pi$
$439$	−5.51524e8	−0.311128	−0.155564	+	0.987826i	$0.549719\pi$
$440$	0	0
$441$	2.16745e8	0.120341
$442$	0	0
$443$	1.80168e9	0.984610	0.492305	−	0.870423i	$-0.336155\pi$
$443$	1.80168e9	0.984610	0.492305	+	0.870423i	$0.336155\pi$
$444$	0	0
$445$	3.24076e9	1.74336
$446$	0	0
$447$	4.40039e8	0.233032
$448$	0	0
$449$	4.05710e8	0.211521	0.105760	−	0.994392i	$-0.466272\pi$
$449$	4.05710e8	0.211521	0.105760	+	0.994392i	$0.466272\pi$
$450$	0	0
$451$	1.39943e9	0.718348
$452$	0	0
$453$	−4.62602e8	−0.233810
$454$	0	0
$455$	−2.15412e9	−1.07209
$456$	0	0
$457$	1.68101e9	0.823877	0.411939	−	0.911212i	$-0.364852\pi$
$457$	1.68101e9	0.823877	0.411939	+	0.911212i	$0.364852\pi$
$458$	0	0
$459$	5.60720e8	0.270646
$460$	0	0
$461$	−2.11400e9	−1.00497	−0.502484	−	0.864586i	$-0.667580\pi$
$461$	−2.11400e9	−1.00497	−0.502484	+	0.864586i	$0.667580\pi$
$462$	0	0
$463$	2.49558e9	1.16852	0.584261	−	0.811566i	$-0.301384\pi$
$463$	2.49558e9	1.16852	0.584261	+	0.811566i	$0.301384\pi$
$464$	0	0
$465$	6.90232e8	0.318353
$466$	0	0
$467$	−2.80333e9	−1.27369	−0.636847	−	0.770990i	$-0.719762\pi$
$467$	−2.80333e9	−1.27369	−0.636847	+	0.770990i	$0.719762\pi$
$468$	0	0
$469$	2.93077e9	1.31183
$470$	0	0
$471$	2.25522e9	0.994524
$472$	0	0
$473$	7.74934e7	0.0336706
$474$	0	0
$475$	7.28667e8	0.311962
$476$	0	0
$477$	1.35233e9	0.570518
$478$	0	0
$479$	−1.17285e8	−0.0487605	−0.0243803	−	0.999703i	$-0.507761\pi$
$479$	−1.17285e8	−0.0487605	−0.0243803	+	0.999703i	$0.507761\pi$
$480$	0	0
$481$	−1.61381e8	−0.0661220
$482$	0	0
$483$	−4.85884e8	−0.196209
$484$	0	0
$485$	−2.52309e9	−1.00424
$486$	0	0
$487$	1.37486e9	0.539393	0.269697	−	0.962945i	$-0.413077\pi$
$487$	1.37486e9	0.539393	0.269697	+	0.962945i	$0.413077\pi$
$488$	0	0
$489$	1.81125e9	0.700484
$490$	0	0
$491$	2.33074e8	0.0888606	0.0444303	−	0.999012i	$-0.485853\pi$
$491$	2.33074e8	0.0888606	0.0444303	+	0.999012i	$0.485853\pi$
$492$	0	0
$493$	2.15607e9	0.810397
$494$	0	0
$495$	−1.27018e9	−0.470703
$496$	0	0
$497$	4.93607e9	1.80357
$498$	0	0
$499$	−8.97141e7	−0.0323228	−0.0161614	−	0.999869i	$-0.505145\pi$
$499$	−8.97141e7	−0.0323228	−0.0161614	+	0.999869i	$0.505145\pi$
$500$	0	0
$501$	−1.59014e9	−0.564943
$502$	0	0
$503$	2.47433e9	0.866901	0.433450	−	0.901177i	$-0.357296\pi$
$503$	2.47433e9	0.866901	0.433450	+	0.901177i	$0.357296\pi$
$504$	0	0
$505$	−3.73594e9	−1.29086
$506$	0	0
$507$	−8.70811e8	−0.296754
$508$	0	0
$509$	1.68523e8	0.0566430	0.0283215	−	0.999599i	$-0.490984\pi$
$509$	1.68523e8	0.0566430	0.0283215	+	0.999599i	$0.490984\pi$
$510$	0	0
$511$	6.75884e9	2.24078
$512$	0	0
$513$	2.48887e8	0.0813938
$514$	0	0
$515$	−5.67798e9	−1.83176
$516$	0	0
$517$	−3.22272e8	−0.102566
$518$	0	0
$519$	−1.56341e9	−0.490892
$520$	0	0
