LMFDB - Embedded newform 150.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	150.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:	$46.8577538226$
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	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +1576.00 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +1576.00 q^{7} -512.000 q^{8} +729.000 q^{9} +7332.00 q^{11} -1728.00 q^{12} +3802.00 q^{13} -12608.0 q^{14} +4096.00 q^{16} +6606.00 q^{17} -5832.00 q^{18} +24860.0 q^{19} -42552.0 q^{21} -58656.0 q^{22} -41448.0 q^{23} +13824.0 q^{24} -30416.0 q^{26} -19683.0 q^{27} +100864. q^{28} -41610.0 q^{29} +33152.0 q^{31} -32768.0 q^{32} -197964. q^{33} -52848.0 q^{34} +46656.0 q^{36} +36466.0 q^{37} -198880. q^{38} -102654. q^{39} -639078. q^{41} +340416. q^{42} +156412. q^{43} +469248. q^{44} +331584. q^{46} +433776. q^{47} -110592. q^{48} +1.66023e6 q^{49} -178362. q^{51} +243328. q^{52} -786078. q^{53} +157464. q^{54} -806912. q^{56} -671220. q^{57} +332880. q^{58} +745140. q^{59} -1.66062e6 q^{61} -265216. q^{62} +1.14890e6 q^{63} +262144. q^{64} +1.58371e6 q^{66} +3.29084e6 q^{67} +422784. q^{68} +1.11910e6 q^{69} +5.71615e6 q^{71} -373248. q^{72} -2.65990e6 q^{73} -291728. q^{74} +1.59104e6 q^{76} +1.15552e7 q^{77} +821232. q^{78} +3.80744e6 q^{79} +531441. q^{81} +5.11262e6 q^{82} -2.22947e6 q^{83} -2.72333e6 q^{84} -1.25130e6 q^{86} +1.12347e6 q^{87} -3.75398e6 q^{88} +5.99121e6 q^{89} +5.99195e6 q^{91} -2.65267e6 q^{92} -895104. q^{93} -3.47021e6 q^{94} +884736. q^{96} +4.06013e6 q^{97} -1.32819e7 q^{98} +5.34503e6 q^{99} +O(q^{100})\)

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	216.000	0.408248
$7$	1576.00	1.73665	0.868327	−	0.495993i	$-0.165196\pi$
$7$	1576.00	1.73665	0.868327	+	0.495993i	$0.165196\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	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	$0.422858\pi$
$14$	−12608.0	−1.22800
$15$	0	0
$16$	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	$-0.363519\pi$
$19$	24860.0	0.831502	0.415751	+	0.909478i	$0.363519\pi$
$20$	0	0
$21$	−42552.0	−1.00266
$22$	−58656.0	−1.17445
$23$	−41448.0	−0.710323	−0.355162	−	0.934805i	$-0.615574\pi$
$23$	−41448.0	−0.710323	−0.355162	+	0.934805i	$0.615574\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	−30416.0	−0.339387
$27$	−19683.0	−0.192450
$28$	100864.	0.868327
$29$	−41610.0	−0.316814	−0.158407	−	0.987374i	$-0.550636\pi$
$29$	−41610.0	−0.316814	−0.158407	+	0.987374i	$0.550636\pi$
$30$	0	0
$31$	33152.0	0.199868	0.0999341	−	0.994994i	$-0.468137\pi$
$31$	33152.0	0.199868	0.0999341	+	0.994994i	$0.468137\pi$
$32$	−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	$-0.481152\pi$
$37$	36466.0	0.118354	0.0591769	+	0.998248i	$0.481152\pi$
$38$	−198880.	−0.587961
$39$	−102654.	−0.277108
$40$	0	0
$41$	−639078.	−1.44814	−0.724070	−	0.689727i	$-0.757731\pi$
$41$	−639078.	−1.44814	−0.724070	+	0.689727i	$0.757731\pi$
$42$	340416.	0.708986
$43$	156412.	0.300006	0.150003	−	0.988686i	$-0.452072\pi$
$43$	156412.	0.300006	0.150003	+	0.988686i	$0.452072\pi$
$44$	469248.	0.830459
$45$	0	0
$46$	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
$49$	1.66023e6	2.01596
$50$	0	0
$51$	−178362.	−0.188281
$52$	243328.	0.239983
$53$	−786078.	−0.725271	−0.362635	−	0.931931i	$-0.618123\pi$
$53$	−786078.	−0.725271	−0.362635	+	0.931931i	$0.618123\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	−806912.	−0.614000
$57$	−671220.	−0.480068
$58$	332880.	0.224022
$59$	745140.	0.472341	0.236171	−	0.971712i	$-0.424108\pi$
$59$	745140.	0.472341	0.236171	+	0.971712i	$0.424108\pi$
$60$	0	0
$61$	−1.66062e6	−0.936732	−0.468366	−	0.883535i	$-0.655157\pi$
$61$	−1.66062e6	−0.936732	−0.468366	+	0.883535i	$0.655157\pi$
$62$	−265216.	−0.141328
$63$	1.14890e6	0.578884
$64$	262144.	0.125000
$65$	0	0
$66$	1.58371e6	0.678067
$67$	3.29084e6	1.33673	0.668366	−	0.743832i	$-0.266994\pi$
