LMFDB - Embedded newform 350.8.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,8,Mod(1,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [1,-8,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$109.334758919$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
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)

$f(q)$

$=$

\(q-8.00000 q^{2} +66.0000 q^{3} +64.0000 q^{4} -528.000 q^{6} +343.000 q^{7} -512.000 q^{8} +2169.00 q^{9} +40.0000 q^{11} +4224.00 q^{12} +4452.00 q^{13} -2744.00 q^{14} +4096.00 q^{16} -36502.0 q^{17} -17352.0 q^{18} -46222.0 q^{19} +22638.0 q^{21} -320.000 q^{22} +105200. q^{23} -33792.0 q^{24} -35616.0 q^{26} -1188.00 q^{27} +21952.0 q^{28} -126334. q^{29} -170964. q^{31} -32768.0 q^{32} +2640.00 q^{33} +292016. q^{34} +138816. q^{36} -20954.0 q^{37} +369776. q^{38} +293832. q^{39} +318486. q^{41} -181104. q^{42} -77744.0 q^{43} +2560.00 q^{44} -841600. q^{46} -703716. q^{47} +270336. q^{48} +117649. q^{49} -2.40913e6 q^{51} +284928. q^{52} -1.60328e6 q^{53} +9504.00 q^{54} -175616. q^{56} -3.05065e6 q^{57} +1.01067e6 q^{58} -1.17189e6 q^{59} -2.06887e6 q^{61} +1.36771e6 q^{62} +743967. q^{63} +262144. q^{64} -21120.0 q^{66} +994268. q^{67} -2.33613e6 q^{68} +6.94320e6 q^{69} +33280.0 q^{71} -1.11053e6 q^{72} +2.97145e6 q^{73} +167632. q^{74} -2.95821e6 q^{76} +13720.0 q^{77} -2.35066e6 q^{78} -2.37617e6 q^{79} -4.82201e6 q^{81} -2.54789e6 q^{82} +2.12236e6 q^{83} +1.44883e6 q^{84} +621952. q^{86} -8.33804e6 q^{87} -20480.0 q^{88} +6.92035e6 q^{89} +1.52704e6 q^{91} +6.73280e6 q^{92} -1.12836e7 q^{93} +5.62973e6 q^{94} -2.16269e6 q^{96} -4.95271e6 q^{97} -941192. q^{98} +86760.0 q^{99} +O(q^{100})\)

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
$3$	66.0000	1.41130	0.705650	−	0.708560i	$-0.250655\pi$
$3$	66.0000	1.41130	0.705650	+	0.708560i	$0.250655\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−528.000	−0.997940
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	2169.00	0.991770
$10$	0	0
$11$	40.0000	0.00906120	0.00453060	−	0.999990i	$-0.498558\pi$
$11$	40.0000	0.00906120	0.00453060	+	0.999990i	$0.498558\pi$
$12$	4224.00	0.705650
$13$	4452.00	0.562022	0.281011	−	0.959705i	$-0.409330\pi$
$13$	4452.00	0.562022	0.281011	+	0.959705i	$0.409330\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	−36502.0	−1.80196	−0.900981	−	0.433859i	$-0.857151\pi$
$17$	−36502.0	−1.80196	−0.900981	+	0.433859i	$0.857151\pi$
$18$	−17352.0	−0.701287
$19$	−46222.0	−1.54601	−0.773003	−	0.634402i	$-0.781246\pi$
