LMFDB - Embedded newform 50.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,8,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	50.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:	$15.6192512742$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+8.00000 q^{2} -12.0000 q^{3} +64.0000 q^{4} -96.0000 q^{6} -1016.00 q^{7} +512.000 q^{8} -2043.00 q^{9} +1092.00 q^{11} -768.000 q^{12} -1382.00 q^{13} -8128.00 q^{14} +4096.00 q^{16} -14706.0 q^{17} -16344.0 q^{18} -39940.0 q^{19} +12192.0 q^{21} +8736.00 q^{22} -68712.0 q^{23} -6144.00 q^{24} -11056.0 q^{26} +50760.0 q^{27} -65024.0 q^{28} -102570. q^{29} +227552. q^{31} +32768.0 q^{32} -13104.0 q^{33} -117648. q^{34} -130752. q^{36} -160526. q^{37} -319520. q^{38} +16584.0 q^{39} +10842.0 q^{41} +97536.0 q^{42} +630748. q^{43} +69888.0 q^{44} -549696. q^{46} -472656. q^{47} -49152.0 q^{48} +208713. q^{49} +176472. q^{51} -88448.0 q^{52} +1.49402e6 q^{53} +406080. q^{54} -520192. q^{56} +479280. q^{57} -820560. q^{58} +2.64066e6 q^{59} +827702. q^{61} +1.82042e6 q^{62} +2.07569e6 q^{63} +262144. q^{64} -104832. q^{66} +126004. q^{67} -941184. q^{68} +824544. q^{69} -1.41473e6 q^{71} -1.04602e6 q^{72} -980282. q^{73} -1.28421e6 q^{74} -2.55616e6 q^{76} -1.10947e6 q^{77} +132672. q^{78} -3.56680e6 q^{79} +3.85892e6 q^{81} +86736.0 q^{82} -5.67289e6 q^{83} +780288. q^{84} +5.04598e6 q^{86} +1.23084e6 q^{87} +559104. q^{88} -1.19512e7 q^{89} +1.40411e6 q^{91} -4.39757e6 q^{92} -2.73062e6 q^{93} -3.78125e6 q^{94} -393216. q^{96} -8.68215e6 q^{97} +1.66970e6 q^{98} -2.23096e6 q^{99} +O(q^{100})\)

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
$3$	−12.0000	−0.256600	−0.128300	−	0.991735i	$-0.540952\pi$
$3$	−12.0000	−0.256600	−0.128300	+	0.991735i	$0.540952\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−96.0000	−0.181444
$7$	−1016.00	−1.11957	−0.559784	−	0.828638i	$-0.689116\pi$
$7$	−1016.00	−1.11957	−0.559784	+	0.828638i	$0.689116\pi$
$8$	512.000	0.353553
$9$	−2043.00	−0.934156
$10$	0	0
$11$	1092.00	0.247371	0.123685	−	0.992321i	$-0.460529\pi$
$11$	1092.00	0.247371	0.123685	+	0.992321i	$0.460529\pi$
$12$	−768.000	−0.128300
$13$	−1382.00	−0.174464	−0.0872321	−	0.996188i	$-0.527802\pi$
$13$	−1382.00	−0.174464	−0.0872321	+	0.996188i	$0.527802\pi$
$14$	−8128.00	−0.791654
$15$	0	0
$16$	4096.00	0.250000
$17$	−14706.0	−0.725978	−0.362989	−	0.931793i	$-0.618244\pi$
$17$	−14706.0	−0.725978	−0.362989	+	0.931793i	$0.618244\pi$
$18$	−16344.0	−0.660548
$19$	−39940.0	−1.33589	−0.667945	−	0.744211i	$-0.732826\pi$
$19$	−39940.0	−1.33589	−0.667945	+	0.744211i	$0.732826\pi$
$20$	0	0
$21$	12192.0	0.287281
$22$	8736.00	0.174917
$23$	−68712.0	−1.17757	−0.588783	−	0.808291i	$-0.700393\pi$
$23$	−68712.0	−1.17757	−0.588783	+	0.808291i	$0.700393\pi$
$24$	−6144.00	−0.0907218
$25$	0	0
$26$	−11056.0	−0.123365
$27$	50760.0	0.496305
$28$	−65024.0	−0.559784
$29$	−102570.	−0.780957	−0.390479	−	0.920612i	$-0.627690\pi$
$29$	−102570.	−0.780957	−0.390479	+	0.920612i	$0.627690\pi$
$30$	0	0
$31$	227552.	1.37188	0.685938	−	0.727660i	$-0.259392\pi$
$31$	227552.	1.37188	0.685938	+	0.727660i	$0.259392\pi$
$32$	32768.0	0.176777
$33$	−13104.0	−0.0634753
$34$	−117648.	−0.513344
$35$	0	0
$36$	−130752.	−0.467078
$37$	−160526.	−0.521002	−0.260501	−	0.965474i	$-0.583888\pi$
$37$	−160526.	−0.521002	−0.260501	+	0.965474i	$0.583888\pi$
$38$	−319520.	−0.944616
$39$	16584.0	0.0447675
$40$	0	0
$41$	10842.0	0.0245678	0.0122839	−	0.999925i	$-0.496090\pi$
$41$	10842.0	0.0245678	0.0122839	+	0.999925i	$0.496090\pi$
$42$	97536.0	0.203139
$43$	630748.	1.20981	0.604904	−	0.796299i	$-0.293212\pi$
$43$	630748.	1.20981	0.604904	+	0.796299i	$0.293212\pi$
$44$	69888.0	0.123685
$45$	0	0
$46$	−549696.	−0.832665
$47$	−472656.	−0.664053	−0.332026	−	0.943270i	$-0.607732\pi$
$47$	−472656.	−0.664053	−0.332026	+	0.943270i	$0.607732\pi$
$48$	−49152.0	−0.0641500
$49$	208713.	0.253433
$50$	0	0
$51$	176472.	0.186286
$52$	−88448.0	−0.0872321
$53$	1.49402e6	1.37845	0.689224	−	0.724548i	$-0.257952\pi$
$53$	1.49402e6	1.37845	0.689224	+	0.724548i	$0.257952\pi$
$54$	406080.	0.350940
$55$	0	0
$56$	−520192.	−0.395827
$57$	479280.	0.342789
$58$	−820560.	−0.552220
$59$	2.64066e6	1.67390	0.836952	−	0.547277i	$-0.184335\pi$
$59$	2.64066e6	1.67390	0.836952	+	0.547277i	$0.184335\pi$
$60$	0	0
$61$	827702.	0.466895	0.233448	−	0.972369i	$-0.424999\pi$
$61$	827702.	0.466895	0.233448	+	0.972369i	$0.424999\pi$
$62$	1.82042e6	0.970063
$63$	2.07569e6	1.04585
$64$	262144.	0.125000
$65$	0	0
$66$	−104832.	−0.0448838
$67$	126004.	0.0511826	0.0255913	−	0.999672i	$-0.491853\pi$
$67$	126004.	0.0511826	0.0255913	+	0.999672i	$0.491853\pi$
$68$	−941184.	−0.362989
$69$	824544.	0.302164
$70$	0	0
$71$	−1.41473e6	−0.469104	−0.234552	−	0.972104i	$-0.575362\pi$
$71$	−1.41473e6	−0.469104	−0.234552	+	0.972104i	$0.575362\pi$
$72$	−1.04602e6	−0.330274
$73$	−980282.	−0.294931	−0.147466	−	0.989067i	$-0.547112\pi$
$73$	−980282.	−0.294931	−0.147466	+	0.989067i	$0.547112\pi$
$74$	−1.28421e6	−0.368404
$75$	0	0
$76$	−2.55616e6	−0.667945
$77$	−1.10947e6	−0.276948
$78$	132672.	0.0316554
$79$	−3.56680e6	−0.813924	−0.406962	−	0.913445i	$-0.633412\pi$
$79$	−3.56680e6	−0.813924	−0.406962	+	0.913445i	$0.633412\pi$
$80$	0	0
$81$	3.85892e6	0.806805
$82$	86736.0	0.0173720
$83$	−5.67289e6	−1.08901	−0.544504	−	0.838758i	$-0.683282\pi$
$83$	−5.67289e6	−1.08901	−0.544504	+	0.838758i	$0.683282\pi$
$84$	780288.	0.143641
$85$	0	0