$67$	3.29084e6	1.33673	0.668366	+	0.743832i	$0.266994\pi$
$68$	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
$75$	0	0
$76$	1.59104e6	0.415751
$77$	1.15552e7	2.88444
$78$	821232.	0.195945
$79$	3.80744e6	0.868837	0.434418	−	0.900711i	$-0.356954\pi$
$79$	3.80744e6	0.868837	0.434418	+	0.900711i	$0.356954\pi$
$80$	0	0
$81$	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	$0.568647\pi$
$84$	−2.72333e6	−0.501329
$85$	0	0
$86$	−1.25130e6	−0.212137
$87$	1.12347e6	0.182913
$88$	−3.75398e6	−0.587223
$89$	5.99121e6	0.900844	0.450422	−	0.892816i	$-0.351274\pi$
$89$	5.99121e6	0.900844	0.450422	+	0.892816i	$0.351274\pi$
$90$	0	0
$91$	5.99195e6	0.833534
$92$	−2.65267e6	−0.355162
$93$	−895104.	−0.115394
$94$	−3.47021e6	−0.430931
$95$	0	0
$96$	884736.	0.102062
$97$	4.06013e6	0.451688	0.225844	−	0.974163i	$-0.427486\pi$
$97$	4.06013e6	0.451688	0.225844	+	0.974163i	$0.427486\pi$
$98$	−1.32819e7	−1.42550
$99$	5.34503e6	0.553639
$100$	0	0
$101$	−1.72819e7	−1.66904	−0.834522	−	0.550975i	$-0.814256\pi$
$101$	−1.72819e7	−1.66904	−0.834522	+	0.550975i	$0.814256\pi$
$102$	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	$-0.581968\pi$
$107$	−6.45440e6	−0.509346	−0.254673	+	0.967027i	$0.581968\pi$
$108$	−1.25971e6	−0.0962250
$109$	−884410.	−0.0654125	−0.0327063	−	0.999465i	$-0.510413\pi$
$109$	−884410.	−0.0654125	−0.0327063	+	0.999465i	$0.510413\pi$
$110$	0	0
$111$	−984582.	−0.0683316
$112$	6.45530e6	0.434163
$113$	−1.21325e7	−0.790999	−0.395499	−	0.918466i	$-0.629428\pi$
$113$	−1.21325e7	−0.790999	−0.395499	+	0.918466i	$0.629428\pi$
$114$	5.36976e6	0.339459
$115$	0	0
$116$	−2.66304e6	−0.158407
$117$	2.77166e6	0.159989
$118$	−5.96112e6	−0.333996
$119$	1.04111e7	0.566344
$120$	0	0
$121$	3.42711e7	1.75865
$122$	1.32849e7	0.662369
$123$	1.72551e7	0.836084
$124$	2.12173e6	0.0999341
$125$	0	0
$126$	−9.19123e6	−0.409333
$127$	−6.86806e6	−0.297524	−0.148762	−	0.988873i	$-0.547529\pi$
$127$	−6.86806e6	−0.297524	−0.148762	+	0.988873i	$0.547529\pi$
$128$	−2.09715e6	−0.0883883
$129$	−4.22312e6	−0.173209
$130$	0	0
$131$	−3.95208e7	−1.53595	−0.767973	−	0.640482i	$-0.778735\pi$
$131$	−3.95208e7	−1.53595	−0.767973	+	0.640482i	$0.778735\pi$
$132$	−1.26697e7	−0.479466
$133$	3.91794e7	1.44403
$134$	−2.63267e7	−0.945212
$135$	0	0
$136$	−3.38227e6	−0.115298
$137$	−1.91741e7	−0.637078	−0.318539	−	0.947910i	$-0.603192\pi$
$137$	−1.91741e7	−0.637078	−0.318539	+	0.947910i	$0.603192\pi$
$138$	−8.95277e6	−0.289988
$139$	1.32449e7	0.418309	0.209154	−	0.977883i	$-0.432929\pi$
$139$	1.32449e7	0.418309	0.209154	+	0.977883i	$0.432929\pi$
$140$	0	0
$141$	−1.17120e7	−0.351854
$142$	−4.57292e7	−1.34025
$143$	2.78763e7	0.797184
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	2.12792e7	0.565874
$147$	−4.48263e7	−1.16392
$148$	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	$0.248556\pi$
$150$	0	0
$151$	−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
$154$	−9.24419e7	−2.03961
$155$	0	0
$156$	−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
$160$	0	0
$161$	−6.53220e7	−1.23359
$162$	−4.25153e6	−0.0785674
$163$	−6.26659e7	−1.13338	−0.566689	−	0.823932i	$-0.691776\pi$
$163$	−6.26659e7	−1.13338	−0.566689	+	0.823932i	$0.691776\pi$
$164$	−4.09010e7	−0.724070
$165$	0	0
$166$	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	$0.674415\pi$
$168$	2.17866e7	0.354493
$169$	−4.82933e7	−0.769633
$170$	0	0
$171$	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	$0.839126\pi$
$180$	0	0
$181$	1.14661e8	1.43727	0.718636	−	0.695386i	$-0.244767\pi$
$181$	1.14661e8	1.43727	0.718636	+	0.695386i	$0.244767\pi$
$182$	−4.79356e7	−0.589398
$183$	4.48367e7	0.540822
$184$	2.12214e7	0.251137
$185$	0	0
$186$	7.16083e6	0.0815959
$187$	4.84352e7	0.541646
$188$	2.77617e7	0.304714
$189$	−3.10204e7	−0.334219
$190$	0	0
$191$	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	$-0.219272\pi$
$193$	1.54198e8	1.54394	0.771968	+	0.635661i	$0.219272\pi$
$194$	−3.24810e7	−0.319392
