LMFDB - Embedded newform 294.8.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,8,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("294.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	294.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:	$91.8411974923$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	294.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +114.000 q^{5} -216.000 q^{6} +512.000 q^{8} +729.000 q^{9} +912.000 q^{10} +7332.00 q^{11} -1728.00 q^{12} +3802.00 q^{13} -3078.00 q^{15} +4096.00 q^{16} +6606.00 q^{17} +5832.00 q^{18} -24860.0 q^{19} +7296.00 q^{20} +58656.0 q^{22} +41448.0 q^{23} -13824.0 q^{24} -65129.0 q^{25} +30416.0 q^{26} -19683.0 q^{27} -41610.0 q^{29} -24624.0 q^{30} -33152.0 q^{31} +32768.0 q^{32} -197964. q^{33} +52848.0 q^{34} +46656.0 q^{36} -36466.0 q^{37} -198880. q^{38} -102654. q^{39} +58368.0 q^{40} +639078. q^{41} -156412. q^{43} +469248. q^{44} +83106.0 q^{45} +331584. q^{46} +433776. q^{47} -110592. q^{48} -521032. q^{50} -178362. q^{51} +243328. q^{52} +786078. q^{53} -157464. q^{54} +835848. q^{55} +671220. q^{57} -332880. q^{58} -745140. q^{59} -196992. q^{60} +1.66062e6 q^{61} -265216. q^{62} +262144. q^{64} +433428. q^{65} -1.58371e6 q^{66} -3.29084e6 q^{67} +422784. q^{68} -1.11910e6 q^{69} +5.71615e6 q^{71} +373248. q^{72} -2.65990e6 q^{73} -291728. q^{74} +1.75848e6 q^{75} -1.59104e6 q^{76} -821232. q^{78} +3.80744e6 q^{79} +466944. q^{80} +531441. q^{81} +5.11262e6 q^{82} -2.22947e6 q^{83} +753084. q^{85} -1.25130e6 q^{86} +1.12347e6 q^{87} +3.75398e6 q^{88} -5.99121e6 q^{89} +664848. q^{90} +2.65267e6 q^{92} +895104. q^{93} +3.47021e6 q^{94} -2.83404e6 q^{95} -884736. q^{96} +4.06013e6 q^{97} +5.34503e6 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	114.000	0.407859	0.203929	−	0.978986i	$-0.434629\pi$
$5$	114.000	0.407859	0.203929	+	0.978986i	$0.434629\pi$
$6$	−216.000	−0.408248
$7$	0	0
$7$	0	0
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	912.000	0.288400
$11$	7332.00	1.66092	0.830459	−	0.557080i	$-0.188078\pi$
$11$	7332.00	1.66092	0.830459	+	0.557080i	$0.188078\pi$
$12$	−1728.00	−0.288675
$13$	3802.00	0.479966	0.239983	−	0.970777i	$-0.422858\pi$
$13$	3802.00	0.479966	0.239983	+	0.970777i	$0.422858\pi$
$14$	0	0
$15$	−3078.00	−0.235477
$16$	4096.00	0.250000
$17$	6606.00	0.326112	0.163056	−	0.986617i	$-0.447865\pi$
$17$	6606.00	0.326112	0.163056	+	0.986617i	$0.447865\pi$
$18$	5832.00	0.235702
$19$	−24860.0	−0.831502	−0.415751	−	0.909478i	$-0.636481\pi$
$19$	−24860.0	−0.831502	−0.415751	+	0.909478i	$0.636481\pi$
$20$	7296.00	0.203929
$21$	0	0
$22$	58656.0	1.17445
$23$	41448.0	0.710323	0.355162	−	0.934805i	$-0.384426\pi$
$23$	41448.0	0.710323	0.355162	+	0.934805i	$0.384426\pi$
$24$	−13824.0	−0.204124
$25$	−65129.0	−0.833651
$26$	30416.0	0.339387
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−41610.0	−0.316814	−0.158407	−	0.987374i	$-0.550636\pi$
$29$	−41610.0	−0.316814	−0.158407	+	0.987374i	$0.550636\pi$
$30$	−24624.0	−0.166508
$31$	−33152.0	−0.199868	−0.0999341	−	0.994994i	$-0.531863\pi$
$31$	−33152.0	−0.199868	−0.0999341	+	0.994994i	$0.531863\pi$
$32$	32768.0	0.176777
$33$	−197964.	−0.958931
$34$	52848.0	0.230596
$35$	0	0
$36$	46656.0	0.166667
$37$	−36466.0	−0.118354	−0.0591769	−	0.998248i	$-0.518848\pi$
$37$	−36466.0	−0.118354	−0.0591769	+	0.998248i	$0.518848\pi$
$38$	−198880.	−0.587961
$39$	−102654.	−0.277108
$40$	58368.0	0.144200
$41$	639078.	1.44814	0.724070	−	0.689727i	$-0.242269\pi$
$41$	639078.	1.44814	0.724070	+	0.689727i	$0.242269\pi$
$42$	0	0
$43$	−156412.	−0.300006	−0.150003	−	0.988686i	$-0.547928\pi$
$43$	−156412.	−0.300006	−0.150003	+	0.988686i	$0.547928\pi$
$44$	469248.	0.830459
$45$	83106.0	0.135953
$46$	331584.	0.502275
$47$	433776.	0.609429	0.304714	−	0.952444i	$-0.401439\pi$
$47$	433776.	0.609429	0.304714	+	0.952444i	$0.401439\pi$
$48$	−110592.	−0.144338
$49$	0	0
$50$	−521032.	−0.589480
$51$	−178362.	−0.188281
$52$	243328.	0.239983
$53$	786078.	0.725271	0.362635	−	0.931931i	$-0.381877\pi$
$53$	786078.	0.725271	0.362635	+	0.931931i	$0.381877\pi$
$54$	−157464.	−0.136083
$55$	835848.	0.677420
$56$	0	0
$57$	671220.	0.480068
$58$	−332880.	−0.224022
$59$	−745140.	−0.472341	−0.236171	−	0.971712i	$-0.575892\pi$
$59$	−745140.	−0.472341	−0.236171	+	0.971712i	$0.575892\pi$
$60$	−196992.	−0.117739
$61$	1.66062e6	0.936732	0.468366	−	0.883535i	$-0.344843\pi$
$61$	1.66062e6	0.936732	0.468366	+	0.883535i	$0.344843\pi$
$62$	−265216.	−0.141328
$63$	0	0
$64$	262144.	0.125000
$65$	433428.	0.195758
$66$	−1.58371e6	−0.678067
$67$	−3.29084e6	−1.33673	−0.668366	−	0.743832i	$-0.733006\pi$
$67$	−3.29084e6	−1.33673	−0.668366	+	0.743832i	$0.733006\pi$
$68$	422784.	0.163056
$69$	−1.11910e6	−0.410105
$70$	0	0
$71$	5.71615e6	1.89539	0.947697	−	0.319171i	$-0.103404\pi$
$71$	5.71615e6	1.89539	0.947697	+	0.319171i	$0.103404\pi$
$72$	373248.	0.117851
$73$	−2.65990e6	−0.800267	−0.400134	−	0.916457i	$-0.631036\pi$
$73$	−2.65990e6	−0.800267	−0.400134	+	0.916457i	$0.631036\pi$
$74$	−291728.	−0.0836888
$75$	1.75848e6	0.481309
$76$	−1.59104e6	−0.415751
$77$	0	0
$78$	−821232.	−0.195945
$79$	3.80744e6	0.868837	0.434418	−	0.900711i	$-0.356954\pi$
$79$	3.80744e6	0.868837	0.434418	+	0.900711i	$0.356954\pi$
$80$	466944.	0.101965
$81$	531441.	0.111111
$82$	5.11262e6	1.02399
$83$	−2.22947e6	−0.427984	−0.213992	−	0.976835i	$-0.568647\pi$
$83$	−2.22947e6	−0.427984	−0.213992	+	0.976835i	$0.568647\pi$
$84$	0	0