$19$	−46222.0	−1.54601	−0.773003	+	0.634402i	$0.781246\pi$
$20$	0	0
$21$	22638.0	0.533422
$22$	−320.000	−0.00640723
$23$	105200.	1.80289	0.901443	−	0.432898i	$-0.142509\pi$
$23$	105200.	1.80289	0.901443	+	0.432898i	$0.142509\pi$
$24$	−33792.0	−0.498970
$25$	0	0
$26$	−35616.0	−0.397410
$27$	−1188.00	−0.0116156
$28$	21952.0	0.188982
$29$	−126334.	−0.961894	−0.480947	−	0.876750i	$-0.659707\pi$
$29$	−126334.	−0.961894	−0.480947	+	0.876750i	$0.659707\pi$
$30$	0	0
$31$	−170964.	−1.03072	−0.515358	−	0.856975i	$-0.672341\pi$
$31$	−170964.	−1.03072	−0.515358	+	0.856975i	$0.672341\pi$
$32$	−32768.0	−0.176777
$33$	2640.00	0.0127881
$34$	292016.	1.27418
$35$	0	0
$36$	138816.	0.495885
$37$	−20954.0	−0.0680081	−0.0340041	−	0.999422i	$-0.510826\pi$
$37$	−20954.0	−0.0680081	−0.0340041	+	0.999422i	$0.510826\pi$
$38$	369776.	1.09319
$39$	293832.	0.793182
$40$	0	0
$41$	318486.	0.721684	0.360842	−	0.932627i	$-0.382489\pi$
$41$	318486.	0.721684	0.360842	+	0.932627i	$0.382489\pi$
$42$	−181104.	−0.377186
$43$	−77744.0	−0.149117	−0.0745585	−	0.997217i	$-0.523755\pi$
$43$	−77744.0	−0.149117	−0.0745585	+	0.997217i	$0.523755\pi$
$44$	2560.00	0.00453060
$45$	0	0
$46$	−841600.	−1.27483
$47$	−703716.	−0.988678	−0.494339	−	0.869269i	$-0.664590\pi$
$47$	−703716.	−0.988678	−0.494339	+	0.869269i	$0.664590\pi$
$48$	270336.	0.352825
$49$	117649.	0.142857
$50$	0	0
$51$	−2.40913e6	−2.54311
$52$	284928.	0.281011
$53$	−1.60328e6	−1.47926	−0.739628	−	0.673016i	$-0.764998\pi$
$53$	−1.60328e6	−1.47926	−0.739628	+	0.673016i	$0.764998\pi$
$54$	9504.00	0.00821350
$55$	0	0
$56$	−175616.	−0.133631
$57$	−3.05065e6	−2.18188
$58$	1.01067e6	0.680162
$59$	−1.17189e6	−0.742859	−0.371429	−	0.928461i	$-0.621132\pi$
$59$	−1.17189e6	−0.742859	−0.371429	+	0.928461i	$0.621132\pi$
$60$	0	0
$61$	−2.06887e6	−1.16702	−0.583511	−	0.812105i	$-0.698322\pi$
$61$	−2.06887e6	−1.16702	−0.583511	+	0.812105i	$0.698322\pi$
$62$	1.36771e6	0.728826
$63$	743967.	0.374854
$64$	262144.	0.125000
$65$	0	0
$66$	−21120.0	−0.00904253
$67$	994268.	0.403870	0.201935	−	0.979399i	$-0.435277\pi$
$67$	994268.	0.403870	0.201935	+	0.979399i	$0.435277\pi$
$68$	−2.33613e6	−0.900981
$69$	6.94320e6	2.54441
$70$	0	0
$71$	33280.0	0.0110352	0.00551759	−	0.999985i	$-0.498244\pi$
$71$	33280.0	0.0110352	0.00551759	+	0.999985i	$0.498244\pi$
$72$	−1.11053e6	−0.350643
$73$	2.97145e6	0.894003	0.447002	−	0.894533i	$-0.352492\pi$
$73$	2.97145e6	0.894003	0.447002	+	0.894533i	$0.352492\pi$
$74$	167632.	0.0480890
$75$	0	0
$76$	−2.95821e6	−0.773003
