LMFDB - Embedded newform 384.10.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,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, 10, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
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:	$197.773761087$
Analytic rank:	$1$
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} - 2070x^{2} - 13768x + 561570 $ x^4 - 2070x^2 - 13768x + 561570
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$14.0249$ 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)

$f(q)$	$=$	$q-81.0000 q^{3} +2687.31 q^{5} -950.516 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} +2687.31 q^{5} -950.516 q^{7} +6561.00 q^{9} +85357.5 q^{11} -124861. q^{13} -217672. q^{15} -21449.0 q^{17} -243033. q^{19} +76991.8 q^{21} +706695. q^{23} +5.26849e6 q^{25} -531441. q^{27} -3.90211e6 q^{29} -8.04528e6 q^{31} -6.91396e6 q^{33} -2.55433e6 q^{35} -1.67047e7 q^{37} +1.01137e7 q^{39} -3.31163e7 q^{41} -3.30635e7 q^{43} +1.76314e7 q^{45} +3.29633e7 q^{47} -3.94501e7 q^{49} +1.73737e6 q^{51} -5.61799e7 q^{53} +2.29382e8 q^{55} +1.96857e7 q^{57} -1.09356e7 q^{59} -3.77590e7 q^{61} -6.23634e6 q^{63} -3.35539e8 q^{65} +9.13783e7 q^{67} -5.72423e7 q^{69} -2.44278e8 q^{71} +1.30973e8 q^{73} -4.26747e8 q^{75} -8.11337e7 q^{77} +4.26074e8 q^{79} +4.30467e7 q^{81} -2.32850e8 q^{83} -5.76401e7 q^{85} +3.16071e8 q^{87} -2.89719e8 q^{89} +1.18682e8 q^{91} +6.51668e8 q^{93} -6.53103e8 q^{95} -2.74576e8 q^{97} +5.60031e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9	$ 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9} + 99664 q^{11} - 60840 q^{13} + 19440 q^{15} - 434952 q^{17} - 631776 q^{19} + 392040 q^{21} - 749392 q^{23} + 5991532 q^{25} - 2125764 q^{27} - 7908544 q^{29} - 11351240 q^{31} - 8072784 q^{33} + 25567008 q^{35} - 13592920 q^{37} + 4928040 q^{39} - 18838888 q^{41} - 14177920 q^{43} - 1574640 q^{45} + 37779120 q^{47} + 9409332 q^{49} + 35231112 q^{51} + 115336512 q^{53} + 184580544 q^{55} + 51173856 q^{57} + 115028080 q^{59} + 173228648 q^{61} - 31755240 q^{63} - 328077984 q^{65} + 231785104 q^{67} + 60700752 q^{69} - 197476208 q^{71} + 44629400 q^{73} - 485314092 q^{75} - 308117920 q^{77} + 355774584 q^{79} + 172186884 q^{81} + 607613328 q^{83} + 1087351392 q^{85} + 640592064 q^{87} - 1157146424 q^{89} + 847629840 q^{91} + 919450440 q^{93} + 329699328 q^{95} - 1599536472 q^{97} + 653895504 q^{99}+O(q^{100}) $ 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9 + 99664 * q^11 - 60840 * q^13 + 19440 * q^15 - 434952 * q^17 - 631776 * q^19 + 392040 * q^21 - 749392 * q^23 + 5991532 * q^25 - 2125764 * q^27 - 7908544 * q^29 - 11351240 * q^31 - 8072784 * q^33 + 25567008 * q^35 - 13592920 * q^37 + 4928040 * q^39 - 18838888 * q^41 - 14177920 * q^43 - 1574640 * q^45 + 37779120 * q^47 + 9409332 * q^49 + 35231112 * q^51 + 115336512 * q^53 + 184580544 * q^55 + 51173856 * q^57 + 115028080 * q^59 + 173228648 * q^61 - 31755240 * q^63 - 328077984 * q^65 + 231785104 * q^67 + 60700752 * q^69 - 197476208 * q^71 + 44629400 * q^73 - 485314092 * q^75 - 308117920 * q^77 + 355774584 * q^79 + 172186884 * q^81 + 607613328 * q^83 + 1087351392 * q^85 + 640592064 * q^87 - 1157146424 * q^89 + 847629840 * q^91 + 919450440 * q^93 + 329699328 * q^95 - 1599536472 * q^97 + 653895504 * 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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	2687.31	1.92288	0.961440	−	0.275016i	$-0.0886831\pi$
$5$	2687.31	1.92288	0.961440	+	0.275016i	$0.0886831\pi$
$6$	0	0
$7$	−950.516	−0.149630	−0.0748149	−	0.997197i	$-0.523837\pi$
$7$	−950.516	−0.149630	−0.0748149	+	0.997197i	$0.523837\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	85357.5	1.75782	0.878911	−	0.476986i	$-0.158271\pi$
$11$	85357.5	1.75782	0.878911	+	0.476986i	$0.158271\pi$
$12$	0	0
$13$	−124861.	−1.21250	−0.606249	−	0.795275i	$-0.707326\pi$
$13$	−124861.	−1.21250	−0.606249	+	0.795275i	$0.707326\pi$
$14$	0	0
$15$	−217672.	−1.11017
$16$	0	0
$17$	−21449.0	−0.0622856	−0.0311428	−	0.999515i	$-0.509915\pi$
$17$	−21449.0	−0.0622856	−0.0311428	+	0.999515i	$0.509915\pi$
$18$	0	0
$19$	−243033.	−0.427832	−0.213916	−	0.976852i	$-0.568622\pi$
$19$	−243033.	−0.427832	−0.213916	+	0.976852i	$0.568622\pi$
$20$	0	0
$21$	76991.8	0.0863888
$22$	0	0
$23$	706695.	0.526570	0.263285	−	0.964718i	$-0.415194\pi$
$23$	706695.	0.526570	0.263285	+	0.964718i	$0.415194\pi$
$24$	0	0
$25$	5.26849e6	2.69747
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	−3.90211e6	−1.02449	−0.512246	−	0.858839i	$-0.671186\pi$
$29$	−3.90211e6	−1.02449	−0.512246	+	0.858839i	$0.671186\pi$
$30$	0	0
$31$	−8.04528e6	−1.56464	−0.782319	−	0.622878i	$-0.785963\pi$
$31$	−8.04528e6	−1.56464	−0.782319	+	0.622878i	$0.785963\pi$
$32$	0	0
$33$	−6.91396e6	−1.01488
$34$	0	0
$35$	−2.55433e6	−0.287720
$36$	0	0
$37$	−1.67047e7	−1.46531	−0.732656	−	0.680599i	$-0.761719\pi$
$37$	−1.67047e7	−1.46531	−0.732656	+	0.680599i	$0.761719\pi$
$38$	0	0
$39$	1.01137e7	0.700036
$40$	0	0
$41$	−3.31163e7	−1.83027	−0.915134	−	0.403149i	$-0.867915\pi$
$41$	−3.31163e7	−1.83027	−0.915134	+	0.403149i	$0.867915\pi$
$42$	0	0
$43$	−3.30635e7	−1.47483	−0.737413	−	0.675442i	$-0.763953\pi$