$521$	−2.41709e9	−0.748792	−0.374396	−	0.927269i	$-0.622150\pi$
$521$	−2.41709e9	−0.748792	−0.374396	+	0.927269i	$0.622150\pi$
$522$	0	0
$523$	−9.40312e8	−0.287419	−0.143710	−	0.989620i	$-0.545903\pi$
$523$	−9.40312e8	−0.287419	−0.143710	+	0.989620i	$0.545903\pi$
$524$	0	0
$525$	−1.64724e9	−0.496821
$526$	0	0
$527$	−1.97658e9	−0.588271
$528$	0	0
$529$	−3.11590e9	−0.915142
$530$	0	0
$531$	−2.00632e9	−0.581526
$532$	0	0
$533$	−1.63422e9	−0.467482
$534$	0	0
$535$	−2.66336e9	−0.751955
$536$	0	0
$537$	−1.27852e9	−0.356284
$538$	0	0
$539$	1.40601e9	0.386747
$540$	0	0
$541$	2.65125e8	0.0719879	0.0359939	−	0.999352i	$-0.488540\pi$
$541$	2.65125e8	0.0719879	0.0359939	+	0.999352i	$0.488540\pi$
$542$	0	0
$543$	−3.59511e9	−0.963636
$544$	0	0
$545$	4.13281e9	1.09360
$546$	0	0
$547$	−6.41977e8	−0.167712	−0.0838559	−	0.996478i	$-0.526724\pi$
$547$	−6.41977e8	−0.167712	−0.0838559	+	0.996478i	$0.526724\pi$
$548$	0	0
$549$	−1.29177e9	−0.333182
$550$	0	0
$551$	9.57015e8	0.243718
$552$	0	0
$553$	−7.75388e9	−1.94976
$554$	0	0
$555$	−2.90714e8	−0.0721839
$556$	0	0
$557$	−2.92485e9	−0.717152	−0.358576	−	0.933501i	$-0.616738\pi$
$557$	−2.92485e9	−0.717152	−0.358576	+	0.933501i	$0.616738\pi$
$558$	0	0
$559$	−9.04944e7	−0.0219119
$560$	0	0
$561$	3.63735e9	0.869790
$562$	0	0
$563$	5.62455e9	1.32834	0.664169	−	0.747583i	$-0.268786\pi$
$563$	5.62455e9	1.32834	0.664169	+	0.747583i	$0.268786\pi$
$564$	0	0
$565$	5.50525e9	1.28413
$566$	0	0
$567$	−5.62640e8	−0.129625
$568$	0	0
$569$	−8.48615e9	−1.93116	−0.965579	−	0.260111i	$-0.916241\pi$
$569$	−8.48615e9	−1.93116	−0.965579	+	0.260111i	$0.916241\pi$
$570$	0	0
$571$	−5.42789e9	−1.22013	−0.610063	−	0.792353i	$-0.708856\pi$
$571$	−5.42789e9	−1.22013	−0.610063	+	0.792353i	$0.708856\pi$
$572$	0	0
$573$	2.46558e9	0.547492
$574$	0	0
$575$	9.79516e8	0.214869
$576$	0	0
$577$	−8.80138e9	−1.90737	−0.953686	−	0.300803i	$-0.902745\pi$
$577$	−8.80138e9	−1.90737	−0.953686	+	0.300803i	$0.902745\pi$
$578$	0	0
$579$	−2.87268e9	−0.615053
$580$	0	0
$581$	3.23237e9	0.683761
$582$	0	0
$583$	8.77248e9	1.83351
$584$	0	0
$585$	1.48328e9	0.306321
$586$	0	0
$587$	−3.50790e8	−0.0715836	−0.0357918	−	0.999359i	$-0.511395\pi$
$587$	−3.50790e8	−0.0715836	−0.0357918	+	0.999359i	$0.511395\pi$
$588$	0	0
$589$	−8.77345e8	−0.176916
$590$	0	0
$591$	−5.16407e9	−1.02905
$592$	0	0
$593$	6.34103e9	1.24873	0.624364	−	0.781133i	$-0.285358\pi$
$593$	6.34103e9	1.24873	0.624364	+	0.781133i	$0.285358\pi$
$594$	0	0
$595$	1.11122e10	2.16268
$596$	0	0
$597$	−1.11642e9	−0.214742
$598$	0	0
$599$	−2.85447e9	−0.542664	−0.271332	−	0.962486i	$-0.587464\pi$
$599$	−2.85447e9	−0.542664	−0.271332	+	0.962486i	$0.587464\pi$
$600$	0	0