$195$	0	0
$196$	1.06255e8	1.00798
$197$	−8.32288e7	−0.775607	−0.387804	−	0.921742i	$-0.626766\pi$
$197$	−8.32288e7	−0.775607	−0.387804	+	0.921742i	$0.626766\pi$
$198$	−4.27602e7	−0.391482
$199$	−7.61722e7	−0.685190	−0.342595	−	0.939483i	$-0.611306\pi$
$199$	−7.61722e7	−0.685190	−0.342595	+	0.939483i	$0.611306\pi$
$200$	0	0
$201$	−8.88526e7	−0.771763
$202$	1.38256e8	1.18019
$203$	−6.55774e7	−0.550196
$204$	−1.14152e7	−0.0941405
$205$	0	0
$206$	−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
$215$	0	0
$216$	1.00777e7	0.0680414
$217$	5.22476e7	0.347102
$218$	7.07528e6	0.0462536
$219$	7.18172e7	0.462034
$220$	0	0
$221$	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	$0.306540\pi$
$224$	−5.16424e7	−0.307000
$225$	0	0
$226$	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	$-0.442731\pi$
$229$	6.50396e7	0.357894	0.178947	+	0.983859i	$0.442731\pi$
$230$	0	0
$231$	−3.11991e8	−1.66533
$232$	2.13043e7	0.112011
$233$	2.51319e8	1.30160	0.650802	−	0.759248i	$-0.274433\pi$
$233$	2.51319e8	1.30160	0.650802	+	0.759248i	$0.274433\pi$
$234$	−2.21733e7	−0.113129
$235$	0	0
$236$	4.76890e7	0.236171
$237$	−1.02801e8	−0.501623
$238$	−8.32884e7	−0.400466
$239$	2.13079e8	1.00960	0.504799	−	0.863237i	$-0.331566\pi$
$239$	2.13079e8	1.00960	0.504799	+	0.863237i	$0.331566\pi$
$240$	0	0
$241$	2.57284e8	1.18400	0.592001	−	0.805937i	$-0.298338\pi$
$241$	2.57284e8	1.18400	0.592001	+	0.805937i	$0.298338\pi$
$242$	−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
$250$	0	0
$251$	1.23058e8	0.491193	0.245596	−	0.969372i	$-0.421016\pi$
$251$	1.23058e8	0.491193	0.245596	+	0.969372i	$0.421016\pi$
$252$	7.35299e7	0.289442
$253$	−3.03897e8	−1.17979
$254$	5.49445e7	0.210381
$255$	0	0
$256$	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
$259$	5.74704e7	0.205539
$260$	0	0
$261$	−3.03337e7	−0.105605
$262$	3.16166e8	1.08608
$263$	−2.98925e8	−1.01325	−0.506625	−	0.862166i	$-0.669107\pi$
$263$	−2.98925e8	−1.01325	−0.506625	+	0.862166i	$0.669107\pi$
$264$	1.01358e8	0.339033
$265$	0	0
$266$	−3.13435e8	−1.02108
$267$	−1.61763e8	−0.520102
$268$	2.10614e8	0.668366
$269$	2.08908e8	0.654368	0.327184	−	0.944961i	$-0.393900\pi$
$269$	2.08908e8	0.654368	0.327184	+	0.944961i	$0.393900\pi$
$270$	0	0
$271$	−1.12749e7	−0.0344129	−0.0172064	−	0.999852i	$-0.505477\pi$
$271$	−1.12749e7	−0.0344129	−0.0172064	+	0.999852i	$0.505477\pi$
$272$	2.70582e7	0.0815281
$273$	−1.61783e8	−0.481241
$274$	1.53393e8	0.450482
$275$	0	0
$276$	7.16221e7	0.205053
$277$	6.58964e8	1.86287	0.931435	−	0.363907i	$-0.118557\pi$
$277$	6.58964e8	1.86287	0.931435	+	0.363907i	$0.118557\pi$
$278$	−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
$285$	0	0
$286$	−2.23010e8	−0.563694
$287$	−1.00719e9	−2.51492
$288$	−2.38879e7	−0.0589256
$289$	−3.66699e8	−0.893651
$290$	0	0
$291$	−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	$0.467749\pi$
$294$	3.58610e8	0.823014
$295$	0	0
$296$	−1.86706e7	−0.0418444
$297$	−1.44316e8	−0.319644
$298$	−4.58899e8	−1.00452
$299$	−1.57585e8	−0.340931
$300$	0	0
$301$	2.46505e8	0.521007
$302$	2.48698e8	0.519575
$303$	4.66612e8	0.963623
$304$	1.01827e8	0.207876
$305$	0	0
$306$	−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	$0.373745\pi$
$308$	7.39535e8	1.44222
$309$	−3.87782e8	−0.747710
$310$	0	0
$311$	−2.04936e8	−0.386328	−0.193164	−	0.981166i	$-0.561875\pi$
$311$	−2.04936e8	−0.386328	−0.193164	+	0.981166i	$0.561875\pi$
$312$	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	$-0.372949\pi$
$317$	4.40831e8	0.777256	0.388628	+	0.921395i	$0.372949\pi$
$318$	−1.69793e8	−0.296090
$319$	−3.05085e8	−0.526202
$320$	0	0
$321$	1.74269e8	0.294071
$322$	5.22576e8	0.872277
$323$	1.64225e8	0.271163
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	5.01327e8	0.801419
$327$	2.38791e7	0.0377659
$328$	3.27208e8	0.511995
$329$	6.83631e8	1.05837
$330$	0	0
$331$	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
$335$	0	0
$336$	−1.74293e8	−0.250664