$85$	753084.	0.133008
$86$	−1.25130e6	−0.212137
$87$	1.12347e6	0.182913
$88$	3.75398e6	0.587223
$89$	−5.99121e6	−0.900844	−0.450422	−	0.892816i	$-0.648726\pi$
$89$	−5.99121e6	−0.900844	−0.450422	+	0.892816i	$0.648726\pi$
$90$	664848.	0.0961332
$91$	0	0
$92$	2.65267e6	0.355162
$93$	895104.	0.115394
$94$	3.47021e6	0.430931
$95$	−2.83404e6	−0.339136
$96$	−884736.	−0.102062
$97$	4.06013e6	0.451688	0.225844	−	0.974163i	$-0.427486\pi$
$97$	4.06013e6	0.451688	0.225844	+	0.974163i	$0.427486\pi$
$98$	0	0
$99$	5.34503e6	0.553639
$100$	−4.16826e6	−0.416826
$101$	1.72819e7	1.66904	0.834522	−	0.550975i	$-0.185744\pi$
$101$	1.72819e7	1.66904	0.834522	+	0.550975i	$0.185744\pi$
$102$	−1.42690e6	−0.133135
$103$	1.43623e7	1.29507	0.647536	−	0.762035i	$-0.275799\pi$
$103$	1.43623e7	1.29507	0.647536	+	0.762035i	$0.275799\pi$
$104$	1.94662e6	0.169694
$105$	0	0
$106$	6.28862e6	0.512844
$107$	6.45440e6	0.509346	0.254673	−	0.967027i	$-0.418032\pi$
$107$	6.45440e6	0.509346	0.254673	+	0.967027i	$0.418032\pi$
$108$	−1.25971e6	−0.0962250
$109$	−884410.	−0.0654125	−0.0327063	−	0.999465i	$-0.510413\pi$
$109$	−884410.	−0.0654125	−0.0327063	+	0.999465i	$0.510413\pi$
$110$	6.68678e6	0.479008
$111$	984582.	0.0683316
$112$	0	0
$113$	1.21325e7	0.790999	0.395499	−	0.918466i	$-0.370572\pi$
$113$	1.21325e7	0.790999	0.395499	+	0.918466i	$0.370572\pi$
$114$	5.36976e6	0.339459
$115$	4.72507e6	0.289712
$116$	−2.66304e6	−0.158407
$117$	2.77166e6	0.159989
$118$	−5.96112e6	−0.333996
$119$	0	0
$120$	−1.57594e6	−0.0832538
$121$	3.42711e7	1.75865
$122$	1.32849e7	0.662369
$123$	−1.72551e7	−0.836084
$124$	−2.12173e6	−0.0999341
$125$	−1.63310e7	−0.747871
$126$	0	0
$127$	6.86806e6	0.297524	0.148762	−	0.988873i	$-0.452471\pi$
$127$	6.86806e6	0.297524	0.148762	+	0.988873i	$0.452471\pi$
$128$	2.09715e6	0.0883883
$129$	4.22312e6	0.173209
$130$	3.46742e6	0.138422
$131$	3.95208e7	1.53595	0.767973	−	0.640482i	$-0.221265\pi$
$131$	3.95208e7	1.53595	0.767973	+	0.640482i	$0.221265\pi$
$132$	−1.26697e7	−0.479466
$133$	0	0
$134$	−2.63267e7	−0.945212
$135$	−2.24386e6	−0.0784925
$136$	3.38227e6	0.115298
$137$	1.91741e7	0.637078	0.318539	−	0.947910i	$-0.396808\pi$
$137$	1.91741e7	0.637078	0.318539	+	0.947910i	$0.396808\pi$
$138$	−8.95277e6	−0.289988
$139$	−1.32449e7	−0.418309	−0.209154	−	0.977883i	$-0.567071\pi$
$139$	−1.32449e7	−0.418309	−0.209154	+	0.977883i	$0.567071\pi$
$140$	0	0
$141$	−1.17120e7	−0.351854
$142$	4.57292e7	1.34025
$143$	2.78763e7	0.797184
$144$	2.98598e6	0.0833333
$145$	−4.74354e6	−0.129215
$146$	−2.12792e7	−0.565874
$147$	0	0
$148$	−2.33382e6	−0.0591769
$149$	5.73624e7	1.42061	0.710306	−	0.703893i	$-0.248556\pi$
$149$	5.73624e7	1.42061	0.710306	+	0.703893i	$0.248556\pi$
$150$	1.40679e7	0.340337
$151$	−3.10873e7	−0.734790	−0.367395	−	0.930065i	$-0.619750\pi$
$151$	−3.10873e7	−0.734790	−0.367395	+	0.930065i	$0.619750\pi$
$152$	−1.27283e7	−0.293981
$153$	4.81577e6	0.108704
$154$	0	0
$155$	−3.77933e6	−0.0815180
$156$	−6.56986e6	−0.138554
$157$	3.37835e7	0.696715	0.348358	−	0.937362i	$-0.386739\pi$
$157$	3.37835e7	0.696715	0.348358	+	0.937362i	$0.386739\pi$
$158$	3.04595e7	0.614360
$159$	−2.12241e7	−0.418735
$160$	3.73555e6	0.0720999
$161$	0	0
$162$	4.25153e6	0.0785674
$163$	6.26659e7	1.13338	0.566689	−	0.823932i	$-0.308224\pi$
$163$	6.26659e7	1.13338	0.566689	+	0.823932i	$0.308224\pi$
$164$	4.09010e7	0.724070
$165$	−2.25679e7	−0.391109
$166$	−1.78357e7	−0.302631
$167$	−6.27072e7	−1.04186	−0.520931	−	0.853599i	$-0.674415\pi$
$167$	−6.27072e7	−1.04186	−0.520931	+	0.853599i	$0.674415\pi$
$168$	0	0
$169$	−4.82933e7	−0.769633
$170$	6.02467e6	0.0940507
$171$	−1.81229e7	−0.277167
$172$	−1.00104e7	−0.150003
$173$	2.70521e7	0.397228	0.198614	−	0.980078i	$-0.436356\pi$
$173$	2.70521e7	0.397228	0.198614	+	0.980078i	$0.436356\pi$
$174$	8.98776e6	0.129339
$175$	0	0
$176$	3.00319e7	0.415229
$177$	2.01188e7	0.272706
$178$	−4.79297e7	−0.636993
$179$	−1.34281e8	−1.74996	−0.874981	−	0.484157i	$-0.839126\pi$
$179$	−1.34281e8	−1.74996	−0.874981	+	0.484157i	$0.839126\pi$
$180$	5.31878e6	0.0679765
$181$	−1.14661e8	−1.43727	−0.718636	−	0.695386i	$-0.755233\pi$
$181$	−1.14661e8	−1.43727	−0.718636	+	0.695386i	$0.755233\pi$
$182$	0	0
$183$	−4.48367e7	−0.540822
$184$	2.12214e7	0.251137
$185$	−4.15712e6	−0.0482716
$186$	7.16083e6	0.0815959
$187$	4.84352e7	0.541646
$188$	2.77617e7	0.304714
$189$	0	0
$190$	−2.26723e7	−0.239805
$191$	1.63605e7	0.169895	0.0849474	−	0.996385i	$-0.472928\pi$
$191$	1.63605e7	0.169895	0.0849474	+	0.996385i	$0.472928\pi$
$192$	−7.07789e6	−0.0721688
$193$	−1.54198e8	−1.54394	−0.771968	−	0.635661i	$-0.780728\pi$
$193$	−1.54198e8	−1.54394	−0.771968	+	0.635661i	$0.780728\pi$
$194$	3.24810e7	0.319392
$195$	−1.17026e7	−0.113021
$196$	0	0
$197$	8.32288e7	0.775607	0.387804	−	0.921742i	$-0.373234\pi$
$197$	8.32288e7	0.775607	0.387804	+	0.921742i	$0.373234\pi$
$198$	4.27602e7	0.391482
$199$	7.61722e7	0.685190	0.342595	−	0.939483i	$-0.388694\pi$
$199$	7.61722e7	0.685190	0.342595	+	0.939483i	$0.388694\pi$
$200$	−3.33460e7	−0.294740
$201$	8.88526e7	0.771763
$202$	1.38256e8	1.18019
$203$	0	0
$204$	−1.14152e7	−0.0941405
$205$	7.28549e7	0.590636
$206$	1.14898e8	0.915755
$207$	3.02156e7	0.236774
$208$	1.55730e7	0.119991
$209$	−1.82274e8	−1.38106
$210$	0	0
$211$	3.52446e7	0.258288	0.129144	−	0.991626i	$-0.458777\pi$
$211$	3.52446e7	0.258288	0.129144	+	0.991626i	$0.458777\pi$
$212$	5.03090e7	0.362635
$213$	−1.54336e8	−1.09431
$214$	5.16352e7	0.360162