$601$	3.22420e9	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$601$	3.22420e9	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$602$	0	0
$603$	−2.01806e9	−0.374820
$604$	0	0
$605$	−1.05963e9	−0.194540
$606$	0	0
$607$	−8.25547e8	−0.149824	−0.0749120	−	0.997190i	$-0.523868\pi$
$607$	−8.25547e8	−0.149824	−0.0749120	+	0.997190i	$0.523868\pi$
$608$	0	0
$609$	−2.16345e9	−0.388139
$610$	0	0
$611$	3.76339e8	0.0667475
$612$	0	0
$613$	−1.26909e9	−0.222526	−0.111263	−	0.993791i	$-0.535490\pi$
$613$	−1.26909e9	−0.222526	−0.111263	+	0.993791i	$0.535490\pi$
$614$	0	0
$615$	−2.94389e9	−0.510339
$616$	0	0
$617$	1.08847e10	1.86560	0.932802	−	0.360389i	$-0.117356\pi$
$617$	1.08847e10	1.86560	0.932802	+	0.360389i	$0.117356\pi$
$618$	0	0
$619$	5.60725e9	0.950239	0.475119	−	0.879921i	$-0.342405\pi$
$619$	5.60725e9	0.950239	0.475119	+	0.879921i	$0.342405\pi$
$620$	0	0
$621$	3.34568e8	0.0560614
$622$	0	0
$623$	9.31219e9	1.54292
$624$	0	0
$625$	−7.28479e9	−1.19354
$626$	0	0
$627$	1.61451e9	0.261580
$628$	0	0
$629$	8.32501e8	0.133385
$630$	0	0
$631$	3.03140e9	0.480330	0.240165	−	0.970732i	$-0.422798\pi$
$631$	3.03140e9	0.480330	0.240165	+	0.970732i	$0.422798\pi$
$632$	0	0
$633$	−1.48223e9	−0.232275
$634$	0	0
$635$	1.07626e10	1.66806
$636$	0	0
$637$	−1.64189e9	−0.251685
$638$	0	0
$639$	−3.39886e9	−0.515323
$640$	0	0
$641$	1.88737e9	0.283045	0.141522	−	0.989935i	$-0.454800\pi$
$641$	1.88737e9	0.283045	0.141522	+	0.989935i	$0.454800\pi$
$642$	0	0
$643$	1.23016e10	1.82483	0.912417	−	0.409263i	$-0.134214\pi$
$643$	1.23016e10	1.82483	0.912417	+	0.409263i	$0.134214\pi$
$644$	0	0
$645$	−1.63017e8	−0.0239207
$646$	0	0
$647$	−5.68136e9	−0.824683	−0.412342	−	0.911029i	$-0.635289\pi$
$647$	−5.68136e9	−0.824683	−0.412342	+	0.911029i	$0.635289\pi$
$648$	0	0
$649$	−1.30148e10	−1.86888
$650$	0	0
$651$	1.98335e9	0.281751
$652$	0	0
$653$	−9.38850e9	−1.31947	−0.659736	−	0.751497i	$-0.729332\pi$
$653$	−9.38850e9	−1.31947	−0.659736	+	0.751497i	$0.729332\pi$
$654$	0	0
$655$	−5.09547e9	−0.708500
$656$	0	0
$657$	−4.65397e9	−0.640243
$658$	0	0
$659$	3.88229e9	0.528432	0.264216	−	0.964463i	$-0.414887\pi$
$659$	3.88229e9	0.528432	0.264216	+	0.964463i	$0.414887\pi$
$660$	0	0
$661$	−3.30387e9	−0.444957	−0.222478	−	0.974938i	$-0.571415\pi$
$661$	−3.30387e9	−0.444957	−0.222478	+	0.974938i	$0.571415\pi$
$662$	0	0
$663$	−4.24758e9	−0.566036
$664$	0	0
$665$	4.93240e9	0.650403
$666$	0	0
$667$	1.28647e9	0.167865
$668$	0	0
$669$	−3.29442e9	−0.425391
$670$	0	0
$671$	−8.37961e9	−1.07077
$672$	0	0
$673$	3.44280e9	0.435371	0.217685	−	0.976019i	$-0.430149\pi$
$673$	3.44280e9	0.435371	0.217685	+	0.976019i	$0.430149\pi$
$674$	0	0
$675$	1.13425e9	0.141954
$676$	0	0
$677$	−1.59211e10	−1.97202	−0.986012	−	0.166677i	$-0.946696\pi$
