LMFDB - Embedded newform 336.8.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,8,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("336.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	336.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:	$104.961368563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{67}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 67 $ x^2 - 67
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.18535$ of defining polynomial
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000 q^{3} -142.966 q^{5} +343.000 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -142.966 q^{5} +343.000 q^{7} +729.000 q^{9} -5645.80 q^{11} +4172.76 q^{13} -3860.07 q^{15} +7767.13 q^{17} +30981.3 q^{19} +9261.00 q^{21} -33765.1 q^{23} -57685.8 q^{25} +19683.0 q^{27} +204926. q^{29} -279998. q^{31} -152437. q^{33} -49037.2 q^{35} -47217.3 q^{37} +112665. q^{39} +488165. q^{41} -531725. q^{43} -104222. q^{45} +247055. q^{47} +117649. q^{49} +209712. q^{51} -651395. q^{53} +807155. q^{55} +836494. q^{57} +147303. q^{59} -1.64426e6 q^{61} +250047. q^{63} -596562. q^{65} -4.25573e6 q^{67} -911658. q^{69} -3.23776e6 q^{71} +2.53933e6 q^{73} -1.55752e6 q^{75} -1.93651e6 q^{77} -5.72525e6 q^{79} +531441. q^{81} +6.06023e6 q^{83} -1.11043e6 q^{85} +5.53300e6 q^{87} -2.65747e6 q^{89} +1.43126e6 q^{91} -7.55994e6 q^{93} -4.42926e6 q^{95} -1.53674e7 q^{97} -4.11579e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{3} - 24 q^{5} + 686 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q + 54 * q^3 - 24 * q^5 + 686 * q^7 + 1458 * q^9	$ 2 q + 54 q^{3} - 24 q^{5} + 686 q^{7} + 1458 q^{9} - 2124 q^{11} - 1084 q^{13} - 648 q^{15} - 29256 q^{17} + 25816 q^{19} + 18522 q^{21} - 68316 q^{23} - 121658 q^{25} + 39366 q^{27} + 211308 q^{29} - 435840 q^{31} - 57348 q^{33} - 8232 q^{35} - 28428 q^{37} - 29268 q^{39} + 749760 q^{41} - 397096 q^{43} - 17496 q^{45} - 840168 q^{47} + 235298 q^{49} - 789912 q^{51} - 246684 q^{53} + 1226128 q^{55} + 697032 q^{57} - 2199504 q^{59} - 1951108 q^{61} + 500094 q^{63} - 1221936 q^{65} - 1532048 q^{67} - 1844532 q^{69} - 2024004 q^{71} - 1709028 q^{73} - 3284766 q^{75} - 728532 q^{77} - 1048168 q^{79} + 1062882 q^{81} + 4894296 q^{83} - 5514912 q^{85} + 5705316 q^{87} - 60864 q^{89} - 371812 q^{91} - 11767680 q^{93} - 5043744 q^{95} - 26046852 q^{97} - 1548396 q^{99}+O(q^{100}) $ 2 * q + 54 * q^3 - 24 * q^5 + 686 * q^7 + 1458 * q^9 - 2124 * q^11 - 1084 * q^13 - 648 * q^15 - 29256 * q^17 + 25816 * q^19 + 18522 * q^21 - 68316 * q^23 - 121658 * q^25 + 39366 * q^27 + 211308 * q^29 - 435840 * q^31 - 57348 * q^33 - 8232 * q^35 - 28428 * q^37 - 29268 * q^39 + 749760 * q^41 - 397096 * q^43 - 17496 * q^45 - 840168 * q^47 + 235298 * q^49 - 789912 * q^51 - 246684 * q^53 + 1226128 * q^55 + 697032 * q^57 - 2199504 * q^59 - 1951108 * q^61 + 500094 * q^63 - 1221936 * q^65 - 1532048 * q^67 - 1844532 * q^69 - 2024004 * q^71 - 1709028 * q^73 - 3284766 * q^75 - 728532 * q^77 - 1048168 * q^79 + 1062882 * q^81 + 4894296 * q^83 - 5514912 * q^85 + 5705316 * q^87 - 60864 * q^89 - 371812 * q^91 - 11767680 * q^93 - 5043744 * q^95 - 26046852 * q^97 - 1548396 * 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
$5$	−142.966	−0.511489	−0.255745	−	0.966744i	$-0.582321\pi$
$5$	−142.966	−0.511489	−0.255745	+	0.966744i	$0.582321\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−5645.80	−1.27894	−0.639471	−	0.768815i	$-0.720847\pi$
$11$	−5645.80	−1.27894	−0.639471	+	0.768815i	$0.720847\pi$
$12$	0	0
$13$	4172.76	0.526771	0.263386	−	0.964691i	$-0.415161\pi$
$13$	4172.76	0.526771	0.263386	+	0.964691i	$0.415161\pi$
$14$	0	0
$15$	−3860.07	−0.295309
$16$	0	0
$17$	7767.13	0.383433	0.191716	−	0.981450i	$-0.438595\pi$
$17$	7767.13	0.383433	0.191716	+	0.981450i	$0.438595\pi$
$18$	0	0
$19$	30981.3	1.03624	0.518121	−	0.855307i	$-0.326632\pi$
$19$	30981.3	1.03624	0.518121	+	0.855307i	$0.326632\pi$
$20$	0	0
$21$	9261.00	0.218218
$22$	0	0
$23$	−33765.1	−0.578656	−0.289328	−	0.957230i	$-0.593432\pi$
$23$	−33765.1	−0.578656	−0.289328	+	0.957230i	$0.593432\pi$
$24$	0	0
$25$	−57685.8	−0.738379
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	204926.	1.56029	0.780143	−	0.625602i	$-0.215146\pi$
$29$	204926.	1.56029	0.780143	+	0.625602i	$0.215146\pi$
$30$	0	0
$31$	−279998.	−1.68806	−0.844031	−	0.536294i	$-0.819824\pi$
$31$	−279998.	−1.68806	−0.844031	+	0.536294i	$0.819824\pi$
$32$	0	0
$33$	−152437.	−0.738398
$34$	0	0
$35$	−49037.2	−0.193325
$36$	0	0
$37$	−47217.3	−0.153248	−0.0766241	−	0.997060i	$-0.524414\pi$
$37$	−47217.3	−0.153248	−0.0766241	+	0.997060i	$0.524414\pi$
$38$	0	0
$39$	112665.	0.304131
$40$	0	0
$41$	488165.	1.10617	0.553087	−	0.833124i	$-0.313450\pi$
$41$	488165.	1.10617	0.553087	+	0.833124i	$0.313450\pi$
$42$	0	0
$43$	−531725.	−1.01988	−0.509938	−	0.860211i	$-0.670332\pi$
$43$	−531725.	−1.01988	−0.509938	+	0.860211i	$0.670332\pi$
$44$	0	0
$45$	−104222.	−0.170496
$46$	0	0
$47$	247055.	0.347097	0.173549	−	0.984825i	$-0.444477\pi$
$47$	247055.	0.347097	0.173549	+	0.984825i	$0.444477\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	209712.	0.221375