$86$	5.04598e6	0.855463
$87$	1.23084e6	0.200394
$88$	559104.	0.0874587
$89$	−1.19512e7	−1.79699	−0.898496	−	0.438982i	$-0.855339\pi$
$89$	−1.19512e7	−1.79699	−0.898496	+	0.438982i	$0.855339\pi$
$90$	0	0
$91$	1.40411e6	0.195325
$92$	−4.39757e6	−0.588783
$93$	−2.73062e6	−0.352023
$94$	−3.78125e6	−0.469556
$95$	0	0
$96$	−393216.	−0.0453609
$97$	−8.68215e6	−0.965886	−0.482943	−	0.875652i	$-0.660432\pi$
$97$	−8.68215e6	−0.965886	−0.482943	+	0.875652i	$0.660432\pi$
$98$	1.66970e6	0.179204
$99$	−2.23096e6	−0.231083
$100$	0	0
$101$	−1.00795e7	−0.973455	−0.486727	−	0.873554i	$-0.661810\pi$
$101$	−1.00795e7	−0.973455	−0.486727	+	0.873554i	$0.661810\pi$
$102$	1.41178e6	0.131724
$103$	−3.74799e6	−0.337962	−0.168981	−	0.985619i	$-0.554048\pi$
$103$	−3.74799e6	−0.337962	−0.168981	+	0.985619i	$0.554048\pi$
$104$	−707584.	−0.0616824
$105$	0	0
$106$	1.19521e7	0.974710
$107$	1.79856e7	1.41932	0.709661	−	0.704543i	$-0.248848\pi$
$107$	1.79856e7	1.41932	0.709661	+	0.704543i	$0.248848\pi$
$108$	3.24864e6	0.248152
$109$	1.22570e7	0.906552	0.453276	−	0.891370i	$-0.350255\pi$
$109$	1.22570e7	0.906552	0.453276	+	0.891370i	$0.350255\pi$
$110$	0	0
$111$	1.92631e6	0.133689
$112$	−4.16154e6	−0.279892
$113$	−1.65950e7	−1.08194	−0.540968	−	0.841043i	$-0.681942\pi$
$113$	−1.65950e7	−1.08194	−0.540968	+	0.841043i	$0.681942\pi$
$114$	3.83424e6	0.242389
$115$	0	0
$116$	−6.56448e6	−0.390479
$117$	2.82343e6	0.162977
$118$	2.11253e7	1.18363
$119$	1.49413e7	0.812782
$120$	0	0
$121$	−1.82947e7	−0.938808
$122$	6.62162e6	0.330145
$123$	−130104.	−0.00630410
$124$	1.45633e7	0.685938
$125$	0	0
$126$	1.66055e7	0.739529
$127$	−1.16826e6	−0.0506087	−0.0253043	−	0.999680i	$-0.508055\pi$
$127$	−1.16826e6	−0.0506087	−0.0253043	+	0.999680i	$0.508055\pi$
$128$	2.09715e6	0.0883883
$129$	−7.56898e6	−0.310437
$130$	0	0
$131$	−7.92383e6	−0.307954	−0.153977	−	0.988074i	$-0.549208\pi$
$131$	−7.92383e6	−0.307954	−0.153977	+	0.988074i	$0.549208\pi$
$132$	−838656.	−0.0317377
$133$	4.05790e7	1.49562
$134$	1.00803e6	0.0361916
$135$	0	0
$136$	−7.52947e6	−0.256672
$137$	315654.	0.0104879	0.00524396	−	0.999986i	$-0.498331\pi$
$137$	315654.	0.0104879	0.00524396	+	0.999986i	$0.498331\pi$
$138$	6.59635e6	0.213662
$139$	3.92038e7	1.23816	0.619079	−	0.785329i	$-0.287506\pi$
$139$	3.92038e7	1.23816	0.619079	+	0.785329i	$0.287506\pi$
$140$	0	0
$141$	5.67187e6	0.170396
$142$	−1.13178e7	−0.331706
$143$	−1.50914e6	−0.0431573
$144$	−8.36813e6	−0.233539
$145$	0	0
$146$	−7.84226e6	−0.208548
$147$	−2.50456e6	−0.0650309
$148$	−1.02737e7	−0.260501
$149$	−2.18860e7	−0.542020	−0.271010	−	0.962577i	$-0.587358\pi$
$149$	−2.18860e7	−0.542020	−0.271010	+	0.962577i	$0.587358\pi$
$150$	0	0
$151$	−2.94154e7	−0.695274	−0.347637	−	0.937629i	$-0.613016\pi$
$151$	−2.94154e7	−0.695274	−0.347637	+	0.937629i	$0.613016\pi$
$152$	−2.04493e7	−0.472308
$153$	3.00444e7	0.678177
$154$	−8.87578e6	−0.195832
$155$	0	0
$156$	1.06138e6	0.0223838
$157$	−6.05550e7	−1.24882	−0.624412	−	0.781095i	$-0.714661\pi$
$157$	−6.05550e7	−1.24882	−0.624412	+	0.781095i	$0.714661\pi$
$158$	−2.85344e7	−0.575531
$159$	−1.79282e7	−0.353710
$160$	0	0
$161$	6.98114e7	1.31837
$162$	3.08714e7	0.570497
$163$	−5.70853e7	−1.03245	−0.516223	−	0.856454i	$-0.672663\pi$
$163$	−5.70853e7	−1.03245	−0.516223	+	0.856454i	$0.672663\pi$
$164$	693888.	0.0122839
$165$	0	0
$166$	−4.53831e7	−0.770045
$167$	8.77265e7	1.45755	0.728775	−	0.684754i	$-0.240090\pi$
$167$	8.77265e7	1.45755	0.728775	+	0.684754i	$0.240090\pi$
$168$	6.24230e6	0.101569
$169$	−6.08386e7	−0.969562
$170$	0	0
$171$	8.15974e7	1.24793
$172$	4.03679e7	0.604904
$173$	−8.56954e6	−0.125833	−0.0629167	−	0.998019i	$-0.520040\pi$
$173$	−8.56954e6	−0.125833	−0.0629167	+	0.998019i	$0.520040\pi$
$174$	9.84672e6	0.141700
$175$	0	0
$176$	4.47283e6	0.0618427
$177$	−3.16879e7	−0.429524
$178$	−9.56095e7	−1.27067
$179$	1.88041e7	0.245056	0.122528	−	0.992465i	$-0.460900\pi$
$179$	1.88041e7	0.245056	0.122528	+	0.992465i	$0.460900\pi$
$180$	0	0
$181$	−5.99625e7	−0.751631	−0.375816	−	0.926694i	$-0.622637\pi$
$181$	−5.99625e7	−0.751631	−0.375816	+	0.926694i	$0.622637\pi$
$182$	1.12329e7	0.138115
$183$	−9.93242e6	−0.119805
$184$	−3.51805e7	−0.416332
$185$	0	0
$186$	−2.18450e7	−0.248918
$187$	−1.60590e7	−0.179586
$188$	−3.02500e7	−0.332026
$189$	−5.15722e7	−0.555647
$190$	0	0
$191$	9.39861e7	0.975993	0.487997	−	0.872845i	$-0.337728\pi$
$191$	9.39861e7	0.975993	0.487997	+	0.872845i	$0.337728\pi$
$192$	−3.14573e6	−0.0320750
$193$	3.51946e7	0.352391	0.176196	−	0.984355i	$-0.443621\pi$
$193$	3.51946e7	0.352391	0.176196	+	0.984355i	$0.443621\pi$
$194$	−6.94572e7	−0.682985
$195$	0	0
$196$	1.33576e7	0.126717
$197$	−1.02985e8	−0.959718	−0.479859	−	0.877346i	$-0.659312\pi$
$197$	−1.02985e8	−0.959718	−0.479859	+	0.877346i	$0.659312\pi$
$198$	−1.78476e7	−0.163400
$199$	8.36376e7	0.752342	0.376171	−	0.926550i	$-0.377240\pi$
$199$	8.36376e7	0.752342	0.376171	+	0.926550i	$0.377240\pi$
$200$	0	0
$201$	−1.51205e6	−0.0131335
$202$	−8.06363e7	−0.688337
$203$	1.04211e8	0.874335
$204$	1.12942e7	0.0931430
$205$	0	0
$206$	−2.99839e7	−0.238975
$207$	1.40379e8	1.10003
$208$	−5.66067e6	−0.0436160
$209$	−4.36145e7	−0.330460
$210$	0	0
$211$	−9.74010e7	−0.713797	−0.356899	−	0.934143i	$-0.616166\pi$
$211$	−9.74010e7	−0.713797	−0.356899	+	0.934143i	$0.616166\pi$
$212$	9.56172e7	0.689224
$213$	1.69767e7	0.120372
$214$	1.43885e8	1.00361
$215$	0	0
$216$	2.59891e7	0.175470
$217$	−2.31193e8	−1.53591