$77$	13720.0	0.00342481
$78$	−2.35066e6	−0.560865
$79$	−2.37617e6	−0.542228	−0.271114	−	0.962547i	$-0.587392\pi$
$79$	−2.37617e6	−0.542228	−0.271114	+	0.962547i	$0.587392\pi$
$80$	0	0
$81$	−4.82201e6	−1.00816
$82$	−2.54789e6	−0.510307
$83$	2.12236e6	0.407423	0.203711	−	0.979031i	$-0.434700\pi$
$83$	2.12236e6	0.407423	0.203711	+	0.979031i	$0.434700\pi$
$84$	1.44883e6	0.266711
$85$	0	0
$86$	621952.	0.105442
$87$	−8.33804e6	−1.35752
$88$	−20480.0	−0.00320362
$89$	6.92035e6	1.04055	0.520275	−	0.853999i	$-0.325830\pi$
$89$	6.92035e6	1.04055	0.520275	+	0.853999i	$0.325830\pi$
$90$	0	0
$91$	1.52704e6	0.212424
$92$	6.73280e6	0.901443
$93$	−1.12836e7	−1.45465
$94$	5.62973e6	0.699101
$95$	0	0
$96$	−2.16269e6	−0.249485
$97$	−4.95271e6	−0.550988	−0.275494	−	0.961303i	$-0.588841\pi$
$97$	−4.95271e6	−0.550988	−0.275494	+	0.961303i	$0.588841\pi$
$98$	−941192.	−0.101015
$99$	86760.0	0.00898662
$100$	0	0
$101$	3.23000e6	0.311945	0.155972	−	0.987761i	$-0.450149\pi$
$101$	3.23000e6	0.311945	0.155972	+	0.987761i	$0.450149\pi$
$102$	1.92731e7	1.79825
$103$	1.79909e6	0.162227	0.0811135	−	0.996705i	$-0.474152\pi$
$103$	1.79909e6	0.162227	0.0811135	+	0.996705i	$0.474152\pi$
$104$	−2.27942e6	−0.198705
$105$	0	0
$106$	1.28262e7	1.04599
$107$	1.56429e7	1.23445	0.617225	−	0.786787i	$-0.288257\pi$
$107$	1.56429e7	1.23445	0.617225	+	0.786787i	$0.288257\pi$
$108$	−76032.0	−0.00580782
$109$	−6.31890e6	−0.467357	−0.233679	−	0.972314i	$-0.575076\pi$
$109$	−6.31890e6	−0.467357	−0.233679	+	0.972314i	$0.575076\pi$
$110$	0	0
$111$	−1.38296e6	−0.0959799
$112$	1.40493e6	0.0944911
$113$	1.02288e7	0.666881	0.333441	−	0.942771i	$-0.391790\pi$
$113$	1.02288e7	0.666881	0.333441	+	0.942771i	$0.391790\pi$
$114$	2.44052e7	1.54282
$115$	0	0
$116$	−8.08538e6	−0.480947
$117$	9.65639e6	0.557396
$118$	9.37515e6	0.525281
$119$	−1.25202e7	−0.681077
$120$	0	0
$121$	−1.94856e7	−0.999918
$122$	1.65510e7	0.825209
$123$	2.10201e7	1.01851
$124$	−1.09417e7	−0.515358
$125$	0	0
$126$	−5.95174e6	−0.265062
$127$	−6.00725e6	−0.260233	−0.130117	−	0.991499i	$-0.541535\pi$
$127$	−6.00725e6	−0.260233	−0.130117	+	0.991499i	$0.541535\pi$
$128$	−2.09715e6	−0.0883883
$129$	−5.13110e6	−0.210449
$130$	0	0
$131$	2.06396e7	0.802144	0.401072	−	0.916047i	$-0.368638\pi$
$131$	2.06396e7	0.802144	0.401072	+	0.916047i	$0.368638\pi$
$132$	168960.	0.00639404
$133$	−1.58541e7	−0.584335
$134$	−7.95414e6	−0.285579
$135$	0	0
$136$	1.86890e7	0.637089
$137$	−4.76199e6	−0.158222	−0.0791109	−	0.996866i	$-0.525208\pi$