$52$	0	0
$53$	−651395.	−0.601007	−0.300503	−	0.953781i	$-0.597155\pi$
$53$	−651395.	−0.601007	−0.300503	+	0.953781i	$0.597155\pi$
$54$	0	0
$55$	807155.	0.654165
$56$	0	0
$57$	836494.	0.598275
$58$	0	0
$59$	147303.	0.0933747	0.0466873	−	0.998910i	$-0.485134\pi$
$59$	147303.	0.0933747	0.0466873	+	0.998910i	$0.485134\pi$
$60$	0	0
$61$	−1.64426e6	−0.927507	−0.463754	−	0.885964i	$-0.653498\pi$
$61$	−1.64426e6	−0.927507	−0.463754	+	0.885964i	$0.653498\pi$
$62$	0	0
$63$	250047.	0.125988
$64$	0	0
$65$	−596562.	−0.269438
$66$	0	0
$67$	−4.25573e6	−1.72867	−0.864336	−	0.502914i	$-0.832261\pi$
$67$	−4.25573e6	−1.72867	−0.864336	+	0.502914i	$0.832261\pi$
$68$	0	0
$69$	−911658.	−0.334087
$70$	0	0
$71$	−3.23776e6	−1.07360	−0.536798	−	0.843711i	$-0.680366\pi$
$71$	−3.23776e6	−1.07360	−0.536798	+	0.843711i	$0.680366\pi$
$72$	0	0
$73$	2.53933e6	0.763992	0.381996	−	0.924164i	$-0.375237\pi$
$73$	2.53933e6	0.763992	0.381996	+	0.924164i	$0.375237\pi$
$74$	0	0
$75$	−1.55752e6	−0.426303
$76$	0	0
$77$	−1.93651e6	−0.483395
$78$	0	0
$79$	−5.72525e6	−1.30647	−0.653236	−	0.757155i	$-0.726589\pi$
$79$	−5.72525e6	−1.30647	−0.653236	+	0.757155i	$0.726589\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	6.06023e6	1.16336	0.581682	−	0.813416i	$-0.302395\pi$
$83$	6.06023e6	1.16336	0.581682	+	0.813416i	$0.302395\pi$
$84$	0	0
$85$	−1.11043e6	−0.196122
$86$	0	0
$87$	5.53300e6	0.900831
$88$	0	0
$89$	−2.65747e6	−0.399580	−0.199790	−	0.979839i	$-0.564026\pi$
$89$	−2.65747e6	−0.399580	−0.199790	+	0.979839i	$0.564026\pi$
$90$	0	0
$91$	1.43126e6	0.199101
$92$	0	0
$93$	−7.55994e6	−0.974603
$94$	0	0
$95$	−4.42926e6	−0.530027
$96$	0	0
$97$	−1.53674e7	−1.70962	−0.854812	−	0.518937i	$-0.826328\pi$
$97$	−1.53674e7	−1.70962	−0.854812	+	0.518937i	$0.826328\pi$
$98$	0	0
$99$	−4.11579e6	−0.426314
$100$	0	0
$101$	2.58594e6	0.249743	0.124872	−	0.992173i	$-0.460148\pi$
$101$	2.58594e6	0.249743	0.124872	+	0.992173i	$0.460148\pi$
$102$	0	0
$103$	1.83745e7	1.65686	0.828430	−	0.560092i	$-0.189234\pi$
$103$	1.83745e7	1.65686	0.828430	+	0.560092i	$0.189234\pi$
$104$	0	0
$105$	−1.32400e6	−0.111616
$106$	0	0
$107$	−8.29651e6	−0.654715	−0.327357	−	0.944901i	$-0.606158\pi$
$107$	−8.29651e6	−0.654715	−0.327357	+	0.944901i	$0.606158\pi$
$108$	0	0
$109$	−1.01784e7	−0.752809	−0.376405	−	0.926455i	$-0.622840\pi$
$109$	−1.01784e7	−0.752809	−0.376405	+	0.926455i	$0.622840\pi$
$110$	0	0
$111$	−1.27487e6	−0.0884779
$112$	0	0
$113$	2.04044e7	1.33030	0.665149	−	0.746710i	$-0.268368\pi$
$113$	2.04044e7	1.33030	0.665149	+	0.746710i	$0.268368\pi$
$114$	0	0
$115$	4.82725e6	0.295977
$116$	0	0
$117$	3.04194e6	0.175590
$118$	0	0
$119$	2.66412e6	0.144924
$120$	0	0
$121$	1.23879e7	0.635693
$122$	0	0
$123$	1.31805e7	0.638650
$124$	0	0
$125$	1.94163e7	0.889162
$126$	0	0
$127$	1.90939e7	0.827146	0.413573	−	0.910471i	$-0.364281\pi$
$127$	1.90939e7	0.827146	0.413573	+	0.910471i	$0.364281\pi$
$128$	0	0
$129$	−1.43566e7	−0.588825
$130$	0	0
$131$	−4.99014e7	−1.93938	−0.969690	−	0.244339i	$-0.921429\pi$
$131$	−4.99014e7	−1.93938	−0.969690	+	0.244339i	$0.921429\pi$
$132$	0	0
$133$	1.06266e7	0.391663
$134$	0	0
$135$	−2.81399e6	−0.0984362
$136$	0	0
$137$	−5.71459e7	−1.89873	−0.949365	−	0.314174i	$-0.898272\pi$
$137$	−5.71459e7	−1.89873	−0.949365	+	0.314174i	$0.898272\pi$
$138$	0	0
$139$	−4.93346e7	−1.55811	−0.779057	−	0.626953i	$-0.784302\pi$
$139$	−4.93346e7	−1.55811	−0.779057	+	0.626953i	$0.784302\pi$
$140$	0	0
$141$	6.67048e6	0.200397
$142$	0	0
$143$	−2.35586e7	−0.673710
$144$	0	0
$145$	−2.92974e7	−0.798069
$146$	0	0
$147$	3.17652e6	0.0824786
$148$	0	0
$149$	4.50335e7	1.11528	0.557640	−	0.830083i	$-0.311707\pi$
$149$	4.50335e7	1.11528	0.557640	+	0.830083i	$0.311707\pi$
$150$	0	0
$151$	−4.58085e7	−1.08275	−0.541374	−	0.840782i	$-0.682096\pi$
$151$	−4.58085e7	−1.08275	−0.541374	+	0.840782i	$0.682096\pi$
$152$	0	0
$153$	5.66223e6	0.127811
$154$	0	0
$155$	4.00301e7	0.863426
$156$	0	0
$157$	−7.07791e7	−1.45968	−0.729838	−	0.683620i	$-0.760404\pi$
$157$	−7.07791e7	−1.45968	−0.729838	+	0.683620i	$0.760404\pi$
$158$	0	0
$159$	−1.75877e7	−0.346991
$160$	0	0
$161$	−1.15814e7	−0.218712
$162$	0	0
$163$	−3.18479e7	−0.576002	−0.288001	−	0.957630i	$-0.592991\pi$
$163$	−3.18479e7	−0.576002	−0.288001	+	0.957630i	$0.592991\pi$
$164$	0	0
$165$	2.17932e7	0.377683
$166$	0	0
$167$	−6.08722e6	−0.101137	−0.0505687	−	0.998721i	$-0.516103\pi$
$167$	−6.08722e6	−0.101137	−0.0505687	+	0.998721i	$0.516103\pi$
$168$	0	0
$169$	−4.53366e7	−0.722512
$170$	0	0
$171$	2.25853e7	0.345414
$172$	0	0
$173$	3.21606e7	0.472241	0.236120	−	0.971724i	$-0.424124\pi$
$173$	3.21606e7	0.472241	0.236120	+	0.971724i	$0.424124\pi$
$174$	0	0
$175$	−1.97862e7	−0.279081
$176$	0	0
$177$	3.97718e6	0.0539099
$178$	0	0
$179$	−8.54379e7	−1.11344	−0.556718	−	0.830702i	$-0.687939\pi$
$179$	−8.54379e7	−1.11344	−0.556718	+	0.830702i	$0.687939\pi$
$180$	0	0
$181$	−3.49348e7	−0.437909	−0.218954	−	0.975735i	$-0.570265\pi$
$181$	−3.49348e7	−0.437909	−0.218954	+	0.975735i	$0.570265\pi$
$182$	0	0
$183$	−4.43951e7	−0.535497
$184$	0	0
$185$	6.75046e6	0.0783849
$186$	0	0
$187$	−4.38516e7	−0.490388
$188$	0	0