$218$	9.80562e7	0.641029
$219$	1.17634e7	0.0756794
$220$	0	0
$221$	2.03237e7	0.126657
$222$	1.54105e7	0.0945325
$223$	1.46457e7	0.0884390	0.0442195	−	0.999022i	$-0.485920\pi$
$223$	1.46457e7	0.0884390	0.0442195	+	0.999022i	$0.485920\pi$
$224$	−3.32923e7	−0.197914
$225$	0	0
$226$	−1.32760e8	−0.765045
$227$	1.84541e8	1.04713	0.523567	−	0.851985i	$-0.324601\pi$
$227$	1.84541e8	1.04713	0.523567	+	0.851985i	$0.324601\pi$
$228$	3.06739e7	0.171395
$229$	−8.75461e6	−0.0481740	−0.0240870	−	0.999710i	$-0.507668\pi$
$229$	−8.75461e6	−0.0481740	−0.0240870	+	0.999710i	$0.507668\pi$
$230$	0	0
$231$	1.33137e7	0.0710650
$232$	−5.25158e7	−0.276110
$233$	1.19556e8	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$233$	1.19556e8	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$234$	2.25874e7	0.115242
$235$	0	0
$236$	1.69002e8	0.836952
$237$	4.28016e7	0.208853
$238$	1.19530e8	0.574723
$239$	3.96209e8	1.87729	0.938646	−	0.344883i	$-0.112081\pi$
$239$	3.96209e8	1.87729	0.938646	+	0.344883i	$0.112081\pi$
$240$	0	0
$241$	−2.56606e8	−1.18089	−0.590443	−	0.807080i	$-0.701047\pi$
$241$	−2.56606e8	−1.18089	−0.590443	+	0.807080i	$0.701047\pi$
$242$	−1.46358e8	−0.663837
$243$	−1.57319e8	−0.703331
$244$	5.29729e7	0.233448
$245$	0	0
$246$	−1.04083e6	−0.00445767
$247$	5.51971e7	0.233065
$248$	1.16507e8	0.485031
$249$	6.80747e7	0.279440
$250$	0	0
$251$	−7.34775e7	−0.293290	−0.146645	−	0.989189i	$-0.546847\pi$
$251$	−7.34775e7	−0.293290	−0.146645	+	0.989189i	$0.546847\pi$
$252$	1.32844e8	0.522926
$253$	−7.50335e7	−0.291295
$254$	−9.34605e6	−0.0357857
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.02701e8	0.744886	0.372443	−	0.928055i	$-0.378520\pi$
$257$	2.02701e8	0.744886	0.372443	+	0.928055i	$0.378520\pi$
$258$	−6.05518e7	−0.219512
$259$	1.63094e8	0.583297
$260$	0	0
$261$	2.09551e8	0.729536
$262$	−6.33906e7	−0.217756
$263$	−1.54254e8	−0.522867	−0.261434	−	0.965221i	$-0.584195\pi$
$263$	−1.54254e8	−0.522867	−0.261434	+	0.965221i	$0.584195\pi$
$264$	−6.70925e6	−0.0224419
$265$	0	0
$266$	3.24632e8	1.05756
$267$	1.43414e8	0.461108
$268$	8.06426e6	0.0255913
$269$	−6.24018e8	−1.95463	−0.977315	−	0.211793i	$-0.932070\pi$
$269$	−6.24018e8	−1.95463	−0.977315	+	0.211793i	$0.932070\pi$
$270$	0	0
$271$	−3.87983e8	−1.18419	−0.592094	−	0.805869i	$-0.701698\pi$
$271$	−3.87983e8	−1.18419	−0.592094	+	0.805869i	$0.701698\pi$
$272$	−6.02358e7	−0.181494
$273$	−1.68493e7	−0.0501203
$274$	2.52523e6	0.00741608
$275$	0	0
$276$	5.27708e7	0.151082
$277$	−4.53952e8	−1.28331	−0.641654	−	0.766994i	$-0.721752\pi$
$277$	−4.53952e8	−1.28331	−0.641654	+	0.766994i	$0.721752\pi$
$278$	3.13630e8	0.875510
$279$	−4.64889e8	−1.28155
$280$	0	0
$281$	3.33770e8	0.897377	0.448689	−	0.893688i	$-0.351891\pi$
$281$	3.33770e8	0.897377	0.448689	+	0.893688i	$0.351891\pi$
$282$	4.53750e7	0.120488
$283$	−5.37695e8	−1.41021	−0.705104	−	0.709104i	$-0.749100\pi$
$283$	−5.37695e8	−1.41021	−0.705104	+	0.709104i	$0.749100\pi$
$284$	−9.05426e7	−0.234552
$285$	0	0
$286$	−1.20732e7	−0.0305168
$287$	−1.10155e7	−0.0275053
$288$	−6.69450e7	−0.165137
$289$	−1.94072e8	−0.472956
$290$	0	0
$291$	1.04186e8	0.247847
$292$	−6.27380e7	−0.147466
$293$	−3.35600e8	−0.779445	−0.389722	−	0.920932i	$-0.627429\pi$
$293$	−3.35600e8	−0.779445	−0.389722	+	0.920932i	$0.627429\pi$
$294$	−2.00364e7	−0.0459838
$295$	0	0
$296$	−8.21893e7	−0.184202
$297$	5.54299e7	0.122771
$298$	−1.75088e8	−0.383266
$299$	9.49600e7	0.205443
$300$	0	0
$301$	−6.40840e8	−1.35446
$302$	−2.35324e8	−0.491633
$303$	1.20954e8	0.249789
$304$	−1.63594e8	−0.333972
$305$	0	0
$306$	2.40355e8	0.479543
$307$	−2.15029e8	−0.424143	−0.212072	−	0.977254i	$-0.568021\pi$
$307$	−2.15029e8	−0.424143	−0.212072	+	0.977254i	$0.568021\pi$
$308$	−7.10062e7	−0.138474
$309$	4.49759e7	0.0867212
$310$	0	0
$311$	7.92062e8	1.49313	0.746565	−	0.665313i	$-0.231702\pi$
$311$	7.92062e8	1.49313	0.746565	+	0.665313i	$0.231702\pi$
$312$	8.49101e6	0.0158277
$313$	1.18457e8	0.218352	0.109176	−	0.994022i	$-0.465179\pi$
$313$	1.18457e8	0.218352	0.109176	+	0.994022i	$0.465179\pi$
$314$	−4.84440e8	−0.883051
$315$	0	0
$316$	−2.28275e8	−0.406962
$317$	5.07310e7	0.0894470	0.0447235	−	0.998999i	$-0.485759\pi$
$317$	5.07310e7	0.0894470	0.0447235	+	0.998999i	$0.485759\pi$
$318$	−1.43426e8	−0.250111
$319$	−1.12006e8	−0.193186
$320$	0	0
$321$	−2.15827e8	−0.364198
$322$	5.58491e8	0.932225
$323$	5.87358e8	0.969826
$324$	2.46971e8	0.403402
$325$	0	0
$326$	−4.56682e8	−0.730050
$327$	−1.47084e8	−0.232621
$328$	5.55110e6	0.00868602
$329$	4.80218e8	0.743453
$330$	0	0
$331$	2.73757e8	0.414923	0.207461	−	0.978243i	$-0.433480\pi$
$331$	2.73757e8	0.414923	0.207461	+	0.978243i	$0.433480\pi$
$332$	−3.63065e8	−0.544504
$333$	3.27955e8	0.486697
$334$	7.01812e8	1.03064
$335$	0	0
$336$	4.99384e7	0.0718203
$337$	9.18512e7	0.130732	0.0653658	−	0.997861i	$-0.479179\pi$
$337$	9.18512e7	0.130732	0.0653658	+	0.997861i	$0.479179\pi$
$338$	−4.86709e8	−0.685584
$339$	1.99140e8	0.277625
$340$	0	0
$341$	2.48487e8	0.339362
$342$	6.52779e8	0.882419
$343$	6.24667e8	0.835833
$344$	3.22943e8	0.427732
$345$	0	0
$346$	−6.85563e7	−0.0889777
$347$	1.36700e9	1.75637	0.878187	−	0.478318i	$-0.158753\pi$
$347$	1.36700e9	1.75637	0.878187	+	0.478318i	$0.158753\pi$
$348$	7.87738e7	0.100197
$349$	1.13143e9	1.42475	0.712377	−	0.701797i	$-0.247619\pi$
$349$	1.13143e9	1.42475	0.712377	+	0.701797i	$0.247619\pi$
$350$	0	0
$351$	−7.01503e7	−0.0865874
$352$	3.57827e7	0.0437294
