LMFDB - Embedded newform 350.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,8,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	350.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:	$109.334758919$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} +93.0000 q^{3} +64.0000 q^{4} -744.000 q^{6} -343.000 q^{7} -512.000 q^{8} +6462.00 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +93.0000 q^{3} +64.0000 q^{4} -744.000 q^{6} -343.000 q^{7} -512.000 q^{8} +6462.00 q^{9} -2167.00 q^{11} +5952.00 q^{12} +1661.00 q^{13} +2744.00 q^{14} +4096.00 q^{16} +35771.0 q^{17} -51696.0 q^{18} +20222.0 q^{19} -31899.0 q^{21} +17336.0 q^{22} +42130.0 q^{23} -47616.0 q^{24} -13288.0 q^{26} +397575. q^{27} -21952.0 q^{28} -111789. q^{29} -269504. q^{31} -32768.0 q^{32} -201531. q^{33} -286168. q^{34} +413568. q^{36} -532774. q^{37} -161776. q^{38} +154473. q^{39} +158056. q^{41} +255192. q^{42} +521874. q^{43} -138688. q^{44} -337040. q^{46} +939733. q^{47} +380928. q^{48} +117649. q^{49} +3.32670e6 q^{51} +106304. q^{52} +408384. q^{53} -3.18060e6 q^{54} +175616. q^{56} +1.88065e6 q^{57} +894312. q^{58} -522172. q^{59} +350080. q^{61} +2.15603e6 q^{62} -2.21647e6 q^{63} +262144. q^{64} +1.61225e6 q^{66} +3.93118e6 q^{67} +2.28934e6 q^{68} +3.91809e6 q^{69} +1.19402e6 q^{71} -3.30854e6 q^{72} -998350. q^{73} +4.26219e6 q^{74} +1.29421e6 q^{76} +743281. q^{77} -1.23578e6 q^{78} -2.12071e6 q^{79} +2.28421e7 q^{81} -1.26445e6 q^{82} +1.74671e6 q^{83} -2.04154e6 q^{84} -4.17499e6 q^{86} -1.03964e7 q^{87} +1.10950e6 q^{88} -1.00777e7 q^{89} -569723. q^{91} +2.69632e6 q^{92} -2.50639e7 q^{93} -7.51786e6 q^{94} -3.04742e6 q^{96} +6.23829e6 q^{97} -941192. q^{98} -1.40032e7 q^{99} +O(q^{100})\)

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
$3$	93.0000	1.98865	0.994325	−	0.106381i	$-0.0339263\pi$
$3$	93.0000	1.98865	0.994325	+	0.106381i	$0.0339263\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−744.000	−1.40619
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	6462.00	2.95473
$10$	0	0
$11$	−2167.00	−0.490890	−0.245445	−	0.969410i	$-0.578934\pi$
$11$	−2167.00	−0.490890	−0.245445	+	0.969410i	$0.578934\pi$
$12$	5952.00	0.994325
$13$	1661.00	0.209685	0.104843	−	0.994489i	$-0.466566\pi$
$13$	1661.00	0.209685	0.104843	+	0.994489i	$0.466566\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	35771.0	1.76587	0.882937	−	0.469491i	$-0.155563\pi$
$17$	35771.0	1.76587	0.882937	+	0.469491i	$0.155563\pi$
$18$	−51696.0	−2.08931
$19$	20222.0	0.676373	0.338187	−	0.941079i	$-0.390186\pi$
$19$	20222.0	0.676373	0.338187	+	0.941079i	$0.390186\pi$
$20$	0	0
$21$	−31899.0	−0.751639
$22$	17336.0	0.347112
$23$	42130.0	0.722011	0.361006	−	0.932564i	$-0.382433\pi$
$23$	42130.0	0.722011	0.361006	+	0.932564i	$0.382433\pi$
$24$	−47616.0	−0.703094
$25$	0	0
$26$	−13288.0	−0.148270
$27$	397575.	3.88728
$28$	−21952.0	−0.188982
$29$	−111789.	−0.851150	−0.425575	−	0.904923i	$-0.639928\pi$
$29$	−111789.	−0.851150	−0.425575	+	0.904923i	$0.639928\pi$
$30$	0	0
$31$	−269504.	−1.62480	−0.812399	−	0.583102i	$-0.801839\pi$
$31$	−269504.	−1.62480	−0.812399	+	0.583102i	$0.801839\pi$
$32$	−32768.0	−0.176777
$33$	−201531.	−0.976210
$34$	−286168.	−1.24866
$35$	0	0
$36$	413568.	1.47737
$37$	−532774.	−1.72917	−0.864584	−	0.502489i	$-0.832418\pi$
$37$	−532774.	−1.72917	−0.864584	+	0.502489i	$0.832418\pi$
$38$	−161776.	−0.478268
$39$	154473.	0.416991
$40$	0	0
$41$	158056.	0.358152	0.179076	−	0.983835i	$-0.442689\pi$
$41$	158056.	0.358152	0.179076	+	0.983835i	$0.442689\pi$
$42$	255192.	0.531489
$43$	521874.	1.00098	0.500491	−	0.865742i	$-0.333153\pi$
$43$	521874.	1.00098	0.500491	+	0.865742i	$0.333153\pi$
$44$	−138688.	−0.245445
$45$	0	0
$46$	−337040.	−0.510539
$47$	939733.	1.32027	0.660134	−	0.751148i	$-0.270500\pi$
$47$	939733.	1.32027	0.660134	+	0.751148i	$0.270500\pi$
$48$	380928.	0.497163
$49$	117649.	0.142857
$50$	0	0
$51$	3.32670e6	3.51171
$52$	106304.	0.104843
$53$	408384.	0.376793	0.188397	−	0.982093i	$-0.439671\pi$
$53$	408384.	0.376793	0.188397	+	0.982093i	$0.439671\pi$
$54$	−3.18060e6	−2.74872
$55$	0	0
$56$	175616.	0.133631
$57$	1.88065e6	1.34507
$58$	894312.	0.601854
$59$	−522172.	−0.331003	−0.165501	−	0.986210i	$-0.552924\pi$
$59$	−522172.	−0.331003	−0.165501	+	0.986210i	$0.552924\pi$
$60$	0	0
$61$	350080.	0.197475	0.0987377	−	0.995113i	$-0.468520\pi$
$61$	350080.	0.197475	0.0987377	+	0.995113i	$0.468520\pi$
$62$	2.15603e6	1.14891
$63$	−2.21647e6	−1.11678
$64$	262144.	0.125000
$65$	0	0
$66$	1.61225e6	0.690284
$67$	3.93118e6	1.59684	0.798419	−	0.602103i	$-0.205670\pi$
$67$	3.93118e6	1.59684	0.798419	+	0.602103i	$0.205670\pi$
$68$	2.28934e6	0.882937
$69$	3.91809e6	1.43583
$70$	0	0
$71$	1.19402e6	0.395919	0.197959	−	0.980210i	$-0.436569\pi$
$71$	1.19402e6	0.395919	0.197959	+	0.980210i	$0.436569\pi$
$72$	−3.30854e6	−1.04466
$73$	−998350.	−0.300367	−0.150184	−	0.988658i	$-0.547987\pi$
$73$	−998350.	−0.300367	−0.150184	+	0.988658i	$0.547987\pi$
$74$	4.26219e6	1.22271
$75$	0	0
$76$	1.29421e6	0.338187
$77$	743281.	0.185539
$78$	−1.23578e6	−0.294857
$79$	−2.12071e6	−0.483934	−0.241967	−	0.970284i	$-0.577793\pi$
$79$	−2.12071e6	−0.483934	−0.241967	+	0.970284i	$0.577793\pi$