$189$	6.75127e6	0.0727393
$190$	0	0
$191$	−1.07543e8	−1.11677	−0.558386	−	0.829581i	$-0.688579\pi$
$191$	−1.07543e8	−1.11677	−0.558386	+	0.829581i	$0.688579\pi$
$192$	0	0
$193$	5.70003e7	0.570724	0.285362	−	0.958420i	$-0.407886\pi$
$193$	5.70003e7	0.570724	0.285362	+	0.958420i	$0.407886\pi$
$194$	0	0
$195$	−1.61072e7	−0.155560
$196$	0	0
$197$	−1.95032e8	−1.81750	−0.908751	−	0.417339i	$-0.862963\pi$
$197$	−1.95032e8	−1.81750	−0.908751	+	0.417339i	$0.862963\pi$
$198$	0	0
$199$	1.76555e8	1.58816	0.794080	−	0.607814i	$-0.207953\pi$
$199$	1.76555e8	1.58816	0.794080	+	0.607814i	$0.207953\pi$
$200$	0	0
$201$	−1.14905e8	−0.998050
$202$	0	0
$203$	7.02896e7	0.589732
$204$	0	0
$205$	−6.97909e7	−0.565796
$206$	0	0
$207$	−2.46148e7	−0.192885
$208$	0	0
$209$	−1.74914e8	−1.32529
$210$	0	0
$211$	−6.98068e7	−0.511575	−0.255788	−	0.966733i	$-0.582335\pi$
$211$	−6.98068e7	−0.511575	−0.255788	+	0.966733i	$0.582335\pi$
$212$	0	0
$213$	−8.74196e7	−0.619841
$214$	0	0
$215$	7.60183e7	0.521656
$216$	0	0
$217$	−9.60392e7	−0.638028
$218$	0	0
$219$	6.85619e7	0.441091
$220$	0	0
$221$	3.24104e7	0.201981
$222$	0	0
$223$	−2.01378e8	−1.21603	−0.608015	−	0.793925i	$-0.708034\pi$
$223$	−2.01378e8	−1.21603	−0.608015	+	0.793925i	$0.708034\pi$
$224$	0	0
$225$	−4.20530e7	−0.246126
$226$	0	0
$227$	−1.75439e8	−0.995490	−0.497745	−	0.867323i	$-0.665838\pi$
$227$	−1.75439e8	−0.995490	−0.497745	+	0.867323i	$0.665838\pi$
$228$	0	0
$229$	2.21291e8	1.21770	0.608849	−	0.793286i	$-0.291632\pi$
$229$	2.21291e8	1.21770	0.608849	+	0.793286i	$0.291632\pi$
$230$	0	0
$231$	−5.22857e7	−0.279088
$232$	0	0
$233$	1.19498e8	0.618893	0.309447	−	0.950917i	$-0.399856\pi$
$233$	1.19498e8	0.618893	0.309447	+	0.950917i	$0.399856\pi$
$234$	0	0
$235$	−3.53204e7	−0.177537
$236$	0	0
$237$	−1.54582e8	−0.754292
$238$	0	0
$239$	−3.40773e7	−0.161463	−0.0807314	−	0.996736i	$-0.525726\pi$
$239$	−3.40773e7	−0.161463	−0.0807314	+	0.996736i	$0.525726\pi$
$240$	0	0
$241$	−1.19911e8	−0.551821	−0.275910	−	0.961183i	$-0.588979\pi$
$241$	−1.19911e8	−0.551821	−0.275910	+	0.961183i	$0.588979\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−1.68198e7	−0.0730699
$246$	0	0
$247$	1.29277e8	0.545863
$248$	0	0
$249$	1.63626e8	0.671669
$250$	0	0
$251$	1.80334e8	0.719812	0.359906	−	0.932988i	$-0.382809\pi$
$251$	1.80334e8	0.719812	0.359906	+	0.932988i	$0.382809\pi$
$252$	0	0
$253$	1.90631e8	0.740068
$254$	0	0
$255$	−2.99817e7	−0.113231
$256$	0	0
$257$	−1.64638e8	−0.605013	−0.302507	−	0.953147i	$-0.597823\pi$
$257$	−1.64638e8	−0.605013	−0.302507	+	0.953147i	$0.597823\pi$
$258$	0	0
$259$	−1.61955e7	−0.0579224
$260$	0	0
$261$	1.49391e8	0.520095
$262$	0	0
$263$	2.15967e8	0.732051	0.366026	−	0.930605i	$-0.380718\pi$
$263$	2.15967e8	0.732051	0.366026	+	0.930605i	$0.380718\pi$
$264$	0	0
$265$	9.31272e7	0.307408
$266$	0	0
$267$	−7.17517e7	−0.230698
$268$	0	0
$269$	4.61919e8	1.44688	0.723441	−	0.690386i	$-0.242559\pi$
$269$	4.61919e8	1.44688	0.723441	+	0.690386i	$0.242559\pi$
$270$	0	0
$271$	1.74186e8	0.531645	0.265822	−	0.964022i	$-0.414357\pi$
$271$	1.74186e8	0.531645	0.265822	+	0.964022i	$0.414357\pi$
$272$	0	0
$273$	3.86440e7	0.114951
$274$	0	0
$275$	3.25682e8	0.944343
$276$	0	0
$277$	1.24627e8	0.352317	0.176159	−	0.984362i	$-0.443633\pi$
$277$	1.24627e8	0.352317	0.176159	+	0.984362i	$0.443633\pi$
$278$	0	0
$279$	−2.04118e8	−0.562688
$280$	0	0
$281$	−6.47088e7	−0.173977	−0.0869884	−	0.996209i	$-0.527724\pi$
$281$	−6.47088e7	−0.173977	−0.0869884	+	0.996209i	$0.527724\pi$
$282$	0	0
$283$	−4.76698e8	−1.25023	−0.625116	−	0.780531i	$-0.714949\pi$
$283$	−4.76698e8	−1.25023	−0.625116	+	0.780531i	$0.714949\pi$
$284$	0	0
$285$	−1.19590e8	−0.306011
$286$	0	0
$287$	1.67441e8	0.418094
$288$	0	0
$289$	−3.50010e8	−0.852979
$290$	0	0
$291$	−4.14921e8	−0.987052
$292$	0	0
$293$	−5.51445e8	−1.28075	−0.640376	−	0.768061i	$-0.721222\pi$
$293$	−5.51445e8	−1.28075	−0.640376	+	0.768061i	$0.721222\pi$
$294$	0	0
$295$	−2.10592e7	−0.0477602
$296$	0	0
$297$	−1.11126e8	−0.246133
$298$	0	0
$299$	−1.40894e8	−0.304819
$300$	0	0
$301$	−1.82382e8	−0.385477
$302$	0	0
$303$	6.98204e7	0.144189
$304$	0	0
$305$	2.35073e8	0.474410
$306$	0	0
$307$	4.77335e8	0.941541	0.470770	−	0.882256i	$-0.343976\pi$
$307$	4.77335e8	0.941541	0.470770	+	0.882256i	$0.343976\pi$
$308$	0	0
$309$	4.96112e8	0.956589
$310$	0	0
$311$	7.81901e8	1.47398	0.736988	−	0.675905i	$-0.236247\pi$
$311$	7.81901e8	1.47398	0.736988	+	0.675905i	$0.236247\pi$
$312$	0	0
$313$	7.88057e8	1.45262	0.726311	−	0.687366i	$-0.241233\pi$
$313$	7.88057e8	1.45262	0.726311	+	0.687366i	$0.241233\pi$
$314$	0	0
$315$	−3.57481e7	−0.0644416
$316$	0	0
$317$	5.58995e8	0.985599	0.492800	−	0.870143i	$-0.335974\pi$
$317$	5.58995e8	0.985599	0.492800	+	0.870143i	$0.335974\pi$
$318$	0	0
$319$	−1.15697e9	−1.99551
$320$	0	0
$321$	−2.24006e8	−0.378000
$322$	0	0
$323$	2.40635e8	0.397329
$324$	0	0
$325$	−2.40709e8	−0.388957
$326$	0	0
$327$	−2.74816e8	−0.434634
$328$	0	0
$329$	8.47399e7	0.131190
$330$	0	0
$331$	3.32247e7	0.0503573	0.0251787	−	0.999683i	$-0.491985\pi$
$331$	3.32247e7	0.0503573	0.0251787	+	0.999683i	$0.491985\pi$
$332$	0	0
$333$	−3.44214e7	−0.0510827
$334$	0	0