$337$	−2.88198e8	−0.410191	−0.205096	−	0.978742i	$-0.565751\pi$
$337$	−2.88198e8	−0.410191	−0.205096	+	0.978742i	$0.565751\pi$
$338$	3.86347e8	0.544213
$339$	3.27577e8	0.456683
$340$	0	0
$341$	2.43070e8	0.331965
$342$	−1.44984e8	−0.195987
$343$	1.31862e9	1.76438
$344$	−8.00829e7	−0.106068
$345$	0	0
$346$	−2.16417e8	−0.280883
$347$	1.10601e9	1.42103	0.710517	−	0.703680i	$-0.248461\pi$
$347$	1.10601e9	1.42103	0.710517	+	0.703680i	$0.248461\pi$
$348$	7.19021e7	0.0914564
$349$	−1.32184e9	−1.66453	−0.832264	−	0.554379i	$-0.812956\pi$
$349$	−1.32184e9	−1.66453	−0.832264	+	0.554379i	$0.812956\pi$
$350$	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
$355$	0	0
$356$	3.83437e8	0.450422
$357$	−2.81099e8	−0.326979
$358$	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	$0.771484\pi$
$360$	0	0
$361$	−2.75852e8	−0.308604
$362$	−9.17284e8	−1.01630
$363$	−9.25318e8	−1.01536
$364$	3.83485e8	0.416767
$365$	0	0
$366$	−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
$370$	0	0
$371$	−1.23886e9	−1.25954
$372$	−5.72867e7	−0.0576970
$373$	4.87945e8	0.486844	0.243422	−	0.969920i	$-0.421730\pi$
$373$	4.87945e8	0.486844	0.243422	+	0.969920i	$0.421730\pi$
$374$	−3.87482e8	−0.383001
$375$	0	0
$376$	−2.22093e8	−0.215466
$377$	−1.58201e8	−0.152060
$378$	2.48163e8	0.236329
$379$	1.11007e9	1.04740	0.523700	−	0.851903i	$-0.324551\pi$
$379$	1.11007e9	1.04740	0.523700	+	0.851903i	$0.324551\pi$
$380$	0	0
$381$	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	$0.537635\pi$
$390$	0	0
$391$	−2.73805e8	−0.231645
$392$	−8.50039e8	−0.712751
$393$	1.06706e9	0.886779
$394$	6.65831e8	0.548437
$395$	0	0
$396$	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
$399$	−1.05784e9	−0.833712
$400$	0	0
$401$	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
$405$	0	0
$406$	5.24619e8	0.389048
$407$	2.67369e8	0.196576
$408$	9.13213e7	0.0665674
$409$	−2.15770e9	−1.55941	−0.779704	−	0.626149i	$-0.784630\pi$
$409$	−2.15770e9	−1.55941	−0.779704	+	0.626149i	$0.784630\pi$
$410$	0	0
$411$	5.17700e8	0.367817
$412$	9.19188e8	0.647536
$413$	1.17434e9	0.820293
$414$	2.41725e8	0.167425
$415$	0	0
$416$	−1.24584e8	−0.0848468
$417$	−3.57612e8	−0.241511
$418$	−1.45819e9	−0.976555
$419$	1.67797e9	1.11438	0.557191	−	0.830384i	$-0.311879\pi$
$419$	1.67797e9	1.11438	0.557191	+	0.830384i	$0.311879\pi$
$420$	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
$425$	0	0
$426$	1.23469e9	0.773791
$427$	−2.61713e9	−1.62678
$428$	−4.13082e8	−0.254673
$429$	−7.52659e8	−0.460254
$430$	0	0
$431$	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	$0.333478\pi$
$434$	−4.17980e8	−0.245438
$435$	0	0
$436$	−5.66022e7	−0.0327063
$437$	−1.03040e9	−0.590636
$438$	−5.74538e8	−0.326708
$439$	−1.17850e9	−0.664817	−0.332409	−	0.943135i	$-0.607861\pi$
$439$	−1.17850e9	−0.664817	−0.332409	+	0.943135i	$0.607861\pi$
$440$	0	0
$441$	1.21031e9	0.671988
$442$	−2.00928e8	−0.110678
$443$	−7.15755e8	−0.391157	−0.195579	−	0.980688i	$-0.562658\pi$
$443$	−7.15755e8	−0.391157	−0.195579	+	0.980688i	$0.562658\pi$
$444$	−6.30132e7	−0.0341658
$445$	0	0
$446$	−1.51305e9	−0.807573
$447$	−1.54879e9	−0.820191
$448$	4.13139e8	0.217082
$449$	−1.37358e9	−0.716132	−0.358066	−	0.933696i	$-0.616564\pi$
$449$	−1.37358e9	−0.716132	−0.358066	+	0.933696i	$0.616564\pi$
$450$	0	0
$451$	−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	$-0.649555\pi$
$457$	−1.84752e9	−0.905488	−0.452744	+	0.891641i	$0.649555\pi$
$458$	−5.20317e8	−0.253069
$459$	−1.30026e8	−0.0627604
$460$	0	0
$461$	3.09414e9	1.47091	0.735455	−	0.677573i	$-0.236968\pi$
$461$	3.09414e9	1.47091	0.735455	+	0.677573i	$0.236968\pi$
$462$	2.49593e9	1.17757
$463$	−3.00451e9	−1.40682	−0.703412	−	0.710782i	$-0.748341\pi$
$463$	−3.00451e9	−1.40682	−0.703412	+	0.710782i	$0.748341\pi$
$464$	−1.70435e8	−0.0792036
$465$	0	0
$466$	−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
$469$	5.18636e9	2.32144
$470$	0	0
$471$	−9.12154e8	−0.402249