$215$	−1.78310e7	−0.122360
$216$	−1.00777e7	−0.0680414
$217$	0	0
$218$	−7.07528e6	−0.0462536
$219$	7.18172e7	0.462034
$220$	5.34943e7	0.338710
$221$	2.51160e7	0.156523
$222$	7.87666e6	0.0483177
$223$	1.89131e8	1.14208	0.571040	−	0.820922i	$-0.306540\pi$
$223$	1.89131e8	1.14208	0.571040	+	0.820922i	$0.306540\pi$
$224$	0	0
$225$	−4.74790e7	−0.277884
$226$	9.70600e7	0.559320
$227$	1.76100e8	0.999239	0.499620	−	0.866245i	$-0.333473\pi$
$227$	1.76100e8	0.999239	0.499620	+	0.866245i	$0.333473\pi$
$228$	4.29581e7	0.240034
$229$	−6.50396e7	−0.357894	−0.178947	−	0.983859i	$-0.557269\pi$
$229$	−6.50396e7	−0.357894	−0.178947	+	0.983859i	$0.557269\pi$
$230$	3.78006e7	0.204857
$231$	0	0
$232$	−2.13043e7	−0.112011
$233$	−2.51319e8	−1.30160	−0.650802	−	0.759248i	$-0.725567\pi$
$233$	−2.51319e8	−1.30160	−0.650802	+	0.759248i	$0.725567\pi$
$234$	2.21733e7	0.113129
$235$	4.94505e7	0.248561
$236$	−4.76890e7	−0.236171
$237$	−1.02801e8	−0.501623
$238$	0	0
$239$	2.13079e8	1.00960	0.504799	−	0.863237i	$-0.331566\pi$
$239$	2.13079e8	1.00960	0.504799	+	0.863237i	$0.331566\pi$
$240$	−1.26075e7	−0.0588693
$241$	−2.57284e8	−1.18400	−0.592001	−	0.805937i	$-0.701662\pi$
$241$	−2.57284e8	−1.18400	−0.592001	+	0.805937i	$0.701662\pi$
$242$	2.74168e8	1.24355
$243$	−1.43489e7	−0.0641500
$244$	1.06280e8	0.468366
$245$	0	0
$246$	−1.38041e8	−0.591200
$247$	−9.45177e7	−0.399093
$248$	−1.69738e7	−0.0706641
$249$	6.01956e7	0.247097
$250$	−1.30648e8	−0.528824
$251$	−1.23058e8	−0.491193	−0.245596	−	0.969372i	$-0.578984\pi$
$251$	−1.23058e8	−0.491193	−0.245596	+	0.969372i	$0.578984\pi$
$252$	0	0
$253$	3.03897e8	1.17979
$254$	5.49445e7	0.210381
$255$	−2.03333e7	−0.0767921
$256$	1.67772e7	0.0625000
$257$	4.43334e8	1.62916	0.814582	−	0.580048i	$-0.196966\pi$
$257$	4.43334e8	1.62916	0.814582	+	0.580048i	$0.196966\pi$
$258$	3.37850e7	0.122477
$259$	0	0
$260$	2.77394e7	0.0978792
$261$	−3.03337e7	−0.105605
$262$	3.16166e8	1.08608
$263$	2.98925e8	1.01325	0.506625	−	0.862166i	$-0.330893\pi$
$263$	2.98925e8	1.01325	0.506625	+	0.862166i	$0.330893\pi$
$264$	−1.01358e8	−0.339033
$265$	8.96129e7	0.295808
$266$	0	0
$267$	1.61763e8	0.520102
$268$	−2.10614e8	−0.668366
$269$	−2.08908e8	−0.654368	−0.327184	−	0.944961i	$-0.606100\pi$
$269$	−2.08908e8	−0.654368	−0.327184	+	0.944961i	$0.606100\pi$
$270$	−1.79509e7	−0.0555026
$271$	1.12749e7	0.0344129	0.0172064	−	0.999852i	$-0.494523\pi$
$271$	1.12749e7	0.0344129	0.0172064	+	0.999852i	$0.494523\pi$
$272$	2.70582e7	0.0815281
$273$	0	0
$274$	1.53393e8	0.450482
$275$	−4.77526e8	−1.38463
$276$	−7.16221e7	−0.205053
$277$	−6.58964e8	−1.86287	−0.931435	−	0.363907i	$-0.881443\pi$
$277$	−6.58964e8	−1.86287	−0.931435	+	0.363907i	$0.881443\pi$
$278$	−1.05959e8	−0.295789
$279$	−2.41678e7	−0.0666227
$280$	0	0
$281$	−1.05123e8	−0.282634	−0.141317	−	0.989964i	$-0.545134\pi$
$281$	−1.05123e8	−0.282634	−0.141317	+	0.989964i	$0.545134\pi$
$282$	−9.36956e7	−0.248798
$283$	−3.30161e8	−0.865911	−0.432956	−	0.901415i	$-0.642529\pi$
$283$	−3.30161e8	−0.865911	−0.432956	+	0.901415i	$0.642529\pi$
$284$	3.65834e8	0.947697
$285$	7.65191e7	0.195800
$286$	2.23010e8	0.563694
$287$	0	0
$288$	2.38879e7	0.0589256
$289$	−3.66699e8	−0.893651
$290$	−3.79483e7	−0.0913691
$291$	−1.09623e8	−0.260782
$292$	−1.70233e8	−0.400134
$293$	8.71002e7	0.202294	0.101147	−	0.994871i	$-0.467749\pi$
$293$	8.71002e7	0.202294	0.101147	+	0.994871i	$0.467749\pi$
$294$	0	0
$295$	−8.49460e7	−0.192649
$296$	−1.86706e7	−0.0418444
$297$	−1.44316e8	−0.319644
$298$	4.58899e8	1.00452
$299$	1.57585e8	0.340931
$300$	1.12543e8	0.240654
$301$	0	0
$302$	−2.48698e8	−0.519575
$303$	−4.66612e8	−0.963623
$304$	−1.01827e8	−0.207876
$305$	1.89310e8	0.382054
$306$	3.85262e7	0.0768654
$307$	3.91709e8	0.772644	0.386322	−	0.922364i	$-0.373745\pi$
$307$	3.91709e8	0.772644	0.386322	+	0.922364i	$0.373745\pi$
$308$	0	0
$309$	−3.87782e8	−0.747710
$310$	−3.02346e7	−0.0576419
$311$	2.04936e8	0.386328	0.193164	−	0.981166i	$-0.438125\pi$
$311$	2.04936e8	0.386328	0.193164	+	0.981166i	$0.438125\pi$
$312$	−5.25588e7	−0.0979726
$313$	−8.77202e8	−1.61694	−0.808471	−	0.588536i	$-0.799705\pi$
$313$	−8.77202e8	−1.61694	−0.808471	+	0.588536i	$0.799705\pi$
$314$	2.70268e8	0.492652
$315$	0	0
$316$	2.43676e8	0.434418
$317$	−4.40831e8	−0.777256	−0.388628	−	0.921395i	$-0.627051\pi$
$317$	−4.40831e8	−0.777256	−0.388628	+	0.921395i	$0.627051\pi$
$318$	−1.69793e8	−0.296090
$319$	−3.05085e8	−0.526202
$320$	2.98844e7	0.0509823
$321$	−1.74269e8	−0.294071
$322$	0	0
$323$	−1.64225e8	−0.271163
$324$	3.40122e7	0.0555556
$325$	−2.47620e8	−0.400124
$326$	5.01327e8	0.801419
$327$	2.38791e7	0.0377659
$328$	3.27208e8	0.511995
$329$	0	0
$330$	−1.80543e8	−0.276555
$331$	1.11223e9	1.68576	0.842882	−	0.538099i	$-0.180857\pi$
$331$	1.11223e9	1.68576	0.842882	+	0.538099i	$0.180857\pi$
$332$	−1.42686e8	−0.213992
$333$	−2.65837e7	−0.0394513
$334$	−5.01658e8	−0.736707
$335$	−3.75155e8	−0.545198
$336$	0	0
$337$	2.88198e8	0.410191	0.205096	−	0.978742i	$-0.434249\pi$
$337$	2.88198e8	0.410191	0.205096	+	0.978742i	$0.434249\pi$
$338$	−3.86347e8	−0.544213
$339$	−3.27577e8	−0.456683
$340$	4.81974e7	0.0665039
$341$	−2.43070e8	−0.331965
$342$	−1.44984e8	−0.195987
$343$	0	0
$344$	−8.00829e7	−0.106068
$345$	−1.27577e8	−0.167265
$346$	2.16417e8	0.280883
$347$	−1.10601e9	−1.42103	−0.710517	−	0.703680i	$-0.751539\pi$
$347$	−1.10601e9	−1.42103	−0.710517	+	0.703680i	$0.751539\pi$
$348$	7.19021e7	0.0914564
$349$	1.32184e9	1.66453	0.832264	−	0.554379i	$-0.187044\pi$