$335$	6.08424e8	0.884198
$336$	0	0
$337$	−4.81153e8	−0.684824	−0.342412	−	0.939550i	$-0.611244\pi$
$337$	−4.81153e8	−0.684824	−0.342412	+	0.939550i	$0.611244\pi$
$338$	0	0
$339$	5.50919e8	0.768048
$340$	0	0
$341$	1.58081e9	2.15893
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	1.30336e8	0.170882
$346$	0	0
$347$	−5.44968e8	−0.700194	−0.350097	−	0.936714i	$-0.613851\pi$
$347$	−5.44968e8	−0.700194	−0.350097	+	0.936714i	$0.613851\pi$
$348$	0	0
$349$	−2.76424e8	−0.348087	−0.174043	−	0.984738i	$-0.555683\pi$
$349$	−2.76424e8	−0.348087	−0.174043	+	0.984738i	$0.555683\pi$
$350$	0	0
$351$	8.21325e7	0.101377
$352$	0	0
$353$	1.57924e8	0.191090	0.0955449	−	0.995425i	$-0.469541\pi$
$353$	1.57924e8	0.191090	0.0955449	+	0.995425i	$0.469541\pi$
$354$	0	0
$355$	4.62889e8	0.549133
$356$	0	0
$357$	7.19313e7	0.0836719
$358$	0	0
$359$	7.12825e8	0.813115	0.406558	−	0.913625i	$-0.366729\pi$
$359$	7.12825e8	0.813115	0.406558	+	0.913625i	$0.366729\pi$
$360$	0	0
$361$	6.59667e7	0.0737988
$362$	0	0
$363$	3.34472e8	0.367018
$364$	0	0
$365$	−3.63037e8	−0.390774
$366$	0	0
$367$	1.14268e9	1.20669	0.603344	−	0.797481i	$-0.293835\pi$
$367$	1.14268e9	1.20669	0.603344	+	0.797481i	$0.293835\pi$
$368$	0	0
$369$	3.55872e8	0.368725
$370$	0	0
$371$	−2.23429e8	−0.227159
$372$	0	0
$373$	1.87850e8	0.187427	0.0937133	−	0.995599i	$-0.470126\pi$
$373$	1.87850e8	0.187427	0.0937133	+	0.995599i	$0.470126\pi$
$374$	0	0
$375$	5.24240e8	0.513358
$376$	0	0
$377$	8.55107e8	0.821913
$378$	0	0
$379$	1.64090e9	1.54827	0.774133	−	0.633022i	$-0.218186\pi$
$379$	1.64090e9	1.54827	0.774133	+	0.633022i	$0.218186\pi$
$380$	0	0
$381$	5.15536e8	0.477553
$382$	0	0
$383$	−7.07482e8	−0.643457	−0.321729	−	0.946832i	$-0.604264\pi$
$383$	−7.07482e8	−0.643457	−0.321729	+	0.946832i	$0.604264\pi$
$384$	0	0
$385$	2.76854e8	0.247251
$386$	0	0
$387$	−3.87627e8	−0.339958
$388$	0	0
$389$	−9.13412e7	−0.0786762	−0.0393381	−	0.999226i	$-0.512525\pi$
$389$	−9.13412e7	−0.0786762	−0.0393381	+	0.999226i	$0.512525\pi$
$390$	0	0
$391$	−2.62258e8	−0.221876
$392$	0	0
$393$	−1.34734e9	−1.11970
$394$	0	0
$395$	8.18515e8	0.668246
$396$	0	0
$397$	−1.43602e9	−1.15185	−0.575923	−	0.817504i	$-0.695357\pi$
$397$	−1.43602e9	−1.15185	−0.575923	+	0.817504i	$0.695357\pi$
$398$	0	0
$399$	2.86917e8	0.226127
$400$	0	0
$401$	9.62314e8	0.745266	0.372633	−	0.927979i	$-0.378455\pi$
$401$	9.62314e8	0.745266	0.372633	+	0.927979i	$0.378455\pi$
$402$	0	0
$403$	−1.16836e9	−0.889223
$404$	0	0
$405$	−7.59778e7	−0.0568322
$406$	0	0
$407$	2.66580e8	0.195996
$408$	0	0
$409$	9.91267e8	0.716406	0.358203	−	0.933644i	$-0.383390\pi$
$409$	9.91267e8	0.716406	0.358203	+	0.933644i	$0.383390\pi$
$410$	0	0
$411$	−1.54294e9	−1.09623
$412$	0	0
$413$	5.05249e7	0.0352923
$414$	0	0
$415$	−8.66404e8	−0.595048
$416$	0	0
$417$	−1.33203e9	−0.899578
$418$	0	0
$419$	5.50029e8	0.365289	0.182645	−	0.983179i	$-0.441534\pi$
$419$	5.50029e8	0.365289	0.182645	+	0.983179i	$0.441534\pi$
$420$	0	0
$421$	−3.40866e8	−0.222636	−0.111318	−	0.993785i	$-0.535507\pi$
$421$	−3.40866e8	−0.222636	−0.111318	+	0.993785i	$0.535507\pi$
$422$	0	0
$423$	1.80103e8	0.115699
$424$	0	0
$425$	−4.48053e8	−0.283118
$426$	0	0
$427$	−5.63983e8	−0.350565
$428$	0	0
$429$	−6.36082e8	−0.388967
$430$	0	0
$431$	2.09919e9	1.26294	0.631468	−	0.775402i	$-0.282453\pi$
$431$	2.09919e9	1.26294	0.631468	+	0.775402i	$0.282453\pi$
$432$	0	0
$433$	8.27062e8	0.489587	0.244794	−	0.969575i	$-0.421280\pi$
$433$	8.27062e8	0.489587	0.244794	+	0.969575i	$0.421280\pi$
$434$	0	0
$435$	−7.91029e8	−0.460766
$436$	0	0
$437$	−1.04609e9	−0.599628
$438$	0	0
$439$	2.25727e9	1.27338	0.636688	−	0.771121i	$-0.280304\pi$
$439$	2.25727e9	1.27338	0.636688	+	0.771121i	$0.280304\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	0	0
$443$	2.55778e9	1.39782	0.698909	−	0.715210i	$-0.253669\pi$
$443$	2.55778e9	1.39782	0.698909	+	0.715210i	$0.253669\pi$
$444$	0	0
$445$	3.79927e8	0.204381
$446$	0	0
$447$	1.21590e9	0.643907
$448$	0	0
$449$	2.53125e9	1.31969	0.659845	−	0.751401i	$-0.270622\pi$
$449$	2.53125e9	1.31969	0.659845	+	0.751401i	$0.270622\pi$
$450$	0	0
$451$	−2.75608e9	−1.41473
$452$	0	0
$453$	−1.23683e9	−0.625124
$454$	0	0
$455$	−2.04621e8	−0.101838
$456$	0	0
$457$	2.63274e8	0.129033	0.0645165	−	0.997917i	$-0.479449\pi$
$457$	2.63274e8	0.129033	0.0645165	+	0.997917i	$0.479449\pi$
$458$	0	0
$459$	1.52880e8	0.0737916
$460$	0	0
$461$	1.56875e9	0.745762	0.372881	−	0.927879i	$-0.378370\pi$
$461$	1.56875e9	0.745762	0.372881	+	0.927879i	$0.378370\pi$
$462$	0	0
$463$	7.10128e8	0.332509	0.166254	−	0.986083i	$-0.446833\pi$
$463$	7.10128e8	0.332509	0.166254	+	0.986083i	$0.446833\pi$
$464$	0	0
$465$	1.08081e9	0.498499
$466$	0	0
$467$	−1.27354e9	−0.578635	−0.289318	−	0.957233i	$-0.593428\pi$
$467$	−1.27354e9	−0.578635	−0.289318	+	0.957233i	$0.593428\pi$
$468$	0	0
$469$	−1.45972e9	−0.653377
$470$	0	0
$471$	−1.91104e9	−0.842744
$472$	0	0
$473$	3.00201e9	1.30436
$474$	0	0
$475$	−1.78718e9	−0.765139
$476$	0	0
$477$	−4.74867e8	−0.200336
$478$	0	0
$479$	3.40882e9	1.41720	0.708598	−	0.705613i	$-0.249328\pi$
$479$	3.40882e9	1.41720	0.708598	+	0.705613i	$0.249328\pi$
$480$	0	0