$472$	−3.81512e8	−0.166998
$473$	1.14681e9	0.498286
$474$	8.22407e8	0.354701
$475$	0	0
$476$	6.66308e8	0.283172
$477$	−5.73051e8	−0.241757
$478$	−1.70464e9	−0.713894
$479$	1.84041e9	0.765141	0.382570	−	0.923926i	$-0.375039\pi$
$479$	1.84041e9	0.765141	0.382570	+	0.923926i	$0.375039\pi$
$480$	0	0
$481$	1.38644e8	0.0568058
$482$	−2.05827e9	−0.837216
$483$	1.76370e9	0.712211
$484$	2.19335e9	0.879323
$485$	0	0
$486$	1.14791e8	0.0453609
$487$	4.26676e8	0.167397	0.0836983	−	0.996491i	$-0.473327\pi$
$487$	4.26676e8	0.167397	0.0836983	+	0.996491i	$0.473327\pi$
$488$	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
$495$	0	0
$496$	1.35791e8	0.0499671
$497$	9.00866e9	3.29164
$498$	−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	$0.301203\pi$
$500$	0	0
$501$	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	$0.458374\pi$
$504$	−5.88239e8	−0.204667
$505$	0	0
$506$	2.43117e9	0.834237
$507$	1.30392e9	0.444348
$508$	−4.39556e8	−0.148762
$509$	−4.44155e8	−0.149287	−0.0746436	−	0.997210i	$-0.523782\pi$
$509$	−4.44155e8	−0.149287	−0.0746436	+	0.997210i	$0.523782\pi$
$510$	0	0
$511$	−4.19200e9	−1.38979
$512$	−1.34218e8	−0.0441942
$513$	−4.89319e8	−0.160023
$514$	−3.54667e9	−1.15199
$515$	0	0
$516$	−2.70280e8	−0.0866044
$517$	3.18045e9	1.01221
$518$	−4.59763e8	−0.145338
$519$	−7.30407e8	−0.229340
$520$	0	0
$521$	3.04963e9	0.944745	0.472372	−	0.881399i	$-0.343398\pi$
$521$	3.04963e9	0.944745	0.472372	+	0.881399i	$0.343398\pi$
$522$	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
$530$	0	0
$531$	5.43207e8	0.157447
$532$	2.50748e9	0.722016
$533$	−2.42977e9	−0.695058
$534$	1.29410e9	0.367768
$535$	0	0
$536$	−1.68491e9	−0.472606
$537$	3.62558e9	1.01034
$538$	−1.67126e9	−0.462708
$539$	1.21728e10	3.34835
$540$	0	0
$541$	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
$545$	0	0
$546$	1.29426e9	0.340289
$547$	−1.99956e9	−0.522371	−0.261185	−	0.965289i	$-0.584113\pi$
$547$	−1.99956e9	−0.522371	−0.261185	+	0.965289i	$0.584113\pi$
$548$	−1.22714e9	−0.318539
$549$	−1.21059e9	−0.312244
$550$	0	0
$551$	−1.03442e9	−0.263432
$552$	−5.72977e8	−0.144994
$553$	6.00053e9	1.50887
$554$	−5.27172e9	−1.31725
$555$	0	0
$556$	8.47674e8	0.209154
$557$	3.37403e9	0.827287	0.413643	−	0.910439i	$-0.364256\pi$
$557$	3.37403e9	0.827287	0.413643	+	0.910439i	$0.364256\pi$
$558$	−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
$565$	0	0
$566$	2.64129e9	0.612292
$567$	8.37551e8	0.192961
$568$	−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	$0.532228\pi$
$570$	0	0
$571$	−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
$574$	8.05750e9	1.77831
$575$	0	0
$576$	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
$580$	0	0
$581$	−3.51364e9	−0.743260
$582$	8.76987e8	0.184401
$583$	−5.76352e9	−1.20461
$584$	1.36187e9	0.282937
$585$	0	0
$586$	−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	$0.646826\pi$
$588$	−2.86888e9	−0.581959
$589$	8.24159e8	0.166191
$590$	0	0
$591$	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	$0.475641\pi$
$600$	0	0
$601$	−4.87162e9	−0.915403	−0.457702	−	0.889106i	$-0.651327\pi$
$601$	−4.87162e9	−0.915403	−0.457702	+	0.889106i	$0.651327\pi$
$602$	−1.97204e9	−0.368408
$603$	2.39902e9	0.445577
$604$	−1.98959e9	−0.367395
$605$	0	0
$606$	−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
$609$	1.77059e9	0.317656
$610$	0	0
$611$	1.64922e9	0.292505
$612$	3.08210e8	0.0543521
$613$	−3.47891e9	−0.610002	−0.305001	−	0.952352i	$-0.598657\pi$
$613$	−3.47891e9	−0.610002	−0.305001	+	0.952352i	$0.598657\pi$
$614$	−3.13367e9	−0.546342
$615$	0	0
$616$	−5.91628e9	−1.01980
$617$	2.39378e8	0.0410286	0.0205143	−	0.999790i	$-0.493470\pi$
$617$	2.39378e8	0.0410286	0.0205143	+	0.999790i	$0.493470\pi$
$618$	3.10226e9	0.528711
$619$	−5.52959e9	−0.937078	−0.468539	−	0.883443i	$-0.655219\pi$
$619$	−5.52959e9	−0.937078	−0.468539	+	0.883443i	$0.655219\pi$
$620$	0	0
$621$	8.15821e8	0.136702