$43$	−3.30635e7	−1.47483	−0.737413	+	0.675442i	$0.763953\pi$
$44$	0	0
$45$	1.76314e7	0.640960
$46$	0	0
$47$	3.29633e7	0.985351	0.492675	−	0.870213i	$-0.336019\pi$
$47$	3.29633e7	0.985351	0.492675	+	0.870213i	$0.336019\pi$
$48$	0	0
$49$	−3.94501e7	−0.977611
$50$	0	0
$51$	1.73737e6	0.0359606
$52$	0	0
$53$	−5.61799e7	−0.978001	−0.489000	−	0.872284i	$-0.662638\pi$
$53$	−5.61799e7	−0.978001	−0.489000	+	0.872284i	$0.662638\pi$
$54$	0	0
$55$	2.29382e8	3.38008
$56$	0	0
$57$	1.96857e7	0.247009
$58$	0	0
$59$	−1.09356e7	−0.117492	−0.0587460	−	0.998273i	$-0.518710\pi$
$59$	−1.09356e7	−0.117492	−0.0587460	+	0.998273i	$0.518710\pi$
$60$	0	0
$61$	−3.77590e7	−0.349170	−0.174585	−	0.984642i	$-0.555858\pi$
$61$	−3.77590e7	−0.349170	−0.174585	+	0.984642i	$0.555858\pi$
$62$	0	0
$63$	−6.23634e6	−0.0498766
$64$	0	0
$65$	−3.35539e8	−2.33149
$66$	0	0
$67$	9.13783e7	0.553996	0.276998	−	0.960871i	$-0.410660\pi$
$67$	9.13783e7	0.553996	0.276998	+	0.960871i	$0.410660\pi$
$68$	0	0
$69$	−5.72423e7	−0.304016
$70$	0	0
$71$	−2.44278e8	−1.14083	−0.570416	−	0.821356i	$-0.693218\pi$
$71$	−2.44278e8	−1.14083	−0.570416	+	0.821356i	$0.693218\pi$
$72$	0	0
$73$	1.30973e8	0.539795	0.269898	−	0.962889i	$-0.413010\pi$
$73$	1.30973e8	0.539795	0.269898	+	0.962889i	$0.413010\pi$
$74$	0	0
$75$	−4.26747e8	−1.55738
$76$	0	0
$77$	−8.11337e7	−0.263023
$78$	0	0
$79$	4.26074e8	1.23073	0.615366	−	0.788241i	$-0.289008\pi$
$79$	4.26074e8	1.23073	0.615366	+	0.788241i	$0.289008\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	−2.32850e8	−0.538547	−0.269274	−	0.963064i	$-0.586784\pi$
$83$	−2.32850e8	−0.538547	−0.269274	+	0.963064i	$0.586784\pi$
$84$	0	0
$85$	−5.76401e7	−0.119768
$86$	0	0
$87$	3.16071e8	0.591490
$88$	0	0
$89$	−2.89719e8	−0.489465	−0.244732	−	0.969591i	$-0.578700\pi$
$89$	−2.89719e8	−0.489465	−0.244732	+	0.969591i	$0.578700\pi$
$90$	0	0
$91$	1.18682e8	0.181426
$92$	0	0
$93$	6.51668e8	0.903344
$94$	0	0
$95$	−6.53103e8	−0.822670
$96$	0	0
$97$	−2.74576e8	−0.314912	−0.157456	−	0.987526i	$-0.550329\pi$
$97$	−2.74576e8	−0.314912	−0.157456	+	0.987526i	$0.550329\pi$
$98$	0	0
$99$	5.60031e8	0.585941
$100$	0	0
$101$	−4.51816e8	−0.432032	−0.216016	−	0.976390i	$-0.569306\pi$
$101$	−4.51816e8	−0.432032	−0.216016	+	0.976390i	$0.569306\pi$
$102$	0	0
$103$	−8.90289e8	−0.779405	−0.389703	−	0.920941i	$-0.627422\pi$
$103$	−8.90289e8	−0.779405	−0.389703	+	0.920941i	$0.627422\pi$
$104$	0	0
$105$	2.06901e8	0.166115
$106$	0	0
$107$	2.56017e9	1.88817	0.944087	−	0.329696i	$-0.106946\pi$
$107$	2.56017e9	1.88817	0.944087	+	0.329696i	$0.106946\pi$
$108$	0	0
$109$	1.33571e9	0.906345	0.453173	−	0.891423i	$-0.350292\pi$
$109$	1.33571e9	0.906345	0.453173	+	0.891423i	$0.350292\pi$
$110$	0	0
$111$	1.35308e9	0.845998
$112$	0	0
$113$	−2.87335e9	−1.65781	−0.828906	−	0.559388i	$-0.811036\pi$
$113$	−2.87335e9	−1.65781	−0.828906	+	0.559388i	$0.811036\pi$
$114$	0	0
$115$	1.89910e9	1.01253
$116$	0	0
$117$	−8.19211e8	−0.404166
$118$	0	0
$119$	2.03876e7	0.00931978
$120$	0	0
$121$	4.92796e9	2.08994
$122$	0	0
$123$	2.68242e9	1.05671
$124$	0	0
$125$	8.90939e9	3.26402
$126$	0	0
$127$	−5.49390e8	−0.187397	−0.0936987	−	0.995601i	$-0.529869\pi$
$127$	−5.49390e8	−0.187397	−0.0936987	+	0.995601i	$0.529869\pi$
$128$	0	0
$129$	2.67815e9	0.851492
$130$	0	0
$131$	−3.50142e8	−0.103878	−0.0519390	−	0.998650i	$-0.516540\pi$
$131$	−3.50142e8	−0.103878	−0.0519390	+	0.998650i	$0.516540\pi$
$132$	0	0
$133$	2.31007e8	0.0640165
$134$	0	0
$135$	−1.42814e9	−0.370058
$136$	0	0
$137$	2.45719e9	0.595930	0.297965	−	0.954577i	$-0.403692\pi$
$137$	2.45719e9	0.595930	0.297965	+	0.954577i	$0.403692\pi$
$138$	0	0
$139$	3.52695e9	0.801368	0.400684	−	0.916216i	$-0.368772\pi$
$139$	3.52695e9	0.801368	0.400684	+	0.916216i	$0.368772\pi$
$140$	0	0
$141$	−2.67003e9	−0.568893
$142$	0	0
$143$	−1.06578e10	−2.13135
$144$	0	0
$145$	−1.04862e10	−1.96997
$146$	0	0
$147$	3.19546e9	0.564424
$148$	0	0
$149$	−4.64409e9	−0.771903	−0.385952	−	0.922519i	$-0.626127\pi$
$149$	−4.64409e9	−0.771903	−0.385952	+	0.922519i	$0.626127\pi$
$150$	0	0
$151$	1.68753e9	0.264152	0.132076	−	0.991240i	$-0.457836\pi$
$151$	1.68753e9	0.264152	0.132076	+	0.991240i	$0.457836\pi$
$152$	0	0
$153$	−1.40727e8	−0.0207619
$154$	0	0
$155$	−2.16201e10	−3.00861
$156$	0	0
$157$	−5.63799e9	−0.740587	−0.370294	−	0.928915i	$-0.620743\pi$
$157$	−5.63799e9	−0.740587	−0.370294	+	0.928915i	$0.620743\pi$
$158$	0	0
$159$	4.55057e9	0.564649
$160$	0	0
$161$	−6.71725e8	−0.0787907
$162$	0	0
$163$	6.19678e9	0.687577	0.343789	−	0.939047i	$-0.388290\pi$
$163$	6.19678e9	0.687577	0.343789	+	0.939047i	$0.388290\pi$
$164$	0	0
$165$	−1.85799e10	−1.95149
$166$	0	0
$167$	9.41917e9	0.937106	0.468553	−	0.883435i	$-0.344776\pi$
$167$	9.41917e9	0.937106	0.468553	+	0.883435i	$0.344776\pi$
$168$	0	0
$169$	4.98570e9	0.470149
$170$	0	0
$171$	−1.59454e9	−0.142611
$172$	0	0
$173$	1.18003e8	0.0100158	0.00500790	−	0.999987i	$-0.498406\pi$
$173$	1.18003e8	0.0100158	0.00500790	+	0.999987i	$0.498406\pi$
$174$	0	0
$175$	−5.00778e9	−0.403621
$176$	0	0
$177$	8.85783e8	0.0678340
$178$	0	0
$179$	−9.46326e9	−0.688973	−0.344486	−	0.938791i	$-0.611947\pi$
$179$	−9.46326e9	−0.688973	−0.344486	+	0.938791i	$0.611947\pi$
$180$	0	0
$181$	−1.75573e10	−1.21592	−0.607960	−	0.793967i	$-0.708012\pi$