$80$	0	0
$81$	2.28421e7	4.77571
$82$	−1.26445e6	−0.253252
$83$	1.74671e6	0.335310	0.167655	−	0.985846i	$-0.446380\pi$
$83$	1.74671e6	0.335310	0.167655	+	0.985846i	$0.446380\pi$
$84$	−2.04154e6	−0.375820
$85$	0	0
$86$	−4.17499e6	−0.707801
$87$	−1.03964e7	−1.69264
$88$	1.10950e6	0.173556
$89$	−1.00777e7	−1.51530	−0.757649	−	0.652662i	$-0.773652\pi$
$89$	−1.00777e7	−1.51530	−0.757649	+	0.652662i	$0.773652\pi$
$90$	0	0
$91$	−569723.	−0.0792536
$92$	2.69632e6	0.361006
$93$	−2.50639e7	−3.23116
$94$	−7.51786e6	−0.933570
$95$	0	0
$96$	−3.04742e6	−0.351547
$97$	6.23829e6	0.694009	0.347004	−	0.937864i	$-0.387199\pi$
$97$	6.23829e6	0.694009	0.347004	+	0.937864i	$0.387199\pi$
$98$	−941192.	−0.101015
$99$	−1.40032e7	−1.45045
$100$	0	0
$101$	4.63305e6	0.447448	0.223724	−	0.974653i	$-0.428179\pi$
$101$	4.63305e6	0.447448	0.223724	+	0.974653i	$0.428179\pi$
$102$	−2.66136e7	−2.48315
$103$	9.64490e6	0.869696	0.434848	−	0.900504i	$-0.356802\pi$
$103$	9.64490e6	0.869696	0.434848	+	0.900504i	$0.356802\pi$
$104$	−850432.	−0.0741349
$105$	0	0
$106$	−3.26707e6	−0.266433
$107$	−5.68201e6	−0.448393	−0.224197	−	0.974544i	$-0.571976\pi$
$107$	−5.68201e6	−0.448393	−0.224197	+	0.974544i	$0.571976\pi$
$108$	2.54448e7	1.94364
$109$	2.42544e7	1.79390	0.896949	−	0.442134i	$-0.145778\pi$
$109$	2.42544e7	1.79390	0.896949	+	0.442134i	$0.145778\pi$
$110$	0	0
$111$	−4.95480e7	−3.43871
$112$	−1.40493e6	−0.0944911
$113$	3.52344e6	0.229716	0.114858	−	0.993382i	$-0.463359\pi$
$113$	3.52344e6	0.229716	0.114858	+	0.993382i	$0.463359\pi$
$114$	−1.50452e7	−0.951108
$115$	0	0
$116$	−7.15450e6	−0.425575
$117$	1.07334e7	0.619564
$118$	4.17738e6	0.234054
$119$	−1.22695e7	−0.667438
$120$	0	0
$121$	−1.47913e7	−0.759027
$122$	−2.80064e6	−0.139636
$123$	1.46992e7	0.712239
$124$	−1.72483e7	−0.812399
$125$	0	0
$126$	1.77317e7	0.789685
$127$	−1.80316e7	−0.781126	−0.390563	−	0.920576i	$-0.627720\pi$
$127$	−1.80316e7	−0.781126	−0.390563	+	0.920576i	$0.627720\pi$
$128$	−2.09715e6	−0.0883883
$129$	4.85343e7	1.99060
$130$	0	0
$131$	1.59177e7	0.618631	0.309315	−	0.950959i	$-0.399900\pi$
$131$	1.59177e7	0.618631	0.309315	+	0.950959i	$0.399900\pi$
$132$	−1.28980e7	−0.488105
$133$	−6.93615e6	−0.255645
$134$	−3.14494e7	−1.12913
$135$	0	0
$136$	−1.83148e7	−0.624331
$137$	2.46119e7	0.817756	0.408878	−	0.912589i	$-0.365920\pi$
$137$	2.46119e7	0.817756	0.408878	+	0.912589i	$0.365920\pi$
$138$	−3.13447e7	−1.01528
$139$	5.58345e7	1.76340	0.881700	−	0.471810i	$-0.156399\pi$
$139$	5.58345e7	1.76340	0.881700	+	0.471810i	$0.156399\pi$
$140$	0	0
$141$	8.73952e7	2.62555
$142$	−9.55213e6	−0.279957
$143$	−3.59939e6	−0.102932
$144$	2.64684e7	0.738683
$145$	0	0
$146$	7.98680e6	0.212392
$147$	1.09414e7	0.284093
$148$	−3.40975e7	−0.864584
$149$	2.31479e7	0.573271	0.286635	−	0.958040i	$-0.407463\pi$
$149$	2.31479e7	0.573271	0.286635	+	0.958040i	$0.407463\pi$
$150$	0	0
$151$	−7.35015e7	−1.73731	−0.868654	−	0.495419i	$-0.835014\pi$
$151$	−7.35015e7	−1.73731	−0.868654	+	0.495419i	$0.835014\pi$
$152$	−1.03537e7	−0.239134
$153$	2.31152e8	5.21769
$154$	−5.94625e6	−0.131196
$155$	0	0
$156$	9.88627e6	0.208495
$157$	7.00924e7	1.44551	0.722757	−	0.691102i	$-0.242875\pi$
$157$	7.00924e7	1.44551	0.722757	+	0.691102i	$0.242875\pi$
$158$	1.69657e7	0.342193
$159$	3.79797e7	0.749310
$160$	0	0
$161$	−1.44506e7	−0.272895
$162$	−1.82737e8	−3.37694
$163$	2.88555e7	0.521882	0.260941	−	0.965355i	$-0.415967\pi$
$163$	2.88555e7	0.521882	0.260941	+	0.965355i	$0.415967\pi$
$164$	1.01156e7	0.179076
$165$	0	0
$166$	−1.39737e7	−0.237100
$167$	−4.70769e7	−0.782168	−0.391084	−	0.920355i	$-0.627900\pi$
$167$	−4.70769e7	−0.782168	−0.391084	+	0.920355i	$0.627900\pi$
$168$	1.63323e7	0.265745
$169$	−5.99896e7	−0.956032
$170$	0	0
$171$	1.30675e8	1.99850
$172$	3.33999e7	0.500491
$173$	−8.09106e7	−1.18807	−0.594037	−	0.804437i	$-0.702467\pi$
$173$	−8.09106e7	−1.18807	−0.594037	+	0.804437i	$0.702467\pi$
$174$	8.31710e7	1.19688
$175$	0	0
$176$	−8.87603e6	−0.122723
$177$	−4.85620e7	−0.658249
$178$	8.06219e7	1.07148
$179$	2.54155e7	0.331217	0.165609	−	0.986192i	$-0.447041\pi$
$179$	2.54155e7	0.331217	0.165609	+	0.986192i	$0.447041\pi$
$180$	0	0
$181$	1.96281e7	0.246038	0.123019	−	0.992404i	$-0.460742\pi$
$181$	1.96281e7	0.246038	0.123019	+	0.992404i	$0.460742\pi$
$182$	4.55778e6	0.0560407
$183$	3.25574e7	0.392710
$184$	−2.15706e7	−0.255270
$185$	0	0
$186$	2.00511e8	2.28477
$187$	−7.75158e7	−0.866851
$188$	6.01429e7	0.660134
$189$	−1.36368e8	−1.46925
$190$	0	0
$191$	−6.06419e7	−0.629732	−0.314866	−	0.949136i	$-0.601960\pi$
$191$	−6.06419e7	−0.629732	−0.314866	+	0.949136i	$0.601960\pi$
$192$	2.43794e7	0.248581
$193$	−7.25552e7	−0.726471	−0.363235	−	0.931697i	$-0.618328\pi$
$193$	−7.25552e7	−0.726471	−0.363235	+	0.931697i	$0.618328\pi$
$194$	−4.99064e7	−0.490738
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	4.99667e7	0.465638	0.232819	−	0.972520i	$-0.425205\pi$
$197$	4.99667e7	0.465638	0.232819	+	0.972520i	$0.425205\pi$
$198$	1.12025e8	1.02562
$199$	4.52659e7	0.407179	0.203590	−	0.979056i	$-0.434739\pi$
$199$	4.52659e7	0.407179	0.203590	+	0.979056i	$0.434739\pi$
$200$	0	0
$201$	3.65599e8	3.17555
$202$	−3.70644e7	−0.316393
$203$	3.83436e7	0.321704