$622$	1.63949e9	0.273175
$623$	9.44215e9	1.56445
$624$	−4.20471e8	−0.0692771
$625$	0	0
$626$	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
$635$	0	0
$636$	1.35834e9	0.209368
$637$	6.31221e9	0.967594
$638$	2.44068e9	0.372081
$639$	4.16707e9	0.631798
$640$	0	0
$641$	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	$0.466934\pi$
$644$	−4.18061e9	−0.616793
$645$	0	0
$646$	−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
$650$	0	0
$651$	−1.41068e9	−0.200399
$652$	−4.01062e9	−0.566689
$653$	−3.24403e9	−0.455921	−0.227960	−	0.973670i	$-0.573206\pi$
$653$	−3.24403e9	−0.455921	−0.227960	+	0.973670i	$0.573206\pi$
$654$	−1.91033e8	−0.0267046
$655$	0	0
$656$	−2.61766e9	−0.362035
$657$	−1.93907e9	−0.266756
$658$	−5.46905e9	−0.748378
$659$	−5.16506e9	−0.703034	−0.351517	−	0.936181i	$-0.614334\pi$
$659$	−5.16506e9	−0.703034	−0.351517	+	0.936181i	$0.614334\pi$
$660$	0	0
$661$	−3.22515e9	−0.434356	−0.217178	−	0.976132i	$-0.569685\pi$
$661$	−3.22515e9	−0.434356	−0.217178	+	0.976132i	$0.569685\pi$
$662$	−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
$670$	0	0
$671$	−1.21757e10	−1.55583
$672$	1.39434e9	0.177246
$673$	2.00633e9	0.253718	0.126859	−	0.991921i	$-0.459510\pi$
$673$	2.00633e9	0.253718	0.126859	+	0.991921i	$0.459510\pi$
$674$	2.30559e9	0.290049
$675$	0	0
$676$	−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
$679$	6.39876e9	0.784425
$680$	0	0
$681$	−4.75471e9	−0.576911
$682$	−1.94456e9	−0.234734
$683$	5.84861e9	0.702393	0.351196	−	0.936302i	$-0.385775\pi$
$683$	5.84861e9	0.702393	0.351196	+	0.936302i	$0.385775\pi$
$684$	1.15987e9	0.138584
$685$	0	0
$686$	−1.05490e10	−1.24760
$687$	−1.75607e9	−0.206630
$688$	6.40664e8	0.0750016
$689$	−2.98867e9	−0.348105
$690$	0	0
$691$	2.58686e9	0.298263	0.149131	−	0.988817i	$-0.452352\pi$
$691$	2.58686e9	0.298263	0.149131	+	0.988817i	$0.452352\pi$
$692$	1.73134e9	0.198614
$693$	8.42376e9	0.961479
$694$	−8.84805e9	−1.00482
$695$	0	0
$696$	−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
$705$	0	0
$706$	9.63161e9	1.03011
$707$	−2.72363e10	−2.89855
$708$	−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	$0.701017\pi$
$710$	0	0
$711$	2.77562e9	0.289612
$712$	−3.06750e9	−0.318496
$713$	−1.37408e9	−0.141971
$714$	2.24879e9	0.231209
$715$	0	0
$716$	−8.59398e9	−0.874981
$717$	−5.75314e9	−0.582892
$718$	1.05646e10	1.06516
$719$	−9.36568e8	−0.0939698	−0.0469849	−	0.998896i	$-0.514961\pi$
$719$	−9.36568e8	−0.0939698	−0.0469849	+	0.998896i	$0.514961\pi$
$720$	0	0
$721$	2.26350e10	2.24909
$722$	2.20682e9	0.218216
$723$	−6.94666e9	−0.683584
$724$	7.33827e9	0.718636
$725$	0	0
$726$	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	$0.565041\pi$
$728$	−3.06788e9	−0.294699
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	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	$0.719602\pi$
$740$	0	0
$741$	−2.55198e9	−0.230416
$742$	9.91087e9	0.890632
$743$	−1.43832e10	−1.28646	−0.643230	−	0.765673i	$-0.722406\pi$
$743$	−1.43832e10	−1.28646	−0.643230	+	0.765673i	$0.722406\pi$
$744$	4.58293e8	0.0407979
$745$	0	0
$746$	−3.90356e9	−0.344251
$747$	−1.62528e9	−0.142661
$748$	3.09985e9	0.270823
$749$	−1.01721e10	−0.884557
$750$	0	0
$751$	−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
$755$	0	0
$756$	−1.98531e9	−0.167110
$757$	1.91569e10	1.60506	0.802529	−	0.596613i	$-0.203487\pi$
$757$	1.91569e10	1.60506	0.802529	+	0.596613i	$0.203487\pi$
$758$	−8.88055e9	−0.740624
$759$	8.20521e9	0.681151
$760$	0	0
$761$	1.79120e10	1.47332	0.736660	−	0.676263i	$-0.236402\pi$
$761$	1.79120e10	1.47332	0.736660	+	0.676263i	$0.236402\pi$
$762$	−1.48350e9	−0.121463
$763$	−1.39383e9	−0.113599
$764$	1.04707e9	0.0849474
$765$	0	0
$766$	1.49529e10	1.20206
$767$	2.83302e9	0.226708
$768$	−4.52985e8	−0.0360844
$769$	−2.14072e10	−1.69753	−0.848765	−	0.528771i	$-0.822653\pi$
$769$	−2.14072e10	−1.69753	−0.848765	+	0.528771i	$0.822653\pi$
$770$	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