$677$	−1.59211e10	−1.97202	−0.986012	+	0.166677i	$0.946696\pi$
$678$	0	0
$679$	−7.24999e9	−0.888778
$680$	0	0
$681$	9.00049e9	1.09207
$682$	0	0
$683$	−7.17757e9	−0.861995	−0.430997	−	0.902353i	$-0.641838\pi$
$683$	−7.17757e9	−0.861995	−0.430997	+	0.902353i	$0.641838\pi$
$684$	0	0
$685$	2.06308e10	2.45244
$686$	0	0
$687$	−3.47979e8	−0.0409453
$688$	0	0
$689$	−1.02442e10	−1.19320
$690$	0	0
$691$	1.18199e10	1.36282	0.681411	−	0.731901i	$-0.261367\pi$
$691$	1.18199e10	1.36282	0.681411	+	0.731901i	$0.261367\pi$
$692$	0	0
$693$	−3.64981e9	−0.416585
$694$	0	0
$695$	−1.92474e10	−2.17483
$696$	0	0
$697$	8.43026e9	0.943032
$698$	0	0
$699$	−9.93033e9	−1.09975
$700$	0	0
$701$	−8.79765e9	−0.964614	−0.482307	−	0.876002i	$-0.660201\pi$
$701$	−8.79765e9	−0.964614	−0.482307	+	0.876002i	$0.660201\pi$
$702$	0	0
$703$	3.69523e8	0.0401142
$704$	0	0
$705$	6.77940e8	0.0728667
$706$	0	0
$707$	−1.07351e10	−1.14245
$708$	0	0
$709$	−4.09495e9	−0.431506	−0.215753	−	0.976448i	$-0.569221\pi$
$709$	−4.09495e9	−0.431506	−0.215753	+	0.976448i	$0.569221\pi$
$710$	0	0
$711$	5.33913e9	0.557092
$712$	0	0
$713$	−1.17938e9	−0.121854
$714$	0	0
$715$	9.62190e9	0.984441
$716$	0	0
$717$	−4.05224e9	−0.410561
$718$	0	0
$719$	−1.16329e10	−1.16718	−0.583589	−	0.812049i	$-0.698352\pi$
$719$	−1.16329e10	−1.16718	−0.583589	+	0.812049i	$0.698352\pi$
$720$	0	0
$721$	−1.63154e10	−1.62116
$722$	0	0
$723$	−2.04554e9	−0.201291
$724$	0	0
$725$	4.36140e9	0.425053
$726$	0	0
$727$	1.69867e10	1.63960	0.819799	−	0.572651i	$-0.194085\pi$
$727$	1.69867e10	1.63960	0.819799	+	0.572651i	$0.194085\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	4.66824e8	0.0442020
$732$	0	0
$733$	−3.28036e9	−0.307650	−0.153825	−	0.988098i	$-0.549159\pi$
$733$	−3.28036e9	−0.307650	−0.153825	+	0.988098i	$0.549159\pi$
$734$	0	0
$735$	−2.95772e9	−0.274758
$736$	0	0
$737$	−1.30910e10	−1.20458
$738$	0	0
$739$	−1.76555e10	−1.60926	−0.804628	−	0.593779i	$-0.797635\pi$
$739$	−1.76555e10	−1.60926	−0.804628	+	0.593779i	$0.797635\pi$
$740$	0	0
$741$	−1.88538e9	−0.170229
$742$	0	0
$743$	−8.33458e9	−0.745458	−0.372729	−	0.927940i	$-0.621578\pi$
$743$	−8.33458e9	−0.745458	−0.372729	+	0.927940i	$0.621578\pi$
$744$	0	0
$745$	−6.00480e9	−0.532049
$746$	0	0
$747$	−2.22573e9	−0.195367
$748$	0	0
$749$	−7.65305e9	−0.665500
$750$	0	0
$751$	−1.15928e10	−0.998729	−0.499365	−	0.866392i	$-0.666433\pi$
$751$	−1.15928e10	−0.998729	−0.499365	+	0.866392i	$0.666433\pi$
$752$	0	0
$753$	6.64338e9	0.567031
$754$	0	0
$755$	6.31270e9	0.533827
$756$	0	0
$757$	2.07716e10	1.74034	0.870170	−	0.492751i	$-0.164009\pi$
$757$	2.07716e10	1.74034	0.870170	+	0.492751i	$0.164009\pi$
$758$	0	0
$759$	2.17032e9	0.180168
$760$	0	0
$761$	6.33390e9	0.520985	0.260492	−	0.965476i	$-0.416115\pi$