$481$	−1.97027e8	−0.0807268
$482$	0	0
$483$	−3.12699e8	−0.126273
$484$	0	0
$485$	2.19702e9	0.874455
$486$	0	0
$487$	−8.22019e8	−0.322501	−0.161250	−	0.986914i	$-0.551553\pi$
$487$	−8.22019e8	−0.322501	−0.161250	+	0.986914i	$0.551553\pi$
$488$	0	0
$489$	−8.59893e8	−0.332555
$490$	0	0
$491$	−4.25483e9	−1.62217	−0.811086	−	0.584927i	$-0.801123\pi$
$491$	−4.25483e9	−1.62217	−0.811086	+	0.584927i	$0.801123\pi$
$492$	0	0
$493$	1.59169e9	0.598264
$494$	0	0
$495$	5.88416e8	0.218055
$496$	0	0
$497$	−1.11055e9	−0.405781
$498$	0	0
$499$	2.24793e9	0.809899	0.404950	−	0.914339i	$-0.367289\pi$
$499$	2.24793e9	0.809899	0.404950	+	0.914339i	$0.367289\pi$
$500$	0	0
$501$	−1.64355e8	−0.0583917
$502$	0	0
$503$	1.04623e9	0.366556	0.183278	−	0.983061i	$-0.441329\pi$
$503$	1.04623e9	0.366556	0.183278	+	0.983061i	$0.441329\pi$
$504$	0	0
$505$	−3.69701e8	−0.127741
$506$	0	0
$507$	−1.22409e9	−0.417143
$508$	0	0
$509$	1.86012e9	0.625213	0.312606	−	0.949883i	$-0.398798\pi$
$509$	1.86012e9	0.625213	0.312606	+	0.949883i	$0.398798\pi$
$510$	0	0
$511$	8.70990e8	0.288762
$512$	0	0
$513$	6.09804e8	0.199425
$514$	0	0
$515$	−2.62693e9	−0.847467
$516$	0	0
$517$	−1.39482e9	−0.443917
$518$	0	0
$519$	8.68337e8	0.272648
$520$	0	0
$521$	1.05570e9	0.327045	0.163523	−	0.986540i	$-0.447714\pi$
$521$	1.05570e9	0.327045	0.163523	+	0.986540i	$0.447714\pi$
$522$	0	0
$523$	−3.40218e9	−1.03992	−0.519962	−	0.854190i	$-0.674054\pi$
$523$	−3.40218e9	−1.03992	−0.519962	+	0.854190i	$0.674054\pi$
$524$	0	0
$525$	−5.34228e8	−0.161127
$526$	0	0
$527$	−2.17478e9	−0.647258
$528$	0	0
$529$	−2.26474e9	−0.665157
$530$	0	0
$531$	1.07384e8	0.0311249
$532$	0	0
$533$	2.03700e9	0.582701
$534$	0	0
$535$	1.18612e9	0.334880
$536$	0	0
$537$	−2.30682e9	−0.642842
$538$	0	0
$539$	−6.64222e8	−0.182706
$540$	0	0
$541$	5.37256e8	0.145878	0.0729391	−	0.997336i	$-0.476762\pi$
$541$	5.37256e8	0.145878	0.0729391	+	0.997336i	$0.476762\pi$
$542$	0	0
$543$	−9.43240e8	−0.252827
$544$	0	0
$545$	1.45515e9	0.385054
$546$	0	0
$547$	−5.21302e9	−1.36186	−0.680932	−	0.732347i	$-0.738425\pi$
$547$	−5.21302e9	−1.36186	−0.680932	+	0.732347i	$0.738425\pi$
$548$	0	0
$549$	−1.19867e9	−0.309169
$550$	0	0
$551$	6.34886e9	1.61683
$552$	0	0
$553$	−1.96376e9	−0.493800
$554$	0	0
$555$	1.82262e8	0.0452555
$556$	0	0
$557$	−1.31361e9	−0.322087	−0.161043	−	0.986947i	$-0.551486\pi$
$557$	−1.31361e9	−0.322087	−0.161043	+	0.986947i	$0.551486\pi$
$558$	0	0
$559$	−2.21876e9	−0.537241
$560$	0	0
$561$	−1.18399e9	−0.283126
$562$	0	0
$563$	4.10376e9	0.969176	0.484588	−	0.874743i	$-0.338970\pi$
$563$	4.10376e9	0.969176	0.484588	+	0.874743i	$0.338970\pi$
$564$	0	0
$565$	−2.91713e9	−0.680434
$566$	0	0
$567$	1.82284e8	0.0419961
$568$	0	0
$569$	−4.86146e9	−1.10630	−0.553151	−	0.833081i	$-0.686575\pi$
$569$	−4.86146e9	−1.10630	−0.553151	+	0.833081i	$0.686575\pi$
$570$	0	0
$571$	5.22410e9	1.17432	0.587159	−	0.809472i	$-0.300247\pi$
$571$	5.22410e9	1.17432	0.587159	+	0.809472i	$0.300247\pi$
$572$	0	0
$573$	−2.90366e9	−0.644769
$574$	0	0
$575$	1.94777e9	0.427267
$576$	0	0
$577$	−3.11762e9	−0.675629	−0.337814	−	0.941213i	$-0.609688\pi$
$577$	−3.11762e9	−0.675629	−0.337814	+	0.941213i	$0.609688\pi$
$578$	0	0
$579$	1.53901e9	0.329508
$580$	0	0
$581$	2.07866e9	0.439710
$582$	0	0
$583$	3.67765e9	0.768653
$584$	0	0
$585$	−4.34894e8	−0.0898126
$586$	0	0
$587$	−5.10674e9	−1.04210	−0.521052	−	0.853525i	$-0.674460\pi$
$587$	−5.10674e9	−1.04210	−0.521052	+	0.853525i	$0.674460\pi$
$588$	0	0
$589$	−8.67468e9	−1.74924
$590$	0	0
$591$	−5.26587e9	−1.04934
$592$	0	0
$593$	−3.32380e9	−0.654551	−0.327276	−	0.944929i	$-0.606131\pi$
$593$	−3.32380e9	−0.654551	−0.327276	+	0.944929i	$0.606131\pi$
$594$	0	0
$595$	−3.80878e8	−0.0741271
$596$	0	0
$597$	4.76698e9	0.916924
$598$	0	0
$599$	6.16012e9	1.17110	0.585552	−	0.810635i	$-0.300878\pi$
$599$	6.16012e9	1.17110	0.585552	+	0.810635i	$0.300878\pi$
$600$	0	0
$601$	3.32673e9	0.625110	0.312555	−	0.949900i	$-0.398815\pi$
$601$	3.32673e9	0.625110	0.312555	+	0.949900i	$0.398815\pi$
$602$	0	0
$603$	−3.10243e9	−0.576224
$604$	0	0
$605$	−1.77104e9	−0.325150
$606$	0	0
$607$	1.43680e9	0.260757	0.130379	−	0.991464i	$-0.458381\pi$
$607$	1.43680e9	0.260757	0.130379	+	0.991464i	$0.458381\pi$
$608$	0	0
$609$	1.89782e9	0.340482
$610$	0	0
$611$	1.03090e9	0.182841
$612$	0	0
$613$	−1.11746e9	−0.195939	−0.0979695	−	0.995189i	$-0.531235\pi$
$613$	−1.11746e9	−0.195939	−0.0979695	+	0.995189i	$0.531235\pi$
$614$	0	0
$615$	−1.88435e9	−0.326663
$616$	0	0
$617$	8.70391e9	1.49182	0.745910	−	0.666047i	$-0.232015\pi$
$617$	8.70391e9	1.49182	0.745910	+	0.666047i	$0.232015\pi$
$618$	0	0
$619$	−9.39371e9	−1.59191	−0.795957	−	0.605353i	$-0.793032\pi$
$619$	−9.39371e9	−1.59191	−0.795957	+	0.605353i	$0.793032\pi$
$620$	0	0
$621$	−6.64599e8	−0.111362
$622$	0	0
$623$	−9.11513e8	−0.151027
$624$	0	0
$625$	1.73084e9	0.283581
$626$	0	0
$627$	−4.72268e9	−0.765159
$628$	0	0
$629$	−3.66743e8	−0.0587604
$630$	0	0
$631$	−4.26923e9	−0.676467	−0.338233	−	0.941062i	$-0.609829\pi$
$631$	−4.26923e9	−0.676467	−0.338233	+	0.941062i	$0.609829\pi$
$632$	0	0
$633$	−1.88478e9	−0.295358
$634$	0	0
$635$	−2.72977e9	−0.423076
$636$	0	0