$775$	0	0
$776$	−2.07878e9	−0.159696
$777$	−1.55170e9	−0.118668
$778$	2.19116e9	0.166819
$779$	−1.58875e10	−1.20413
$780$	0	0
$781$	4.19108e10	3.14809
$782$	2.19044e9	0.163798
$783$	8.19010e8	0.0609709
$784$	6.80031e9	0.503991
$785$	0	0
$786$	−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
$790$	0	0
$791$	−1.91208e10	−1.37369
$792$	−2.73665e9	−0.195741
$793$	−6.31367e9	−0.449599
$794$	4.99641e9	0.354231
$795$	0	0
$796$	−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	$0.382547\pi$
$798$	8.46274e9	0.589523
$799$	2.86552e9	0.198742
$800$	0	0
$801$	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	$0.504401\pi$
$810$	0	0
$811$	−5.82687e9	−0.383586	−0.191793	−	0.981435i	$-0.561430\pi$
$811$	−5.82687e9	−0.383586	−0.191793	+	0.981435i	$0.561430\pi$
$812$	−4.19695e9	−0.275098
$813$	3.04423e8	0.0198683
$814$	−2.13895e9	−0.139000
$815$	0	0
$816$	−7.30571e8	−0.0470703
$817$	3.88840e9	0.249456
$818$	1.72616e10	1.10267
$819$	4.36813e9	0.277845
$820$	0	0
$821$	−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	$-0.542262\pi$
$823$	−4.23403e9	−0.264761	−0.132381	+	0.991199i	$0.542262\pi$
$824$	−7.35350e9	−0.457877
$825$	0	0
$826$	−9.39473e9	−0.580035
$827$	−5.70597e9	−0.350800	−0.175400	−	0.984497i	$-0.556122\pi$
$827$	−5.70597e9	−0.350800	−0.175400	+	0.984497i	$0.556122\pi$
$828$	−1.93380e9	−0.118387
$829$	2.51612e10	1.53388	0.766940	−	0.641719i	$-0.221779\pi$
$829$	2.51612e10	1.53388	0.766940	+	0.641719i	$0.221779\pi$
$830$	0	0
$831$	−1.77920e10	−1.07553
$832$	9.96671e8	0.0599957
$833$	1.09675e10	0.657431
$834$	2.86090e9	0.170774
$835$	0	0
$836$	1.16655e10	0.690528
$837$	−6.52531e8	−0.0384647
$838$	−1.34237e10	−0.787988
$839$	2.27048e10	1.32724	0.663622	−	0.748068i	$-0.269018\pi$
$839$	2.27048e10	1.32724	0.663622	+	0.748068i	$0.269018\pi$
$840$	0	0
$841$	−1.55185e10	−0.899629
$842$	4.20187e9	0.242577
$843$	2.83832e9	0.163179
$844$	2.25565e9	0.129144
$845$	0	0
$846$	−2.52978e9	−0.143644
$847$	5.40112e10	3.05416
$848$	−3.21978e9	−0.181318
$849$	8.91435e9	0.499934
$850$	0	0
$851$	−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	$0.790591\pi$
$854$	2.09371e10	1.15031
$855$	0	0
$856$	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	$-0.623284\pi$
$859$	−1.40330e10	−0.755394	−0.377697	+	0.925929i	$0.623284\pi$
$860$	0	0
$861$	2.71940e10	1.45199
$862$	−1.36474e9	−0.0725731
$863$	−3.24994e10	−1.72122	−0.860612	−	0.509261i	$-0.829919\pi$
$863$	−3.24994e10	−1.72122	−0.860612	+	0.509261i	$0.829919\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	−1.35038e10	−0.706549
$867$	9.90088e9	0.515949
$868$	3.34384e9	0.173551
$869$	2.79162e10	1.44307
$870$	0	0
$871$	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	$-0.177944\pi$
$877$	3.38694e10	1.69554	0.847772	+	0.530361i	$0.177944\pi$
$878$	9.42797e9	0.470097
$879$	−2.35171e9	−0.116794
$880$	0	0
$881$	−1.52708e10	−0.752397	−0.376198	−	0.926539i	$-0.622769\pi$
$881$	−1.52708e10	−0.752397	−0.376198	+	0.926539i	$0.622769\pi$
$882$	−9.68248e9	−0.475167
$883$	1.12045e10	0.547685	0.273842	−	0.961775i	$-0.411705\pi$
$883$	1.12045e10	0.547685	0.273842	+	0.961775i	$0.411705\pi$
$884$	1.60742e9	0.0782614
$885$	0	0
$886$	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
$889$	−1.08241e10	−0.516695
$890$	0	0
$891$	3.89653e9	0.184546
$892$	1.21044e10	0.571040
$893$	1.07837e10	0.506742
$894$	1.23903e10	0.579963
$895$	0	0
$896$	−3.30511e9	−0.153500
$897$	4.25480e9	0.196837
$898$	1.09887e10	0.506382
$899$	−1.37945e9	−0.0633211
$900$	0	0
$901$	−5.19283e9	−0.236520
$902$	3.74858e10	1.70076
$903$	−6.65564e9	−0.300804
$904$	6.21184e9	0.279660
$905$	0	0
$906$	−6.71485e9	−0.299977
$907$	−1.18095e9	−0.0525539	−0.0262770	−	0.999655i	$-0.508365\pi$
$907$	−1.18095e9	−0.0525539	−0.0262770	+	0.999655i	$0.508365\pi$
$908$	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
$915$	0	0
$916$	4.16254e9	0.178947
$917$	−6.22848e10	−2.66741