$761$	6.33390e9	0.520985	0.260492	+	0.965476i	$0.416115\pi$
$762$	0	0
$763$	1.18755e10	0.967864
$764$	0	0
$765$	−7.65162e9	−0.617929
$766$	0	0
$767$	1.51983e10	1.21622
$768$	0	0
$769$	−6.52316e9	−0.517268	−0.258634	−	0.965975i	$-0.583272\pi$
$769$	−6.52316e9	−0.517268	−0.258634	+	0.965975i	$0.583272\pi$
$770$	0	0
$771$	8.67439e9	0.681630
$772$	0	0
$773$	−6.77839e9	−0.527835	−0.263918	−	0.964545i	$-0.585015\pi$
$773$	−6.77839e9	−0.527835	−0.263918	+	0.964545i	$0.585015\pi$
$774$	0	0
$775$	−3.99832e9	−0.308547
$776$	0	0
$777$	−8.35353e8	−0.0638846
$778$	0	0
$779$	3.74195e9	0.283607
$780$	0	0
$781$	−2.20481e10	−1.65612
$782$	0	0
$783$	1.48970e9	0.110900
$784$	0	0
$785$	−3.07748e10	−2.27066
$786$	0	0
$787$	−2.82358e9	−0.206485	−0.103242	−	0.994656i	$-0.532922\pi$
$787$	−2.82358e9	−0.206485	−0.103242	+	0.994656i	$0.532922\pi$
$788$	0	0
$789$	−4.84500e9	−0.351175
$790$	0	0
$791$	1.58191e10	1.13649
$792$	0	0
$793$	9.78546e9	0.696827
$794$	0	0
$795$	−1.84540e10	−1.30259
$796$	0	0
$797$	−5.30653e9	−0.371284	−0.185642	−	0.982617i	$-0.559436\pi$
$797$	−5.30653e9	−0.371284	−0.185642	+	0.982617i	$0.559436\pi$
$798$	0	0
$799$	−1.94138e9	−0.134647
$800$	0	0
$801$	−6.41215e9	−0.440849
$802$	0	0
$803$	−3.01899e10	−2.05759
$804$	0	0
$805$	6.63041e9	0.447976
$806$	0	0
$807$	1.94149e9	0.130040
$808$	0	0
$809$	−6.64448e9	−0.441206	−0.220603	−	0.975364i	$-0.570803\pi$
$809$	−6.64448e9	−0.441206	−0.220603	+	0.975364i	$0.570803\pi$
$810$	0	0
$811$	1.83502e10	1.20800	0.604000	−	0.796984i	$-0.293573\pi$
$811$	1.83502e10	1.20800	0.604000	+	0.796984i	$0.293573\pi$
$812$	0	0
$813$	−1.45220e10	−0.947784
$814$	0	0
$815$	−2.47165e10	−1.59932
$816$	0	0
$817$	2.07209e8	0.0132933
$818$	0	0
$819$	4.26213e9	0.271102
$820$	0	0
$821$	1.10009e10	0.693788	0.346894	−	0.937904i	$-0.387236\pi$
$821$	1.10009e10	0.693788	0.346894	+	0.937904i	$0.387236\pi$
$822$	0	0
$823$	1.35279e10	0.845926	0.422963	−	0.906147i	$-0.360990\pi$
$823$	1.35279e10	0.845926	0.422963	+	0.906147i	$0.360990\pi$
$824$	0	0
$825$	7.35780e9	0.456204
$826$	0	0
$827$	1.77028e9	0.108836	0.0544181	−	0.998518i	$-0.482670\pi$
$827$	1.77028e9	0.108836	0.0544181	+	0.998518i	$0.482670\pi$
$828$	0	0
$829$	−8.54339e9	−0.520822	−0.260411	−	0.965498i	$-0.583858\pi$
$829$	−8.54339e9	−0.520822	−0.260411	+	0.965498i	$0.583858\pi$
$830$	0	0
$831$	5.97871e9	0.361413
$832$	0	0
$833$	8.46986e9	0.507713
$834$	0	0
$835$	2.16992e10	1.28986
$836$	0	0
$837$	−1.36569e9	−0.0805030
$838$	0	0
$839$	−1.23734e10	−0.723307	−0.361653	−	0.932313i	$-0.617788\pi$
$839$	−1.23734e10	−0.723307	−0.361653	+	0.932313i	$0.617788\pi$
$840$	0	0
$841$	−1.15217e10	−0.667930
$842$	0	0
$843$	5.41138e9	0.311108
$844$	0	0