$181$	−1.75573e10	−1.21592	−0.607960	+	0.793967i	$0.708012\pi$
$182$	0	0
$183$	3.05848e9	0.201593
$184$	0	0
$185$	−4.48905e10	−2.81762
$186$	0	0
$187$	−1.83084e9	−0.109487
$188$	0	0
$189$	5.05143e8	0.0287963
$190$	0	0
$191$	−1.11901e10	−0.608394	−0.304197	−	0.952609i	$-0.598388\pi$
$191$	−1.11901e10	−0.608394	−0.304197	+	0.952609i	$0.598388\pi$
$192$	0	0
$193$	−2.78678e9	−0.144575	−0.0722877	−	0.997384i	$-0.523030\pi$
$193$	−2.78678e9	−0.144575	−0.0722877	+	0.997384i	$0.523030\pi$
$194$	0	0
$195$	2.71787e10	1.34608
$196$	0	0
$197$	3.22790e10	1.52694	0.763470	−	0.645843i	$-0.223494\pi$
$197$	3.22790e10	1.52694	0.763470	+	0.645843i	$0.223494\pi$
$198$	0	0
$199$	2.55007e10	1.15269	0.576345	−	0.817206i	$-0.304478\pi$
$199$	2.55007e10	1.15269	0.576345	+	0.817206i	$0.304478\pi$
$200$	0	0
$201$	−7.40164e9	−0.319850
$202$	0	0
$203$	3.70902e9	0.153294
$204$	0	0
$205$	−8.89937e10	−3.51939
$206$	0	0
$207$	4.63662e9	0.175523
$208$	0	0
$209$	−2.07447e10	−0.752053
$210$	0	0
$211$	8.47230e9	0.294259	0.147130	−	0.989117i	$-0.452997\pi$
$211$	8.47230e9	0.294259	0.147130	+	0.989117i	$0.452997\pi$
$212$	0	0
$213$	1.97865e10	0.658659
$214$	0	0
$215$	−8.88518e10	−2.83591
$216$	0	0
$217$	7.64717e9	0.234116
$218$	0	0
$219$	−1.06088e10	−0.311651
$220$	0	0
$221$	2.67814e9	0.0755211
$222$	0	0
$223$	−4.56578e10	−1.23636	−0.618178	−	0.786038i	$-0.712129\pi$
$223$	−4.56578e10	−1.23636	−0.618178	+	0.786038i	$0.712129\pi$
$224$	0	0
$225$	3.45665e10	0.899155
$226$	0	0
$227$	−3.68244e10	−0.920490	−0.460245	−	0.887792i	$-0.652238\pi$
$227$	−3.68244e10	−0.920490	−0.460245	+	0.887792i	$0.652238\pi$
$228$	0	0
$229$	2.88405e10	0.693015	0.346507	−	0.938047i	$-0.387368\pi$
$229$	2.88405e10	0.693015	0.346507	+	0.938047i	$0.387368\pi$
$230$	0	0
$231$	6.57183e9	0.151856
$232$	0	0
$233$	−3.88931e10	−0.864513	−0.432256	−	0.901751i	$-0.642282\pi$
$233$	−3.88931e10	−0.864513	−0.432256	+	0.901751i	$0.642282\pi$
$234$	0	0
$235$	8.85826e10	1.89471
$236$	0	0
$237$	−3.45120e10	−0.710564
$238$	0	0
$239$	−4.42661e10	−0.877568	−0.438784	−	0.898593i	$-0.644591\pi$
$239$	−4.42661e10	−0.877568	−0.438784	+	0.898593i	$0.644591\pi$
$240$	0	0
$241$	9.41867e10	1.79851	0.899255	−	0.437426i	$-0.144110\pi$
$241$	9.41867e10	1.79851	0.899255	+	0.437426i	$0.144110\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−1.06015e11	−1.87983
$246$	0	0
$247$	3.03452e10	0.518746
$248$	0	0
$249$	1.88608e10	0.310931
$250$	0	0
$251$	2.84495e10	0.452421	0.226211	−	0.974078i	$-0.427366\pi$
$251$	2.84495e10	0.452421	0.226211	+	0.974078i	$0.427366\pi$
$252$	0	0
$253$	6.03217e10	0.925617
$254$	0	0
$255$	4.66885e9	0.0691479
$256$	0	0
$257$	−1.16667e10	−0.166820	−0.0834100	−	0.996515i	$-0.526581\pi$
$257$	−1.16667e10	−0.166820	−0.0834100	+	0.996515i	$0.526581\pi$
$258$	0	0
$259$	1.58781e10	0.219254
$260$	0	0
$261$	−2.56017e10	−0.341497
$262$	0	0
$263$	−1.37197e11	−1.76825	−0.884127	−	0.467247i	$-0.845246\pi$
$263$	−1.37197e11	−1.76825	−0.884127	+	0.467247i	$0.845246\pi$
$264$	0	0
$265$	−1.50972e11	−1.88058
$266$	0	0
$267$	2.34672e10	0.282593
$268$	0	0
$269$	−5.76644e10	−0.671463	−0.335732	−	0.941958i	$-0.608984\pi$
$269$	−5.76644e10	−0.671463	−0.335732	+	0.941958i	$0.608984\pi$
$270$	0	0
$271$	−1.16504e11	−1.31213	−0.656067	−	0.754703i	$-0.727781\pi$
$271$	−1.16504e11	−1.31213	−0.656067	+	0.754703i	$0.727781\pi$
$272$	0	0
$273$	−9.61325e9	−0.104746
$274$	0	0
$275$	4.49705e11	4.74166
$276$	0	0
$277$	1.33146e10	0.135884	0.0679419	−	0.997689i	$-0.478357\pi$
$277$	1.33146e10	0.135884	0.0679419	+	0.997689i	$0.478357\pi$
$278$	0	0
$279$	−5.27851e10	−0.521546
$280$	0	0
$281$	−3.34219e10	−0.319781	−0.159891	−	0.987135i	$-0.551114\pi$
$281$	−3.34219e10	−0.319781	−0.159891	+	0.987135i	$0.551114\pi$
$282$	0	0
$283$	5.11954e10	0.474452	0.237226	−	0.971455i	$-0.423762\pi$
$283$	5.11954e10	0.474452	0.237226	+	0.971455i	$0.423762\pi$
$284$	0	0
$285$	5.29014e10	0.474969
$286$	0	0
$287$	3.14776e10	0.273863
$288$	0	0
$289$	−1.18128e11	−0.996121
$290$	0	0
$291$	2.22407e10	0.181815
$292$	0	0
$293$	1.69283e11	1.34187	0.670934	−	0.741517i	$-0.265893\pi$
$293$	1.69283e11	1.34187	0.670934	+	0.741517i	$0.265893\pi$
$294$	0	0
$295$	−2.93873e10	−0.225923
$296$	0	0
$297$	−4.53625e10	−0.338293
$298$	0	0
$299$	−8.82384e10	−0.638465
$300$	0	0
$301$	3.14274e10	0.220678
$302$	0	0
$303$	3.65971e10	0.249434
$304$	0	0
$305$	−1.01470e11	−0.671411
$306$	0	0
$307$	−1.67510e11	−1.07626	−0.538130	−	0.842862i	$-0.680869\pi$
$307$	−1.67510e11	−1.07626	−0.538130	+	0.842862i	$0.680869\pi$
$308$	0	0
$309$	7.21134e10	0.449990
$310$	0	0
$311$	2.42940e11	1.47257	0.736287	−	0.676669i	$-0.236577\pi$
$311$	2.42940e11	1.47257	0.736287	+	0.676669i	$0.236577\pi$
$312$	0	0
$313$	−2.59566e11	−1.52862	−0.764308	−	0.644851i	$-0.776919\pi$
$313$	−2.59566e11	−1.52862	−0.764308	+	0.644851i	$0.776919\pi$
$314$	0	0
$315$	−1.67589e10	−0.0959067
$316$	0	0
$317$	−2.98973e11	−1.66290	−0.831448	−	0.555603i	$-0.812488\pi$
$317$	−2.98973e11	−1.66290	−0.831448	+	0.555603i	$0.812488\pi$
$318$	0	0
$319$	−3.33074e11	−1.80087
$320$	0	0
$321$	−2.07374e11	−1.09014
$322$	0	0
$323$	5.21282e9	0.0266478
$324$	0	0
$325$	−6.57827e11	−3.27067
$326$	0	0
$327$	−1.08193e11	−0.523279