$137$	−4.76199e6	−0.158222	−0.0791109	+	0.996866i	$0.525208\pi$
$138$	−5.55456e7	−1.79917
$139$	−5.05723e6	−0.159721	−0.0798604	−	0.996806i	$-0.525447\pi$
$139$	−5.05723e6	−0.159721	−0.0798604	+	0.996806i	$0.525447\pi$
$140$	0	0
$141$	−4.64453e7	−1.39532
$142$	−266240.	−0.00780304
$143$	178080.	0.00509259
$144$	8.88422e6	0.247942
$145$	0	0
$146$	−2.37716e7	−0.632156
$147$	7.76483e6	0.201614
$148$	−1.34106e6	−0.0340041
$149$	−2.72736e7	−0.675447	−0.337723	−	0.941245i	$-0.609657\pi$
$149$	−2.72736e7	−0.675447	−0.337723	+	0.941245i	$0.609657\pi$
$150$	0	0
$151$	6.48921e6	0.153381	0.0766906	−	0.997055i	$-0.475565\pi$
$151$	6.48921e6	0.153381	0.0766906	+	0.997055i	$0.475565\pi$
$152$	2.36657e7	0.546596
$153$	−7.91728e7	−1.78713
$154$	−109760.	−0.00242171
$155$	0	0
$156$	1.88052e7	0.396591
$157$	6.30810e7	1.30092	0.650459	−	0.759541i	$-0.274577\pi$
$157$	6.30810e7	1.30092	0.650459	+	0.759541i	$0.274577\pi$
$158$	1.90093e7	0.383413
$159$	−1.05816e8	−2.08767
$160$	0	0
$161$	3.60836e7	0.681427
$162$	3.85761e7	0.712879
$163$	−8.32271e7	−1.50525	−0.752624	−	0.658450i	$-0.771212\pi$
$163$	−8.32271e7	−1.50525	−0.752624	+	0.658450i	$0.771212\pi$
$164$	2.03831e7	0.360842
$165$	0	0
$166$	−1.69789e7	−0.288091
$167$	−3.06916e7	−0.509931	−0.254965	−	0.966950i	$-0.582064\pi$
$167$	−3.06916e7	−0.509931	−0.254965	+	0.966950i	$0.582064\pi$
$168$	−1.15907e7	−0.188593
$169$	−4.29282e7	−0.684131
$170$	0	0
$171$	−1.00256e8	−1.53328
$172$	−4.97562e6	−0.0745585
$173$	5.27338e7	0.774333	0.387167	−	0.922010i	$-0.373454\pi$
$173$	5.27338e7	0.774333	0.387167	+	0.922010i	$0.373454\pi$
$174$	6.67044e7	0.959913
$175$	0	0
$176$	163840.	0.00226530
$177$	−7.73450e7	−1.04840
$178$	−5.53628e7	−0.735780
$179$	8.42739e7	1.09827	0.549133	−	0.835735i	$-0.314958\pi$
$179$	8.42739e7	1.09827	0.549133	+	0.835735i	$0.314958\pi$
$180$	0	0
$181$	−1.03956e8	−1.30309	−0.651547	−	0.758608i	$-0.725880\pi$
$181$	−1.03956e8	−1.30309	−0.651547	+	0.758608i	$0.725880\pi$
$182$	−1.22163e7	−0.150207
$183$	−1.36546e8	−1.64702
$184$	−5.38624e7	−0.637417
$185$	0	0
$186$	9.02690e7	1.02859
$187$	−1.46008e6	−0.0163279
$188$	−4.50378e7	−0.494339
$189$	−407484.	−0.00439030
$190$	0	0
$191$	−1.24775e8	−1.29572	−0.647861	−	0.761759i	$-0.724336\pi$
$191$	−1.24775e8	−1.29572	−0.647861	+	0.761759i	$0.724336\pi$
$192$	1.73015e7	0.176413
$193$	−1.47589e8	−1.47776	−0.738878	−	0.673839i	$-0.764644\pi$
$193$	−1.47589e8	−1.47776	−0.738878	+	0.673839i	$0.764644\pi$
$194$	3.96217e7	0.389607
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	−1.55812e8	−1.45200	−0.726002	−	0.687692i	$-0.758624\pi$