$353$	4.48395e7	0.0542562	0.0271281	−	0.999632i	$-0.491364\pi$
$353$	4.48395e7	0.0542562	0.0271281	+	0.999632i	$0.491364\pi$
$354$	−2.53503e8	−0.303719
$355$	0	0
$356$	−7.64876e8	−0.898496
$357$	−1.79296e8	−0.208560
$358$	1.50432e8	0.173281
$359$	3.98281e8	0.454317	0.227158	−	0.973858i	$-0.427057\pi$
$359$	3.98281e8	0.454317	0.227158	+	0.973858i	$0.427057\pi$
$360$	0	0
$361$	7.01332e8	0.784600
$362$	−4.79700e8	−0.531483
$363$	2.19536e8	0.240898
$364$	8.98632e7	0.0976623
$365$	0	0
$366$	−7.94594e7	−0.0847152
$367$	−1.63472e9	−1.72628	−0.863140	−	0.504964i	$-0.831506\pi$
$367$	−1.63472e9	−1.72628	−0.863140	+	0.504964i	$0.831506\pi$
$368$	−2.81444e8	−0.294391
$369$	−2.21502e7	−0.0229501
$370$	0	0
$371$	−1.51792e9	−1.54327
$372$	−1.74760e8	−0.176012
$373$	1.54633e9	1.54284	0.771421	−	0.636325i	$-0.219546\pi$
$373$	1.54633e9	1.54284	0.771421	+	0.636325i	$0.219546\pi$
$374$	−1.28472e8	−0.126986
$375$	0	0
$376$	−2.42000e8	−0.234778
$377$	1.41752e8	0.136249
$378$	−4.12577e8	−0.392902
$379$	−1.05688e9	−0.997216	−0.498608	−	0.866828i	$-0.666155\pi$
$379$	−1.05688e9	−0.997216	−0.498608	+	0.866828i	$0.666155\pi$
$380$	0	0
$381$	1.40191e7	0.0129862
$382$	7.51889e8	0.690132
$383$	−2.24910e8	−0.204556	−0.102278	−	0.994756i	$-0.532613\pi$
$383$	−2.24910e8	−0.204556	−0.102278	+	0.994756i	$0.532613\pi$
$384$	−2.51658e7	−0.0226805
$385$	0	0
$386$	2.81556e8	0.249178
$387$	−1.28862e9	−1.13015
$388$	−5.55657e8	−0.482943
$389$	1.01788e9	0.876746	0.438373	−	0.898793i	$-0.355555\pi$
$389$	1.01788e9	0.876746	0.438373	+	0.898793i	$0.355555\pi$
$390$	0	0
$391$	1.01048e9	0.854887
$392$	1.06861e8	0.0896021
$393$	9.50859e7	0.0790210
$394$	−8.23883e8	−0.678623
$395$	0	0
$396$	−1.42781e8	−0.115541
$397$	1.47565e9	1.18363	0.591817	−	0.806072i	$-0.298411\pi$
$397$	1.47565e9	1.18363	0.591817	+	0.806072i	$0.298411\pi$
$398$	6.69100e8	0.531986
$399$	−4.86948e8	−0.383776
$400$	0	0
$401$	2.74912e8	0.212906	0.106453	−	0.994318i	$-0.466051\pi$
$401$	2.74912e8	0.212906	0.106453	+	0.994318i	$0.466051\pi$
$402$	−1.20964e7	−0.00928676
$403$	−3.14477e8	−0.239343
$404$	−6.45090e8	−0.486727
$405$	0	0
$406$	8.33689e8	0.618248
$407$	−1.75294e8	−0.128881
$408$	9.03537e7	0.0658620
$409$	−1.63427e9	−1.18112	−0.590558	−	0.806995i	$-0.701092\pi$
$409$	−1.63427e9	−1.18112	−0.590558	+	0.806995i	$0.701092\pi$
$410$	0	0
$411$	−3.78785e6	−0.00269120
$412$	−2.39871e8	−0.168981
$413$	−2.68291e9	−1.87405
$414$	1.12303e9	0.777839
$415$	0	0
$416$	−4.52854e7	−0.0308412
$417$	−4.70445e8	−0.317712
$418$	−3.48916e8	−0.233670
$419$	−1.11280e9	−0.739039	−0.369519	−	0.929223i	$-0.620478\pi$
$419$	−1.11280e9	−0.739039	−0.369519	+	0.929223i	$0.620478\pi$
$420$	0	0
$421$	9.22528e8	0.602549	0.301274	−	0.953537i	$-0.402588\pi$
$421$	9.22528e8	0.602549	0.301274	+	0.953537i	$0.402588\pi$
$422$	−7.79208e8	−0.504731
$423$	9.65636e8	0.620329
$424$	7.64937e8	0.487355
$425$	0	0
$426$	1.35814e8	0.0851159
$427$	−8.40945e8	−0.522721
$428$	1.15108e9	0.709661
$429$	1.81097e7	0.0110742
$430$	0	0
$431$	−9.81508e8	−0.590505	−0.295252	−	0.955419i	$-0.595404\pi$
$431$	−9.81508e8	−0.590505	−0.295252	+	0.955419i	$0.595404\pi$
$432$	2.07913e8	0.124076
$433$	−2.84998e9	−1.68707	−0.843537	−	0.537071i	$-0.819531\pi$
$433$	−2.84998e9	−1.68707	−0.843537	+	0.537071i	$0.819531\pi$
$434$	−1.84954e9	−1.08605
$435$	0	0
$436$	7.84450e8	0.453276
$437$	2.74436e9	1.57310
$438$	9.41071e7	0.0535134
$439$	−1.05622e9	−0.595838	−0.297919	−	0.954591i	$-0.596292\pi$
$439$	−1.05622e9	−0.595838	−0.297919	+	0.954591i	$0.596292\pi$
$440$	0	0
$441$	−4.26401e8	−0.236746
$442$	1.62590e8	0.0895601
$443$	−1.82325e9	−0.996401	−0.498201	−	0.867062i	$-0.666006\pi$
$443$	−1.82325e9	−0.996401	−0.498201	+	0.867062i	$0.666006\pi$
$444$	1.23284e8	0.0668446
$445$	0	0
$446$	1.17166e8	0.0625358
$447$	2.62633e8	0.139082
$448$	−2.66338e8	−0.139946
$449$	1.84846e9	0.963713	0.481856	−	0.876250i	$-0.339963\pi$
$449$	1.84846e9	0.963713	0.481856	+	0.876250i	$0.339963\pi$
$450$	0	0
$451$	1.18395e7	0.00607735
$452$	−1.06208e9	−0.540968
$453$	3.52985e8	0.178407
$454$	1.47633e9	0.740435
$455$	0	0
$456$	2.45391e8	0.121194
$457$	2.98066e9	1.46085	0.730425	−	0.682993i	$-0.239322\pi$
$457$	2.98066e9	1.46085	0.730425	+	0.682993i	$0.239322\pi$
$458$	−7.00369e7	−0.0340642
$459$	−7.46477e8	−0.360306
$460$	0	0
$461$	−2.52781e9	−1.20169	−0.600843	−	0.799367i	$-0.705168\pi$
$461$	−2.52781e9	−1.20169	−0.600843	+	0.799367i	$0.705168\pi$
$462$	1.06509e8	0.0502505
$463$	8.90291e8	0.416868	0.208434	−	0.978036i	$-0.433163\pi$
$463$	8.90291e8	0.416868	0.208434	+	0.978036i	$0.433163\pi$
$464$	−4.20127e8	−0.195239
$465$	0	0
$466$	9.56450e8	0.437836
$467$	−2.65667e9	−1.20706	−0.603529	−	0.797341i	$-0.706239\pi$
$467$	−2.65667e9	−1.20706	−0.603529	+	0.797341i	$0.706239\pi$
$468$	1.80699e8	0.0814884
$469$	−1.28020e8	−0.0573024
$470$	0	0
$471$	7.26660e8	0.320448
$472$	1.35202e9	0.591814
$473$	6.88777e8	0.299271
$474$	3.42413e8	0.147681
$475$	0	0
$476$	9.56243e8	0.406391
$477$	−3.05228e9	−1.28769
$478$	3.16967e9	1.32745
$479$	1.30093e9	0.540855	0.270428	−	0.962740i	$-0.412835\pi$
$479$	1.30093e9	0.540855	0.270428	+	0.962740i	$0.412835\pi$
$480$	0	0
$481$	2.21847e8	0.0908962
$482$	−2.05285e9	−0.835012
$483$	−8.37737e8	−0.338293
$484$	−1.17086e9	−0.469404
$485$	0	0
$486$	−1.25855e9	−0.497330
$487$	1.07447e9	0.421542	0.210771	−	0.977535i	$-0.432402\pi$
$487$	1.07447e9	0.421542	0.210771	+	0.977535i	$0.432402\pi$
$488$	4.23783e8	0.165072