$637$	4.90921e8	0.0752530
$638$	0	0
$639$	−2.36033e9	−0.357865
$640$	0	0
$641$	−1.15550e10	−1.73288	−0.866438	−	0.499284i	$-0.833596\pi$
$641$	−1.15550e10	−1.73288	−0.866438	+	0.499284i	$0.833596\pi$
$642$	0	0
$643$	−5.16858e9	−0.766714	−0.383357	−	0.923600i	$-0.625232\pi$
$643$	−5.16858e9	−0.766714	−0.383357	+	0.923600i	$0.625232\pi$
$644$	0	0
$645$	2.05250e9	0.301178
$646$	0	0
$647$	−7.10838e9	−1.03182	−0.515912	−	0.856641i	$-0.672547\pi$
$647$	−7.10838e9	−1.03182	−0.515912	+	0.856641i	$0.672547\pi$
$648$	0	0
$649$	−8.31642e8	−0.119421
$650$	0	0
$651$	−2.59306e9	−0.368365
$652$	0	0
$653$	−6.07024e9	−0.853120	−0.426560	−	0.904459i	$-0.640275\pi$
$653$	−6.07024e9	−0.853120	−0.426560	+	0.904459i	$0.640275\pi$
$654$	0	0
$655$	7.13418e9	0.991972
$656$	0	0
$657$	1.85117e9	0.254664
$658$	0	0
$659$	5.66589e9	0.771204	0.385602	−	0.922665i	$-0.373994\pi$
$659$	5.66589e9	0.771204	0.385602	+	0.922665i	$0.373994\pi$
$660$	0	0
$661$	−3.55489e9	−0.478763	−0.239382	−	0.970926i	$-0.576945\pi$
$661$	−3.55489e9	−0.478763	−0.239382	+	0.970926i	$0.576945\pi$
$662$	0	0
$663$	8.75080e8	0.116614
$664$	0	0
$665$	−1.51923e9	−0.200331
$666$	0	0
$667$	−6.91935e9	−0.902869
$668$	0	0
$669$	−5.43720e9	−0.702076
$670$	0	0
$671$	9.28318e9	1.18623
$672$	0	0
$673$	−4.52928e9	−0.572765	−0.286382	−	0.958115i	$-0.592453\pi$
$673$	−4.52928e9	−0.572765	−0.286382	+	0.958115i	$0.592453\pi$
$674$	0	0
$675$	−1.13543e9	−0.142101
$676$	0	0
$677$	5.34668e9	0.662253	0.331126	−	0.943586i	$-0.392571\pi$
$677$	5.34668e9	0.662253	0.331126	+	0.943586i	$0.392571\pi$
$678$	0	0
$679$	−5.27104e9	−0.646177
$680$	0	0
$681$	−4.73687e9	−0.574746
$682$	0	0
$683$	9.38294e8	0.112685	0.0563425	−	0.998411i	$-0.482056\pi$
$683$	9.38294e8	0.112685	0.0563425	+	0.998411i	$0.482056\pi$
$684$	0	0
$685$	8.16991e9	0.971181
$686$	0	0
$687$	5.97486e9	0.703038
$688$	0	0
$689$	−2.71812e9	−0.316593
$690$	0	0
$691$	−1.20205e10	−1.38596	−0.692979	−	0.720958i	$-0.743702\pi$
$691$	−1.20205e10	−1.38596	−0.692979	+	0.720958i	$0.743702\pi$
$692$	0	0
$693$	−1.41171e9	−0.161132
$694$	0	0
$695$	7.05315e9	0.796959
$696$	0	0
$697$	3.79164e9	0.424143
$698$	0	0
$699$	3.22645e9	0.357318
$700$	0	0
$701$	−7.34130e9	−0.804933	−0.402467	−	0.915435i	$-0.631847\pi$
$701$	−7.34130e9	−0.804933	−0.402467	+	0.915435i	$0.631847\pi$
$702$	0	0
$703$	−1.46285e9	−0.158802
$704$	0	0
$705$	−9.53650e8	−0.102501
$706$	0	0
$707$	8.86977e8	0.0943940
$708$	0	0
$709$	5.97761e9	0.629891	0.314946	−	0.949110i	$-0.398014\pi$
$709$	5.97761e9	0.629891	0.314946	+	0.949110i	$0.398014\pi$
$710$	0	0
$711$	−4.17371e9	−0.435490
$712$	0	0
$713$	9.45415e9	0.976808
$714$	0	0
$715$	3.36807e9	0.344595
$716$	0	0
$717$	−9.20086e8	−0.0932205
$718$	0	0
$719$	−1.70963e10	−1.71534	−0.857670	−	0.514200i	$-0.828089\pi$
$719$	−1.70963e10	−1.71534	−0.857670	+	0.514200i	$0.828089\pi$
$720$	0	0
$721$	6.30246e9	0.626235
$722$	0	0
$723$	−3.23759e9	−0.318594
$724$	0	0
$725$	−1.18213e10	−1.15208
$726$	0	0
$727$	4.26982e9	0.412135	0.206067	−	0.978538i	$-0.433933\pi$
$727$	4.26982e9	0.412135	0.206067	+	0.978538i	$0.433933\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−4.12997e9	−0.391054
$732$	0	0
$733$	−1.42785e10	−1.33912	−0.669561	−	0.742757i	$-0.733518\pi$
$733$	−1.42785e10	−1.33912	−0.669561	+	0.742757i	$0.733518\pi$
$734$	0	0
$735$	−4.54134e8	−0.0421869
$736$	0	0
$737$	2.40270e10	2.21087
$738$	0	0
$739$	−1.71996e10	−1.56770	−0.783851	−	0.620949i	$-0.786747\pi$
$739$	−1.71996e10	−1.56770	−0.783851	+	0.620949i	$0.786747\pi$
$740$	0	0
$741$	3.49049e9	0.315154
$742$	0	0
$743$	4.25404e9	0.380488	0.190244	−	0.981737i	$-0.439072\pi$
$743$	4.25404e9	0.380488	0.190244	+	0.981737i	$0.439072\pi$
$744$	0	0
$745$	−6.43824e9	−0.570454
$746$	0	0
$747$	4.41791e9	0.387788
$748$	0	0
$749$	−2.84570e9	−0.247459
$750$	0	0
$751$	1.18402e10	1.02004	0.510020	−	0.860162i	$-0.329638\pi$
$751$	1.18402e10	1.02004	0.510020	+	0.860162i	$0.329638\pi$
$752$	0	0
$753$	4.86901e9	0.415584
$754$	0	0
$755$	6.54905e9	0.553814
$756$	0	0
$757$	5.70274e9	0.477802	0.238901	−	0.971044i	$-0.423213\pi$
$757$	5.70274e9	0.477802	0.238901	+	0.971044i	$0.423213\pi$
$758$	0	0
$759$	5.14704e9	0.427278
$760$	0	0
$761$	4.96655e9	0.408516	0.204258	−	0.978917i	$-0.434522\pi$
$761$	4.96655e9	0.408516	0.204258	+	0.978917i	$0.434522\pi$
$762$	0	0
$763$	−3.49118e9	−0.284535
$764$	0	0
$765$	−8.09505e8	−0.0653739
$766$	0	0
$767$	6.14660e8	0.0491871
$768$	0	0
$769$	−1.61711e10	−1.28232	−0.641162	−	0.767405i	$-0.721547\pi$
$769$	−1.61711e10	−1.28232	−0.641162	+	0.767405i	$0.721547\pi$
$770$	0	0
$771$	−4.44523e9	−0.349305
$772$	0	0
$773$	2.14116e10	1.66733	0.833664	−	0.552272i	$-0.186239\pi$
$773$	2.14116e10	1.66733	0.833664	+	0.552272i	$0.186239\pi$
$774$	0	0
$775$	1.61519e10	1.24643
$776$	0	0
$777$	−4.37280e8	−0.0334415
$778$	0	0
$779$	1.51240e10	1.14626
$780$	0	0
$781$	1.82798e10	1.37307
$782$	0	0
$783$	4.03356e9	0.300277
$784$	0	0
$785$	1.01190e10	0.746609
$786$	0	0
$787$	−5.21932e8	−0.0381683	−0.0190841	−	0.999818i	$-0.506075\pi$
$787$	−5.21932e8	−0.0381683	−0.0190841	+	0.999818i	$0.506075\pi$
$788$	0	0
$789$	5.83110e9	0.422650
$790$	0	0
$791$	6.99871e9	0.502806