$197$	−1.55812e8	−1.45200	−0.726002	+	0.687692i	$0.758624\pi$
$198$	−694080.	−0.00635450
$199$	−1.33193e7	−0.119810	−0.0599052	−	0.998204i	$-0.519080\pi$
$199$	−1.33193e7	−0.119810	−0.0599052	+	0.998204i	$0.519080\pi$
$200$	0	0
$201$	6.56217e7	0.569982
$202$	−2.58400e7	−0.220578
$203$	−4.33326e7	−0.363562
$204$	−1.54184e8	−1.27155
$205$	0	0
$206$	−1.43927e7	−0.114712
$207$	2.28179e8	1.78805
$208$	1.82354e7	0.140506
$209$	−1.84888e6	−0.0140087
$210$	0	0
$211$	−2.04940e8	−1.50189	−0.750945	−	0.660365i	$-0.770402\pi$
$211$	−2.04940e8	−1.50189	−0.750945	+	0.660365i	$0.770402\pi$
$212$	−1.02610e8	−0.739628
$213$	2.19648e6	0.0155739
$214$	−1.25143e8	−0.872888
$215$	0	0
$216$	608256.	0.00410675
$217$	−5.86407e7	−0.389574
$218$	5.05512e7	0.330471
$219$	1.96116e8	1.26171
$220$	0	0
$221$	−1.62507e8	−1.01274
$222$	1.10637e7	0.0678681
$223$	6.84858e7	0.413555	0.206778	−	0.978388i	$-0.433702\pi$
$223$	6.84858e7	0.413555	0.206778	+	0.978388i	$0.433702\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	−8.18301e7	−0.471556
$227$	1.93627e7	0.109869	0.0549344	−	0.998490i	$-0.482505\pi$
$227$	1.93627e7	0.109869	0.0549344	+	0.998490i	$0.482505\pi$
$228$	−1.95242e8	−1.09094
$229$	3.08157e8	1.69569	0.847847	−	0.530241i	$-0.177898\pi$
$229$	3.08157e8	1.69569	0.847847	+	0.530241i	$0.177898\pi$
$230$	0	0
$231$	905520.	0.00483344
$232$	6.46830e7	0.340081
$233$	−3.55797e7	−0.184271	−0.0921355	−	0.995746i	$-0.529369\pi$
$233$	−3.55797e7	−0.184271	−0.0921355	+	0.995746i	$0.529369\pi$
$234$	−7.72511e7	−0.394139
$235$	0	0
$236$	−7.50012e7	−0.371429
$237$	−1.56827e8	−0.765247
$238$	1.00161e8	0.481594
$239$	−2.30056e8	−1.09004	−0.545018	−	0.838424i	$-0.683477\pi$
$239$	−2.30056e8	−1.09004	−0.545018	+	0.838424i	$0.683477\pi$
$240$	0	0
$241$	5.03495e6	0.0231705	0.0115853	−	0.999933i	$-0.496312\pi$
$241$	5.03495e6	0.0231705	0.0115853	+	0.999933i	$0.496312\pi$
$242$	1.55885e8	0.707049
$243$	−3.15655e8	−1.41121
$244$	−1.32408e8	−0.583511
$245$	0	0
$246$	−1.68161e8	−0.720197
$247$	−2.05780e8	−0.868890
$248$	8.75336e7	0.364413
$249$	1.40076e8	0.574996
$250$	0	0
$251$	−1.03283e8	−0.412258	−0.206129	−	0.978525i	$-0.566087\pi$
$251$	−1.03283e8	−0.412258	−0.206129	+	0.978525i	$0.566087\pi$
$252$	4.76139e7	0.187427
$253$	4.20800e6	0.0163363
$254$	4.80580e7	0.184013
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.32282e8	0.853592	0.426796	−	0.904348i	$-0.359642\pi$
$257$	2.32282e8	0.853592	0.426796	+	0.904348i	$0.359642\pi$
$258$	4.10488e7	0.148810