$489$	6.85024e8	0.264926
$490$	0	0
$491$	−7.83344e8	−0.298653	−0.149327	−	0.988788i	$-0.547711\pi$
$491$	−7.83344e8	−0.298653	−0.149327	+	0.988788i	$0.547711\pi$
$492$	−8.32666e6	−0.00315205
$493$	1.50839e9	0.566958
$494$	4.41577e8	0.164802
$495$	0	0
$496$	9.32053e8	0.342969
$497$	1.43736e9	0.525193
$498$	5.44598e8	0.197594
$499$	−6.23188e8	−0.224526	−0.112263	−	0.993679i	$-0.535810\pi$
$499$	−6.23188e8	−0.224526	−0.112263	+	0.993679i	$0.535810\pi$
$500$	0	0
$501$	−1.05272e9	−0.374007
$502$	−5.87820e8	−0.207387
$503$	2.70927e9	0.949215	0.474607	−	0.880198i	$-0.342590\pi$
$503$	2.70927e9	0.949215	0.474607	+	0.880198i	$0.342590\pi$
$504$	1.06275e9	0.369764
$505$	0	0
$506$	−6.00268e8	−0.205977
$507$	7.30063e8	0.248790
$508$	−7.47684e7	−0.0253043
$509$	3.49943e9	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$509$	3.49943e9	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$510$	0	0
$511$	9.95967e8	0.330196
$512$	1.34218e8	0.0441942
$513$	−2.02735e9	−0.663008
$514$	1.62161e9	0.526714
$515$	0	0
$516$	−4.84414e8	−0.155218
$517$	−5.16140e8	−0.164267
$518$	1.30476e9	0.412453
$519$	1.02835e8	0.0322889
$520$	0	0
$521$	−1.37683e9	−0.426530	−0.213265	−	0.976994i	$-0.568410\pi$
$521$	−1.37683e9	−0.426530	−0.213265	+	0.976994i	$0.568410\pi$
$522$	1.67640e9	0.515860
$523$	2.86154e9	0.874669	0.437334	−	0.899299i	$-0.355923\pi$
$523$	2.86154e9	0.874669	0.437334	+	0.899299i	$0.355923\pi$
$524$	−5.07125e8	−0.153977
$525$	0	0
$526$	−1.23403e9	−0.369723
$527$	−3.34638e9	−0.995951
$528$	−5.36740e7	−0.0158688
$529$	1.31651e9	0.386661
$530$	0	0
$531$	−5.39487e9	−1.56369
$532$	2.59706e9	0.747810
$533$	−1.49836e7	−0.00428620
$534$	1.14731e9	0.326053
$535$	0	0
$536$	6.45140e7	0.0180958
$537$	−2.25649e8	−0.0628815
$538$	−4.99215e9	−1.38213
$539$	2.27915e8	0.0626919
$540$	0	0
$541$	5.34467e9	1.45121	0.725605	−	0.688111i	$-0.241560\pi$
$541$	5.34467e9	1.45121	0.725605	+	0.688111i	$0.241560\pi$
$542$	−3.10387e9	−0.837347
$543$	7.19550e8	0.192869
$544$	−4.81886e8	−0.128336
$545$	0	0
$546$	−1.34795e8	−0.0354404
$547$	3.37135e9	0.880740	0.440370	−	0.897816i	$-0.354847\pi$
$547$	3.37135e9	0.880740	0.440370	+	0.897816i	$0.354847\pi$
$548$	2.02019e7	0.00524396
$549$	−1.69100e9	−0.436153
$550$	0	0
$551$	4.09665e9	1.04327
$552$	4.22167e8	0.106831
$553$	3.62387e9	0.911244
$554$	−3.63162e9	−0.907436
$555$	0	0
$556$	2.50904e9	0.619079
$557$	5.61106e9	1.37579	0.687894	−	0.725811i	$-0.258535\pi$
$557$	5.61106e9	1.37579	0.687894	+	0.725811i	$0.258535\pi$
$558$	−3.71911e9	−0.906190
$559$	−8.71694e8	−0.211068
$560$	0	0
$561$	1.92707e8	0.0460817
$562$	2.67016e9	0.634542
$563$	−6.69690e9	−1.58159	−0.790795	−	0.612081i	$-0.790333\pi$
$563$	−6.69690e9	−1.58159	−0.790795	+	0.612081i	$0.790333\pi$
$564$	3.63000e8	0.0851980
$565$	0	0
$566$	−4.30156e9	−0.997168
$567$	−3.92066e9	−0.903273
$568$	−7.24341e8	−0.165853
$569$	1.96850e9	0.447964	0.223982	−	0.974593i	$-0.428094\pi$
$569$	1.96850e9	0.447964	0.223982	+	0.974593i	$0.428094\pi$
$570$	0	0
$571$	1.02926e9	0.231365	0.115682	−	0.993286i	$-0.463094\pi$
$571$	1.02926e9	0.231365	0.115682	+	0.993286i	$0.463094\pi$
$572$	−9.65852e7	−0.0215787
$573$	−1.12783e9	−0.250440
$574$	−8.81238e7	−0.0194492
$575$	0	0
$576$	−5.35560e8	−0.116770
$577$	−3.31179e9	−0.717708	−0.358854	−	0.933394i	$-0.616832\pi$
$577$	−3.31179e9	−0.717708	−0.358854	+	0.933394i	$0.616832\pi$
$578$	−1.55258e9	−0.334431
$579$	−4.22335e8	−0.0904236
$580$	0	0
$581$	5.76366e9	1.21922
$582$	8.33486e8	0.175254
$583$	1.63147e9	0.340988
$584$	−5.01904e8	−0.104274
$585$	0	0
$586$	−2.68480e9	−0.551151
$587$	5.59411e8	0.114156	0.0570778	−	0.998370i	$-0.481822\pi$
$587$	5.59411e8	0.114156	0.0570778	+	0.998370i	$0.481822\pi$
$588$	−1.60292e8	−0.0325155
$589$	−9.08843e9	−1.83267
$590$	0	0
$591$	1.23582e9	0.246264
$592$	−6.57514e8	−0.130250
$593$	3.02459e9	0.595628	0.297814	−	0.954624i	$-0.403742\pi$
$593$	3.02459e9	0.595628	0.297814	+	0.954624i	$0.403742\pi$
$594$	4.43439e8	0.0868124
$595$	0	0
$596$	−1.40071e9	−0.271010
$597$	−1.00365e9	−0.193051
$598$	7.59680e8	0.145270
$599$	−5.63246e9	−1.07079	−0.535395	−	0.844602i	$-0.679837\pi$
$599$	−5.63246e9	−1.07079	−0.535395	+	0.844602i	$0.679837\pi$
$600$	0	0
$601$	3.40792e8	0.0640366	0.0320183	−	0.999487i	$-0.489807\pi$
$601$	3.40792e8	0.0640366	0.0320183	+	0.999487i	$0.489807\pi$
$602$	−5.12672e9	−0.957749
$603$	−2.57426e8	−0.0478126
$604$	−1.88259e9	−0.347637
$605$	0	0
$606$	9.67636e8	0.176627
$607$	−3.85420e9	−0.699477	−0.349739	−	0.936847i	$-0.613730\pi$
$607$	−3.85420e9	−0.699477	−0.349739	+	0.936847i	$0.613730\pi$
$608$	−1.30875e9	−0.236154
$609$	−1.25053e9	−0.224355
$610$	0	0
$611$	6.53211e8	0.115853
$612$	1.92284e9	0.339088
$613$	−9.22245e9	−1.61709	−0.808545	−	0.588434i	$-0.799745\pi$
$613$	−9.22245e9	−1.61709	−0.808545	+	0.588434i	$0.799745\pi$
$614$	−1.72023e9	−0.299915
$615$	0	0
$616$	−5.68050e8	−0.0979160
$617$	−6.53611e9	−1.12027	−0.560133	−	0.828402i	$-0.689250\pi$
$617$	−6.53611e9	−1.12027	−0.560133	+	0.828402i	$0.689250\pi$
$618$	3.59807e8	0.0613211
$619$	1.36559e9	0.231420	0.115710	−	0.993283i	$-0.463086\pi$
$619$	1.36559e9	0.231420	0.115710	+	0.993283i	$0.463086\pi$
$620$	0	0
$621$	−3.48782e9	−0.584431
$622$	6.33649e9	1.05580
$623$	1.21424e10	2.01186
$624$	6.79281e7	0.0111919
$625$	0	0
$626$	9.47659e8	0.154398
$627$	5.23374e8	0.0847960
$628$	−3.87552e9	−0.624412
$629$	2.36070e9	0.378236
$630$	0	0
$631$	1.54079e9	0.244141	0.122070	−	0.992521i	$-0.461047\pi$