$349$	1.32184e9	1.66453	0.832264	+	0.554379i	$0.187044\pi$
$350$	0	0
$351$	−7.48348e7	−0.0923695
$352$	2.40255e8	0.293612
$353$	−1.20395e9	−1.45679	−0.728396	−	0.685157i	$-0.759734\pi$
$353$	−1.20395e9	−1.45679	−0.728396	+	0.685157i	$0.759734\pi$
$354$	1.60950e8	0.192832
$355$	6.51641e8	0.773053
$356$	−3.83437e8	−0.450422
$357$	0	0
$358$	−1.07425e9	−1.23741
$359$	−1.32057e9	−1.50637	−0.753185	−	0.657809i	$-0.771484\pi$
$359$	−1.32057e9	−1.50637	−0.753185	+	0.657809i	$0.771484\pi$
$360$	4.25503e7	0.0480666
$361$	−2.75852e8	−0.308604
$362$	−9.17284e8	−1.01630
$363$	−9.25318e8	−1.01536
$364$	0	0
$365$	−3.03228e8	−0.326396
$366$	−3.58693e8	−0.382419
$367$	−1.75107e9	−1.84915	−0.924575	−	0.381000i	$-0.875580\pi$
$367$	−1.75107e9	−1.84915	−0.924575	+	0.381000i	$0.875580\pi$
$368$	1.69771e8	0.177581
$369$	4.65888e8	0.482713
$370$	−3.32570e7	−0.0341332
$371$	0	0
$372$	5.72867e7	0.0576970
$373$	−4.87945e8	−0.486844	−0.243422	−	0.969920i	$-0.578270\pi$
$373$	−4.87945e8	−0.486844	−0.243422	+	0.969920i	$0.578270\pi$
$374$	3.87482e8	0.383001
$375$	4.40936e8	0.431783
$376$	2.22093e8	0.215466
$377$	−1.58201e8	−0.152060
$378$	0	0
$379$	1.11007e9	1.04740	0.523700	−	0.851903i	$-0.324551\pi$
$379$	1.11007e9	1.04740	0.523700	+	0.851903i	$0.324551\pi$
$380$	−1.81379e8	−0.169568
$381$	−1.85438e8	−0.171775
$382$	1.30884e8	0.120134
$383$	−1.86912e9	−1.69997	−0.849983	−	0.526810i	$-0.823388\pi$
$383$	−1.86912e9	−1.69997	−0.849983	+	0.526810i	$0.823388\pi$
$384$	−5.66231e7	−0.0510310
$385$	0	0
$386$	−1.23359e9	−1.09173
$387$	−1.14024e8	−0.100002
$388$	2.59848e8	0.225844
$389$	−2.73895e8	−0.235918	−0.117959	−	0.993018i	$-0.537635\pi$
$389$	−2.73895e8	−0.235918	−0.117959	+	0.993018i	$0.537635\pi$
$390$	−9.36204e7	−0.0799180
$391$	2.73805e8	0.231645
$392$	0	0
$393$	−1.06706e9	−0.886779
$394$	6.65831e8	0.548437
$395$	4.34048e8	0.354363
$396$	3.42082e8	0.276820
$397$	−6.24552e8	−0.500958	−0.250479	−	0.968122i	$-0.580588\pi$
$397$	−6.24552e8	−0.500958	−0.250479	+	0.968122i	$0.580588\pi$
$398$	6.09378e8	0.484502
$399$	0	0
$400$	−2.66768e8	−0.208413
$401$	5.55500e8	0.430208	0.215104	−	0.976591i	$-0.430991\pi$
$401$	5.55500e8	0.430208	0.215104	+	0.976591i	$0.430991\pi$
$402$	7.10821e8	0.545719
$403$	−1.26044e8	−0.0959299
$404$	1.10604e9	0.834522
$405$	6.05843e7	0.0453176
$406$	0	0
$407$	−2.67369e8	−0.196576
$408$	−9.13213e7	−0.0665674
$409$	2.15770e9	1.55941	0.779704	−	0.626149i	$-0.215370\pi$
$409$	2.15770e9	1.55941	0.779704	+	0.626149i	$0.215370\pi$
$410$	5.82839e8	0.417643
$411$	−5.17700e8	−0.367817
$412$	9.19188e8	0.647536
$413$	0	0
$414$	2.41725e8	0.167425
$415$	−2.54159e8	−0.174557
$416$	1.24584e8	0.0848468
$417$	3.57612e8	0.241511
$418$	−1.45819e9	−0.976555
$419$	−1.67797e9	−1.11438	−0.557191	−	0.830384i	$-0.688121\pi$
$419$	−1.67797e9	−1.11438	−0.557191	+	0.830384i	$0.688121\pi$
$420$	0	0
$421$	−5.25233e8	−0.343056	−0.171528	−	0.985179i	$-0.554870\pi$
$421$	−5.25233e8	−0.343056	−0.171528	+	0.985179i	$0.554870\pi$
$422$	2.81957e8	0.182637
$423$	3.16223e8	0.203143
$424$	4.02472e8	0.256422
$425$	−4.30242e8	−0.271864
$426$	−1.23469e9	−0.773791
$427$	0	0
$428$	4.13082e8	0.254673
$429$	−7.52659e8	−0.460254
$430$	−1.42648e8	−0.0865218
$431$	1.70593e8	0.102634	0.0513169	−	0.998682i	$-0.483658\pi$
$431$	1.70593e8	0.102634	0.0513169	+	0.998682i	$0.483658\pi$
$432$	−8.06216e7	−0.0481125
$433$	1.68797e9	0.999210	0.499605	−	0.866253i	$-0.333478\pi$
$433$	1.68797e9	0.999210	0.499605	+	0.866253i	$0.333478\pi$
$434$	0	0
$435$	1.28076e8	0.0746026
$436$	−5.66022e7	−0.0327063
$437$	−1.03040e9	−0.590636
$438$	5.74538e8	0.326708
$439$	1.17850e9	0.664817	0.332409	−	0.943135i	$-0.392139\pi$
$439$	1.17850e9	0.664817	0.332409	+	0.943135i	$0.392139\pi$
$440$	4.27954e8	0.239504
$441$	0	0
$442$	2.00928e8	0.110678
$443$	7.15755e8	0.391157	0.195579	−	0.980688i	$-0.437342\pi$
$443$	7.15755e8	0.391157	0.195579	+	0.980688i	$0.437342\pi$
$444$	6.30132e7	0.0341658
$445$	−6.82998e8	−0.367417
$446$	1.51305e9	0.807573
$447$	−1.54879e9	−0.820191
$448$	0	0
$449$	−1.37358e9	−0.716132	−0.358066	−	0.933696i	$-0.616564\pi$
$449$	−1.37358e9	−0.716132	−0.358066	+	0.933696i	$0.616564\pi$
$450$	−3.79832e8	−0.196493
$451$	4.68572e9	2.40524
$452$	7.76480e8	0.395499
$453$	8.39357e8	0.424231
$454$	1.40880e9	0.706569
$455$	0	0
$456$	3.43665e8	0.169730
$457$	1.84752e9	0.905488	0.452744	−	0.891641i	$-0.350445\pi$
$457$	1.84752e9	0.905488	0.452744	+	0.891641i	$0.350445\pi$
$458$	−5.20317e8	−0.253069
$459$	−1.30026e8	−0.0627604
$460$	3.02405e8	0.144856
$461$	−3.09414e9	−1.47091	−0.735455	−	0.677573i	$-0.763032\pi$
$461$	−3.09414e9	−1.47091	−0.735455	+	0.677573i	$0.763032\pi$
$462$	0	0
$463$	3.00451e9	1.40682	0.703412	−	0.710782i	$-0.251659\pi$
$463$	3.00451e9	1.40682	0.703412	+	0.710782i	$0.251659\pi$
$464$	−1.70435e8	−0.0792036
$465$	1.02042e8	0.0470645
$466$	−2.01055e9	−0.920373
$467$	2.99252e9	1.35965	0.679825	−	0.733374i	$-0.262056\pi$
$467$	2.99252e9	1.35965	0.679825	+	0.733374i	$0.262056\pi$
$468$	1.77386e8	0.0799943
$469$	0	0
$470$	3.95604e8	0.175759
$471$	−9.12154e8	−0.402249
$472$	−3.81512e8	−0.166998
$473$	−1.14681e9	−0.498286
$474$	−8.22407e8	−0.354701
$475$	1.61911e9	0.693183
$476$	0	0
$477$	5.73051e8	0.241757
$478$	1.70464e9	0.713894
$479$	−1.84041e9	−0.765141	−0.382570	−	0.923926i	$-0.624961\pi$
$479$	−1.84041e9	−0.765141	−0.382570	+	0.923926i	$0.624961\pi$
$480$	−1.00860e8	−0.0416269
$481$	−1.38644e8	−0.0568058
$482$	−2.05827e9	−0.837216