$259$	−7.18722e6	−0.0257047
$260$	0	0
$261$	−2.74018e8	−0.953977
$262$	−1.65117e8	−0.567201
$263$	−4.16749e8	−1.41263	−0.706317	−	0.707896i	$-0.749644\pi$
$263$	−4.16749e8	−1.41263	−0.706317	+	0.707896i	$0.749644\pi$
$264$	−1.35168e6	−0.00452127
$265$	0	0
$266$	1.26833e8	0.413187
$267$	4.56743e8	1.46853
$268$	6.36332e7	0.201935
$269$	−3.14679e8	−0.985676	−0.492838	−	0.870121i	$-0.664041\pi$
$269$	−3.14679e8	−0.985676	−0.492838	+	0.870121i	$0.664041\pi$
$270$	0	0
$271$	1.92137e8	0.586433	0.293216	−	0.956046i	$-0.405274\pi$
$271$	1.92137e8	0.586433	0.293216	+	0.956046i	$0.405274\pi$
$272$	−1.49512e8	−0.450490
$273$	1.00784e8	0.299795
$274$	3.80959e7	0.111880
$275$	0	0
$276$	4.44365e8	1.27221
$277$	4.40393e8	1.24498	0.622489	−	0.782629i	$-0.286122\pi$
$277$	4.40393e8	1.24498	0.622489	+	0.782629i	$0.286122\pi$
$278$	4.04579e7	0.112940
$279$	−3.70821e8	−1.02223
$280$	0	0
$281$	3.59235e8	0.965842	0.482921	−	0.875664i	$-0.339576\pi$
$281$	3.59235e8	0.965842	0.482921	+	0.875664i	$0.339576\pi$
$282$	3.71562e8	0.986642
$283$	8.11467e7	0.212823	0.106411	−	0.994322i	$-0.466064\pi$
$283$	8.11467e7	0.212823	0.106411	+	0.994322i	$0.466064\pi$
$284$	2.12992e6	0.00551759
$285$	0	0
$286$	−1.42464e6	−0.00360101
$287$	1.09241e8	0.272771
$288$	−7.10738e7	−0.175322
$289$	9.22057e8	2.24706
$290$	0	0
$291$	−3.26879e8	−0.777609
$292$	1.90173e8	0.447002
$293$	−2.53416e8	−0.588569	−0.294285	−	0.955718i	$-0.595081\pi$
$293$	−2.53416e8	−0.588569	−0.294285	+	0.955718i	$0.595081\pi$
$294$	−6.21187e7	−0.142563
$295$	0	0
$296$	1.07284e7	0.0240445
$297$	−47520.0	−0.000105252 0
$298$	2.18189e8	0.477613
$299$	4.68350e8	1.01326
$300$	0	0
$301$	−2.66662e7	−0.0563609
$302$	−5.19137e7	−0.108457
$303$	2.13180e8	0.440248
$304$	−1.89325e8	−0.386501
$305$	0	0
$306$	6.33383e8	1.26369
$307$	−8.72706e8	−1.72141	−0.860703	−	0.509107i	$-0.829976\pi$
$307$	−8.72706e8	−1.72141	−0.860703	+	0.509107i	$0.829976\pi$
$308$	878080.	0.00171241
$309$	1.18740e8	0.228951
$310$	0	0
$311$	−6.71611e8	−1.26607	−0.633033	−	0.774124i	$-0.718190\pi$
$311$	−6.71611e8	−1.26607	−0.633033	+	0.774124i	$0.718190\pi$
$312$	−1.50442e8	−0.280432
$313$	1.92216e8	0.354312	0.177156	−	0.984183i	$-0.443310\pi$
$313$	1.92216e8	0.354312	0.177156	+	0.984183i	$0.443310\pi$
$314$	−5.04648e8	−0.919888
$315$	0	0
$316$	−1.52075e8	−0.271114
$317$	1.33837e8	0.235977	0.117988	−	0.993015i	$-0.462355\pi$
$317$	1.33837e8	0.235977	0.117988	+	0.993015i	$0.462355\pi$
$318$	8.46531e8	1.47621
$319$	−5.05336e6	−0.00871591
$320$	0	0