$631$	1.54079e9	0.244141	0.122070	+	0.992521i	$0.461047\pi$
$632$	−1.82620e9	−0.287766
$633$	1.16881e9	0.183160
$634$	4.05848e8	0.0632486
$635$	0	0
$636$	−1.14741e9	−0.176855
$637$	−2.88441e8	−0.0442150
$638$	−8.96052e8	−0.136603
$639$	2.89029e9	0.438216
$640$	0	0
$641$	−4.54018e9	−0.680879	−0.340440	−	0.940266i	$-0.610576\pi$
$641$	−4.54018e9	−0.680879	−0.340440	+	0.940266i	$0.610576\pi$
$642$	−1.72661e9	−0.257527
$643$	−1.14054e10	−1.69189	−0.845944	−	0.533272i	$-0.820962\pi$
$643$	−1.14054e10	−1.69189	−0.845944	+	0.533272i	$0.820962\pi$
$644$	4.46793e9	0.659183
$645$	0	0
$646$	4.69886e9	0.685770
$647$	1.26393e10	1.83468	0.917338	−	0.398109i	$-0.130334\pi$
$647$	1.26393e10	1.83468	0.917338	+	0.398109i	$0.130334\pi$
$648$	1.97577e9	0.285248
$649$	2.88360e9	0.414075
$650$	0	0
$651$	2.77431e9	0.394114
$652$	−3.65346e9	−0.516223
$653$	1.05004e10	1.47575	0.737873	−	0.674940i	$-0.235830\pi$
$653$	1.05004e10	1.47575	0.737873	+	0.674940i	$0.235830\pi$
$654$	−1.17667e9	−0.164488
$655$	0	0
$656$	4.44088e7	0.00614194
$657$	2.00272e9	0.275512
$658$	3.84175e9	0.525700
$659$	9.64818e9	1.31325	0.656624	−	0.754219i	$-0.271984\pi$
$659$	9.64818e9	1.31325	0.656624	+	0.754219i	$0.271984\pi$
$660$	0	0
$661$	−6.58299e9	−0.886580	−0.443290	−	0.896378i	$-0.646189\pi$
$661$	−6.58299e9	−0.886580	−0.443290	+	0.896378i	$0.646189\pi$
$662$	2.19006e9	0.293395
$663$	−2.43884e8	−0.0325002
$664$	−2.90452e9	−0.385023
$665$	0	0
$666$	2.62364e9	0.344147
$667$	7.04779e9	0.919629
$668$	5.61450e9	0.728775
$669$	−1.75749e8	−0.0226935
$670$	0	0
$671$	9.03851e8	0.115496
$672$	3.99507e8	0.0507846
$673$	8.54649e9	1.08077	0.540387	−	0.841416i	$-0.318278\pi$
$673$	8.54649e9	1.08077	0.540387	+	0.841416i	$0.318278\pi$
$674$	7.34810e8	0.0924411
$675$	0	0
$676$	−3.89367e9	−0.484781
$677$	−8.71305e9	−1.07922	−0.539610	−	0.841915i	$-0.681428\pi$
$677$	−8.71305e9	−1.07922	−0.539610	+	0.841915i	$0.681428\pi$
$678$	1.59312e9	0.196311
$679$	8.82106e9	1.08138
$680$	0	0
$681$	−2.21449e9	−0.268695
$682$	1.98789e9	0.239965
$683$	−1.46109e10	−1.75470	−0.877351	−	0.479849i	$-0.840692\pi$
$683$	−1.46109e10	−1.75470	−0.877351	+	0.479849i	$0.840692\pi$
$684$	5.22223e9	0.623965
$685$	0	0
$686$	4.99734e9	0.591023
$687$	1.05055e8	0.0123615
$688$	2.58354e9	0.302452
$689$	−2.06473e9	−0.240490
$690$	0	0
$691$	−1.47348e10	−1.69891	−0.849454	−	0.527662i	$-0.823069\pi$
$691$	−1.47348e10	−1.69891	−0.849454	+	0.527662i	$0.823069\pi$
$692$	−5.48451e8	−0.0629167
$693$	2.26665e9	0.258713
$694$	1.09360e10	1.24194
$695$	0	0
$696$	6.30190e8	0.0708499
$697$	−1.59442e8	−0.0178357
$698$	9.05146e9	1.00745
$699$	−1.43467e9	−0.158885
$700$	0	0
$701$	1.31502e9	0.144185	0.0720923	−	0.997398i	$-0.477032\pi$
$701$	1.31502e9	0.144185	0.0720923	+	0.997398i	$0.477032\pi$
$702$	−5.61203e8	−0.0612265
$703$	6.41141e9	0.696001
$704$	2.86261e8	0.0309213
$705$	0	0
$706$	3.58716e8	0.0383649
$707$	1.02408e10	1.08985
$708$	−2.02803e9	−0.214762
$709$	6.64028e8	0.0699721	0.0349860	−	0.999388i	$-0.488861\pi$
$709$	6.64028e8	0.0699721	0.0349860	+	0.999388i	$0.488861\pi$
$710$	0	0
$711$	7.28697e9	0.760332
$712$	−6.11901e9	−0.635333
$713$	−1.56356e10	−1.61547
$714$	−1.43436e9	−0.147474
$715$	0	0
$716$	1.20346e9	0.122528
$717$	−4.75451e9	−0.481713
$718$	3.18624e9	0.321250
$719$	4.95034e9	0.496689	0.248344	−	0.968672i	$-0.420114\pi$
$719$	4.95034e9	0.496689	0.248344	+	0.968672i	$0.420114\pi$
$720$	0	0
$721$	3.80796e9	0.378372
$722$	5.61065e9	0.554796
$723$	3.07928e9	0.303015
$724$	−3.83760e9	−0.375816
$725$	0	0
$726$	1.75629e9	0.170341
$727$	−8.81101e9	−0.850463	−0.425231	−	0.905085i	$-0.639807\pi$
$727$	−8.81101e9	−0.850463	−0.425231	+	0.905085i	$0.639807\pi$
$728$	7.18905e8	0.0690577
$729$	−6.55163e9	−0.626330
$730$	0	0
$731$	−9.27578e9	−0.878293
$732$	−6.35675e8	−0.0599027
$733$	1.49414e8	0.0140129	0.00700643	−	0.999975i	$-0.497770\pi$
$733$	1.49414e8	0.0140129	0.00700643	+	0.999975i	$0.497770\pi$
$734$	−1.30777e10	−1.22066
$735$	0	0
$736$	−2.25155e9	−0.208166
$737$	1.37596e8	0.0126611
$738$	−1.77202e8	−0.0162282
$739$	−4.70806e9	−0.429127	−0.214564	−	0.976710i	$-0.568833\pi$
$739$	−4.70806e9	−0.429127	−0.214564	+	0.976710i	$0.568833\pi$
$740$	0	0
$741$	−6.62365e8	−0.0598045
$742$	−1.21434e10	−1.09125
$743$	−1.69676e9	−0.151761	−0.0758805	−	0.997117i	$-0.524177\pi$
$743$	−1.69676e9	−0.151761	−0.0758805	+	0.997117i	$0.524177\pi$
$744$	−1.39808e9	−0.124459
$745$	0	0
$746$	1.23707e10	1.09095
$747$	1.15897e10	1.01730
$748$	−1.02777e9	−0.0897928
$749$	−1.82733e10	−1.58903
$750$	0	0
$751$	1.06650e10	0.918800	0.459400	−	0.888229i	$-0.348064\pi$
$751$	1.06650e10	0.918800	0.459400	+	0.888229i	$0.348064\pi$
$752$	−1.93600e9	−0.166013
$753$	8.81731e8	0.0752581
$754$	1.13401e9	0.0963427
$755$	0	0
$756$	−3.30062e9	−0.277824
$757$	−6.22876e9	−0.521874	−0.260937	−	0.965356i	$-0.584032\pi$
$757$	−6.22876e9	−0.521874	−0.260937	+	0.965356i	$0.584032\pi$
$758$	−8.45506e9	−0.705138
$759$	9.00402e8	0.0747464
$760$	0	0
$761$	−8.38334e9	−0.689558	−0.344779	−	0.938684i	$-0.612046\pi$
$761$	−8.38334e9	−0.689558	−0.344779	+	0.938684i	$0.612046\pi$
$762$	1.12153e8	0.00918263
$763$	−1.24531e10	−1.01495
$764$	6.01511e9	0.487997
$765$	0	0
$766$	−1.79928e9	−0.144643
$767$	−3.64939e9	−0.292036
$768$	−2.01327e8	−0.0160375
$769$	−1.18649e10	−0.940852	−0.470426	−	0.882439i	$-0.655900\pi$
$769$	−1.18649e10	−0.940852	−0.470426	+	0.882439i	$0.655900\pi$
$770$	0	0
$771$	−2.43241e9	−0.191138