$204$	2.12909e8	1.75585
$205$	0	0
$206$	−7.71592e7	−0.614968
$207$	2.72244e8	2.13335
$208$	6.80346e6	0.0524213
$209$	−4.38211e7	−0.332025
$210$	0	0
$211$	1.70784e8	1.25158	0.625790	−	0.779992i	$-0.284777\pi$
$211$	1.70784e8	1.25158	0.625790	+	0.779992i	$0.284777\pi$
$212$	2.61366e7	0.188397
$213$	1.11043e8	0.787344
$214$	4.54561e7	0.317062
$215$	0	0
$216$	−2.03558e8	−1.37436
$217$	9.24399e7	0.614116
$218$	−1.94035e8	−1.26848
$219$	−9.28466e7	−0.597326
$220$	0	0
$221$	5.94156e7	0.370278
$222$	3.96384e8	2.43154
$223$	−1.44305e8	−0.871396	−0.435698	−	0.900093i	$-0.643498\pi$
$223$	−1.44305e8	−0.871396	−0.435698	+	0.900093i	$0.643498\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	−2.81875e7	−0.162434
$227$	−1.52289e8	−0.864125	−0.432063	−	0.901844i	$-0.642214\pi$
$227$	−1.52289e8	−0.864125	−0.432063	+	0.901844i	$0.642214\pi$
$228$	1.20361e8	0.672535
$229$	−9.31342e7	−0.512490	−0.256245	−	0.966612i	$-0.582485\pi$
$229$	−9.31342e7	−0.512490	−0.256245	+	0.966612i	$0.582485\pi$
$230$	0	0
$231$	6.91251e7	0.368973
$232$	5.72360e7	0.300927
$233$	−2.87478e7	−0.148888	−0.0744439	−	0.997225i	$-0.523718\pi$
$233$	−2.87478e7	−0.148888	−0.0744439	+	0.997225i	$0.523718\pi$
$234$	−8.58671e7	−0.438098
$235$	0	0
$236$	−3.34190e7	−0.165501
$237$	−1.97226e8	−0.962376
$238$	9.81556e7	0.471950
$239$	1.43827e8	0.681473	0.340736	−	0.940159i	$-0.389324\pi$
$239$	1.43827e8	0.681473	0.340736	+	0.940159i	$0.389324\pi$
$240$	0	0
$241$	4.18177e8	1.92442	0.962210	−	0.272308i	$-0.0877868\pi$
$241$	4.18177e8	1.92442	0.962210	+	0.272308i	$0.0877868\pi$
$242$	1.18330e8	0.536713
$243$	1.25482e9	5.60994
$244$	2.24051e7	0.0987377
$245$	0	0
$246$	−1.17594e8	−0.503629
$247$	3.35887e7	0.141826
$248$	1.37986e8	0.574453
$249$	1.62444e8	0.666815
$250$	0	0
$251$	9.14002e7	0.364829	0.182414	−	0.983222i	$-0.441609\pi$
$251$	9.14002e7	0.364829	0.182414	+	0.983222i	$0.441609\pi$
$252$	−1.41854e8	−0.558392
$253$	−9.12957e7	−0.354428
$254$	1.44253e8	0.552340
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.08667e8	0.766810	0.383405	−	0.923580i	$-0.374751\pi$
$257$	2.08667e8	0.766810	0.383405	+	0.923580i	$0.374751\pi$
$258$	−3.88274e8	−1.40757
$259$	1.82741e8	0.653564
$260$	0	0
$261$	−7.22381e8	−2.51492
$262$	−1.27342e8	−0.437438
$263$	7.31090e6	0.0247814	0.0123907	−	0.999923i	$-0.496056\pi$
$263$	7.31090e6	0.0247814	0.0123907	+	0.999923i	$0.496056\pi$
$264$	1.03184e8	0.345142
$265$	0	0
$266$	5.54892e7	0.180768
$267$	−9.37230e8	−3.01340
$268$	2.51595e8	0.798419
$269$	−4.04584e8	−1.26729	−0.633645	−	0.773624i	$-0.718442\pi$
$269$	−4.04584e8	−1.26729	−0.633645	+	0.773624i	$0.718442\pi$
$270$	0	0
$271$	1.32167e8	0.403395	0.201697	−	0.979448i	$-0.435354\pi$
$271$	1.32167e8	0.403395	0.201697	+	0.979448i	$0.435354\pi$
$272$	1.46518e8	0.441469
$273$	−5.29842e7	−0.157608
$274$	−1.96895e8	−0.578241
$275$	0	0
$276$	2.50758e8	0.717914
$277$	1.56748e8	0.443121	0.221560	−	0.975147i	$-0.428885\pi$
$277$	1.56748e8	0.443121	0.221560	+	0.975147i	$0.428885\pi$
$278$	−4.46676e8	−1.24691
$279$	−1.74153e9	−4.80084
$280$	0	0
$281$	−8.45381e7	−0.227290	−0.113645	−	0.993521i	$-0.536253\pi$
$281$	−8.45381e7	−0.227290	−0.113645	+	0.993521i	$0.536253\pi$
$282$	−6.99161e8	−1.85655
$283$	−4.30617e8	−1.12938	−0.564689	−	0.825304i	$-0.691004\pi$
$283$	−4.30617e8	−1.12938	−0.564689	+	0.825304i	$0.691004\pi$
$284$	7.64170e7	0.197959
$285$	0	0
$286$	2.87951e7	0.0727843
$287$	−5.42132e7	−0.135369
$288$	−2.11747e8	−0.522328
$289$	8.69226e8	2.11831
$290$	0	0
$291$	5.80161e8	1.38014
$292$	−6.38944e7	−0.150184
$293$	−4.98926e7	−0.115878	−0.0579388	−	0.998320i	$-0.518453\pi$
$293$	−4.98926e7	−0.115878	−0.0579388	+	0.998320i	$0.518453\pi$
$294$	−8.75309e7	−0.200884
$295$	0	0
$296$	2.72780e8	0.611353
$297$	−8.61545e8	−1.90823
$298$	−1.85183e8	−0.405363
$299$	6.99779e7	0.151395
$300$	0	0
$301$	−1.79003e8	−0.378335
$302$	5.88012e8	1.22846
$303$	4.30874e8	0.889817
$304$	8.28293e7	0.169093
$305$	0	0
$306$	−1.84922e9	−3.68946
$307$	−4.78055e8	−0.942960	−0.471480	−	0.881877i	$-0.656280\pi$
$307$	−4.78055e8	−0.942960	−0.471480	+	0.881877i	$0.656280\pi$
$308$	4.75700e7	0.0927696
$309$	8.96976e8	1.72952
$310$	0	0
$311$	−7.56687e8	−1.42644	−0.713222	−	0.700938i	$-0.752765\pi$
$311$	−7.56687e8	−1.42644	−0.713222	+	0.700938i	$0.752765\pi$
$312$	−7.90902e7	−0.147429
$313$	−4.71868e8	−0.869792	−0.434896	−	0.900481i	$-0.643215\pi$
$313$	−4.71868e8	−0.869792	−0.434896	+	0.900481i	$0.643215\pi$
$314$	−5.60739e8	−1.02213
$315$	0	0
$316$	−1.35725e8	−0.241967
$317$	7.11191e8	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$317$	7.11191e8	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$318$	−3.03838e8	−0.529842
$319$	2.42247e8	0.417821
$320$	0	0
$321$	−5.28427e8	−0.891698
$322$	1.15605e8	0.192966
$323$	7.23361e8	1.19439
$324$	1.46189e9	2.38786
$325$	0	0
$326$	−2.30844e8	−0.369026
$327$	2.25566e9	3.56744
$328$	−8.09247e7	−0.126626
$329$	−3.22328e8	−0.499014
$330$	0	0
$331$	4.07293e8	0.617318	0.308659	−	0.951173i	$-0.400120\pi$
$331$	4.07293e8	0.617318	0.308659	+	0.951173i	$0.400120\pi$
$332$	1.11789e8	0.167655
$333$	−3.44279e9	−5.10923
$334$	3.76615e8	0.553076
$335$	0	0
$336$	−1.30658e8	−0.187910