$321$	1.03243e9	1.74218
$322$	−2.88669e8	−0.481842
$323$	1.68720e9	2.78584
$324$	−3.08609e8	−0.504081
$325$	0	0
$326$	6.65817e8	1.06437
$327$	−4.17048e8	−0.659581
$328$	−1.63065e8	−0.255154
$329$	−2.41375e8	−0.373685
$330$	0	0
$331$	4.25298e8	0.644608	0.322304	−	0.946636i	$-0.395543\pi$
$331$	4.25298e8	0.644608	0.322304	+	0.946636i	$0.395543\pi$
$332$	1.35831e8	0.203711
$333$	−4.54492e7	−0.0674484
$334$	2.45532e8	0.360576
$335$	0	0
$336$	9.27252e7	0.133355
$337$	−1.07703e9	−1.53293	−0.766463	−	0.642288i	$-0.777985\pi$
$337$	−1.07703e9	−1.53293	−0.766463	+	0.642288i	$0.777985\pi$
$338$	3.43426e8	0.483754
$339$	6.75098e8	0.941170
$340$	0	0
$341$	−6.83856e6	−0.00933952
$342$	8.02044e8	1.08419
$343$	4.03536e7	0.0539949
$344$	3.98049e7	0.0527208
$345$	0	0
$346$	−4.21871e8	−0.547536
$347$	7.23764e8	0.929916	0.464958	−	0.885333i	$-0.346069\pi$
$347$	7.23764e8	0.929916	0.464958	+	0.885333i	$0.346069\pi$
$348$	−5.33635e8	−0.678761
$349$	−4.48132e8	−0.564310	−0.282155	−	0.959369i	$-0.591049\pi$
$349$	−4.48132e8	−0.564310	−0.282155	+	0.959369i	$0.591049\pi$
$350$	0	0
$351$	−5.28898e6	−0.00652825
$352$	−1.31072e6	−0.00160181
$353$	1.49946e9	1.81435	0.907177	−	0.420749i	$-0.138233\pi$
$353$	1.49946e9	1.81435	0.907177	+	0.420749i	$0.138233\pi$
$354$	6.18760e8	0.741329
$355$	0	0
$356$	4.42902e8	0.520275
$357$	−8.26332e8	−0.961205
$358$	−6.74191e8	−0.776591
$359$	−2.56890e8	−0.293033	−0.146516	−	0.989208i	$-0.546806\pi$
$359$	−2.56890e8	−0.293033	−0.146516	+	0.989208i	$0.546806\pi$
$360$	0	0
$361$	1.24260e9	1.39013
$362$	8.31650e8	0.921427
$363$	−1.28605e9	−1.41118
$364$	9.77303e7	0.106212
$365$	0	0
$366$	1.09236e9	1.16462
$367$	−6.50424e8	−0.686856	−0.343428	−	0.939179i	$-0.611588\pi$
$367$	−6.50424e8	−0.686856	−0.343428	+	0.939179i	$0.611588\pi$
$368$	4.30899e8	0.450722
$369$	6.90796e8	0.715744
$370$	0	0
$371$	−5.49924e8	−0.559106
$372$	−7.22152e8	−0.727325
$373$	4.66127e8	0.465076	0.232538	−	0.972587i	$-0.425297\pi$
$373$	4.66127e8	0.465076	0.232538	+	0.972587i	$0.425297\pi$
$374$	1.16806e7	0.0115456
$375$	0	0
$376$	3.60303e8	0.349551
$377$	−5.62439e8	−0.540606
$378$	3.25987e6	0.00310441
$379$	2.85860e8	0.269721	0.134861	−	0.990865i	$-0.456941\pi$
$379$	2.85860e8	0.269721	0.134861	+	0.990865i	$0.456941\pi$
$380$	0	0
$381$	−3.96478e8	−0.367267
$382$	9.98202e8	0.916213
$383$	1.65075e9	1.50136	0.750681	−	0.660665i	$-0.229725\pi$
$383$	1.65075e9	1.50136	0.750681	+	0.660665i	$0.229725\pi$
$384$	−1.38412e8	−0.124743
$385$	0	0
$386$	1.18071e9	1.04493
$387$	−1.68627e8	−0.147890