$772$	2.25245e9	0.176196
$773$	−5.56680e9	−0.433488	−0.216744	−	0.976228i	$-0.569544\pi$
$773$	−5.56680e9	−0.433488	−0.216744	+	0.976228i	$0.569544\pi$
$774$	−1.03089e10	−0.799136
$775$	0	0
$776$	−4.44526e9	−0.341492
$777$	−1.95713e9	−0.149674
$778$	8.14306e9	0.619953
$779$	−4.33029e8	−0.0328198
$780$	0	0
$781$	−1.54488e9	−0.116042
$782$	8.08383e9	0.604496
$783$	−5.20645e9	−0.387593
$784$	8.54888e8	0.0633583
$785$	0	0
$786$	7.60687e8	0.0558763
$787$	−1.34611e8	−0.00984395	−0.00492198	−	0.999988i	$-0.501567\pi$
$787$	−1.34611e8	−0.00984395	−0.00492198	+	0.999988i	$0.501567\pi$
$788$	−6.59106e9	−0.479859
$789$	1.85105e9	0.134168
$790$	0	0
$791$	1.68605e10	1.21130
$792$	−1.14225e9	−0.0817001
$793$	−1.14388e9	−0.0814565
$794$	1.18052e10	0.836955
$795$	0	0
$796$	5.35280e9	0.376171
$797$	7.41548e9	0.518842	0.259421	−	0.965764i	$-0.416468\pi$
$797$	7.41548e9	0.518842	0.259421	+	0.965764i	$0.416468\pi$
$798$	−3.89559e9	−0.271371
$799$	6.95088e9	0.482088
$800$	0	0
$801$	2.44163e10	1.67867
$802$	2.19930e9	0.150548
$803$	−1.07047e9	−0.0729574
$804$	−9.67711e7	−0.00656673
$805$	0	0
$806$	−2.51581e9	−0.169241
$807$	7.48822e9	0.501558
$808$	−5.16072e9	−0.344168
$809$	−1.41542e10	−0.939863	−0.469932	−	0.882703i	$-0.655721\pi$
$809$	−1.41542e10	−0.939863	−0.469932	+	0.882703i	$0.655721\pi$
$810$	0	0
$811$	−2.63708e10	−1.73600	−0.868001	−	0.496563i	$-0.834595\pi$
$811$	−2.63708e10	−1.73600	−0.868001	+	0.496563i	$0.834595\pi$
$812$	6.66951e9	0.437168
$813$	4.65580e9	0.303863
$814$	−1.40236e9	−0.0911324
$815$	0	0
$816$	7.22829e8	0.0465715
$817$	−2.51921e10	−1.61617
$818$	−1.30742e10	−0.835176
$819$	−2.86860e9	−0.182464
$820$	0	0
$821$	8.06264e9	0.508483	0.254241	−	0.967141i	$-0.418174\pi$
$821$	8.06264e9	0.508483	0.254241	+	0.967141i	$0.418174\pi$
$822$	−3.03028e7	−0.00190297
$823$	2.34202e10	1.46451	0.732253	−	0.681033i	$-0.238469\pi$
$823$	2.34202e10	1.46451	0.732253	+	0.681033i	$0.238469\pi$
$824$	−1.91897e9	−0.119488
$825$	0	0
$826$	−2.14633e10	−1.32515
$827$	−5.55722e9	−0.341655	−0.170828	−	0.985301i	$-0.554644\pi$
$827$	−5.55722e9	−0.341655	−0.170828	+	0.985301i	$0.554644\pi$
$828$	8.98423e9	0.550015
$829$	2.84256e10	1.73288	0.866440	−	0.499281i	$-0.166403\pi$
$829$	2.84256e10	1.73288	0.866440	+	0.499281i	$0.166403\pi$
$830$	0	0
$831$	5.44743e9	0.329297
$832$	−3.62283e8	−0.0218080
$833$	−3.06933e9	−0.183987
$834$	−3.76356e9	−0.224656
$835$	0	0
$836$	−2.79133e9	−0.165230
$837$	1.15505e10	0.680868
$838$	−8.90238e9	−0.522579
$839$	1.04036e10	0.608156	0.304078	−	0.952647i	$-0.401652\pi$
$839$	1.04036e10	0.608156	0.304078	+	0.952647i	$0.401652\pi$
$840$	0	0
$841$	−6.72927e9	−0.390105
$842$	7.38023e9	0.426066
$843$	−4.00524e9	−0.230267
$844$	−6.23367e9	−0.356899
$845$	0	0
$846$	7.72509e9	0.438639
$847$	1.85874e10	1.05106
$848$	6.11950e9	0.344612
$849$	6.45234e9	0.361860
$850$	0	0
$851$	1.10301e10	0.613514
$852$	1.08651e9	0.0601860
$853$	1.80580e10	0.996205	0.498102	−	0.867118i	$-0.334030\pi$
$853$	1.80580e10	0.996205	0.498102	+	0.867118i	$0.334030\pi$
$854$	−6.72756e9	−0.369620
$855$	0	0
$856$	9.20861e9	0.501806
$857$	6.34034e9	0.344096	0.172048	−	0.985089i	$-0.444962\pi$
$857$	6.34034e9	0.344096	0.172048	+	0.985089i	$0.444962\pi$
$858$	1.44878e8	0.00783062
$859$	1.21489e10	0.653973	0.326987	−	0.945029i	$-0.393967\pi$
$859$	1.21489e10	0.653973	0.326987	+	0.945029i	$0.393967\pi$
$860$	0	0
$861$	1.32186e8	0.00705786
$862$	−7.85206e9	−0.417550
$863$	2.87111e10	1.52059	0.760295	−	0.649578i	$-0.225054\pi$
$863$	2.87111e10	1.52059	0.760295	+	0.649578i	$0.225054\pi$
$864$	1.66330e9	0.0877351
$865$	0	0
$866$	−2.27998e10	−1.19294
$867$	2.32887e9	0.121361
$868$	−1.47963e10	−0.767954
$869$	−3.89495e9	−0.201341
$870$	0	0
$871$	−1.74138e8	−0.00892953
$872$	6.27560e9	0.320514
$873$	1.77376e10	0.902289
$874$	2.19549e10	1.11235
$875$	0	0
$876$	7.52857e8	0.0378397
$877$	−2.46021e10	−1.23161	−0.615806	−	0.787898i	$-0.711169\pi$
$877$	−2.46021e10	−1.23161	−0.615806	+	0.787898i	$0.711169\pi$
$878$	−8.44975e9	−0.421321
$879$	4.02720e9	0.200006
$880$	0	0
$881$	−1.25378e10	−0.617738	−0.308869	−	0.951105i	$-0.599951\pi$
$881$	−1.25378e10	−0.617738	−0.308869	+	0.951105i	$0.599951\pi$
$882$	−3.41121e9	−0.167405
$883$	−1.93097e10	−0.943873	−0.471937	−	0.881633i	$-0.656445\pi$
$883$	−1.93097e10	−0.943873	−0.471937	+	0.881633i	$0.656445\pi$
$884$	1.30072e9	0.0633286
$885$	0	0
$886$	−1.45860e10	−0.704562
$887$	−3.20268e10	−1.54092	−0.770462	−	0.637486i	$-0.779974\pi$
$887$	−3.20268e10	−1.54092	−0.770462	+	0.637486i	$0.779974\pi$
$888$	9.86272e8	0.0472663
$889$	1.18695e9	0.0566599
$890$	0	0
$891$	4.21394e9	0.199580
$892$	9.37327e8	0.0442195
$893$	1.88779e10	0.887101
$894$	2.10106e9	0.0983461
$895$	0	0
$896$	−2.13071e9	−0.0989568
$897$	−1.13952e9	−0.0527167
$898$	1.47877e10	0.681448
$899$	−2.33400e10	−1.07138
$900$	0	0
$901$	−2.19710e10	−1.00072
$902$	9.47157e7	0.00429733
$903$	7.69008e9	0.347555
$904$	−8.49662e9	−0.382522
$905$	0	0
$906$	2.82388e9	0.126153
$907$	−2.33703e9	−0.104002	−0.0520008	−	0.998647i	$-0.516560\pi$
$907$	−2.33703e9	−0.104002	−0.0520008	+	0.998647i	$0.516560\pi$
$908$	1.18106e10	0.523567
$909$	2.05925e10	0.909359
$910$	0	0
$911$	2.20343e10	0.965573	0.482786	−	0.875738i	$-0.339625\pi$
$911$	2.20343e10	0.965573	0.482786	+	0.875738i	$0.339625\pi$
$912$	1.96313e9	0.0856973
$913$	−6.19480e9	−0.269389
$914$	2.38453e10	1.03298
$915$	0	0
$916$	−5.60295e8	−0.0240870
$917$	8.05061e9	0.344775