$337$	−9.18486e8	−1.30728	−0.653639	−	0.756807i	$-0.726758\pi$
$337$	−9.18486e8	−1.30728	−0.653639	+	0.756807i	$0.726758\pi$
$338$	4.79917e8	0.676017
$339$	3.27680e8	0.456826
$340$	0	0
$341$	5.84015e8	0.797598
$342$	−1.04540e9	−1.41315
$343$	−4.03536e7	−0.0539949
$344$	−2.67199e8	−0.353900
$345$	0	0
$346$	6.47284e8	0.840096
$347$	1.43801e9	1.84760	0.923802	−	0.382870i	$-0.125064\pi$
$347$	1.43801e9	1.84760	0.923802	+	0.382870i	$0.125064\pi$
$348$	−6.65368e8	−0.846320
$349$	−7.70970e8	−0.970842	−0.485421	−	0.874281i	$-0.661334\pi$
$349$	−7.70970e8	−0.970842	−0.485421	+	0.874281i	$0.661334\pi$
$350$	0	0
$351$	6.60372e8	0.815105
$352$	7.10083e7	0.0867780
$353$	−9.62808e8	−1.16501	−0.582503	−	0.812829i	$-0.697927\pi$
$353$	−9.62808e8	−1.16501	−0.582503	+	0.812829i	$0.697927\pi$
$354$	3.88496e8	0.465452
$355$	0	0
$356$	−6.44975e8	−0.757649
$357$	−1.14106e9	−1.32730
$358$	−2.03324e8	−0.234206
$359$	−1.26150e9	−1.43898	−0.719492	−	0.694500i	$-0.755625\pi$
$359$	−1.26150e9	−1.43898	−0.719492	+	0.694500i	$0.755625\pi$
$360$	0	0
$361$	−4.84942e8	−0.542519
$362$	−1.57025e8	−0.173975
$363$	−1.37559e9	−1.50944
$364$	−3.64623e7	−0.0396268
$365$	0	0
$366$	−2.60460e8	−0.277688
$367$	1.58431e7	0.0167305	0.00836525	−	0.999965i	$-0.497337\pi$
$367$	1.58431e7	0.0167305	0.00836525	+	0.999965i	$0.497337\pi$
$368$	1.72564e8	0.180503
$369$	1.02136e9	1.05824
$370$	0	0
$371$	−1.40076e8	−0.142414
$372$	−1.60409e9	−1.61558
$373$	1.07114e8	0.106873	0.0534363	−	0.998571i	$-0.482983\pi$
$373$	1.07114e8	0.106873	0.0534363	+	0.998571i	$0.482983\pi$
$374$	6.20126e8	0.612956
$375$	0	0
$376$	−4.81143e8	−0.466785
$377$	−1.85682e8	−0.178474
$378$	1.09095e9	1.03892
$379$	−1.88930e9	−1.78264	−0.891318	−	0.453379i	$-0.850219\pi$
$379$	−1.88930e9	−1.78264	−0.891318	+	0.453379i	$0.850219\pi$
$380$	0	0
$381$	−1.67694e9	−1.55339
$382$	4.85135e8	0.445288
$383$	8.66602e8	0.788178	0.394089	−	0.919072i	$-0.371060\pi$
$383$	8.66602e8	0.788178	0.394089	+	0.919072i	$0.371060\pi$
$384$	−1.95035e8	−0.175774
$385$	0	0
$386$	5.80442e8	0.513692
$387$	3.37235e9	2.95763
$388$	3.99251e8	0.347004
$389$	−9.65943e8	−0.832009	−0.416004	−	0.909363i	$-0.636570\pi$
$389$	−9.65943e8	−0.832009	−0.416004	+	0.909363i	$0.636570\pi$
$390$	0	0
$391$	1.50703e9	1.27498
$392$	−6.02363e7	−0.0505076
$393$	1.48035e9	1.23024
$394$	−3.99733e8	−0.329256
$395$	0	0
$396$	−8.96202e8	−0.725225
$397$	−2.34951e8	−0.188456	−0.0942280	−	0.995551i	$-0.530038\pi$
$397$	−2.34951e8	−0.188456	−0.0942280	+	0.995551i	$0.530038\pi$
$398$	−3.62127e8	−0.287919
$399$	−6.45062e8	−0.508389
$400$	0	0
$401$	−1.92546e9	−1.49118	−0.745590	−	0.666405i	$-0.767832\pi$
$401$	−1.92546e9	−1.49118	−0.745590	+	0.666405i	$0.767832\pi$
$402$	−2.92479e9	−2.24545
$403$	−4.47646e8	−0.340696
$404$	2.96515e8	0.223724
$405$	0	0
$406$	−3.06749e8	−0.227479
$407$	1.15452e9	0.848832
$408$	−1.70327e9	−1.24158
$409$	−1.04881e9	−0.757990	−0.378995	−	0.925399i	$-0.623730\pi$
$409$	−1.04881e9	−0.757990	−0.378995	+	0.925399i	$0.623730\pi$
$410$	0	0
$411$	2.28891e9	1.62623
$412$	6.17274e8	0.434848
$413$	1.79105e8	0.125107
$414$	−2.17795e9	−1.50851
$415$	0	0
$416$	−5.44276e7	−0.0370675
$417$	5.19261e9	3.50679
$418$	3.50569e8	0.234777
$419$	−1.96198e9	−1.30300	−0.651502	−	0.758647i	$-0.725861\pi$
$419$	−1.96198e9	−1.30300	−0.651502	+	0.758647i	$0.725861\pi$
$420$	0	0
$421$	2.19735e9	1.43520	0.717601	−	0.696455i	$-0.245240\pi$
$421$	2.19735e9	1.43520	0.717601	+	0.696455i	$0.245240\pi$
$422$	−1.36627e9	−0.885000
$423$	6.07255e9	3.90104
$424$	−2.09093e8	−0.133217
$425$	0	0
$426$	−8.88348e8	−0.556736
$427$	−1.20077e8	−0.0746387
$428$	−3.63649e8	−0.224197
$429$	−3.34743e8	−0.204697
$430$	0	0
$431$	3.15560e7	0.0189851	0.00949253	−	0.999955i	$-0.496978\pi$
$431$	3.15560e7	0.0189851	0.00949253	+	0.999955i	$0.496978\pi$
$432$	1.62847e9	0.971820
$433$	1.02447e9	0.606447	0.303223	−	0.952920i	$-0.401937\pi$
$433$	1.02447e9	0.606447	0.303223	+	0.952920i	$0.401937\pi$
$434$	−7.39519e8	−0.434245
$435$	0	0
$436$	1.55228e9	0.896949
$437$	8.51953e8	0.488349
$438$	7.42772e8	0.422373
$439$	2.86699e9	1.61734	0.808668	−	0.588265i	$-0.200189\pi$
$439$	2.86699e9	1.61734	0.808668	+	0.588265i	$0.200189\pi$
$440$	0	0
$441$	7.60248e8	0.422105
$442$	−4.75325e8	−0.261826
$443$	−2.68927e9	−1.46967	−0.734837	−	0.678244i	$-0.762741\pi$
$443$	−2.68927e9	−1.46967	−0.734837	+	0.678244i	$0.762741\pi$
$444$	−3.17107e9	−1.71935
$445$	0	0
$446$	1.15444e9	0.616170
$447$	2.15275e9	1.14003
$448$	−8.99154e7	−0.0472456
$449$	1.76468e9	0.920031	0.460016	−	0.887911i	$-0.347844\pi$
$449$	1.76468e9	0.920031	0.460016	+	0.887911i	$0.347844\pi$
$450$	0	0
$451$	−3.42507e8	−0.175813
$452$	2.25500e8	0.114858
$453$	−6.83564e9	−3.45490
$454$	1.21831e9	0.611029
$455$	0	0
$456$	−9.62891e8	−0.475554
$457$	3.01751e9	1.47891	0.739455	−	0.673206i	$-0.235083\pi$
$457$	3.01751e9	1.47891	0.739455	+	0.673206i	$0.235083\pi$
$458$	7.45074e8	0.362385
$459$	1.42217e10	6.86445
$460$	0	0
$461$	−2.12400e9	−1.00972	−0.504861	−	0.863200i	$-0.668456\pi$
$461$	−2.12400e9	−1.00972	−0.504861	+	0.863200i	$0.668456\pi$
$462$	−5.53001e8	−0.260903
$463$	−6.57271e8	−0.307759	−0.153880	−	0.988090i	$-0.549177\pi$
$463$	−6.57271e8	−0.307759	−0.153880	+	0.988090i	$0.549177\pi$