$328$	0	0
$329$	−3.13322e10	−0.147438
$330$	0	0
$331$	3.64913e11	1.67095	0.835475	−	0.549528i	$-0.185193\pi$
$331$	3.64913e11	1.67095	0.835475	+	0.549528i	$0.185193\pi$
$332$	0	0
$333$	−1.09599e11	−0.488437
$334$	0	0
$335$	2.45561e11	1.06527
$336$	0	0
$337$	−1.45199e11	−0.613237	−0.306618	−	0.951833i	$-0.599197\pi$
$337$	−1.45199e11	−0.613237	−0.306618	+	0.951833i	$0.599197\pi$
$338$	0	0
$339$	2.32741e11	0.957139
$340$	0	0
$341$	−6.86725e11	−2.75035
$342$	0	0
$343$	7.58547e10	0.295910
$344$	0	0
$345$	−1.53827e11	−0.584585
$346$	0	0
$347$	1.28543e10	0.0475955	0.0237978	−	0.999717i	$-0.492424\pi$
$347$	1.28543e10	0.0475955	0.0237978	+	0.999717i	$0.492424\pi$
$348$	0	0
$349$	1.59013e11	0.573744	0.286872	−	0.957969i	$-0.407385\pi$
$349$	1.59013e11	0.573744	0.286872	+	0.957969i	$0.407385\pi$
$350$	0	0
$351$	6.63561e10	0.233345
$352$	0	0
$353$	1.03715e11	0.355512	0.177756	−	0.984075i	$-0.443116\pi$
$353$	1.03715e11	0.355512	0.177756	+	0.984075i	$0.443116\pi$
$354$	0	0
$355$	−6.56449e11	−2.19368
$356$	0	0
$357$	−1.65140e9	−0.00538078
$358$	0	0
$359$	4.16136e10	0.132224	0.0661120	−	0.997812i	$-0.478941\pi$
$359$	4.16136e10	0.132224	0.0661120	+	0.997812i	$0.478941\pi$
$360$	0	0
$361$	−2.63623e11	−0.816959
$362$	0	0
$363$	−3.99165e11	−1.20663
$364$	0	0
$365$	3.51965e11	1.03796
$366$	0	0
$367$	−3.78771e11	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$367$	−3.78771e11	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$368$	0	0
$369$	−2.17276e11	−0.610089
$370$	0	0
$371$	5.33999e10	0.146338
$372$	0	0
$373$	−7.90824e10	−0.211539	−0.105769	−	0.994391i	$-0.533731\pi$
$373$	−7.90824e10	−0.211539	−0.105769	+	0.994391i	$0.533731\pi$
$374$	0	0
$375$	−7.21661e11	−1.88448
$376$	0	0
$377$	4.87220e11	1.24219
$378$	0	0
$379$	5.43207e11	1.35235	0.676175	−	0.736741i	$-0.263636\pi$
$379$	5.43207e11	1.35235	0.676175	+	0.736741i	$0.263636\pi$
$380$	0	0
$381$	4.45006e10	0.108194
$382$	0	0
$383$	6.88749e11	1.63556	0.817780	−	0.575531i	$-0.195204\pi$
$383$	6.88749e11	1.63556	0.817780	+	0.575531i	$0.195204\pi$
$384$	0	0
$385$	−2.18031e11	−0.505761
$386$	0	0
$387$	−2.16930e11	−0.491609
$388$	0	0
$389$	−4.70395e11	−1.04157	−0.520786	−	0.853687i	$-0.674361\pi$
$389$	−4.70395e11	−1.04157	−0.520786	+	0.853687i	$0.674361\pi$
$390$	0	0
$391$	−1.51579e10	−0.0327977
$392$	0	0
$393$	2.83615e10	0.0599740
$394$	0	0
$395$	1.14499e12	2.36655
$396$	0	0
$397$	5.27728e11	1.06623	0.533117	−	0.846042i	$-0.321021\pi$
$397$	5.27728e11	1.06623	0.533117	+	0.846042i	$0.321021\pi$
$398$	0	0
$399$	−1.87115e10	−0.0369599
$400$	0	0
$401$	−1.78849e11	−0.345411	−0.172705	−	0.984974i	$-0.555251\pi$
$401$	−1.78849e11	−0.345411	−0.172705	+	0.984974i	$0.555251\pi$
$402$	0	0
$403$	1.00454e12	1.89712
$404$	0	0
$405$	1.15680e11	0.213653
$406$	0	0
$407$	−1.42587e12	−2.57576
$408$	0	0
$409$	4.40686e11	0.778707	0.389353	−	0.921088i	$-0.372698\pi$
$409$	4.40686e11	0.778707	0.389353	+	0.921088i	$0.372698\pi$
$410$	0	0
$411$	−1.99032e11	−0.344061
$412$	0	0
$413$	1.03945e10	0.0175803
$414$	0	0
$415$	−6.25738e11	−1.03556
$416$	0	0
$417$	−2.85683e11	−0.462670
$418$	0	0
$419$	−3.08642e10	−0.0489207	−0.0244603	−	0.999701i	$-0.507787\pi$
$419$	−3.08642e10	−0.0489207	−0.0244603	+	0.999701i	$0.507787\pi$
$420$	0	0
$421$	−2.60022e11	−0.403405	−0.201703	−	0.979447i	$-0.564647\pi$
$421$	−2.60022e11	−0.403405	−0.201703	+	0.979447i	$0.564647\pi$
$422$	0	0
$423$	2.16272e11	0.328450
$424$	0	0
$425$	−1.13004e11	−0.168013
$426$	0	0
$427$	3.58906e10	0.0522462
$428$	0	0
$429$	8.63282e11	1.23054
$430$	0	0
$431$	−5.19190e11	−0.724734	−0.362367	−	0.932035i	$-0.618031\pi$
$431$	−5.19190e11	−0.724734	−0.362367	+	0.932035i	$0.618031\pi$
$432$	0	0
$433$	1.00829e12	1.37844	0.689222	−	0.724550i	$-0.257953\pi$
$433$	1.00829e12	1.37844	0.689222	+	0.724550i	$0.257953\pi$
$434$	0	0
$435$	8.49379e11	1.13736
$436$	0	0
$437$	−1.71750e11	−0.225284
$438$	0	0
$439$	2.62358e10	0.0337135	0.0168568	−	0.999858i	$-0.494634\pi$
$439$	2.62358e10	0.0337135	0.0168568	+	0.999858i	$0.494634\pi$
$440$	0	0
$441$	−2.58832e11	−0.325870
$442$	0	0
$443$	−6.82095e10	−0.0841450	−0.0420725	−	0.999115i	$-0.513396\pi$
$443$	−6.82095e10	−0.0841450	−0.0420725	+	0.999115i	$0.513396\pi$
$444$	0	0
$445$	−7.78563e11	−0.941182
$446$	0	0
$447$	3.76172e11	0.445658
$448$	0	0
$449$	1.23102e12	1.42941	0.714707	−	0.699424i	$-0.246560\pi$
$449$	1.23102e12	1.42941	0.714707	+	0.699424i	$0.246560\pi$
$450$	0	0
$451$	−2.82673e12	−3.21728
$452$	0	0
$453$	−1.36690e11	−0.152508
$454$	0	0
$455$	3.18935e11	0.348860
$456$	0	0
$457$	2.33313e11	0.250216	0.125108	−	0.992143i	$-0.460072\pi$
$457$	2.33313e11	0.250216	0.125108	+	0.992143i	$0.460072\pi$
$458$	0	0
$459$	1.13989e10	0.0119869
$460$	0	0
$461$	7.98394e11	0.823310	0.411655	−	0.911340i	$-0.364951\pi$
$461$	7.98394e11	0.823310	0.411655	+	0.911340i	$0.364951\pi$
$462$	0	0
$463$	−1.46529e11	−0.148186	−0.0740931	−	0.997251i	$-0.523606\pi$
$463$	−1.46529e11	−0.148186	−0.0740931	+	0.997251i	$0.523606\pi$
$464$	0	0
$465$	1.75123e12	1.73702
$466$	0	0
$467$	1.53277e12	1.49125	0.745627	−	0.666364i	$-0.232150\pi$
$467$	1.53277e12	1.49125	0.745627	+	0.666364i	$0.232150\pi$
$468$	0	0
$469$	−8.68566e10	−0.0828943
$470$	0	0
$471$	4.56677e11	0.427578