$918$	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	$0.230649\pi$
$920$	0	0
$921$	−1.05761e10	−0.446086
$922$	−2.47531e10	−1.04009
$923$	2.17328e10	0.909725
$924$	−1.99674e10	−0.832665
$925$	0	0
$926$	2.40361e10	0.994775
$927$	1.04701e10	0.431691
$928$	1.36348e9	0.0560054
$929$	−2.17764e10	−0.891111	−0.445556	−	0.895254i	$-0.646994\pi$
$929$	−2.17764e10	−0.891111	−0.445556	+	0.895254i	$0.646994\pi$
$930$	0	0
$931$	4.12734e10	1.67628
$932$	1.60844e10	0.650802
$933$	5.53327e9	0.223047
$934$	−2.39401e10	−0.961418
$935$	0	0
$936$	−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	$0.573845\pi$
$938$	−4.14909e10	−1.64151
$939$	2.36845e10	0.933542
$940$	0	0
$941$	−3.83930e10	−1.50207	−0.751033	−	0.660265i	$-0.770444\pi$
$941$	−3.83930e10	−1.50207	−0.751033	+	0.660265i	$0.770444\pi$
$942$	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	$-0.568638\pi$
$947$	−1.11841e10	−0.427933	−0.213966	+	0.976841i	$0.568638\pi$
$948$	−6.57926e9	−0.250812
$949$	−1.01129e10	−0.384101
$950$	0	0
$951$	−1.19024e10	−0.448749
$952$	−5.33046e9	−0.200233
$953$	5.19835e9	0.194554	0.0972770	−	0.995257i	$-0.468987\pi$
$953$	5.19835e9	0.194554	0.0972770	+	0.995257i	$0.468987\pi$
$954$	4.58441e9	0.170948
$955$	0	0
$956$	1.36371e10	0.504799
$957$	8.23728e9	0.303803
$958$	−1.47233e10	−0.541036
$959$	−3.02183e10	−1.10638
$960$	0	0
$961$	−2.64136e10	−0.960053
$962$	−1.10915e9	−0.0401677
$963$	−4.70526e9	−0.169782
$964$	1.64662e10	0.592001
$965$	0	0
$966$	−1.41096e10	−0.503609
$967$	2.69243e8	0.00957529	0.00478765	−	0.999989i	$-0.498476\pi$
$967$	2.69243e8	0.00957529	0.00478765	+	0.999989i	$0.498476\pi$
$968$	−1.75468e10	−0.621776
$969$	−4.43408e9	−0.156556
$970$	0	0
$971$	4.37283e9	0.153284	0.0766418	−	0.997059i	$-0.475580\pi$
$971$	4.37283e9	0.153284	0.0766418	+	0.997059i	$0.475580\pi$
$972$	−9.18330e8	−0.0320750
$973$	2.08740e10	0.726457
$974$	−3.41341e9	−0.118367
$975$	0	0
$976$	−6.80189e9	−0.234183
$977$	3.74991e10	1.28644	0.643220	−	0.765681i	$-0.277598\pi$
$977$	3.74991e10	1.28644	0.643220	+	0.765681i	$0.277598\pi$
$978$	−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
$985$	0	0
$986$	2.19901e9	0.0730562
$987$	−1.84580e10	−0.611048
$988$	6.04913e9	0.199546
$989$	−6.48296e9	−0.213102
$990$	0	0
$991$	−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
$994$	−7.20692e10	−2.32754
$995$	0	0
$996$	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	150.8.a.e.1.1		1
3.2	odd	2		450.8.a.ba.1.1		1
5.2	odd	4		150.8.c.k.49.1		2
5.3	odd	4		150.8.c.k.49.2		2
5.4	even	2		6.8.a.a.1.1	✓	1
15.2	even	4		450.8.c.a.199.2		2
15.8	even	4		450.8.c.a.199.1		2
15.14	odd	2		18.8.a.a.1.1		1
20.19	odd	2		48.8.a.b.1.1		1
35.4	even	6		294.8.e.c.79.1		2
35.9	even	6		294.8.e.c.67.1		2
35.19	odd	6		294.8.e.d.67.1		2
35.24	odd	6		294.8.e.d.79.1		2
35.34	odd	2		294.8.a.l.1.1		1
40.19	odd	2		192.8.a.n.1.1		1
40.29	even	2		192.8.a.f.1.1		1
45.4	even	6		162.8.c.d.55.1		2
45.14	odd	6		162.8.c.i.55.1		2
45.29	odd	6		162.8.c.i.109.1		2
45.34	even	6		162.8.c.d.109.1		2
60.59	even	2		144.8.a.h.1.1		1
120.29	odd	2		576.8.a.h.1.1		1
120.59	even	2		576.8.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.8.a.a.1.1	✓	1	5.4	even	2
18.8.a.a.1.1		1	15.14	odd	2
48.8.a.b.1.1		1	20.19	odd	2
144.8.a.h.1.1		1	60.59	even	2
150.8.a.e.1.1		1	1.1	even	1	trivial
150.8.c.k.49.1		2	5.2	odd	4
150.8.c.k.49.2		2	5.3	odd	4
162.8.c.d.55.1		2	45.4	even	6
162.8.c.d.109.1		2	45.34	even	6
162.8.c.i.55.1		2	45.14	odd	6
162.8.c.i.109.1		2	45.29	odd	6
192.8.a.f.1.1		1	40.29	even	2
192.8.a.n.1.1		1	40.19	odd	2
294.8.a.l.1.1		1	35.34	odd	2
294.8.e.c.67.1		2	35.9	even	6
294.8.e.c.79.1		2	35.4	even	6
294.8.e.d.67.1		2	35.19	odd	6
294.8.e.d.79.1		2	35.24	odd	6
450.8.a.ba.1.1		1	3.2	odd	2
450.8.c.a.199.1		2	15.8	even	4
450.8.c.a.199.2		2	15.2	even	4
576.8.a.h.1.1		1	120.29	odd	2
576.8.a.i.1.1		1	120.59	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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