$464$	−4.57888e8	−0.212787
$465$	0	0
$466$	2.29983e8	0.105280
$467$	−3.60517e9	−1.63801	−0.819005	−	0.573787i	$-0.805474\pi$
$467$	−3.60517e9	−1.63801	−0.819005	+	0.573787i	$0.805474\pi$
$468$	6.86936e8	0.309782
$469$	−1.34839e9	−0.603548
$470$	0	0
$471$	6.51860e9	2.87462
$472$	2.67352e8	0.117027
$473$	−1.13090e9	−0.491372
$474$	1.57781e9	0.680503
$475$	0	0
$476$	−7.85245e8	−0.333719
$477$	2.63898e9	1.11332
$478$	−1.15062e9	−0.481874
$479$	4.52579e7	0.0188157	0.00940785	−	0.999956i	$-0.497005\pi$
$479$	4.52579e7	0.0188157	0.00940785	+	0.999956i	$0.497005\pi$
$480$	0	0
$481$	−8.84938e8	−0.362581
$482$	−3.34541e9	−1.36077
$483$	−1.34390e9	−0.542692
$484$	−9.46642e8	−0.379513
$485$	0	0
$486$	−1.00385e10	−3.96683
$487$	−6.04177e8	−0.237035	−0.118518	−	0.992952i	$-0.537814\pi$
$487$	−6.04177e8	−0.237035	−0.118518	+	0.992952i	$0.537814\pi$
$488$	−1.79241e8	−0.0698181
$489$	2.68356e9	1.03784
$490$	0	0
$491$	−8.64289e8	−0.329514	−0.164757	−	0.986334i	$-0.552684\pi$
$491$	−8.64289e8	−0.329514	−0.164757	+	0.986334i	$0.552684\pi$
$492$	9.40749e8	0.356120
$493$	−3.99880e9	−1.50302
$494$	−2.68710e8	−0.100286
$495$	0	0
$496$	−1.10389e9	−0.406199
$497$	−4.09547e8	−0.149643
$498$	−1.29955e9	−0.471510
$499$	−3.18150e9	−1.14625	−0.573126	−	0.819467i	$-0.694269\pi$
$499$	−3.18150e9	−1.14625	−0.573126	+	0.819467i	$0.694269\pi$
$500$	0	0
$501$	−4.37815e9	−1.55546
$502$	−7.31202e8	−0.257973
$503$	3.09156e9	1.08315	0.541577	−	0.840651i	$-0.317827\pi$
$503$	3.09156e9	1.08315	0.541577	+	0.840651i	$0.317827\pi$
$504$	1.13483e9	0.394843
$505$	0	0
$506$	7.30366e8	0.250619
$507$	−5.57903e9	−1.90121
$508$	−1.15402e9	−0.390563
$509$	−6.78827e8	−0.228164	−0.114082	−	0.993471i	$-0.536393\pi$
$509$	−6.78827e8	−0.228164	−0.114082	+	0.993471i	$0.536393\pi$
$510$	0	0
$511$	3.42434e8	0.113528
$512$	−1.34218e8	−0.0441942
$513$	8.03976e9	2.62925
$514$	−1.66934e9	−0.542217
$515$	0	0
$516$	3.10619e9	0.995301
$517$	−2.03640e9	−0.648107
$518$	−1.46193e9	−0.462139
$519$	−7.52468e9	−2.36267
$520$	0	0
$521$	−2.11415e9	−0.654943	−0.327471	−	0.944861i	$-0.606197\pi$
$521$	−2.11415e9	−0.654943	−0.327471	+	0.944861i	$0.606197\pi$
$522$	5.77904e9	1.77832
$523$	−1.76464e9	−0.539387	−0.269694	−	0.962946i	$-0.586922\pi$
$523$	−1.76464e9	−0.539387	−0.269694	+	0.962946i	$0.586922\pi$
$524$	1.01873e9	0.309315
$525$	0	0
$526$	−5.84872e7	−0.0175231
$527$	−9.64043e9	−2.86919
$528$	−8.25471e8	−0.244052
$529$	−1.62989e9	−0.478700
$530$	0	0
$531$	−3.37428e9	−0.978024
$532$	−4.43913e8	−0.127823
$533$	2.62531e8	0.0750992
$534$	7.49784e9	2.13080
$535$	0	0
$536$	−2.01276e9	−0.564567
$537$	2.36364e9	0.658675
$538$	3.23667e9	0.896110
$539$	−2.54945e8	−0.0701272
$540$	0	0
$541$	4.76266e9	1.29318	0.646590	−	0.762838i	$-0.276195\pi$
$541$	4.76266e9	1.29318	0.646590	+	0.762838i	$0.276195\pi$
$542$	−1.05733e9	−0.285243
$543$	1.82541e9	0.489284
$544$	−1.17214e9	−0.312165
$545$	0	0
$546$	4.23874e8	0.111445
$547$	−5.09834e8	−0.133190	−0.0665952	−	0.997780i	$-0.521214\pi$
$547$	−5.09834e8	−0.133190	−0.0665952	+	0.997780i	$0.521214\pi$
$548$	1.57516e9	0.408878
$549$	2.26222e9	0.583487
$550$	0	0
$551$	−2.26060e9	−0.575695
$552$	−2.00606e9	−0.507642
$553$	7.27403e8	0.182910
$554$	−1.25398e9	−0.313334
$555$	0	0
$556$	3.57341e9	0.881700
$557$	3.25470e9	0.798026	0.399013	−	0.916945i	$-0.369353\pi$
$557$	3.25470e9	0.798026	0.399013	+	0.916945i	$0.369353\pi$
$558$	1.39323e10	3.39471
$559$	8.66833e8	0.209891
$560$	0	0
$561$	−7.20897e9	−1.72386
$562$	6.76305e8	0.160718
$563$	−5.20284e9	−1.22874	−0.614372	−	0.789017i	$-0.710591\pi$
$563$	−5.20284e9	−1.22874	−0.614372	+	0.789017i	$0.710591\pi$
$564$	5.59329e9	1.31278
$565$	0	0
$566$	3.44494e9	0.798590
$567$	−7.83483e9	−1.80505
$568$	−6.11336e8	−0.139978
$569$	−2.20275e9	−0.501272	−0.250636	−	0.968081i	$-0.580640\pi$
$569$	−2.20275e9	−0.501272	−0.250636	+	0.968081i	$0.580640\pi$
$570$	0	0
$571$	4.09030e9	0.919450	0.459725	−	0.888061i	$-0.347948\pi$
$571$	4.09030e9	0.919450	0.459725	+	0.888061i	$0.347948\pi$
$572$	−2.30361e8	−0.0514662
$573$	−5.63970e9	−1.25232
$574$	4.33706e8	0.0957202
$575$	0	0
$576$	1.69397e9	0.369342
$577$	−5.60809e9	−1.21535	−0.607673	−	0.794187i	$-0.707897\pi$
$577$	−5.60809e9	−1.21535	−0.607673	+	0.794187i	$0.707897\pi$
$578$	−6.95381e9	−1.49787
$579$	−6.74764e9	−1.44470
$580$	0	0
$581$	−5.99121e8	−0.126735
$582$	−4.64129e9	−0.975907
$583$	−8.84968e8	−0.184964
$584$	5.11155e8	0.106196
$585$	0	0
$586$	3.99141e8	0.0819379
$587$	−2.90597e9	−0.593005	−0.296502	−	0.955032i	$-0.595820\pi$
$587$	−2.90597e9	−0.593005	−0.296502	+	0.955032i	$0.595820\pi$
$588$	7.00247e8	0.142046
$589$	−5.44991e9	−1.09897
$590$	0	0
$591$	4.64690e9	0.925992
$592$	−2.18224e9	−0.432292
$593$	−4.70019e9	−0.925601	−0.462800	−	0.886463i	$-0.653155\pi$
$593$	−4.70019e9	−0.925601	−0.462800	+	0.886463i	$0.653155\pi$
$594$	6.89236e9	1.34932
$595$	0	0
$596$	1.48147e9	0.286635
$597$	4.20973e9	0.809737
$598$	−5.59823e8	−0.107053
$599$	2.54569e9	0.483962	0.241981	−	0.970281i	$-0.422203\pi$
$599$	2.54569e9	0.483962	0.241981	+	0.970281i	$0.422203\pi$
$600$	0	0
$601$	−1.91667e9	−0.360152	−0.180076	−	0.983653i	$-0.557634\pi$
$601$	−1.91667e9	−0.360152	−0.180076	+	0.983653i	$0.557634\pi$
$602$	1.43202e9	0.267524