$792$	0	0
$793$	−6.86113e9	−0.488584
$794$	0	0
$795$	2.51443e9	0.177482
$796$	0	0
$797$	4.66230e9	0.326209	0.163105	−	0.986609i	$-0.447849\pi$
$797$	4.66230e9	0.326209	0.163105	+	0.986609i	$0.447849\pi$
$798$	0	0
$799$	1.91891e9	0.133088
$800$	0	0
$801$	−1.93730e9	−0.133193
$802$	0	0
$803$	−1.43365e10	−0.977102
$804$	0	0
$805$	1.65575e9	0.111869
$806$	0	0
$807$	1.24718e10	0.835358
$808$	0	0
$809$	−1.95275e10	−1.29666	−0.648332	−	0.761358i	$-0.724533\pi$
$809$	−1.95275e10	−1.29666	−0.648332	+	0.761358i	$0.724533\pi$
$810$	0	0
$811$	8.41907e8	0.0554231	0.0277116	−	0.999616i	$-0.491178\pi$
$811$	8.41907e8	0.0554231	0.0277116	+	0.999616i	$0.491178\pi$
$812$	0	0
$813$	4.70303e9	0.306945
$814$	0	0
$815$	4.55315e9	0.294619
$816$	0	0
$817$	−1.64735e10	−1.05684
$818$	0	0
$819$	1.04339e9	0.0663669
$820$	0	0
$821$	−1.04566e10	−0.659462	−0.329731	−	0.944075i	$-0.606958\pi$
$821$	−1.04566e10	−0.659462	−0.329731	+	0.944075i	$0.606958\pi$
$822$	0	0
$823$	9.58935e9	0.599639	0.299819	−	0.953996i	$-0.403074\pi$
$823$	9.58935e9	0.599639	0.299819	+	0.953996i	$0.403074\pi$
$824$	0	0
$825$	8.79343e9	0.545217
$826$	0	0
$827$	3.77708e9	0.232213	0.116107	−	0.993237i	$-0.462959\pi$
$827$	3.77708e9	0.232213	0.116107	+	0.993237i	$0.462959\pi$
$828$	0	0
$829$	1.53685e10	0.936892	0.468446	−	0.883492i	$-0.344814\pi$
$829$	1.53685e10	0.936892	0.468446	+	0.883492i	$0.344814\pi$
$830$	0	0
$831$	3.36494e9	0.203411
$832$	0	0
$833$	9.13795e8	0.0547761
$834$	0	0
$835$	8.70264e8	0.0517307
$836$	0	0
$837$	−5.51120e9	−0.324868
$838$	0	0
$839$	6.40989e9	0.374700	0.187350	−	0.982293i	$-0.440010\pi$
$839$	6.40989e9	0.374700	0.187350	+	0.982293i	$0.440010\pi$
$840$	0	0
$841$	2.47448e10	1.43449
$842$	0	0
$843$	−1.74714e9	−0.100446
$844$	0	0
$845$	6.48157e9	0.369557
$846$	0	0
$847$	4.24904e9	0.240269
$848$	0	0
$849$	−1.28709e10	−0.721822
$850$	0	0
$851$	1.59430e9	0.0886781
$852$	0	0
$853$	−3.02885e9	−0.167092	−0.0835460	−	0.996504i	$-0.526625\pi$
$853$	−3.02885e9	−0.167092	−0.0835460	+	0.996504i	$0.526625\pi$
$854$	0	0
$855$	−3.22893e9	−0.176676
$856$	0	0
$857$	−1.81606e10	−0.985595	−0.492797	−	0.870144i	$-0.664026\pi$
$857$	−1.81606e10	−0.985595	−0.492797	+	0.870144i	$0.664026\pi$
$858$	0	0
$859$	1.26039e10	0.678467	0.339233	−	0.940702i	$-0.389832\pi$
$859$	1.26039e10	0.678467	0.339233	+	0.940702i	$0.389832\pi$
$860$	0	0
$861$	4.52090e9	0.241387
$862$	0	0
$863$	3.26139e10	1.72729	0.863643	−	0.504103i	$-0.168177\pi$
$863$	3.26139e10	1.72729	0.863643	+	0.504103i	$0.168177\pi$
$864$	0	0
$865$	−4.59787e9	−0.241546
$866$	0	0
$867$	−9.45028e9	−0.492468
$868$	0	0
$869$	3.23236e10	1.67090
$870$	0	0
$871$	−1.77582e10	−0.910615
$872$	0	0
$873$	−1.12029e10	−0.569875
$874$	0	0
$875$	6.65978e9	0.336072
$876$	0	0
$877$	2.60232e10	1.30275	0.651377	−	0.758754i	$-0.274192\pi$
$877$	2.60232e10	1.30275	0.651377	+	0.758754i	$0.274192\pi$
$878$	0	0
$879$	−1.48890e10	−0.739443
$880$	0	0
$881$	1.33519e10	0.657849	0.328925	−	0.944356i	$-0.393314\pi$
$881$	1.33519e10	0.657849	0.328925	+	0.944356i	$0.393314\pi$
$882$	0	0
$883$	7.44408e9	0.363872	0.181936	−	0.983310i	$-0.441764\pi$
$883$	7.44408e9	0.363872	0.181936	+	0.983310i	$0.441764\pi$
$884$	0	0
$885$	−5.68600e8	−0.0275743
$886$	0	0
$887$	5.03665e9	0.242331	0.121166	−	0.992632i	$-0.461337\pi$
$887$	5.03665e9	0.242331	0.121166	+	0.992632i	$0.461337\pi$
$888$	0	0
$889$	6.54921e9	0.312632
$890$	0	0
$891$	−3.00041e9	−0.142105
$892$	0	0
$893$	7.65407e9	0.359677
$894$	0	0
$895$	1.22147e10	0.569510
$896$	0	0
$897$	−3.80413e9	−0.175988
$898$	0	0
$899$	−5.73788e10	−2.63386
$900$	0	0
$901$	−5.05947e9	−0.230446
$902$	0	0
$903$	−4.92430e9	−0.222555
$904$	0	0
$905$	4.99448e9	0.223986
$906$	0	0
$907$	2.22674e10	0.990933	0.495466	−	0.868627i	$-0.334997\pi$
$907$	2.22674e10	0.990933	0.495466	+	0.868627i	$0.334997\pi$
$908$	0	0
$909$	1.88515e9	0.0832477
$910$	0	0
$911$	7.62429e9	0.334107	0.167053	−	0.985948i	$-0.446575\pi$
$911$	7.62429e9	0.334107	0.167053	+	0.985948i	$0.446575\pi$
$912$	0	0
$913$	−3.42148e10	−1.48788
$914$	0	0
$915$	6.34698e9	0.273901
$916$	0	0
$917$	−1.71162e10	−0.733017
$918$	0	0
$919$	3.89828e10	1.65680	0.828398	−	0.560140i	$-0.189253\pi$
$919$	3.89828e10	1.65680	0.828398	+	0.560140i	$0.189253\pi$
$920$	0	0
$921$	1.28880e10	0.543599
$922$	0	0
$923$	−1.35104e10	−0.565539
$924$	0	0
$925$	2.72377e9	0.113155
$926$	0	0
$927$	1.33950e10	0.552287
$928$	0	0
$929$	1.62040e10	0.663084	0.331542	−	0.943441i	$-0.392431\pi$
$929$	1.62040e10	0.663084	0.331542	+	0.943441i	$0.392431\pi$
$930$	0	0
$931$	3.64491e9	0.148035
$932$	0	0
$933$	2.11113e10	0.851001
$934$	0	0
$935$	6.26927e9	0.250828
$936$	0	0
$937$	−2.16005e10	−0.857778	−0.428889	−	0.903357i	$-0.641095\pi$
$937$	−2.16005e10	−0.857778	−0.428889	+	0.903357i	$0.641095\pi$
$938$	0	0
$939$	2.12775e10	0.838672
$940$	0	0
$941$	−3.61422e10	−1.41401	−0.707003	−	0.707211i	$-0.749953\pi$
$941$	−3.61422e10	−1.41401	−0.707003	+	0.707211i	$0.749953\pi$
$942$	0	0
$943$	−1.64830e10	−0.640094
$944$	0	0
$945$	−9.65200e8	−0.0372054
$946$	0	0
$947$	−3.21432e10	−1.22988	−0.614942	−	0.788572i	$-0.710821\pi$
$947$	−3.21432e10	−1.22988	−0.614942	+	0.788572i	$0.710821\pi$