$845$	1.18832e10	0.677537
$846$	0	0
$847$	−3.04479e9	−0.172173
$848$	0	0
$849$	1.51228e10	0.848117
$850$	0	0
$851$	4.96734e8	0.0276293
$852$	0	0
$853$	−1.67180e10	−0.922282	−0.461141	−	0.887327i	$-0.652560\pi$
$853$	−1.67180e10	−0.922282	−0.461141	+	0.887327i	$0.652560\pi$
$854$	0	0
$855$	−3.39633e9	−0.185835
$856$	0	0
$857$	3.33011e9	0.180728	0.0903642	−	0.995909i	$-0.471197\pi$
$857$	3.33011e9	0.180728	0.0903642	+	0.995909i	$0.471197\pi$
$858$	0	0
$859$	−1.10846e10	−0.596682	−0.298341	−	0.954459i	$-0.596433\pi$
$859$	−1.10846e10	−0.596682	−0.298341	+	0.954459i	$0.596433\pi$
$860$	0	0
$861$	−8.45914e9	−0.451664
$862$	0	0
$863$	2.94912e10	1.56191	0.780954	−	0.624589i	$-0.214734\pi$
$863$	2.94912e10	1.56191	0.780954	+	0.624589i	$0.214734\pi$
$864$	0	0
$865$	2.13344e10	1.12079
$866$	0	0
$867$	1.08324e10	0.564492
$868$	0	0
$869$	3.46345e10	1.79036
$870$	0	0
$871$	1.52872e10	0.783909
$872$	0	0
$873$	4.99217e9	0.253945
$874$	0	0
$875$	−7.99615e9	−0.403508
$876$	0	0
$877$	−3.29461e9	−0.164932	−0.0824660	−	0.996594i	$-0.526280\pi$
$877$	−3.29461e9	−0.164932	−0.0824660	+	0.996594i	$0.526280\pi$
$878$	0	0
$879$	−1.34734e10	−0.669139
$880$	0	0
$881$	1.93619e10	0.953966	0.476983	−	0.878913i	$-0.341730\pi$
$881$	1.93619e10	0.953966	0.476983	+	0.878913i	$0.341730\pi$
$882$	0	0
$883$	2.16110e9	0.105636	0.0528181	−	0.998604i	$-0.483180\pi$
$883$	2.16110e9	0.105636	0.0528181	+	0.998604i	$0.483180\pi$
$884$	0	0
$885$	2.73784e10	1.32772
$886$	0	0
$887$	−3.57751e10	−1.72127	−0.860633	−	0.509226i	$-0.829932\pi$
$887$	−3.57751e10	−1.72127	−0.860633	+	0.509226i	$0.829932\pi$
$888$	0	0
$889$	3.09259e10	1.47627
$890$	0	0
$891$	2.51317e9	0.119028
$892$	0	0
$893$	−8.61721e8	−0.0404936
$894$	0	0
$895$	1.74467e10	0.813454
$896$	0	0
$897$	−2.53443e9	−0.117248
$898$	0	0
$899$	−5.25131e9	−0.241051
$900$	0	0
$901$	5.28458e10	2.40699
$902$	0	0
$903$	−4.68423e8	−0.0211705
$904$	0	0
$905$	4.90592e10	2.20014
$906$	0	0
$907$	1.96026e10	0.872347	0.436173	−	0.899863i	$-0.356333\pi$
$907$	1.96026e10	0.872347	0.436173	+	0.899863i	$0.356333\pi$
$908$	0	0
$909$	7.39191e9	0.326425
$910$	0	0
$911$	3.00474e10	1.31672	0.658359	−	0.752704i	$-0.271251\pi$
$911$	3.00474e10	1.31672	0.658359	+	0.752704i	$0.271251\pi$
$912$	0	0
$913$	−1.44381e10	−0.627861
$914$	0	0
$915$	1.76276e10	0.760710
$916$	0	0
$917$	−1.46416e10	−0.627041
$918$	0	0
$919$	2.48018e10	1.05409	0.527046	−	0.849837i	$-0.323299\pi$
$919$	2.48018e10	1.05409	0.527046	+	0.849837i	$0.323299\pi$
$920$	0	0
$921$	−2.21698e10	−0.935091
$922$	0	0
$923$	2.57471e10	1.07776
$924$	0	0
$925$	1.68403e9	0.0699604
$926$	0	0
$927$	1.12344e10	0.463202
$928$	0	0
$929$	−2.43504e10	−0.996438	−0.498219	−	0.867051i	$-0.666012\pi$