$472$	0	0
$473$	−2.82222e12	−2.59248
$474$	0	0
$475$	−1.28041e12	−1.15406
$476$	0	0
$477$	−3.68596e11	−0.326000
$478$	0	0
$479$	−7.94475e11	−0.689558	−0.344779	−	0.938684i	$-0.612046\pi$
$479$	−7.94475e11	−0.689558	−0.344779	+	0.938684i	$0.612046\pi$
$480$	0	0
$481$	2.08576e12	1.77669
$482$	0	0
$483$	5.44097e10	0.0454898
$484$	0	0
$485$	−7.37870e11	−0.605539
$486$	0	0
$487$	−1.34532e12	−1.08379	−0.541893	−	0.840447i	$-0.682292\pi$
$487$	−1.34532e12	−1.08379	−0.541893	+	0.840447i	$0.682292\pi$
$488$	0	0
$489$	−5.01939e11	−0.396973
$490$	0	0
$491$	−1.75188e12	−1.36031	−0.680154	−	0.733069i	$-0.738087\pi$
$491$	−1.75188e12	−1.36031	−0.680154	+	0.733069i	$0.738087\pi$
$492$	0	0
$493$	8.36964e10	0.0638110
$494$	0	0
$495$	1.50497e12	1.12669
$496$	0	0
$497$	2.32190e11	0.170702
$498$	0	0
$499$	7.66101e11	0.553138	0.276569	−	0.960994i	$-0.410803\pi$
$499$	7.66101e11	0.553138	0.276569	+	0.960994i	$0.410803\pi$
$500$	0	0
$501$	−7.62953e11	−0.541038
$502$	0	0
$503$	2.02872e12	1.41308	0.706540	−	0.707673i	$-0.250255\pi$
$503$	2.02872e12	1.41308	0.706540	+	0.707673i	$0.250255\pi$
$504$	0	0
$505$	−1.21417e12	−0.830745
$506$	0	0
$507$	−4.03842e11	−0.271441
$508$	0	0
$509$	3.08049e11	0.203418	0.101709	−	0.994814i	$-0.467569\pi$
$509$	3.08049e11	0.203418	0.101709	+	0.994814i	$0.467569\pi$
$510$	0	0
$511$	−1.24492e11	−0.0807695
$512$	0	0
$513$	1.29158e11	0.0823364
$514$	0	0
$515$	−2.39248e12	−1.49870
$516$	0	0
$517$	2.81367e12	1.73207
$518$	0	0
$519$	−9.55824e9	−0.00578262
$520$	0	0
$521$	−2.45259e12	−1.45833	−0.729165	−	0.684338i	$-0.760091\pi$
$521$	−2.45259e12	−1.45833	−0.729165	+	0.684338i	$0.760091\pi$
$522$	0	0
$523$	2.41650e12	1.41231	0.706154	−	0.708059i	$-0.250429\pi$
$523$	2.41650e12	1.41231	0.706154	+	0.708059i	$0.250429\pi$
$524$	0	0
$525$	4.05630e11	0.233031
$526$	0	0
$527$	1.72564e11	0.0974543
$528$	0	0
$529$	−1.30174e12	−0.722724
$530$	0	0
$531$	−7.17484e10	−0.0391640
$532$	0	0
$533$	4.13493e12	2.21920
$534$	0	0
$535$	6.87996e12	3.63073
$536$	0	0
$537$	7.66524e11	0.397779
$538$	0	0
$539$	−3.36737e12	−1.71847
$540$	0	0
$541$	−1.14349e12	−0.573911	−0.286955	−	0.957944i	$-0.592643\pi$
$541$	−1.14349e12	−0.573911	−0.286955	+	0.957944i	$0.592643\pi$
$542$	0	0
$543$	1.42215e12	0.702012
$544$	0	0
$545$	3.58947e12	1.74279
$546$	0	0
$547$	−1.20917e12	−0.577492	−0.288746	−	0.957406i	$-0.593238\pi$
$547$	−1.20917e12	−0.577492	−0.288746	+	0.957406i	$0.593238\pi$
$548$	0	0
$549$	−2.47737e11	−0.116390
$550$	0	0
$551$	9.48340e11	0.438310
$552$	0	0
$553$	−4.04991e11	−0.184154
$554$	0	0
$555$	3.63613e12	1.62675
$556$	0	0
$557$	2.18500e12	0.961838	0.480919	−	0.876765i	$-0.340303\pi$
$557$	2.18500e12	0.961838	0.480919	+	0.876765i	$0.340303\pi$
$558$	0	0
$559$	4.12833e12	1.78822
$560$	0	0
$561$	1.48298e11	0.0632123
$562$	0	0
$563$	1.68753e11	0.0707886	0.0353943	−	0.999373i	$-0.488731\pi$
$563$	1.68753e11	0.0707886	0.0353943	+	0.999373i	$0.488731\pi$
$564$	0	0
$565$	−7.72157e12	−3.18777
$566$	0	0
$567$	−4.09166e10	−0.0166255
$568$	0	0
$569$	−3.00268e12	−1.20089	−0.600446	−	0.799665i	$-0.705010\pi$
$569$	−3.00268e12	−1.20089	−0.600446	+	0.799665i	$0.705010\pi$
$570$	0	0
$571$	−4.51453e12	−1.77726	−0.888628	−	0.458628i	$-0.848341\pi$
$571$	−4.51453e12	−1.77726	−0.888628	+	0.458628i	$0.848341\pi$
$572$	0	0
$573$	9.06401e11	0.351257
$574$	0	0
$575$	3.72321e12	1.42041
$576$	0	0
$577$	3.10024e12	1.16441	0.582203	−	0.813044i	$-0.302191\pi$
$577$	3.10024e12	1.16441	0.582203	+	0.813044i	$0.302191\pi$
$578$	0	0
$579$	2.25729e11	0.0834706
$580$	0	0
$581$	2.21327e11	0.0805828
$582$	0	0
$583$	−4.79537e12	−1.71915
$584$	0	0
$585$	−2.20147e12	−0.777162
$586$	0	0
$587$	−2.50051e11	−0.0869275	−0.0434637	−	0.999055i	$-0.513839\pi$
$587$	−2.50051e11	−0.0869275	−0.0434637	+	0.999055i	$0.513839\pi$
$588$	0	0
$589$	1.95527e12	0.669402
$590$	0	0
$591$	−2.61460e12	−0.881580
$592$	0	0
$593$	−1.58185e12	−0.525314	−0.262657	−	0.964889i	$-0.584599\pi$
$593$	−1.58185e12	−0.525314	−0.262657	+	0.964889i	$0.584599\pi$
$594$	0	0
$595$	5.47878e10	0.0179208
$596$	0	0
$597$	−2.06555e12	−0.665506
$598$	0	0
$599$	−4.55230e12	−1.44481	−0.722404	−	0.691471i	$-0.756963\pi$
$599$	−4.55230e12	−1.44481	−0.722404	+	0.691471i	$0.756963\pi$
$600$	0	0
$601$	5.54016e12	1.73216	0.866079	−	0.499908i	$-0.166633\pi$
$601$	5.54016e12	1.73216	0.866079	+	0.499908i	$0.166633\pi$
$602$	0	0
$603$	5.99533e11	0.184665
$604$	0	0
$605$	1.32429e13	4.01870
$606$	0	0
$607$	−2.77998e12	−0.831175	−0.415587	−	0.909553i	$-0.636424\pi$
$607$	−2.77998e12	−0.831175	−0.415587	+	0.909553i	$0.636424\pi$
$608$	0	0
$609$	−3.00430e11	−0.0885046
$610$	0	0
$611$	−4.11583e12	−1.19474
$612$	0	0
$613$	−1.59888e11	−0.0457346	−0.0228673	−	0.999739i	$-0.507280\pi$
$613$	−1.59888e11	−0.0457346	−0.0228673	+	0.999739i	$0.507280\pi$
$614$	0	0
$615$	7.20849e12	2.03192
$616$	0	0
$617$	6.35346e12	1.76493	0.882464	−	0.470380i	$-0.155883\pi$
$617$	6.35346e12	1.76493	0.882464	+	0.470380i	$0.155883\pi$
$618$	0	0
$619$	−1.67188e12	−0.457716	−0.228858	−	0.973460i	$-0.573499\pi$
$619$	−1.67188e12	−0.457716	−0.228858	+	0.973460i	$0.573499\pi$
$620$	0	0
$621$	−3.75566e11	−0.101339
$622$	0	0
$623$	2.75382e11	0.0732386
$624$	0	0
$625$	1.36522e13	3.57885
$626$	0	0