$948$	0	0
$949$	1.05960e10	0.402449
$950$	0	0
$951$	1.50929e10	0.569036
$952$	0	0
$953$	−3.56860e10	−1.33559	−0.667795	−	0.744345i	$-0.732762\pi$
$953$	−3.56860e10	−1.33559	−0.667795	+	0.744345i	$0.732762\pi$
$954$	0	0
$955$	1.53749e10	0.571217
$956$	0	0
$957$	−3.12382e10	−1.15211
$958$	0	0
$959$	−1.96011e10	−0.717653
$960$	0	0
$961$	5.08861e10	1.84956
$962$	0	0
$963$	−6.04815e9	−0.218238
$964$	0	0
$965$	−8.14908e9	−0.291919
$966$	0	0
$967$	3.52443e10	1.25342	0.626710	−	0.779253i	$-0.284401\pi$
$967$	3.52443e10	1.25342	0.626710	+	0.779253i	$0.284401\pi$
$968$	0	0
$969$	6.49715e9	0.229398
$970$	0	0
$971$	−4.01550e10	−1.40758	−0.703789	−	0.710409i	$-0.748510\pi$
$971$	−4.01550e10	−1.40758	−0.703789	+	0.710409i	$0.748510\pi$
$972$	0	0
$973$	−1.69218e10	−0.588912
$974$	0	0
$975$	−6.49915e9	−0.224564
$976$	0	0
$977$	5.11365e10	1.75428	0.877142	−	0.480231i	$-0.159447\pi$
$977$	5.11365e10	1.75428	0.877142	+	0.480231i	$0.159447\pi$
$978$	0	0
$979$	1.50035e10	0.511040
$980$	0	0
$981$	−7.42002e9	−0.250936
$982$	0	0
$983$	−4.49914e9	−0.151075	−0.0755374	−	0.997143i	$-0.524067\pi$
$983$	−4.49914e9	−0.151075	−0.0755374	+	0.997143i	$0.524067\pi$
$984$	0	0
$985$	2.78829e10	0.929633
$986$	0	0
$987$	2.28798e9	0.0757428
$988$	0	0
$989$	1.79537e10	0.590157
$990$	0	0
$991$	−3.81773e10	−1.24608	−0.623042	−	0.782188i	$-0.714104\pi$
$991$	−3.81773e10	−1.24608	−0.623042	+	0.782188i	$0.714104\pi$
$992$	0	0
$993$	8.97066e8	0.0290738
$994$	0	0
$995$	−2.52413e10	−0.812327
$996$	0	0
$997$	−2.50727e10	−0.801251	−0.400626	−	0.916242i	$-0.631207\pi$
$997$	−2.50727e10	−0.801251	−0.400626	+	0.916242i	$0.631207\pi$
$998$	0	0
$999$	−9.29379e8	−0.0294926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.8.a.p.1.1		2
4.3	odd	2		21.8.a.c.1.1	✓	2
12.11	even	2		63.8.a.c.1.2		2
20.19	odd	2		525.8.a.d.1.2		2
28.3	even	6		147.8.e.e.79.2		4
28.11	odd	6		147.8.e.f.79.2		4
28.19	even	6		147.8.e.e.67.2		4
28.23	odd	6		147.8.e.f.67.2		4
28.27	even	2		147.8.a.d.1.1		2
84.83	odd	2		441.8.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.a.c.1.1	✓	2	4.3	odd	2
63.8.a.c.1.2		2	12.11	even	2
147.8.a.d.1.1		2	28.27	even	2
147.8.e.e.67.2		4	28.19	even	6
147.8.e.e.79.2		4	28.3	even	6
147.8.e.f.67.2		4	28.23	odd	6
147.8.e.f.79.2		4	28.11	odd	6
336.8.a.p.1.1		2	1.1	even	1	trivial
441.8.a.h.1.2		2	84.83	odd	2
525.8.a.d.1.2		2	20.19	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -142.966 q^{5} +343.000 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -142.966 q^{5} +343.000 q^{7} +729.000 q^{9} -5645.80 q^{11} +4172.76 q^{13} -3860.07 q^{15} +7767.13 q^{17} +30981.3 q^{19} +9261.00 q^{21} -33765.1 q^{23} -57685.8 q^{25} +19683.0 q^{27} +204926. q^{29} -279998. q^{31} -152437. q^{33} -49037.2 q^{35} -47217.3 q^{37} +112665. q^{39} +488165. q^{41} -531725. q^{43} -104222. q^{45} +247055. q^{47} +117649. q^{49} +209712. q^{51} -651395. q^{53} +807155. q^{55} +836494. q^{57} +147303. q^{59} -1.64426e6 q^{61} +250047. q^{63} -596562. q^{65} -4.25573e6 q^{67} -911658. q^{69} -3.23776e6 q^{71} +2.53933e6 q^{73} -1.55752e6 q^{75} -1.93651e6 q^{77} -5.72525e6 q^{79} +531441. q^{81} +6.06023e6 q^{83} -1.11043e6 q^{85} +5.53300e6 q^{87} -2.65747e6 q^{89} +1.43126e6 q^{91} -7.55994e6 q^{93} -4.42926e6 q^{95} -1.53674e7 q^{97} -4.11579e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{3} - 24 q^{5} + 686 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q + 54 * q^3 - 24 * q^5 + 686 * q^7 + 1458 * q^9	\( 2 q + 54 q^{3} - 24 q^{5} + 686 q^{7} + 1458 q^{9} - 2124 q^{11} - 1084 q^{13} - 648 q^{15} - 29256 q^{17} + 25816 q^{19} + 18522 q^{21} - 68316 q^{23} - 121658 q^{25} + 39366 q^{27} + 211308 q^{29} - 435840 q^{31} - 57348 q^{33} - 8232 q^{35} - 28428 q^{37} - 29268 q^{39} + 749760 q^{41} - 397096 q^{43} - 17496 q^{45} - 840168 q^{47} + 235298 q^{49} - 789912 q^{51} - 246684 q^{53} + 1226128 q^{55} + 697032 q^{57} - 2199504 q^{59} - 1951108 q^{61} + 500094 q^{63} - 1221936 q^{65} - 1532048 q^{67} - 1844532 q^{69} - 2024004 q^{71} - 1709028 q^{73} - 3284766 q^{75} - 728532 q^{77} - 1048168 q^{79} + 1062882 q^{81} + 4894296 q^{83} - 5514912 q^{85} + 5705316 q^{87} - 60864 q^{89} - 371812 q^{91} - 11767680 q^{93} - 5043744 q^{95} - 26046852 q^{97} - 1548396 q^{99}+O(q^{100}) \) 2 * q + 54 * q^3 - 24 * q^5 + 686 * q^7 + 1458 * q^9 - 2124 * q^11 - 1084 * q^13 - 648 * q^15 - 29256 * q^17 + 25816 * q^19 + 18522 * q^21 - 68316 * q^23 - 121658 * q^25 + 39366 * q^27 + 211308 * q^29 - 435840 * q^31 - 57348 * q^33 - 8232 * q^35 - 28428 * q^37 - 29268 * q^39 + 749760 * q^41 - 397096 * q^43 - 17496 * q^45 - 840168 * q^47 + 235298 * q^49 - 789912 * q^51 - 246684 * q^53 + 1226128 * q^55 + 697032 * q^57 - 2199504 * q^59 - 1951108 * q^61 + 500094 * q^63 - 1221936 * q^65 - 1532048 * q^67 - 1844532 * q^69 - 2024004 * q^71 - 1709028 * q^73 - 3284766 * q^75 - 728532 * q^77 - 1048168 * q^79 + 1062882 * q^81 + 4894296 * q^83 - 5514912 * q^85 + 5705316 * q^87 - 60864 * q^89 - 371812 * q^91 - 11767680 * q^93 - 5043744 * q^95 - 26046852 * q^97 - 1548396 * q^99

Introduction

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