$918$	−5.97181e9	−0.254775
$919$	−1.43277e10	−0.608938	−0.304469	−	0.952522i	$-0.598479\pi$
$919$	−1.43277e10	−0.608938	−0.304469	+	0.952522i	$0.598479\pi$
$920$	0	0
$921$	2.58035e9	0.108835
$922$	−2.02225e10	−0.849720
$923$	1.95515e9	0.0818418
$924$	8.52074e8	0.0355325
$925$	0	0
$926$	7.12233e9	0.294770
$927$	7.65715e9	0.315710
$928$	−3.36101e9	−0.138055
$929$	1.31280e10	0.537208	0.268604	−	0.963251i	$-0.413438\pi$
$929$	1.31280e10	0.537208	0.268604	+	0.963251i	$0.413438\pi$
$930$	0	0
$931$	−8.33600e9	−0.338558
$932$	7.65160e9	0.309597
$933$	−9.50474e9	−0.383137
$934$	−2.12533e10	−0.853519
$935$	0	0
$936$	1.44559e9	0.0576210
$937$	3.87626e10	1.53930	0.769652	−	0.638463i	$-0.220429\pi$
$937$	3.87626e10	1.53930	0.769652	+	0.638463i	$0.220429\pi$
$938$	−1.02416e9	−0.0405189
$939$	−1.42149e9	−0.0560291
$940$	0	0
$941$	2.06279e10	0.807035	0.403517	−	0.914972i	$-0.367788\pi$
$941$	2.06279e10	0.807035	0.403517	+	0.914972i	$0.367788\pi$
$942$	5.81328e9	0.226591
$943$	−7.44976e8	−0.0289302
$944$	1.08161e10	0.418476
$945$	0	0
$946$	5.51021e9	0.211617
$947$	2.11705e10	0.810040	0.405020	−	0.914308i	$-0.367264\pi$
$947$	2.11705e10	0.810040	0.405020	+	0.914308i	$0.367264\pi$
$948$	2.73930e9	0.104427
$949$	1.35475e9	0.0514550
$950$	0	0
$951$	−6.08771e8	−0.0229521
$952$	7.64994e9	0.287362
$953$	−2.14876e10	−0.804196	−0.402098	−	0.915597i	$-0.631719\pi$
$953$	−2.14876e10	−0.804196	−0.402098	+	0.915597i	$0.631719\pi$
$954$	−2.44182e10	−0.910531
$955$	0	0
$956$	2.53574e10	0.938646
$957$	1.34408e9	0.0495715
$958$	1.04075e10	0.382442
$959$	−3.20704e8	−0.0117419
$960$	0	0
$961$	2.42673e10	0.882043
$962$	1.77478e9	0.0642733
$963$	−3.67445e10	−1.32587
$964$	−1.64228e10	−0.590443
$965$	0	0
$966$	−6.70189e9	−0.239209
$967$	−3.92625e10	−1.39632	−0.698161	−	0.715941i	$-0.745998\pi$
$967$	−3.92625e10	−1.39632	−0.698161	+	0.715941i	$0.745998\pi$
$968$	−9.36689e9	−0.331919
$969$	−7.04829e9	−0.248857
$970$	0	0
$971$	−5.62647e10	−1.97228	−0.986140	−	0.165917i	$-0.946941\pi$
$971$	−5.62647e10	−1.97228	−0.986140	+	0.165917i	$0.946941\pi$
$972$	−1.00684e10	−0.351665
$973$	−3.98310e10	−1.38620
$974$	8.59573e9	0.298076
$975$	0	0
$976$	3.39027e9	0.116724
$977$	8.43437e9	0.289349	0.144674	−	0.989479i	$-0.453787\pi$
$977$	8.43437e9	0.289349	0.144674	+	0.989479i	$0.453787\pi$
$978$	5.48019e9	0.187331
$979$	−1.30507e10	−0.444523
$980$	0	0
$981$	−2.50411e10	−0.846861
$982$	−6.26675e9	−0.211180
$983$	2.24230e10	0.752932	0.376466	−	0.926430i	$-0.377139\pi$
$983$	2.24230e10	0.752932	0.376466	+	0.926430i	$0.377139\pi$
$984$	−6.66132e7	−0.00222883
$985$	0	0
$986$	1.20672e10	0.400900
$987$	−5.76262e9	−0.190770
$988$	3.53261e9	0.116532
$989$	−4.33400e10	−1.42463
$990$	0	0
$991$	3.46728e10	1.13170	0.565849	−	0.824509i	$-0.308548\pi$
$991$	3.46728e10	1.13170	0.565849	+	0.824509i	$0.308548\pi$
$992$	7.45642e9	0.242516
$993$	−3.28508e9	−0.106469
$994$	1.14989e10	0.371368
$995$	0	0
$996$	4.35678e9	0.139720
$997$	2.96474e10	0.947444	0.473722	−	0.880674i	$-0.342910\pi$
$997$	2.96474e10	0.947444	0.473722	+	0.880674i	$0.342910\pi$
$998$	−4.98550e9	−0.158764
$999$	−8.14830e9	−0.258576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.8.a.g.1.1		1
3.2	odd	2		450.8.a.c.1.1		1
4.3	odd	2		400.8.a.l.1.1		1
5.2	odd	4		50.8.b.c.49.2		2
5.3	odd	4		50.8.b.c.49.1		2
5.4	even	2		2.8.a.a.1.1	✓	1
15.2	even	4		450.8.c.g.199.1		2
15.8	even	4		450.8.c.g.199.2		2
15.14	odd	2		18.8.a.b.1.1		1
20.3	even	4		400.8.c.j.49.2		2
20.7	even	4		400.8.c.j.49.1		2
20.19	odd	2		16.8.a.b.1.1		1
35.4	even	6		98.8.c.d.79.1		2
35.9	even	6		98.8.c.d.67.1		2
35.19	odd	6		98.8.c.e.67.1		2
35.24	odd	6		98.8.c.e.79.1		2
35.34	odd	2		98.8.a.a.1.1		1
40.19	odd	2		64.8.a.e.1.1		1
40.29	even	2		64.8.a.c.1.1		1
45.4	even	6		162.8.c.l.55.1		2
45.14	odd	6		162.8.c.a.55.1		2
45.29	odd	6		162.8.c.a.109.1		2
45.34	even	6		162.8.c.l.109.1		2
55.54	odd	2		242.8.a.e.1.1		1
60.59	even	2		144.8.a.i.1.1		1
65.34	odd	4		338.8.b.d.337.1		2
65.44	odd	4		338.8.b.d.337.2		2
65.64	even	2		338.8.a.d.1.1		1
80.19	odd	4		256.8.b.f.129.1		2
80.29	even	4		256.8.b.b.129.2		2
80.59	odd	4		256.8.b.f.129.2		2
80.69	even	4		256.8.b.b.129.1		2
85.84	even	2		578.8.a.b.1.1		1
120.29	odd	2		576.8.a.g.1.1		1
120.59	even	2		576.8.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	5.4	even	2
16.8.a.b.1.1		1	20.19	odd	2
18.8.a.b.1.1		1	15.14	odd	2
50.8.a.g.1.1		1	1.1	even	1	trivial
50.8.b.c.49.1		2	5.3	odd	4
50.8.b.c.49.2		2	5.2	odd	4
64.8.a.c.1.1		1	40.29	even	2
64.8.a.e.1.1		1	40.19	odd	2
98.8.a.a.1.1		1	35.34	odd	2
98.8.c.d.67.1		2	35.9	even	6
98.8.c.d.79.1		2	35.4	even	6
98.8.c.e.67.1		2	35.19	odd	6
98.8.c.e.79.1		2	35.24	odd	6
144.8.a.i.1.1		1	60.59	even	2
162.8.c.a.55.1		2	45.14	odd	6
162.8.c.a.109.1		2	45.29	odd	6
162.8.c.l.55.1		2	45.4	even	6
162.8.c.l.109.1		2	45.34	even	6
242.8.a.e.1.1		1	55.54	odd	2
256.8.b.b.129.1		2	80.69	even	4
256.8.b.b.129.2		2	80.29	even	4
256.8.b.f.129.1		2	80.19	odd	4
256.8.b.f.129.2		2	80.59	odd	4
338.8.a.d.1.1		1	65.64	even	2
338.8.b.d.337.1		2	65.34	odd	4
338.8.b.d.337.2		2	65.44	odd	4
400.8.a.l.1.1		1	4.3	odd	2
400.8.c.j.49.1		2	20.7	even	4
400.8.c.j.49.2		2	20.3	even	4
450.8.a.c.1.1		1	3.2	odd	2
450.8.c.g.199.1		2	15.2	even	4
450.8.c.g.199.2		2	15.8	even	4
576.8.a.f.1.1		1	120.59	even	2
576.8.a.g.1.1		1	120.29	odd	2
578.8.a.b.1.1		1	85.84	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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