$483$	0	0
$484$	2.19335e9	0.879323
$485$	4.62854e8	0.184225
$486$	−1.14791e8	−0.0453609
$487$	−4.26676e8	−0.167397	−0.0836983	−	0.996491i	$-0.526673\pi$
$487$	−4.26676e8	−0.167397	−0.0836983	+	0.996491i	$0.526673\pi$
$488$	8.50236e8	0.331185
$489$	−1.69198e9	−0.654356
$490$	0	0
$491$	6.07547e7	0.0231630	0.0115815	−	0.999933i	$-0.496313\pi$
$491$	6.07547e7	0.0231630	0.0115815	+	0.999933i	$0.496313\pi$
$492$	−1.10433e9	−0.418042
$493$	−2.74876e8	−0.103317
$494$	−7.56142e8	−0.282201
$495$	6.09333e8	0.225807
$496$	−1.35791e8	−0.0499671
$497$	0	0
$498$	4.81565e8	0.174724
$499$	3.24588e9	1.16945	0.584723	−	0.811233i	$-0.301203\pi$
$499$	3.24588e9	1.16945	0.584723	+	0.811233i	$0.301203\pi$
$500$	−1.04518e9	−0.373935
$501$	1.69310e9	0.601519
$502$	−9.84464e8	−0.347326
$503$	7.44381e8	0.260800	0.130400	−	0.991461i	$-0.458374\pi$
$503$	7.44381e8	0.260800	0.130400	+	0.991461i	$0.458374\pi$
$504$	0	0
$505$	1.97014e9	0.680734
$506$	2.43117e9	0.834237
$507$	1.30392e9	0.444348
$508$	4.39556e8	0.148762
$509$	4.44155e8	0.149287	0.0746436	−	0.997210i	$-0.476218\pi$
$509$	4.44155e8	0.149287	0.0746436	+	0.997210i	$0.476218\pi$
$510$	−1.62666e8	−0.0543002
$511$	0	0
$512$	1.34218e8	0.0441942
$513$	4.89319e8	0.160023
$514$	3.54667e9	1.15199
$515$	1.63730e9	0.528207
$516$	2.70280e8	0.0866044
$517$	3.18045e9	1.01221
$518$	0	0
$519$	−7.30407e8	−0.229340
$520$	2.21915e8	0.0692110
$521$	−3.04963e9	−0.944745	−0.472372	−	0.881399i	$-0.656602\pi$
$521$	−3.04963e9	−0.944745	−0.472372	+	0.881399i	$0.656602\pi$
$522$	−2.42670e8	−0.0746738
$523$	1.40306e9	0.428866	0.214433	−	0.976739i	$-0.431210\pi$
$523$	1.40306e9	0.428866	0.214433	+	0.976739i	$0.431210\pi$
$524$	2.52933e9	0.767973
$525$	0	0
$526$	2.39140e9	0.716476
$527$	−2.19002e8	−0.0651795
$528$	−8.10861e8	−0.239733
$529$	−1.68689e9	−0.495441
$530$	7.16903e8	0.209168
$531$	−5.43207e8	−0.157447
$532$	0	0
$533$	2.42977e9	0.695058
$534$	1.29410e9	0.367768
$535$	7.35802e8	0.207741
$536$	−1.68491e9	−0.472606
$537$	3.62558e9	1.01034
$538$	−1.67126e9	−0.462708
$539$	0	0
$540$	−1.43607e8	−0.0392462
$541$	4.21106e9	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$541$	4.21106e9	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$542$	9.01994e7	0.0243336
$543$	3.09583e9	0.829809
$544$	2.16465e8	0.0576491
$545$	−1.00823e8	−0.0266791
$546$	0	0
$547$	1.99956e9	0.522371	0.261185	−	0.965289i	$-0.415887\pi$
$547$	1.99956e9	0.522371	0.261185	+	0.965289i	$0.415887\pi$
$548$	1.22714e9	0.318539
$549$	1.21059e9	0.312244
$550$	−3.82021e9	−0.979078
$551$	1.03442e9	0.263432
$552$	−5.72977e8	−0.144994
$553$	0	0
$554$	−5.27172e9	−1.31725
$555$	1.12242e8	0.0278696
$556$	−8.47674e8	−0.209154
$557$	−3.37403e9	−0.827287	−0.413643	−	0.910439i	$-0.635744\pi$
$557$	−3.37403e9	−0.827287	−0.413643	+	0.910439i	$0.635744\pi$
$558$	−1.93342e8	−0.0471094
$559$	−5.94678e8	−0.143993
$560$	0	0
$561$	−1.30775e9	−0.312719
$562$	−8.40983e8	−0.199853
$563$	5.58021e9	1.31787	0.658933	−	0.752201i	$-0.271008\pi$
$563$	5.58021e9	1.31787	0.658933	+	0.752201i	$0.271008\pi$
$564$	−7.49565e8	−0.175927
$565$	1.38310e9	0.322616
$566$	−2.64129e9	−0.612292
$567$	0	0
$568$	2.92667e9	0.670123
$569$	−8.88310e8	−0.202149	−0.101074	−	0.994879i	$-0.532228\pi$
$569$	−8.88310e8	−0.202149	−0.101074	+	0.994879i	$0.532228\pi$
$570$	6.12153e8	0.138452
$571$	−1.79171e9	−0.402755	−0.201377	−	0.979514i	$-0.564542\pi$
$571$	−1.79171e9	−0.402755	−0.201377	+	0.979514i	$0.564542\pi$
$572$	1.78408e9	0.398592
$573$	−4.41734e8	−0.0980888
$574$	0	0
$575$	−2.69947e9	−0.592162
$576$	1.91103e8	0.0416667
$577$	−3.82103e9	−0.828066	−0.414033	−	0.910262i	$-0.635880\pi$
$577$	−3.82103e9	−0.828066	−0.414033	+	0.910262i	$0.635880\pi$
$578$	−2.93360e9	−0.631906
$579$	4.16336e9	0.891392
$580$	−3.03587e8	−0.0646077
$581$	0	0
$582$	−8.76987e8	−0.184401
$583$	5.76352e9	1.20461
$584$	−1.36187e9	−0.282937
$585$	3.15969e8	0.0652528
$586$	6.96802e8	0.143043
$587$	−4.36219e9	−0.890166	−0.445083	−	0.895489i	$-0.646826\pi$
$587$	−4.36219e9	−0.890166	−0.445083	+	0.895489i	$0.646826\pi$
$588$	0	0
$589$	8.24159e8	0.166191
$590$	−6.79568e8	−0.136223
$591$	−2.24718e9	−0.447797
$592$	−1.49365e8	−0.0295884
$593$	−6.38531e9	−1.25745	−0.628724	−	0.777628i	$-0.716423\pi$
$593$	−6.38531e9	−1.25745	−0.628724	+	0.777628i	$0.716423\pi$
$594$	−1.15453e9	−0.226022
$595$	0	0
$596$	3.67120e9	0.710306
$597$	−2.05665e9	−0.395594
$598$	1.26068e9	0.241075
$599$	8.04297e8	0.152905	0.0764527	−	0.997073i	$-0.475641\pi$
$599$	8.04297e8	0.152905	0.0764527	+	0.997073i	$0.475641\pi$
$600$	9.00343e8	0.170168
$601$	4.87162e9	0.915403	0.457702	−	0.889106i	$-0.348673\pi$
$601$	4.87162e9	0.915403	0.457702	+	0.889106i	$0.348673\pi$
$602$	0	0
$603$	−2.39902e9	−0.445577
$604$	−1.98959e9	−0.367395
$605$	3.90690e9	0.717280
$606$	−3.73290e9	−0.681384
$607$	−7.17517e9	−1.30218	−0.651091	−	0.759000i	$-0.725688\pi$
$607$	−7.17517e9	−1.30218	−0.651091	+	0.759000i	$0.725688\pi$
$608$	−8.14612e8	−0.146990
$609$	0	0
$610$	1.51448e9	0.270153
$611$	1.64922e9	0.292505
$612$	3.08210e8	0.0543521
$613$	3.47891e9	0.610002	0.305001	−	0.952352i	$-0.401343\pi$
$613$	3.47891e9	0.610002	0.305001	+	0.952352i	$0.401343\pi$
$614$	3.13367e9	0.546342
$615$	−1.96708e9	−0.341004
$616$	0	0
$617$	−2.39378e8	−0.0410286	−0.0205143	−	0.999790i	$-0.506530\pi$
$617$	−2.39378e8	−0.0410286	−0.0205143	+	0.999790i	$0.506530\pi$
$618$	−3.10226e9	−0.528711
$619$	5.52959e9	0.937078	0.468539	−	0.883443i	$-0.344781\pi$