$627$	1.68032e12	0.434198
$628$	0	0
$629$	3.58299e11	0.0912678
$630$	0	0
$631$	−3.69214e12	−0.927141	−0.463571	−	0.886060i	$-0.653432\pi$
$631$	−3.69214e12	−0.927141	−0.463571	+	0.886060i	$0.653432\pi$
$632$	0	0
$633$	−6.86256e11	−0.169891
$634$	0	0
$635$	−1.47638e12	−0.360343
$636$	0	0
$637$	4.92577e12	1.18535
$638$	0	0
$639$	−1.60271e12	−0.380277
$640$	0	0
$641$	−2.43986e12	−0.570826	−0.285413	−	0.958405i	$-0.592131\pi$
$641$	−2.43986e12	−0.570826	−0.285413	+	0.958405i	$0.592131\pi$
$642$	0	0
$643$	1.91171e12	0.441034	0.220517	−	0.975383i	$-0.429226\pi$
$643$	1.91171e12	0.441034	0.220517	+	0.975383i	$0.429226\pi$
$644$	0	0
$645$	7.19699e12	1.63732
$646$	0	0
$647$	6.33401e12	1.42105	0.710525	−	0.703672i	$-0.248458\pi$
$647$	6.33401e12	1.42105	0.710525	+	0.703672i	$0.248458\pi$
$648$	0	0
$649$	−9.33435e11	−0.206530
$650$	0	0
$651$	−6.19421e11	−0.135167
$652$	0	0
$653$	2.21499e12	0.476718	0.238359	−	0.971177i	$-0.423390\pi$
$653$	2.21499e12	0.476718	0.238359	+	0.971177i	$0.423390\pi$
$654$	0	0
$655$	−9.40939e11	−0.199745
$656$	0	0
$657$	8.59314e11	0.179932
$658$	0	0
$659$	3.59753e12	0.743054	0.371527	−	0.928422i	$-0.378834\pi$
$659$	3.59753e12	0.743054	0.371527	+	0.928422i	$0.378834\pi$
$660$	0	0
$661$	−2.41730e12	−0.492521	−0.246260	−	0.969204i	$-0.579202\pi$
$661$	−2.41730e12	−0.492521	−0.246260	+	0.969204i	$0.579202\pi$
$662$	0	0
$663$	−2.16929e11	−0.0436021
$664$	0	0
$665$	6.20785e11	0.123096
$666$	0	0
$667$	−2.75760e12	−0.539467
$668$	0	0
$669$	3.69828e12	0.713810
$670$	0	0
$671$	−3.22302e12	−0.613778
$672$	0	0
$673$	−1.32596e12	−0.249152	−0.124576	−	0.992210i	$-0.539757\pi$
$673$	−1.32596e12	−0.249152	−0.124576	+	0.992210i	$0.539757\pi$
$674$	0	0
$675$	−2.79989e12	−0.519127
$676$	0	0
$677$	−2.49781e12	−0.456995	−0.228497	−	0.973545i	$-0.573381\pi$
$677$	−2.49781e12	−0.456995	−0.228497	+	0.973545i	$0.573381\pi$
$678$	0	0
$679$	2.60989e11	0.0471203
$680$	0	0
$681$	2.98277e12	0.531445
$682$	0	0
$683$	1.02091e13	1.79512	0.897558	−	0.440895i	$-0.145339\pi$
$683$	1.02091e13	1.79512	0.897558	+	0.440895i	$0.145339\pi$
$684$	0	0
$685$	6.60321e12	1.14590
$686$	0	0
$687$	−2.33608e12	−0.400112
$688$	0	0
$689$	7.01466e12	1.18582
$690$	0	0
$691$	5.33760e12	0.890625	0.445313	−	0.895375i	$-0.353093\pi$
$691$	5.33760e12	0.890625	0.445313	+	0.895375i	$0.353093\pi$
$692$	0	0
$693$	−5.32318e11	−0.0876742
$694$	0	0
$695$	9.47798e12	1.54093
$696$	0	0
$697$	7.10313e11	0.113999
$698$	0	0
$699$	3.15034e12	0.499127
$700$	0	0
$701$	−6.96175e12	−1.08890	−0.544449	−	0.838794i	$-0.683261\pi$
$701$	−6.96175e12	−1.08890	−0.544449	+	0.838794i	$0.683261\pi$
$702$	0	0
$703$	4.05978e12	0.626908
$704$	0	0
$705$	−7.17519e12	−1.09391
$706$	0	0
$707$	4.29459e11	0.0646449
$708$	0	0
$709$	−1.14423e13	−1.70061	−0.850307	−	0.526287i	$-0.823584\pi$
$709$	−1.14423e13	−1.70061	−0.850307	+	0.526287i	$0.823584\pi$
$710$	0	0
$711$	2.79547e12	0.410244
$712$	0	0
$713$	−5.68556e12	−0.823892
$714$	0	0
$715$	−2.86408e13	−4.09834
$716$	0	0
$717$	3.58555e12	0.506664
$718$	0	0
$719$	−5.43048e12	−0.757806	−0.378903	−	0.925436i	$-0.623699\pi$
$719$	−5.43048e12	−0.757806	−0.378903	+	0.925436i	$0.623699\pi$
$720$	0	0
$721$	8.46234e11	0.116622
$722$	0	0
$723$	−7.62912e12	−1.03837
$724$	0	0
$725$	−2.05582e13	−2.76353
$726$	0	0
$727$	−6.18258e12	−0.820852	−0.410426	−	0.911894i	$-0.634620\pi$
$727$	−6.18258e12	−0.820852	−0.410426	+	0.911894i	$0.634620\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	7.09180e11	0.0918604
$732$	0	0
$733$	1.14558e13	1.46574	0.732870	−	0.680368i	$-0.238180\pi$
$733$	1.14558e13	1.46574	0.732870	+	0.680368i	$0.238180\pi$
$734$	0	0
$735$	8.58718e12	1.08532
$736$	0	0
$737$	7.79983e12	0.973826
$738$	0	0
$739$	−1.13658e13	−1.40184	−0.700920	−	0.713240i	$-0.747227\pi$
$739$	−1.13658e13	−1.40184	−0.700920	+	0.713240i	$0.747227\pi$
$740$	0	0
$741$	−2.45796e12	−0.299498
$742$	0	0
$743$	1.42813e13	1.71917	0.859584	−	0.510995i	$-0.170723\pi$
$743$	1.42813e13	1.71917	0.859584	+	0.510995i	$0.170723\pi$
$744$	0	0
$745$	−1.24801e13	−1.48428
$746$	0	0
$747$	−1.52773e12	−0.179516
$748$	0	0
$749$	−2.43348e12	−0.282527
$750$	0	0
$751$	2.50025e12	0.286816	0.143408	−	0.989664i	$-0.454194\pi$
$751$	2.50025e12	0.286816	0.143408	+	0.989664i	$0.454194\pi$
$752$	0	0
$753$	−2.30441e12	−0.261205
$754$	0	0
$755$	4.53490e12	0.507933
$756$	0	0
$757$	−1.03072e13	−1.14080	−0.570402	−	0.821366i	$-0.693213\pi$
$757$	−1.03072e13	−1.14080	−0.570402	+	0.821366i	$0.693213\pi$
$758$	0	0
$759$	−4.88606e12	−0.534405
$760$	0	0
$761$	−8.02116e12	−0.866974	−0.433487	−	0.901160i	$-0.642717\pi$
$761$	−8.02116e12	−0.866974	−0.433487	+	0.901160i	$0.642717\pi$
$762$	0	0
$763$	−1.26962e12	−0.135616
$764$	0	0
$765$	−3.78177e11	−0.0399226
$766$	0	0
$767$	1.36543e12	0.142459
$768$	0	0
$769$	1.09614e13	1.13031	0.565154	−	0.824985i	$-0.308817\pi$
$769$	1.09614e13	1.13031	0.565154	+	0.824985i	$0.308817\pi$
$770$	0	0
$771$	9.45001e11	0.0963136
$772$	0	0
$773$	7.65013e12	0.770657	0.385329	−	0.922779i	$-0.374088\pi$
$773$	7.65013e12	0.770657	0.385329	+	0.922779i	$0.374088\pi$
$774$	0	0
$775$	−4.23865e13	−4.22055
$776$	0	0
$777$	−1.28612e12	−0.126587
$778$	0	0
$779$	8.04835e12	0.783048
$780$	0	0
$781$	−2.08509e13	−2.00538
$782$	0	0