$603$	2.54033e10	4.71823
$604$	−4.70409e9	−0.868654
$605$	0	0
$606$	−3.44699e9	−0.629196
$607$	8.45961e9	1.53529	0.767644	−	0.640877i	$-0.221429\pi$
$607$	8.45961e9	1.53529	0.767644	+	0.640877i	$0.221429\pi$
$608$	−6.62634e8	−0.119567
$609$	3.56596e9	0.639758
$610$	0	0
$611$	1.56090e9	0.276841
$612$	1.47937e10	2.60884
$613$	7.93248e9	1.39090	0.695452	−	0.718573i	$-0.255204\pi$
$613$	7.93248e9	1.39090	0.695452	+	0.718573i	$0.255204\pi$
$614$	3.82444e9	0.666773
$615$	0	0
$616$	−3.80560e8	−0.0655980
$617$	3.77436e9	0.646912	0.323456	−	0.946243i	$-0.395155\pi$
$617$	3.77436e9	0.646912	0.323456	+	0.946243i	$0.395155\pi$
$618$	−7.17581e9	−1.22296
$619$	3.48711e9	0.590947	0.295473	−	0.955351i	$-0.404523\pi$
$619$	3.48711e9	0.590947	0.295473	+	0.955351i	$0.404523\pi$
$620$	0	0
$621$	1.67498e10	2.80666
$622$	6.05349e9	1.00865
$623$	3.45666e9	0.572729
$624$	6.32721e8	0.104248
$625$	0	0
$626$	3.77495e9	0.615036
$627$	−4.07536e9	−0.660282
$628$	4.48592e9	0.722757
$629$	−1.90579e10	−3.05349
$630$	0	0
$631$	−3.44806e9	−0.546352	−0.273176	−	0.961964i	$-0.588074\pi$
$631$	−3.44806e9	−0.546352	−0.273176	+	0.961964i	$0.588074\pi$
$632$	1.08580e9	0.171097
$633$	1.58829e10	2.48895
$634$	−5.68953e9	−0.886674
$635$	0	0
$636$	2.43070e9	0.374655
$637$	1.95415e8	0.0299550
$638$	−1.93797e9	−0.295444
$639$	7.71573e9	1.16983
$640$	0	0
$641$	3.81588e9	0.572257	0.286129	−	0.958191i	$-0.407632\pi$
$641$	3.81588e9	0.572257	0.286129	+	0.958191i	$0.407632\pi$
$642$	4.22742e9	0.630525
$643$	−1.02692e10	−1.52335	−0.761674	−	0.647960i	$-0.775622\pi$
$643$	−1.02692e10	−1.52335	−0.761674	+	0.647960i	$0.775622\pi$
$644$	−9.24838e8	−0.136447
$645$	0	0
$646$	−5.78689e9	−0.844562
$647$	−4.08497e9	−0.592958	−0.296479	−	0.955039i	$-0.595813\pi$
$647$	−4.08497e9	−0.592958	−0.296479	+	0.955039i	$0.595813\pi$
$648$	−1.16951e10	−1.68847
$649$	1.13155e9	0.162486
$650$	0	0
$651$	8.59691e9	1.22126
$652$	1.84675e9	0.260941
$653$	−1.00155e10	−1.40759	−0.703793	−	0.710405i	$-0.748512\pi$
$653$	−1.00155e10	−1.40759	−0.703793	+	0.710405i	$0.748512\pi$
$654$	−1.80453e10	−2.52256
$655$	0	0
$656$	6.47397e8	0.0895380
$657$	−6.45134e9	−0.887505
$658$	2.57863e9	0.352856
$659$	−3.08151e8	−0.0419434	−0.0209717	−	0.999780i	$-0.506676\pi$
$659$	−3.08151e8	−0.0419434	−0.0209717	+	0.999780i	$0.506676\pi$
$660$	0	0
$661$	6.46819e9	0.871120	0.435560	−	0.900160i	$-0.356550\pi$
$661$	6.46819e9	0.871120	0.435560	+	0.900160i	$0.356550\pi$
$662$	−3.25835e9	−0.436510
$663$	5.52565e9	0.736353
$664$	−8.94314e8	−0.118550
$665$	0	0
$666$	2.75423e10	3.61277
$667$	−4.70967e9	−0.614540
$668$	−3.01292e9	−0.391084
$669$	−1.34204e10	−1.73290
$670$	0	0
$671$	−7.58623e8	−0.0969387
$672$	1.04527e9	0.132872
$673$	5.92891e9	0.749759	0.374880	−	0.927074i	$-0.377684\pi$
$673$	5.92891e9	0.749759	0.374880	+	0.927074i	$0.377684\pi$
$674$	7.34789e9	0.924385
$675$	0	0
$676$	−3.83933e9	−0.478016
$677$	−8.56663e9	−1.06108	−0.530542	−	0.847659i	$-0.678011\pi$
$677$	−8.56663e9	−1.06108	−0.530542	+	0.847659i	$0.678011\pi$
$678$	−2.62144e9	−0.323025
$679$	−2.13974e9	−0.262311
$680$	0	0
$681$	−1.41628e10	−1.71844
$682$	−4.67212e9	−0.563987
$683$	−1.14190e9	−0.137137	−0.0685685	−	0.997646i	$-0.521843\pi$
$683$	−1.14190e9	−0.137137	−0.0685685	+	0.997646i	$0.521843\pi$
$684$	8.36317e9	0.999251
$685$	0	0
$686$	3.22829e8	0.0381802
$687$	−8.66148e9	−1.01916
$688$	2.13760e9	0.250245
$689$	6.78326e8	0.0790080
$690$	0	0
$691$	2.27766e9	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$691$	2.27766e9	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$692$	−5.17828e9	−0.594037
$693$	4.80308e9	0.548218
$694$	−1.15041e10	−1.30645
$695$	0	0
$696$	5.32295e9	0.598439
$697$	5.65382e9	0.632452
$698$	6.16776e9	0.686489
$699$	−2.67355e9	−0.296086
$700$	0	0
$701$	−4.90493e9	−0.537799	−0.268899	−	0.963168i	$-0.586660\pi$
$701$	−4.90493e9	−0.537799	−0.268899	+	0.963168i	$0.586660\pi$
$702$	−5.28298e9	−0.576367
$703$	−1.07738e10	−1.16956
$704$	−5.68066e8	−0.0613613
$705$	0	0
$706$	7.70246e9	0.823784
$707$	−1.58914e9	−0.169119
$708$	−3.10797e9	−0.329124
$709$	3.69147e9	0.388990	0.194495	−	0.980904i	$-0.437693\pi$
$709$	3.69147e9	0.388990	0.194495	+	0.980904i	$0.437693\pi$
$710$	0	0
$711$	−1.37040e10	−1.42990
$712$	5.15980e9	0.535739
$713$	−1.13542e10	−1.17312
$714$	9.12847e9	0.938543
$715$	0	0
$716$	1.62659e9	0.165609
$717$	1.33759e10	1.35521
$718$	1.00920e10	1.01752
$719$	−1.38999e10	−1.39463	−0.697316	−	0.716764i	$-0.745623\pi$
$719$	−1.38999e10	−1.39463	−0.697316	+	0.716764i	$0.745623\pi$
$720$	0	0
$721$	−3.30820e9	−0.328714
$722$	3.87954e9	0.383619
$723$	3.88904e10	3.82700
$724$	1.25620e9	0.123019
$725$	0	0
$726$	1.10047e10	1.06733
$727$	1.38352e10	1.33542	0.667708	−	0.744424i	$-0.267276\pi$
$727$	1.38352e10	1.33542	0.667708	+	0.744424i	$0.267276\pi$
$728$	2.91698e8	0.0280204
$729$	6.67424e10	6.38051
$730$	0	0
$731$	1.86680e10	1.76761
$732$	2.08368e9	0.196355
$733$	−1.91632e10	−1.79723	−0.898617	−	0.438733i	$-0.855427\pi$
$733$	−1.91632e10	−1.79723	−0.898617	+	0.438733i	$0.855427\pi$
$734$	−1.26745e8	−0.0118303
$735$	0	0
$736$	−1.38052e9	−0.127635
$737$	−8.51886e9	−0.783872
$738$	−8.17086e9	−0.748291
$739$	−6.12743e9	−0.558500	−0.279250	−	0.960218i	$-0.590086\pi$