$619$	5.52959e9	0.937078	0.468539	+	0.883443i	$0.344781\pi$
$620$	−2.41877e8	−0.0407590
$621$	−8.15821e8	−0.136702
$622$	1.63949e9	0.273175
$623$	0	0
$624$	−4.20471e8	−0.0692771
$625$	3.22647e9	0.528626
$626$	−7.01762e9	−1.14335
$627$	4.92139e9	0.797354
$628$	2.16214e9	0.348358
$629$	−2.40894e8	−0.0385966
$630$	0	0
$631$	−6.13683e9	−0.972392	−0.486196	−	0.873850i	$-0.661616\pi$
$631$	−6.13683e9	−0.972392	−0.486196	+	0.873850i	$0.661616\pi$
$632$	1.94941e9	0.307180
$633$	−9.51603e8	−0.149122
$634$	−3.52664e9	−0.549603
$635$	7.82959e8	0.121348
$636$	−1.35834e9	−0.209368
$637$	0	0
$638$	−2.44068e9	−0.372081
$639$	4.16707e9	0.631798
$640$	2.39075e8	0.0360500
$641$	1.07038e10	1.60522	0.802611	−	0.596503i	$-0.203444\pi$
$641$	1.07038e10	1.60522	0.802611	+	0.596503i	$0.203444\pi$
$642$	−1.39415e9	−0.207940
$643$	1.39803e9	0.207385	0.103692	−	0.994609i	$-0.466934\pi$
$643$	1.39803e9	0.207385	0.103692	+	0.994609i	$0.466934\pi$
$644$	0	0
$645$	4.81436e8	0.0706447
$646$	−1.31380e9	−0.191741
$647$	5.31605e9	0.771656	0.385828	−	0.922571i	$-0.373916\pi$
$647$	5.31605e9	0.771656	0.385828	+	0.922571i	$0.373916\pi$
$648$	2.72098e8	0.0392837
$649$	−5.46337e9	−0.784520
$650$	−1.98096e9	−0.282930
$651$	0	0
$652$	4.01062e9	0.566689
$653$	3.24403e9	0.455921	0.227960	−	0.973670i	$-0.426794\pi$
$653$	3.24403e9	0.455921	0.227960	+	0.973670i	$0.426794\pi$
$654$	1.91033e8	0.0267046
$655$	4.50537e9	0.626449
$656$	2.61766e9	0.362035
$657$	−1.93907e9	−0.266756
$658$	0	0
$659$	−5.16506e9	−0.703034	−0.351517	−	0.936181i	$-0.614334\pi$
$659$	−5.16506e9	−0.703034	−0.351517	+	0.936181i	$0.614334\pi$
$660$	−1.44435e9	−0.195554
$661$	3.22515e9	0.434356	0.217178	−	0.976132i	$-0.430315\pi$
$661$	3.22515e9	0.434356	0.217178	+	0.976132i	$0.430315\pi$
$662$	8.89784e9	1.19201
$663$	−6.78132e8	−0.0903685
$664$	−1.14149e9	−0.151315
$665$	0	0
$666$	−2.12670e8	−0.0278963
$667$	−1.72465e9	−0.225041
$668$	−4.01326e9	−0.520931
$669$	−5.10655e9	−0.659380
$670$	−3.00124e9	−0.385513
$671$	1.21757e10	1.55583
$672$	0	0
$673$	−2.00633e9	−0.253718	−0.126859	−	0.991921i	$-0.540490\pi$
$673$	−2.00633e9	−0.253718	−0.126859	+	0.991921i	$0.540490\pi$
$674$	2.30559e9	0.290049
$675$	1.28193e9	0.160436
$676$	−3.09077e9	−0.384816
$677$	−1.00211e10	−1.24124	−0.620619	−	0.784112i	$-0.713119\pi$
$677$	−1.00211e10	−1.24124	−0.620619	+	0.784112i	$0.713119\pi$
$678$	−2.62062e9	−0.322924
$679$	0	0
$680$	3.85579e8	0.0470254
$681$	−4.75471e9	−0.576911
$682$	−1.94456e9	−0.234734
$683$	−5.84861e9	−0.702393	−0.351196	−	0.936302i	$-0.614225\pi$
$683$	−5.84861e9	−0.702393	−0.351196	+	0.936302i	$0.614225\pi$
$684$	−1.15987e9	−0.138584
$685$	2.18584e9	0.259838
$686$	0	0
$687$	1.75607e9	0.206630
$688$	−6.40664e8	−0.0750016
$689$	2.98867e9	0.348105
$690$	−1.02062e9	−0.118274
$691$	−2.58686e9	−0.298263	−0.149131	−	0.988817i	$-0.547648\pi$
$691$	−2.58686e9	−0.298263	−0.149131	+	0.988817i	$0.547648\pi$
$692$	1.73134e9	0.198614
$693$	0	0
$694$	−8.84805e9	−1.00482
$695$	−1.50992e9	−0.170611
$696$	5.75217e8	0.0646694
$697$	4.22175e9	0.472256
$698$	1.05747e10	1.17700
$699$	6.78560e9	0.751481
$700$	0	0
$701$	−1.74460e9	−0.191286	−0.0956429	−	0.995416i	$-0.530491\pi$
$701$	−1.74460e9	−0.191286	−0.0956429	+	0.995416i	$0.530491\pi$
$702$	−5.98678e8	−0.0653151
$703$	9.06545e8	0.0984114
$704$	1.92204e9	0.207615
$705$	−1.33516e9	−0.143507
$706$	−9.63161e9	−1.03011
$707$	0	0
$708$	1.28760e9	0.136353
$709$	−1.12051e10	−1.18074	−0.590368	−	0.807134i	$-0.701017\pi$
$709$	−1.12051e10	−1.18074	−0.590368	+	0.807134i	$0.701017\pi$
$710$	5.21313e9	0.546631
$711$	2.77562e9	0.289612
$712$	−3.06750e9	−0.318496
$713$	−1.37408e9	−0.141971
$714$	0	0
$715$	3.17789e9	0.325138
$716$	−8.59398e9	−0.874981
$717$	−5.75314e9	−0.582892
$718$	−1.05646e10	−1.06516
$719$	9.36568e8	0.0939698	0.0469849	−	0.998896i	$-0.485039\pi$
$719$	9.36568e8	0.0939698	0.0469849	+	0.998896i	$0.485039\pi$
$720$	3.40402e8	0.0339882
$721$	0	0
$722$	−2.20682e9	−0.218216
$723$	6.94666e9	0.683584
$724$	−7.33827e9	−0.718636
$725$	2.71002e9	0.264113
$726$	−7.40255e9	−0.717965
$727$	−4.20445e9	−0.405825	−0.202913	−	0.979197i	$-0.565041\pi$
$727$	−4.20445e9	−0.405825	−0.202913	+	0.979197i	$0.565041\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	−2.42583e9	−0.230797
$731$	−1.03326e9	−0.0978358
$732$	−2.86955e9	−0.270411
$733$	1.15491e10	1.08314	0.541571	−	0.840655i	$-0.317830\pi$
$733$	1.15491e10	1.08314	0.541571	+	0.840655i	$0.317830\pi$
$734$	−1.40086e10	−1.30755
$735$	0	0
$736$	1.35817e9	0.125569
$737$	−2.41284e10	−2.22020
$738$	3.72710e9	0.341330
$739$	−1.39655e10	−1.27292	−0.636460	−	0.771310i	$-0.719602\pi$
$739$	−1.39655e10	−1.27292	−0.636460	+	0.771310i	$0.719602\pi$
$740$	−2.66056e8	−0.0241358
$741$	2.55198e9	0.230416
$742$	0	0
$743$	1.43832e10	1.28646	0.643230	−	0.765673i	$-0.277594\pi$
$743$	1.43832e10	1.28646	0.643230	+	0.765673i	$0.277594\pi$
$744$	4.58293e8	0.0407979
$745$	6.53932e9	0.579409
$746$	−3.90356e9	−0.344251
$747$	−1.62528e9	−0.142661
$748$	3.09985e9	0.270823
$749$	0	0
$750$	3.52749e9	0.305317
$751$	−6.70841e8	−0.0577936	−0.0288968	−	0.999582i	$-0.509199\pi$
$751$	−6.70841e8	−0.0577936	−0.0288968	+	0.999582i	$0.509199\pi$
$752$	1.77675e9	0.152357
$753$	3.32257e9	0.283590
$754$	−1.26561e9	−0.107523
$755$	−3.54395e9	−0.299691
$756$	0	0
$757$	−1.91569e10	−1.60506	−0.802529	−	0.596613i	$-0.796513\pi$
$757$	−1.91569e10	−1.60506	−0.802529	+	0.596613i	$0.796513\pi$
$758$	8.88055e9	0.740624
$759$	−8.20521e9	−0.681151