$783$	2.07374e12	0.197163
$784$	0	0
$785$	−1.51510e13	−1.42406
$786$	0	0
$787$	−2.07846e13	−1.93133	−0.965663	−	0.259797i	$-0.916344\pi$
$787$	−2.07846e13	−1.93133	−0.965663	+	0.259797i	$0.916344\pi$
$788$	0	0
$789$	1.11130e13	1.02090
$790$	0	0
$791$	2.73116e12	0.248058
$792$	0	0
$793$	4.71462e12	0.423367
$794$	0	0
$795$	1.22288e13	1.08575
$796$	0	0
$797$	−5.87904e12	−0.516112	−0.258056	−	0.966130i	$-0.583082\pi$
$797$	−5.87904e12	−0.516112	−0.258056	+	0.966130i	$0.583082\pi$
$798$	0	0
$799$	−7.07032e11	−0.0613731
$800$	0	0
$801$	−1.90084e12	−0.163155
$802$	0	0
$803$	1.11795e13	0.948863
$804$	0	0
$805$	−1.80513e12	−0.151505
$806$	0	0
$807$	4.67081e12	0.387669
$808$	0	0
$809$	1.17127e13	0.961363	0.480682	−	0.876895i	$-0.340389\pi$
$809$	1.17127e13	0.961363	0.480682	+	0.876895i	$0.340389\pi$
$810$	0	0
$811$	2.27415e12	0.184597	0.0922987	−	0.995731i	$-0.470579\pi$
$811$	2.27415e12	0.184597	0.0922987	+	0.995731i	$0.470579\pi$
$812$	0	0
$813$	9.43681e12	0.757561
$814$	0	0
$815$	1.66526e13	1.32213
$816$	0	0
$817$	8.03552e12	0.630979
$818$	0	0
$819$	7.78673e11	0.0604753
$820$	0	0
$821$	−8.39279e12	−0.644707	−0.322353	−	0.946619i	$-0.604474\pi$
$821$	−8.39279e12	−0.644707	−0.322353	+	0.946619i	$0.604474\pi$
$822$	0	0
$823$	3.70938e12	0.281839	0.140920	−	0.990021i	$-0.454994\pi$
$823$	3.70938e12	0.281839	0.140920	+	0.990021i	$0.454994\pi$
$824$	0	0
$825$	−3.64261e13	−2.73760
$826$	0	0
$827$	2.53080e13	1.88141	0.940706	−	0.339224i	$-0.110164\pi$
$827$	2.53080e13	1.88141	0.940706	+	0.339224i	$0.110164\pi$
$828$	0	0
$829$	−1.59061e13	−1.16968	−0.584842	−	0.811147i	$-0.698844\pi$
$829$	−1.59061e13	−1.16968	−0.584842	+	0.811147i	$0.698844\pi$
$830$	0	0
$831$	−1.07848e12	−0.0784525
$832$	0	0
$833$	8.46167e11	0.0608911
$834$	0	0
$835$	2.53122e13	1.80194
$836$	0	0
$837$	4.27559e12	0.301115
$838$	0	0
$839$	2.57050e12	0.179097	0.0895487	−	0.995982i	$-0.471458\pi$
$839$	2.57050e12	0.179097	0.0895487	+	0.995982i	$0.471458\pi$
$840$	0	0
$841$	7.19297e11	0.0495822
$842$	0	0
$843$	2.70718e12	0.184626
$844$	0	0
$845$	1.33981e13	0.904041
$846$	0	0
$847$	−4.68411e12	−0.312717
$848$	0	0
$849$	−4.14683e12	−0.273925
$850$	0	0
$851$	−1.18051e13	−0.771590
$852$	0	0
$853$	6.41468e12	0.414863	0.207431	−	0.978250i	$-0.433490\pi$
$853$	6.41468e12	0.414863	0.207431	+	0.978250i	$0.433490\pi$
$854$	0	0
$855$	−4.28501e12	−0.274223
$856$	0	0
$857$	−1.31074e13	−0.830046	−0.415023	−	0.909811i	$-0.636226\pi$
$857$	−1.31074e13	−0.830046	−0.415023	+	0.909811i	$0.636226\pi$
$858$	0	0
$859$	1.06234e13	0.665723	0.332861	−	0.942976i	$-0.391986\pi$
$859$	1.06234e13	0.665723	0.332861	+	0.942976i	$0.391986\pi$
$860$	0	0
$861$	−2.54969e12	−0.158115
$862$	0	0
$863$	4.71438e12	0.289318	0.144659	−	0.989482i	$-0.453791\pi$
$863$	4.71438e12	0.289318	0.144659	+	0.989482i	$0.453791\pi$
$864$	0	0
$865$	3.17110e11	0.0192592
$866$	0	0
$867$	9.56835e12	0.575110
$868$	0	0
$869$	3.63687e13	2.16341
$870$	0	0
$871$	−1.14096e13	−0.671719
$872$	0	0
$873$	−1.80149e12	−0.104971
$874$	0	0
$875$	−8.46852e12	−0.488395
$876$	0	0
$877$	4.59207e12	0.262126	0.131063	−	0.991374i	$-0.458161\pi$
$877$	4.59207e12	0.262126	0.131063	+	0.991374i	$0.458161\pi$
$878$	0	0
$879$	−1.37119e13	−0.774728
$880$	0	0
$881$	1.82236e13	1.01916	0.509580	−	0.860423i	$-0.329801\pi$
$881$	1.82236e13	1.01916	0.509580	+	0.860423i	$0.329801\pi$
$882$	0	0
$883$	−3.47414e12	−0.192320	−0.0961598	−	0.995366i	$-0.530656\pi$
$883$	−3.47414e12	−0.192320	−0.0961598	+	0.995366i	$0.530656\pi$
$884$	0	0
$885$	2.38037e12	0.130437
$886$	0	0
$887$	−7.73676e12	−0.419665	−0.209833	−	0.977737i	$-0.567292\pi$
$887$	−7.73676e12	−0.419665	−0.209833	+	0.977737i	$0.567292\pi$
$888$	0	0
$889$	5.22204e11	0.0280403
$890$	0	0
$891$	3.67436e12	0.195314
$892$	0	0
$893$	−8.01117e12	−0.421565
$894$	0	0
$895$	−2.54307e13	−1.32481
$896$	0	0
$897$	7.14731e12	0.368618
$898$	0	0
$899$	3.13936e13	1.60296
$900$	0	0
$901$	1.20500e12	0.0609153
$902$	0	0
$903$	−2.54562e12	−0.127409
$904$	0	0
$905$	−4.71820e13	−2.33807
$906$	0	0
$907$	2.47553e13	1.21461	0.607303	−	0.794470i	$-0.292251\pi$
$907$	2.47553e13	1.21461	0.607303	+	0.794470i	$0.292251\pi$
$908$	0	0
$909$	−2.96437e12	−0.144011
$910$	0	0
$911$	−3.49811e13	−1.68268	−0.841339	−	0.540508i	$-0.818232\pi$
$911$	−3.49811e13	−1.68268	−0.841339	+	0.540508i	$0.818232\pi$
$912$	0	0
$913$	−1.98755e13	−0.946670
$914$	0	0
$915$	8.21907e12	0.387639
$916$	0	0
$917$	3.32816e11	0.0155432
$918$	0	0
$919$	−1.70135e13	−0.786816	−0.393408	−	0.919364i	$-0.628704\pi$
$919$	−1.70135e13	−0.786816	−0.393408	+	0.919364i	$0.628704\pi$
$920$	0	0
$921$	1.35683e13	0.621379
$922$	0	0
$923$	3.05007e13	1.38325
$924$	0	0
$925$	−8.80083e13	−3.95263
$926$	0	0
$927$	−5.84118e12	−0.259802
$928$	0	0
$929$	7.63520e12	0.336318	0.168159	−	0.985760i	$-0.446218\pi$
$929$	7.63520e12	0.336318	0.168159	+	0.985760i	$0.446218\pi$
$930$	0	0
$931$	9.58767e12	0.418254
$932$	0	0
$933$	−1.96781e13	−0.850191
$934$	0	0
$935$	−4.92002e12	−0.210530
$936$	0	0
$937$	−3.23096e12	−0.136932	−0.0684658	−	0.997653i	$-0.521810\pi$
$937$	−3.23096e12	−0.136932	−0.0684658	+	0.997653i	$0.521810\pi$
$938$	0	0
$939$	2.10249e13	0.882547
$940$	0	0
$941$	3.68554e13	1.53232	0.766158	−	0.642652i	$-0.222166\pi$