$929$	−2.43504e10	−0.996438	−0.498219	+	0.867051i	$0.666012\pi$
$930$	0	0
$931$	3.75952e9	0.152689
$932$	0	0
$933$	1.46831e9	0.0591877
$934$	0	0
$935$	−4.96355e10	−1.98587
$936$	0	0
$937$	−4.85512e10	−1.92802	−0.964011	−	0.265864i	$-0.914343\pi$
$937$	−4.85512e10	−1.92802	−0.964011	+	0.265864i	$0.914343\pi$
$938$	0	0
$939$	−1.81437e10	−0.715150
$940$	0	0
$941$	2.87097e10	1.12322	0.561610	−	0.827402i	$-0.310182\pi$
$941$	2.87097e10	1.12322	0.561610	+	0.827402i	$0.310182\pi$
$942$	0	0
$943$	5.03014e9	0.195339
$944$	0	0
$945$	7.67783e9	0.295956
$946$	0	0
$947$	−4.85339e10	−1.85703	−0.928517	−	0.371290i	$-0.878916\pi$
$947$	−4.85339e10	−1.85703	−0.928517	+	0.371290i	$0.878916\pi$
$948$	0	0
$949$	3.52549e10	1.33902
$950$	0	0
$951$	−4.04209e9	−0.152396
$952$	0	0
$953$	−5.62731e9	−0.210608	−0.105304	−	0.994440i	$-0.533582\pi$
$953$	−5.62731e9	−0.210608	−0.105304	+	0.994440i	$0.533582\pi$
$954$	0	0
$955$	−3.36454e10	−1.25001
$956$	0	0
$957$	9.66357e9	0.356407
$958$	0	0
$959$	5.92816e10	2.17047
$960$	0	0
$961$	−2.26985e10	−0.825020
$962$	0	0
$963$	5.26970e9	0.190149
$964$	0	0
$965$	3.92008e10	1.40427
$966$	0	0
$967$	1.43610e10	0.510732	0.255366	−	0.966844i	$-0.417804\pi$
$967$	1.43610e10	0.510732	0.255366	+	0.966844i	$0.417804\pi$
$968$	0	0
$969$	9.72589e9	0.343397
$970$	0	0
$971$	3.71664e9	0.130281	0.0651407	−	0.997876i	$-0.479250\pi$
$971$	3.71664e9	0.130281	0.0651407	+	0.997876i	$0.479250\pi$
$972$	0	0
$973$	−5.53066e10	−1.92478
$974$	0	0
$975$	−8.59222e9	−0.296886
$976$	0	0
$977$	1.35573e10	0.465096	0.232548	−	0.972585i	$-0.425294\pi$
$977$	1.35573e10	0.465096	0.232548	+	0.972585i	$0.425294\pi$
$978$	0	0
$979$	−4.15951e10	−1.41678
$980$	0	0
$981$	−8.17715e9	−0.276542
$982$	0	0
$983$	7.75077e9	0.260260	0.130130	−	0.991497i	$-0.458461\pi$
$983$	7.75077e9	0.260260	0.130130	+	0.991497i	$0.458461\pi$
$984$	0	0
$985$	7.04693e10	2.34949
$986$	0	0
$987$	1.94803e9	0.0644890
$988$	0	0
$989$	2.78543e8	0.00915597
$990$	0	0
$991$	−4.43046e10	−1.44608	−0.723039	−	0.690808i	$-0.757255\pi$
$991$	−4.43046e10	−1.44608	−0.723039	+	0.690808i	$0.757255\pi$
$992$	0	0
$993$	−6.18824e9	−0.200560
$994$	0	0
$995$	1.52348e10	0.490292
$996$	0	0
$997$	−3.71271e9	−0.118647	−0.0593236	−	0.998239i	$-0.518894\pi$
$997$	−3.71271e9	−0.118647	−0.0593236	+	0.998239i	$0.518894\pi$
$998$	0	0
$999$	5.75204e8	0.0182533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.k.1.1	yes	3
4.3	odd	2		384.8.a.i.1.1	✓	3
8.3	odd	2		384.8.a.l.1.3	yes	3
8.5	even	2		384.8.a.j.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.i.1.1	✓	3	4.3	odd	2
384.8.a.j.1.3	yes	3	8.5	even	2
384.8.a.k.1.1	yes	3	1.1	even	1	trivial