$760$	−1.45103e9	−0.119903
$761$	−1.79120e10	−1.47332	−0.736660	−	0.676263i	$-0.763598\pi$
$761$	−1.79120e10	−1.47332	−0.736660	+	0.676263i	$0.763598\pi$
$762$	−1.48350e9	−0.121463
$763$	0	0
$764$	1.04707e9	0.0849474
$765$	5.48998e8	0.0443359
$766$	−1.49529e10	−1.20206
$767$	−2.83302e9	−0.226708
$768$	−4.52985e8	−0.0360844
$769$	2.14072e10	1.69753	0.848765	−	0.528771i	$-0.177347\pi$
$769$	2.14072e10	1.69753	0.848765	+	0.528771i	$0.177347\pi$
$770$	0	0
$771$	−1.19700e10	−0.940599
$772$	−9.86870e9	−0.771968
$773$	7.55163e8	0.0588047	0.0294024	−	0.999568i	$-0.490640\pi$
$773$	7.55163e8	0.0588047	0.0294024	+	0.999568i	$0.490640\pi$
$774$	−9.12195e8	−0.0707122
$775$	2.15916e9	0.166620
$776$	2.07878e9	0.159696
$777$	0	0
$778$	−2.19116e9	−0.166819
$779$	−1.58875e10	−1.20413
$780$	−7.48964e8	−0.0565106
$781$	4.19108e10	3.14809
$782$	2.19044e9	0.163798
$783$	8.19010e8	0.0609709
$784$	0	0
$785$	3.85132e9	0.284162
$786$	−8.53649e9	−0.627047
$787$	−2.04665e10	−1.49669	−0.748347	−	0.663308i	$-0.769152\pi$
$787$	−2.04665e10	−1.49669	−0.748347	+	0.663308i	$0.769152\pi$
$788$	5.32664e9	0.387804
$789$	−8.07097e9	−0.585000
$790$	3.47239e9	0.250572
$791$	0	0
$792$	2.73665e9	0.195741
$793$	6.31367e9	0.449599
$794$	−4.99641e9	−0.354231
$795$	−2.41955e9	−0.170785
$796$	4.87502e9	0.342595
$797$	1.03098e10	0.721348	0.360674	−	0.932692i	$-0.382547\pi$
$797$	1.03098e10	0.721348	0.360674	+	0.932692i	$0.382547\pi$
$798$	0	0
$799$	2.86552e9	0.198742
$800$	−2.13415e9	−0.147370
$801$	−4.36759e9	−0.300281
$802$	4.44400e9	0.304203
$803$	−1.95024e10	−1.32918
$804$	5.68656e9	0.385881
$805$	0	0
$806$	−1.00835e9	−0.0678327
$807$	5.64051e9	0.377799
$808$	8.84835e9	0.590096
$809$	−4.16428e8	−0.0276516	−0.0138258	−	0.999904i	$-0.504401\pi$
$809$	−4.16428e8	−0.0276516	−0.0138258	+	0.999904i	$0.504401\pi$
$810$	4.84674e8	0.0320444
$811$	5.82687e9	0.383586	0.191793	−	0.981435i	$-0.438570\pi$
$811$	5.82687e9	0.383586	0.191793	+	0.981435i	$0.438570\pi$
$812$	0	0
$813$	−3.04423e8	−0.0198683
$814$	−2.13895e9	−0.139000
$815$	7.14391e9	0.462258
$816$	−7.30571e8	−0.0470703
$817$	3.88840e9	0.249456
$818$	1.72616e10	1.10267
$819$	0	0
$820$	4.66271e9	0.295318
$821$	−2.08333e10	−1.31388	−0.656941	−	0.753942i	$-0.728150\pi$
$821$	−2.08333e10	−1.31388	−0.656941	+	0.753942i	$0.728150\pi$
$822$	−4.14160e9	−0.260086
$823$	4.23403e9	0.264761	0.132381	−	0.991199i	$-0.457738\pi$
$823$	4.23403e9	0.264761	0.132381	+	0.991199i	$0.457738\pi$
$824$	7.35350e9	0.457877
$825$	1.28932e10	0.799414
$826$	0	0
$827$	5.70597e9	0.350800	0.175400	−	0.984497i	$-0.443878\pi$
$827$	5.70597e9	0.350800	0.175400	+	0.984497i	$0.443878\pi$
$828$	1.93380e9	0.118387
$829$	−2.51612e10	−1.53388	−0.766940	−	0.641719i	$-0.778221\pi$
$829$	−2.51612e10	−1.53388	−0.766940	+	0.641719i	$0.778221\pi$
$830$	−2.03327e9	−0.123431
$831$	1.77920e10	1.07553
$832$	9.96671e8	0.0599957
$833$	0	0
$834$	2.86090e9	0.170774
$835$	−7.14862e9	−0.424932
$836$	−1.16655e10	−0.690528
$837$	6.52531e8	0.0384647
$838$	−1.34237e10	−0.787988
$839$	−2.27048e10	−1.32724	−0.663622	−	0.748068i	$-0.730982\pi$
$839$	−2.27048e10	−1.32724	−0.663622	+	0.748068i	$0.730982\pi$
$840$	0	0
$841$	−1.55185e10	−0.899629
$842$	−4.20187e9	−0.242577
$843$	2.83832e9	0.163179
$844$	2.25565e9	0.129144
$845$	−5.50544e9	−0.313901
$846$	2.52978e9	0.143644
$847$	0	0
$848$	3.21978e9	0.181318
$849$	8.91435e9	0.499934
$850$	−3.44194e9	−0.192237
$851$	−1.51144e9	−0.0840695
$852$	−9.87751e9	−0.547153
$853$	−2.86872e10	−1.58258	−0.791292	−	0.611439i	$-0.790591\pi$
$853$	−2.86872e10	−1.58258	−0.791292	+	0.611439i	$0.790591\pi$
$854$	0	0
$855$	−2.06602e9	−0.113045
$856$	3.30465e9	0.180081
$857$	−5.34950e9	−0.290322	−0.145161	−	0.989408i	$-0.546370\pi$
$857$	−5.34950e9	−0.290322	−0.145161	+	0.989408i	$0.546370\pi$
$858$	−6.02127e9	−0.325449
$859$	1.40330e10	0.755394	0.377697	−	0.925929i	$-0.376716\pi$
$859$	1.40330e10	0.755394	0.377697	+	0.925929i	$0.376716\pi$
$860$	−1.14118e9	−0.0611801
$861$	0	0
$862$	1.36474e9	0.0725731
$863$	3.24994e10	1.72122	0.860612	−	0.509261i	$-0.170081\pi$
$863$	3.24994e10	1.72122	0.860612	+	0.509261i	$0.170081\pi$
$864$	−6.44973e8	−0.0340207
$865$	3.08394e9	0.162013
$866$	1.35038e10	0.706549
$867$	9.90088e9	0.515949
$868$	0	0
$869$	2.79162e10	1.44307
$870$	1.02460e9	0.0527520
$871$	−1.25118e10	−0.641586
$872$	−4.52818e8	−0.0231268
$873$	2.95983e9	0.150563
$874$	−8.24318e9	−0.417642
$875$	0	0
$876$	4.59630e9	0.231017
$877$	−3.38694e10	−1.69554	−0.847772	−	0.530361i	$-0.822056\pi$
$877$	−3.38694e10	−1.69554	−0.847772	+	0.530361i	$0.822056\pi$
$878$	9.42797e9	0.470097
$879$	−2.35171e9	−0.116794
$880$	3.42363e9	0.169355
$881$	1.52708e10	0.752397	0.376198	−	0.926539i	$-0.377231\pi$
$881$	1.52708e10	0.752397	0.376198	+	0.926539i	$0.377231\pi$
$882$	0	0
$883$	−1.12045e10	−0.547685	−0.273842	−	0.961775i	$-0.588295\pi$
$883$	−1.12045e10	−0.547685	−0.273842	+	0.961775i	$0.588295\pi$
$884$	1.60742e9	0.0782614
$885$	2.29354e9	0.111226
$886$	5.72604e9	0.276590
$887$	−6.97232e9	−0.335463	−0.167732	−	0.985833i	$-0.553644\pi$
$887$	−6.97232e9	−0.335463	−0.167732	+	0.985833i	$0.553644\pi$
$888$	5.04106e8	0.0241589
$889$	0	0
$890$	−5.46398e9	−0.259803
$891$	3.89653e9	0.184546
$892$	1.21044e10	0.571040
$893$	−1.07837e10	−0.506742
$894$	−1.23903e10	−0.579963
$895$	−1.53080e10	−0.713737
$896$	0	0
$897$	−4.25480e9	−0.196837
$898$	−1.09887e10	−0.506382
$899$	1.37945e9	0.0633211
$900$	−3.03866e9	−0.138942
$901$	5.19283e9	0.236520