$941$	3.68554e13	1.53232	0.766158	+	0.642652i	$0.222166\pi$
$942$	0	0
$943$	−2.34031e13	−0.963765
$944$	0	0
$945$	1.35747e12	0.0553718
$946$	0	0
$947$	−2.16984e13	−0.876703	−0.438352	−	0.898804i	$-0.644437\pi$
$947$	−2.16984e13	−0.876703	−0.438352	+	0.898804i	$0.644437\pi$
$948$	0	0
$949$	−1.63534e13	−0.654500
$950$	0	0
$951$	2.42168e13	0.960073
$952$	0	0
$953$	−9.33996e12	−0.366798	−0.183399	−	0.983039i	$-0.558710\pi$
$953$	−9.33996e12	−0.366798	−0.183399	+	0.983039i	$0.558710\pi$
$954$	0	0
$955$	−3.00713e13	−1.16987
$956$	0	0
$957$	2.69790e13	1.03973
$958$	0	0
$959$	−2.33560e12	−0.0891690
$960$	0	0
$961$	3.82869e13	1.44809
$962$	0	0
$963$	1.67973e13	0.629392
$964$	0	0
$965$	−7.48892e12	−0.278001
$966$	0	0
$967$	6.17891e12	0.227244	0.113622	−	0.993524i	$-0.463755\pi$
$967$	6.17891e12	0.227244	0.113622	+	0.993524i	$0.463755\pi$
$968$	0	0
$969$	−4.22238e11	−0.0153851
$970$	0	0
$971$	−4.66682e13	−1.68475	−0.842373	−	0.538894i	$-0.818842\pi$
$971$	−4.66682e13	−1.68475	−0.842373	+	0.538894i	$0.818842\pi$
$972$	0	0
$973$	−3.35242e12	−0.119909
$974$	0	0
$975$	5.32840e13	1.88832
$976$	0	0
$977$	−4.66780e13	−1.63903	−0.819515	−	0.573058i	$-0.805757\pi$
$977$	−4.66780e13	−1.63903	−0.819515	+	0.573058i	$0.805757\pi$
$978$	0	0
$979$	−2.47297e13	−0.860392
$980$	0	0
$981$	8.76361e12	0.302115
$982$	0	0
$983$	−3.93231e13	−1.34325	−0.671625	−	0.740891i	$-0.734404\pi$
$983$	−3.93231e13	−1.34325	−0.671625	+	0.740891i	$0.734404\pi$
$984$	0	0
$985$	8.67436e13	2.93612
$986$	0	0
$987$	2.53791e12	0.0851233
$988$	0	0
$989$	−2.33658e13	−0.776600
$990$	0	0
$991$	−1.82257e13	−0.600280	−0.300140	−	0.953895i	$-0.597033\pi$
$991$	−1.82257e13	−0.600280	−0.300140	+	0.953895i	$0.597033\pi$
$992$	0	0
$993$	−2.95580e13	−0.964724
$994$	0	0
$995$	6.85281e13	2.21648
$996$	0	0
$997$	−3.16514e13	−1.01453	−0.507264	−	0.861791i	$-0.669343\pi$
$997$	−3.16514e13	−1.01453	−0.507264	+	0.861791i	$0.669343\pi$
$998$	0	0
$999$	8.87754e12	0.281999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.b.1.4	✓	4
4.3	odd	2		384.10.a.f.1.4	yes	4
8.3	odd	2		384.10.a.c.1.1	yes	4
8.5	even	2		384.10.a.g.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.b.1.4	✓	4	1.1	even	1	trivial
384.10.a.c.1.1	yes	4	8.3	odd	2
384.10.a.f.1.4	yes	4	4.3	odd	2
384.10.a.g.1.1	yes	4	8.5	even	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} +2687.31 q^{5} -950.516 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} +2687.31 q^{5} -950.516 q^{7} +6561.00 q^{9} +85357.5 q^{11} -124861. q^{13} -217672. q^{15} -21449.0 q^{17} -243033. q^{19} +76991.8 q^{21} +706695. q^{23} +5.26849e6 q^{25} -531441. q^{27} -3.90211e6 q^{29} -8.04528e6 q^{31} -6.91396e6 q^{33} -2.55433e6 q^{35} -1.67047e7 q^{37} +1.01137e7 q^{39} -3.31163e7 q^{41} -3.30635e7 q^{43} +1.76314e7 q^{45} +3.29633e7 q^{47} -3.94501e7 q^{49} +1.73737e6 q^{51} -5.61799e7 q^{53} +2.29382e8 q^{55} +1.96857e7 q^{57} -1.09356e7 q^{59} -3.77590e7 q^{61} -6.23634e6 q^{63} -3.35539e8 q^{65} +9.13783e7 q^{67} -5.72423e7 q^{69} -2.44278e8 q^{71} +1.30973e8 q^{73} -4.26747e8 q^{75} -8.11337e7 q^{77} +4.26074e8 q^{79} +4.30467e7 q^{81} -2.32850e8 q^{83} -5.76401e7 q^{85} +3.16071e8 q^{87} -2.89719e8 q^{89} +1.18682e8 q^{91} +6.51668e8 q^{93} -6.53103e8 q^{95} -2.74576e8 q^{97} +5.60031e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9	\( 4 q - 324 q^{3} - 240 q^{5} - 4840 q^{7} + 26244 q^{9} + 99664 q^{11} - 60840 q^{13} + 19440 q^{15} - 434952 q^{17} - 631776 q^{19} + 392040 q^{21} - 749392 q^{23} + 5991532 q^{25} - 2125764 q^{27} - 7908544 q^{29} - 11351240 q^{31} - 8072784 q^{33} + 25567008 q^{35} - 13592920 q^{37} + 4928040 q^{39} - 18838888 q^{41} - 14177920 q^{43} - 1574640 q^{45} + 37779120 q^{47} + 9409332 q^{49} + 35231112 q^{51} + 115336512 q^{53} + 184580544 q^{55} + 51173856 q^{57} + 115028080 q^{59} + 173228648 q^{61} - 31755240 q^{63} - 328077984 q^{65} + 231785104 q^{67} + 60700752 q^{69} - 197476208 q^{71} + 44629400 q^{73} - 485314092 q^{75} - 308117920 q^{77} + 355774584 q^{79} + 172186884 q^{81} + 607613328 q^{83} + 1087351392 q^{85} + 640592064 q^{87} - 1157146424 q^{89} + 847629840 q^{91} + 919450440 q^{93} + 329699328 q^{95} - 1599536472 q^{97} + 653895504 q^{99}+O(q^{100}) \) 4 * q - 324 * q^3 - 240 * q^5 - 4840 * q^7 + 26244 * q^9 + 99664 * q^11 - 60840 * q^13 + 19440 * q^15 - 434952 * q^17 - 631776 * q^19 + 392040 * q^21 - 749392 * q^23 + 5991532 * q^25 - 2125764 * q^27 - 7908544 * q^29 - 11351240 * q^31 - 8072784 * q^33 + 25567008 * q^35 - 13592920 * q^37 + 4928040 * q^39 - 18838888 * q^41 - 14177920 * q^43 - 1574640 * q^45 + 37779120 * q^47 + 9409332 * q^49 + 35231112 * q^51 + 115336512 * q^53 + 184580544 * q^55 + 51173856 * q^57 + 115028080 * q^59 + 173228648 * q^61 - 31755240 * q^63 - 328077984 * q^65 + 231785104 * q^67 + 60700752 * q^69 - 197476208 * q^71 + 44629400 * q^73 - 485314092 * q^75 - 308117920 * q^77 + 355774584 * q^79 + 172186884 * q^81 + 607613328 * q^83 + 1087351392 * q^85 + 640592064 * q^87 - 1157146424 * q^89 + 847629840 * q^91 + 919450440 * q^93 + 329699328 * q^95 - 1599536472 * q^97 + 653895504 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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