$739$	−6.12743e9	−0.558500	−0.279250	+	0.960218i	$0.590086\pi$
$740$	0	0
$741$	3.12375e9	0.282041
$742$	1.12061e9	0.100702
$743$	2.04245e9	0.182680	0.0913399	−	0.995820i	$-0.470885\pi$
$743$	2.04245e9	0.182680	0.0913399	+	0.995820i	$0.470885\pi$
$744$	1.28327e10	1.14239
$745$	0	0
$746$	−8.56914e8	−0.0755703
$747$	1.12872e10	0.990752
$748$	−4.96101e9	−0.433425
$749$	1.94893e9	0.169477
$750$	0	0
$751$	−1.16753e9	−0.100584	−0.0502919	−	0.998735i	$-0.516015\pi$
$751$	−1.16753e9	−0.100584	−0.0502919	+	0.998735i	$0.516015\pi$
$752$	3.84915e9	0.330067
$753$	8.50022e9	0.725517
$754$	1.48545e9	0.126200
$755$	0	0
$756$	−8.72757e9	−0.734627
$757$	2.27882e9	0.190930	0.0954649	−	0.995433i	$-0.469566\pi$
$757$	2.27882e9	0.190930	0.0954649	+	0.995433i	$0.469566\pi$
$758$	1.51144e10	1.26051
$759$	−8.49050e9	−0.704834
$760$	0	0
$761$	7.99008e9	0.657211	0.328605	−	0.944467i	$-0.393421\pi$
$761$	7.99008e9	0.657211	0.328605	+	0.944467i	$0.393421\pi$
$762$	1.34155e10	1.09841
$763$	−8.31926e9	−0.678030
$764$	−3.88108e9	−0.314866
$765$	0	0
$766$	−6.93282e9	−0.557326
$767$	−8.67328e8	−0.0694064
$768$	1.56028e9	0.124291
$769$	−1.95565e10	−1.55078	−0.775389	−	0.631484i	$-0.782446\pi$
$769$	−1.95565e10	−1.55078	−0.775389	+	0.631484i	$0.782446\pi$
$770$	0	0
$771$	1.94060e10	1.52492
$772$	−4.64353e9	−0.363235
$773$	7.59461e8	0.0591394	0.0295697	−	0.999563i	$-0.490586\pi$
$773$	7.59461e8	0.0591394	0.0295697	+	0.999563i	$0.490586\pi$
$774$	−2.69788e10	−2.09136
$775$	0	0
$776$	−3.19401e9	−0.245369
$777$	1.69950e10	1.29971
$778$	7.72754e9	0.588319
$779$	3.19621e9	0.242245
$780$	0	0
$781$	−2.58743e9	−0.194353
$782$	−1.20563e10	−0.901548
$783$	−4.44445e10	−3.30866
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	−1.18428e10	−0.869912
$787$	−5.82450e9	−0.425938	−0.212969	−	0.977059i	$-0.568313\pi$
$787$	−5.82450e9	−0.425938	−0.212969	+	0.977059i	$0.568313\pi$
$788$	3.19787e9	0.232819
$789$	6.79914e8	0.0492815
$790$	0	0
$791$	−1.20854e9	−0.0868247
$792$	7.16961e9	0.512811
$793$	5.81483e8	0.0414077
$794$	1.87961e9	0.133258
$795$	0	0
$796$	2.89702e9	0.203590
$797$	−1.11184e10	−0.777924	−0.388962	−	0.921254i	$-0.627166\pi$
$797$	−1.11184e10	−0.777924	−0.388962	+	0.921254i	$0.627166\pi$
$798$	5.16049e9	0.359485
$799$	3.36152e10	2.33143
$800$	0	0
$801$	−6.51224e10	−4.47730
$802$	1.54037e10	1.05442
$803$	2.16342e9	0.147447
$804$	2.33984e10	1.58778
$805$	0	0
$806$	3.58117e9	0.240909
$807$	−3.76263e10	−2.52020
$808$	−2.37212e9	−0.158197
$809$	1.98746e10	1.31971	0.659856	−	0.751392i	$-0.270617\pi$
$809$	1.98746e10	1.31971	0.659856	+	0.751392i	$0.270617\pi$
$810$	0	0
$811$	−1.10898e10	−0.730048	−0.365024	−	0.930998i	$-0.618939\pi$
$811$	−1.10898e10	−0.730048	−0.365024	+	0.930998i	$0.618939\pi$
$812$	2.45399e9	0.160852
$813$	1.22915e10	0.802211
$814$	−9.23617e9	−0.600215
$815$	0	0
$816$	1.36262e10	0.877927
$817$	1.05533e10	0.677037
$818$	8.39045e9	0.535980
$819$	−3.68155e9	−0.234173
$820$	0	0
$821$	−2.18555e10	−1.37835	−0.689176	−	0.724594i	$-0.742027\pi$
$821$	−2.18555e10	−1.37835	−0.689176	+	0.724594i	$0.742027\pi$
$822$	−1.83113e10	−1.14992
$823$	−1.94834e10	−1.21833	−0.609167	−	0.793042i	$-0.708496\pi$
$823$	−1.94834e10	−1.21833	−0.609167	+	0.793042i	$0.708496\pi$
$824$	−4.93819e9	−0.307484
$825$	0	0
$826$	−1.43284e9	−0.0884642
$827$	1.61558e10	0.993253	0.496627	−	0.867964i	$-0.334572\pi$
$827$	1.61558e10	0.993253	0.496627	+	0.867964i	$0.334572\pi$
$828$	1.74236e10	1.06668
$829$	2.59312e10	1.58082	0.790410	−	0.612578i	$-0.209868\pi$
$829$	2.59312e10	1.58082	0.790410	+	0.612578i	$0.209868\pi$
$830$	0	0
$831$	1.45775e10	0.881212
$832$	4.35421e8	0.0262107
$833$	4.20842e9	0.252268
$834$	−4.15409e10	−2.47967
$835$	0	0
$836$	−2.80455e9	−0.166013
$837$	−1.07148e11	−6.31604
$838$	1.56959e10	0.921363
$839$	2.43163e10	1.42145	0.710724	−	0.703471i	$-0.248368\pi$
$839$	2.43163e10	1.42145	0.710724	+	0.703471i	$0.248368\pi$
$840$	0	0
$841$	−4.75310e9	−0.275544
$842$	−1.75788e10	−1.01484
$843$	−7.86204e9	−0.452000
$844$	1.09302e10	0.625790
$845$	0	0
$846$	−4.85804e10	−2.75845
$847$	5.07341e9	0.286885
$848$	1.67274e9	0.0941983
$849$	−4.00474e10	−2.24594
$850$	0	0
$851$	−2.24458e10	−1.24848
$852$	7.10678e9	0.393672
$853$	1.56779e10	0.864902	0.432451	−	0.901658i	$-0.357649\pi$
$853$	1.56779e10	0.864902	0.432451	+	0.901658i	$0.357649\pi$
$854$	9.60620e8	0.0527775
$855$	0	0
$856$	2.90919e9	0.158531
$857$	8.89023e9	0.482481	0.241240	−	0.970465i	$-0.422446\pi$
$857$	8.89023e9	0.482481	0.241240	+	0.970465i	$0.422446\pi$
$858$	2.67794e9	0.144742
$859$	−1.92711e10	−1.03736	−0.518681	−	0.854968i	$-0.673577\pi$
$859$	−1.92711e10	−1.03736	−0.518681	+	0.854968i	$0.673577\pi$
$860$	0	0
$861$	−5.04183e9	−0.269201
$862$	−2.52448e8	−0.0134245
$863$	−3.01710e10	−1.59791	−0.798954	−	0.601392i	$-0.794613\pi$
$863$	−3.01710e10	−1.59791	−0.798954	+	0.601392i	$0.794613\pi$
$864$	−1.30277e10	−0.687181
$865$	0	0
$866$	−8.19578e9	−0.428822
$867$	8.08380e10	4.21259
$868$	5.91615e9	0.307058
$869$	4.59558e9	0.237559
$870$	0	0
$871$	6.52968e9	0.334833
$872$	−1.24183e10	−0.634239
$873$	4.03119e10	2.05061
$874$	−6.81562e9	−0.345315
$875$	0	0
$876$	−5.94218e9	−0.298663