384.8.a.l.1.3	yes	3	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)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -368.444 q^{5} -1058.71 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -368.444 q^{5} -1058.71 q^{7} +729.000 q^{9} +4728.97 q^{11} -5522.34 q^{13} -9947.99 q^{15} +28487.5 q^{17} +12644.8 q^{19} -28585.1 q^{21} +16997.8 q^{23} +57626.0 q^{25} +19683.0 q^{27} +75684.6 q^{29} -69384.1 q^{31} +127682. q^{33} +390074. q^{35} +29223.4 q^{37} -149103. q^{39} +295928. q^{41} +16387.0 q^{43} -268596. q^{45} -68148.4 q^{47} +297318. q^{49} +769163. q^{51} +1.85505e6 q^{53} -1.74236e6 q^{55} +341409. q^{57} -2.75215e6 q^{59} -1.77198e6 q^{61} -771798. q^{63} +2.03467e6 q^{65} -2.76825e6 q^{67} +458941. q^{69} -4.66235e6 q^{71} -6.38405e6 q^{73} +1.55590e6 q^{75} -5.00659e6 q^{77} +7.32391e6 q^{79} +531441. q^{81} -3.05313e6 q^{83} -1.04961e7 q^{85} +2.04349e6 q^{87} -8.79581e6 q^{89} +5.84654e6 q^{91} -1.87337e6 q^{93} -4.65889e6 q^{95} +6.84797e6 q^{97} +3.44742e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 81 q^{3} - 308 q^{5} - 6 q^{7} + 2187 q^{9}+O(q^{10}) \) 3 * q + 81 * q^3 - 308 * q^5 - 6 * q^7 + 2187 * q^9	\( 3 q + 81 q^{3} - 308 q^{5} - 6 q^{7} + 2187 q^{9} + 1468 q^{11} - 3070 q^{13} - 8316 q^{15} - 5126 q^{17} + 14168 q^{19} - 162 q^{21} - 59980 q^{23} + 31529 q^{25} + 59049 q^{27} - 71984 q^{29} - 177470 q^{31} + 39636 q^{33} - 84760 q^{35} + 184014 q^{37} - 82890 q^{39} + 657250 q^{41} - 120544 q^{43} - 224532 q^{45} - 665132 q^{47} + 1204647 q^{49} - 138402 q^{51} + 111888 q^{53} - 1854224 q^{55} + 382536 q^{57} - 1033308 q^{59} - 3584306 q^{61} - 4374 q^{63} + 1230472 q^{65} - 2344900 q^{67} - 1619460 q^{69} - 6311524 q^{71} - 2112174 q^{73} + 851283 q^{75} - 6670456 q^{77} - 2018830 q^{79} + 1594323 q^{81} - 7430484 q^{83} - 9821848 q^{85} - 1943568 q^{87} - 13300410 q^{89} + 10605052 q^{91} - 4791690 q^{93} - 303008 q^{95} + 2960414 q^{97} + 1070172 q^{99}+O(q^{100}) \) 3 * q + 81 * q^3 - 308 * q^5 - 6 * q^7 + 2187 * q^9 + 1468 * q^11 - 3070 * q^13 - 8316 * q^15 - 5126 * q^17 + 14168 * q^19 - 162 * q^21 - 59980 * q^23 + 31529 * q^25 + 59049 * q^27 - 71984 * q^29 - 177470 * q^31 + 39636 * q^33 - 84760 * q^35 + 184014 * q^37 - 82890 * q^39 + 657250 * q^41 - 120544 * q^43 - 224532 * q^45 - 665132 * q^47 + 1204647 * q^49 - 138402 * q^51 + 111888 * q^53 - 1854224 * q^55 + 382536 * q^57 - 1033308 * q^59 - 3584306 * q^61 - 4374 * q^63 + 1230472 * q^65 - 2344900 * q^67 - 1619460 * q^69 - 6311524 * q^71 - 2112174 * q^73 + 851283 * q^75 - 6670456 * q^77 - 2018830 * q^79 + 1594323 * q^81 - 7430484 * q^83 - 9821848 * q^85 - 1943568 * q^87 - 13300410 * q^89 + 10605052 * q^91 - 4791690 * q^93 - 303008 * q^95 + 2960414 * q^97 + 1070172 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more