$902$	3.74858e10	1.70076
$903$	0	0
$904$	6.21184e9	0.279660
$905$	−1.30713e10	−0.586204
$906$	6.71485e9	0.299977
$907$	1.18095e9	0.0525539	0.0262770	−	0.999655i	$-0.491635\pi$
$907$	1.18095e9	0.0525539	0.0262770	+	0.999655i	$0.491635\pi$
$908$	1.12704e10	0.499620
$909$	1.25985e10	0.556348
$910$	0	0
$911$	1.27915e10	0.560541	0.280271	−	0.959921i	$-0.409576\pi$
$911$	1.27915e10	0.560541	0.280271	+	0.959921i	$0.409576\pi$
$912$	2.74932e9	0.120017
$913$	−1.63465e10	−0.710847
$914$	1.47802e10	0.640276
$915$	−5.11138e9	−0.220579
$916$	−4.16254e9	−0.178947
$917$	0	0
$918$	−1.04021e9	−0.0443783
$919$	3.52353e10	1.49752	0.748761	−	0.662840i	$-0.230649\pi$
$919$	3.52353e10	1.49752	0.748761	+	0.662840i	$0.230649\pi$
$920$	2.41924e9	0.102429
$921$	−1.05761e10	−0.446086
$922$	−2.47531e10	−1.04009
$923$	2.17328e10	0.909725
$924$	0	0
$925$	2.37499e9	0.0986658
$926$	2.40361e10	0.994775
$927$	1.04701e10	0.431691
$928$	−1.36348e9	−0.0560054
$929$	2.17764e10	0.891111	0.445556	−	0.895254i	$-0.353006\pi$
$929$	2.17764e10	0.891111	0.445556	+	0.895254i	$0.353006\pi$
$930$	8.16335e8	0.0332796
$931$	0	0
$932$	−1.60844e10	−0.650802
$933$	−5.53327e9	−0.223047
$934$	2.39401e10	0.961418
$935$	5.52161e9	0.220915
$936$	1.41909e9	0.0565645
$937$	−1.15795e10	−0.459833	−0.229916	−	0.973210i	$-0.573845\pi$
$937$	−1.15795e10	−0.459833	−0.229916	+	0.973210i	$0.573845\pi$
$938$	0	0
$939$	2.36845e10	0.933542
$940$	3.16483e9	0.124280
$941$	3.83930e10	1.50207	0.751033	−	0.660265i	$-0.229556\pi$
$941$	3.83930e10	1.50207	0.751033	+	0.660265i	$0.229556\pi$
$942$	−7.29723e9	−0.284433
$943$	2.64885e10	1.02865
$944$	−3.05209e9	−0.118085
$945$	0	0
$946$	−9.17450e9	−0.352341
$947$	1.11841e10	0.427933	0.213966	−	0.976841i	$-0.431362\pi$
$947$	1.11841e10	0.427933	0.213966	+	0.976841i	$0.431362\pi$
$948$	−6.57926e9	−0.250812
$949$	−1.01129e10	−0.384101
$950$	1.29529e10	0.490154
$951$	1.19024e10	0.448749
$952$	0	0
$953$	−5.19835e9	−0.194554	−0.0972770	−	0.995257i	$-0.531013\pi$
$953$	−5.19835e9	−0.194554	−0.0972770	+	0.995257i	$0.531013\pi$
$954$	4.58441e9	0.170948
$955$	1.86510e9	0.0692931
$956$	1.36371e10	0.504799
$957$	8.23728e9	0.303803
$958$	−1.47233e10	−0.541036
$959$	0	0
$960$	−8.06879e8	−0.0294347
$961$	−2.64136e10	−0.960053
$962$	−1.10915e9	−0.0401677
$963$	4.70526e9	0.169782
$964$	−1.64662e10	−0.592001
$965$	−1.75786e10	−0.629708
$966$	0	0
$967$	−2.69243e8	−0.00957529	−0.00478765	−	0.999989i	$-0.501524\pi$
$967$	−2.69243e8	−0.00957529	−0.00478765	+	0.999989i	$0.501524\pi$
$968$	1.75468e10	0.621776
$969$	4.43408e9	0.156556
$970$	3.70283e9	0.130267
$971$	−4.37283e9	−0.153284	−0.0766418	−	0.997059i	$-0.524420\pi$
$971$	−4.37283e9	−0.153284	−0.0766418	+	0.997059i	$0.524420\pi$
$972$	−9.18330e8	−0.0320750
$973$	0	0
$974$	−3.41341e9	−0.118367
$975$	6.68575e9	0.231012
$976$	6.80189e9	0.234183
$977$	−3.74991e10	−1.28644	−0.643220	−	0.765681i	$-0.722402\pi$
$977$	−3.74991e10	−1.28644	−0.643220	+	0.765681i	$0.722402\pi$
$978$	−1.35358e10	−0.462699
$979$	−4.39276e10	−1.49623
$980$	0	0
$981$	−6.44735e8	−0.0218042
$982$	4.86038e8	0.0163787
$983$	3.06190e10	1.02814	0.514071	−	0.857748i	$-0.328137\pi$
$983$	3.06190e10	1.02814	0.514071	+	0.857748i	$0.328137\pi$
$984$	−8.83461e9	−0.295600
$985$	9.48808e9	0.316338
$986$	−2.19901e9	−0.0730562
$987$	0	0
$988$	−6.04913e9	−0.199546
$989$	−6.48296e9	−0.213102
$990$	4.87467e9	0.159669
$991$	−1.72703e10	−0.563693	−0.281847	−	0.959459i	$-0.590947\pi$
$991$	−1.72703e10	−0.563693	−0.281847	+	0.959459i	$0.590947\pi$
$992$	−1.08632e9	−0.0353320
$993$	−3.00302e10	−0.973276
$994$	0	0
$995$	8.68363e9	0.279461
$996$	3.85252e9	0.123548
$997$	3.75077e9	0.119864	0.0599319	−	0.998202i	$-0.480912\pi$
$997$	3.75077e9	0.119864	0.0599319	+	0.998202i	$0.480912\pi$
$998$	2.59670e10	0.826923
$999$	7.17760e8	0.0227772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.8.a.l.1.1		1
7.2	even	3		294.8.e.d.67.1		2
7.3	odd	6		294.8.e.c.79.1		2
7.4	even	3		294.8.e.d.79.1		2
7.5	odd	6		294.8.e.c.67.1		2
7.6	odd	2		6.8.a.a.1.1	✓	1
21.20	even	2		18.8.a.a.1.1		1
28.27	even	2		48.8.a.b.1.1		1
35.13	even	4		150.8.c.k.49.1		2
35.27	even	4		150.8.c.k.49.2		2
35.34	odd	2		150.8.a.e.1.1		1
56.13	odd	2		192.8.a.f.1.1		1
56.27	even	2		192.8.a.n.1.1		1
63.13	odd	6		162.8.c.d.55.1		2
63.20	even	6		162.8.c.i.109.1		2
63.34	odd	6		162.8.c.d.109.1		2
63.41	even	6		162.8.c.i.55.1		2
84.83	odd	2		144.8.a.h.1.1		1
105.62	odd	4		450.8.c.a.199.1		2
105.83	odd	4		450.8.c.a.199.2		2
105.104	even	2		450.8.a.ba.1.1		1
168.83	odd	2		576.8.a.i.1.1		1
168.125	even	2		576.8.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.8.a.a.1.1	✓	1	7.6	odd	2
18.8.a.a.1.1		1	21.20	even	2
48.8.a.b.1.1		1	28.27	even	2
144.8.a.h.1.1		1	84.83	odd	2
150.8.a.e.1.1		1	35.34	odd	2
150.8.c.k.49.1		2	35.13	even	4
150.8.c.k.49.2		2	35.27	even	4
162.8.c.d.55.1		2	63.13	odd	6
162.8.c.d.109.1		2	63.34	odd	6
162.8.c.i.55.1		2	63.41	even	6
162.8.c.i.109.1		2	63.20	even	6
192.8.a.f.1.1		1	56.13	odd	2
192.8.a.n.1.1		1	56.27	even	2
294.8.a.l.1.1		1	1.1	even	1	trivial
294.8.e.c.67.1		2	7.5	odd	6
294.8.e.c.79.1		2	7.3	odd	6
294.8.e.d.67.1		2	7.2	even	3
294.8.e.d.79.1		2	7.4	even	3
450.8.a.ba.1.1		1	105.104	even	2
450.8.c.a.199.1		2	105.62	odd	4
450.8.c.a.199.2		2	105.83	odd	4
576.8.a.h.1.1		1	168.125	even	2
576.8.a.i.1.1		1	168.83	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:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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