$877$	1.62526e10	0.813625	0.406813	−	0.913512i	$-0.366640\pi$
$877$	1.62526e10	0.813625	0.406813	+	0.913512i	$0.366640\pi$
$878$	−2.29359e10	−1.14363
$879$	−4.64001e9	−0.230440
$880$	0	0
$881$	−2.39281e10	−1.17894	−0.589471	−	0.807790i	$-0.700664\pi$
$881$	−2.39281e10	−1.17894	−0.589471	+	0.807790i	$0.700664\pi$
$882$	−6.08198e9	−0.298473
$883$	−2.78433e10	−1.36100	−0.680500	−	0.732748i	$-0.738237\pi$
$883$	−2.78433e10	−1.36100	−0.680500	+	0.732748i	$0.738237\pi$
$884$	3.80260e9	0.185139
$885$	0	0
$886$	2.15141e10	1.03922
$887$	−1.37057e10	−0.659430	−0.329715	−	0.944080i	$-0.606953\pi$
$887$	−1.37057e10	−0.659430	−0.329715	+	0.944080i	$0.606953\pi$
$888$	2.53686e10	1.21577
$889$	6.18484e9	0.295238
$890$	0	0
$891$	−4.94988e10	−2.34435
$892$	−9.23555e9	−0.435698
$893$	1.90033e10	0.892994
$894$	−1.72220e10	−0.806126
$895$	0	0
$896$	7.19323e8	0.0334077
$897$	6.50795e9	0.301072
$898$	−1.41174e10	−0.650560
$899$	3.01276e10	1.38295
$900$	0	0
$901$	1.46083e10	0.665370
$902$	2.74006e9	0.124319
$903$	−1.66473e10	−0.752377
$904$	−1.80400e9	−0.0812170
$905$	0	0
$906$	5.46851e10	2.44298
$907$	−8.76888e9	−0.390228	−0.195114	−	0.980781i	$-0.562508\pi$
$907$	−8.76888e9	−0.390228	−0.195114	+	0.980781i	$0.562508\pi$
$908$	−9.74646e9	−0.432063
$909$	2.99388e10	1.32209
$910$	0	0
$911$	−4.31141e10	−1.88932	−0.944659	−	0.328055i	$-0.893607\pi$
$911$	−4.31141e10	−1.88932	−0.944659	+	0.328055i	$0.893607\pi$
$912$	7.70313e9	0.336268
$913$	−3.78512e9	−0.164601
$914$	−2.41401e10	−1.04575
$915$	0	0
$916$	−5.96059e9	−0.256245
$917$	−5.45978e9	−0.233820
$918$	−1.13773e11	−4.85390
$919$	5.24951e9	0.223108	0.111554	−	0.993758i	$-0.464417\pi$
$919$	5.24951e9	0.223108	0.111554	+	0.993758i	$0.464417\pi$
$920$	0	0
$921$	−4.44591e10	−1.87522
$922$	1.69920e10	0.713982
$923$	1.98326e9	0.0830183
$924$	4.42401e9	0.184486
$925$	0	0
$926$	5.25817e9	0.217619
$927$	6.23254e10	2.56972
$928$	3.66310e9	0.150463
$929$	−1.10864e10	−0.453666	−0.226833	−	0.973934i	$-0.572837\pi$
$929$	−1.10864e10	−0.453666	−0.226833	+	0.973934i	$0.572837\pi$
$930$	0	0
$931$	2.37910e9	0.0966248
$932$	−1.83986e9	−0.0744439
$933$	−7.03719e10	−2.83670
$934$	2.88413e10	1.15825
$935$	0	0
$936$	−5.49549e9	−0.219049
$937$	3.00244e9	0.119230	0.0596151	−	0.998221i	$-0.481013\pi$
$937$	3.00244e9	0.119230	0.0596151	+	0.998221i	$0.481013\pi$
$938$	1.07871e10	0.426773
$939$	−4.38837e10	−1.72971
$940$	0	0
$941$	−6.58676e9	−0.257696	−0.128848	−	0.991664i	$-0.541128\pi$
$941$	−6.58676e9	−0.257696	−0.128848	+	0.991664i	$0.541128\pi$
$942$	−5.21488e10	−2.03267
$943$	6.65890e9	0.258590
$944$	−2.13882e9	−0.0827507
$945$	0	0
$946$	9.04721e9	0.347453
$947$	−1.86035e10	−0.711818	−0.355909	−	0.934521i	$-0.615829\pi$
$947$	−1.86035e10	−0.711818	−0.355909	+	0.934521i	$0.615829\pi$
$948$	−1.26225e10	−0.481188
$949$	−1.65826e9	−0.0629826
$950$	0	0
$951$	6.61407e10	2.49366
$952$	6.28196e9	0.235975
$953$	−5.05453e9	−0.189172	−0.0945858	−	0.995517i	$-0.530153\pi$
$953$	−5.05453e9	−0.189172	−0.0945858	+	0.995517i	$0.530153\pi$
$954$	−2.11118e10	−0.787239
$955$	0	0
$956$	9.20494e9	0.340736
$957$	2.25289e10	0.830901
$958$	−3.62063e8	−0.0133047
$959$	−8.44189e9	−0.309083
$960$	0	0
$961$	4.51198e10	1.63997
$962$	7.07950e9	0.256383
$963$	−3.67172e10	−1.32488
$964$	2.67633e10	0.962210
$965$	0	0
$966$	1.07512e10	0.383741
$967$	−4.72101e9	−0.167897	−0.0839484	−	0.996470i	$-0.526753\pi$
$967$	−4.72101e9	−0.167897	−0.0839484	+	0.996470i	$0.526753\pi$
$968$	7.57314e9	0.268356
$969$	6.72726e10	2.37523
$970$	0	0
$971$	−5.91365e9	−0.207295	−0.103647	−	0.994614i	$-0.533051\pi$
$971$	−5.91365e9	−0.207295	−0.103647	+	0.994614i	$0.533051\pi$
$972$	8.03083e10	2.80497
$973$	−1.91512e10	−0.666503
$974$	4.83341e9	0.167609
$975$	0	0
$976$	1.43393e9	0.0493688
$977$	8.41284e8	0.0288610	0.0144305	−	0.999896i	$-0.495406\pi$
$977$	8.41284e8	0.0288610	0.0144305	+	0.999896i	$0.495406\pi$
$978$	−2.14685e10	−0.733865
$979$	2.18385e10	0.743845
$980$	0	0
$981$	1.56732e11	5.30049
$982$	6.91431e9	0.233001
$983$	−2.43741e10	−0.818448	−0.409224	−	0.912434i	$-0.634201\pi$
$983$	−2.43741e10	−0.818448	−0.409224	+	0.912434i	$0.634201\pi$
$984$	−7.52599e9	−0.251815
$985$	0	0
$986$	3.19904e10	1.06280
$987$	−2.99765e10	−0.992365
$988$	2.14968e9	0.0709128
$989$	2.19866e10	0.722720
$990$	0	0
$991$	−2.89614e10	−0.945281	−0.472641	−	0.881255i	$-0.656699\pi$
$991$	−2.89614e10	−0.945281	−0.472641	+	0.881255i	$0.656699\pi$
$992$	8.83111e9	0.287226
$993$	3.78783e10	1.22763
$994$	3.27638e9	0.105814
$995$	0	0
$996$	1.03964e10	0.333408
$997$	3.36073e10	1.07399	0.536996	−	0.843585i	$-0.319559\pi$
$997$	3.36073e10	1.07399	0.536996	+	0.843585i	$0.319559\pi$
$998$	2.54520e10	0.810523
$999$	−2.11818e11	−6.72176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.e.1.1		1
5.2	odd	4		350.8.c.a.99.1		2
5.3	odd	4		350.8.c.a.99.2		2
5.4	even	2		70.8.a.a.1.1	✓	1
20.19	odd	2		560.8.a.b.1.1		1
35.34	odd	2		490.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.8.a.a.1.1	✓	1	5.4	even	2
350.8.a.e.1.1		1	1.1	even	1	trivial
350.8.c.a.99.1		2	5.2	odd	4
350.8.c.a.99.2		2	5.3	odd	4
490.8.a.e.1.1		1	35.34	odd	2
560.8.a.b.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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