$388$	−3.16973e8	−0.275494
$389$	1.51304e9	1.30325	0.651624	−	0.758542i	$-0.274088\pi$
$389$	1.51304e9	1.30325	0.651624	+	0.758542i	$0.274088\pi$
$390$	0	0
$391$	−3.84001e9	−3.24873
$392$	−6.02363e7	−0.0505076
$393$	1.36221e9	1.13207
$394$	1.24649e9	1.02672
$395$	0	0
$396$	5.55264e6	0.00449331
$397$	8.63794e8	0.692857	0.346428	−	0.938076i	$-0.387394\pi$
$397$	8.63794e8	0.692857	0.346428	+	0.938076i	$0.387394\pi$
$398$	1.06554e8	0.0847187
$399$	−1.04637e9	−0.824673
$400$	0	0
$401$	1.14042e8	0.0883199	0.0441599	−	0.999024i	$-0.485939\pi$
$401$	1.14042e8	0.0883199	0.0441599	+	0.999024i	$0.485939\pi$
$402$	−5.24974e8	−0.403038
$403$	−7.61132e8	−0.579285
$404$	2.06720e8	0.155972
$405$	0	0
$406$	3.46660e8	0.257077
$407$	−838160.	−0.000616235 0
$408$	1.23348e9	0.899125
$409$	−1.18328e9	−0.855176	−0.427588	−	0.903974i	$-0.640637\pi$
$409$	−1.18328e9	−0.855176	−0.427588	+	0.903974i	$0.640637\pi$
$410$	0	0
$411$	−3.14291e8	−0.223298
$412$	1.15142e8	0.0811135
$413$	−4.01960e8	−0.280774
$414$	−1.82543e9	−1.26434
$415$	0	0
$416$	−1.45883e8	−0.0993524
$417$	−3.33777e8	−0.225414
$418$	1.47910e7	0.00990562
$419$	2.27959e8	0.151394	0.0756970	−	0.997131i	$-0.475882\pi$
$419$	2.27959e8	0.151394	0.0756970	+	0.997131i	$0.475882\pi$
$420$	0	0
$421$	−3.90700e7	−0.0255186	−0.0127593	−	0.999919i	$-0.504062\pi$
$421$	−3.90700e7	−0.0255186	−0.0127593	+	0.999919i	$0.504062\pi$
$422$	1.63952e9	1.06200
$423$	−1.52636e9	−0.980541
$424$	8.20878e8	0.522996
$425$	0	0
$426$	−1.75718e7	−0.0110124
$427$	−7.09623e8	−0.441093
$428$	1.00114e9	0.617225
$429$	1.17533e7	0.00718718
$430$	0	0
$431$	−2.58620e9	−1.55594	−0.777968	−	0.628304i	$-0.783749\pi$
$431$	−2.58620e9	−1.55594	−0.777968	+	0.628304i	$0.783749\pi$
$432$	−4.86605e6	−0.00290391
$433$	1.78893e9	1.05897	0.529486	−	0.848318i	$-0.322385\pi$
$433$	1.78893e9	1.05897	0.529486	+	0.848318i	$0.322385\pi$
$434$	4.69125e8	0.275470
$435$	0	0
$436$	−4.04410e8	−0.233679
$437$	−4.86255e9	−2.78727
$438$	−1.56893e9	−0.892162
$439$	−4.58905e8	−0.258879	−0.129440	−	0.991587i	$-0.541318\pi$
$439$	−4.58905e8	−0.258879	−0.129440	+	0.991587i	$0.541318\pi$
$440$	0	0
$441$	2.55181e8	0.141681
$442$	1.30006e9	0.716117
$443$	−1.38459e9	−0.756672	−0.378336	−	0.925668i	$-0.623504\pi$
$443$	−1.38459e9	−0.756672	−0.378336	+	0.925668i	$0.623504\pi$
$444$	−8.85097e7	−0.0479900
$445$	0	0
$446$	−5.47887e8	−0.292428
$447$	−1.80006e9	−0.953259
$448$	8.99154e7	0.0472456
$449$	−2.73611e9	−1.42650	−0.713248	−	0.700911i	$-0.752777\pi$
$449$	−2.73611e9	−1.42650	−0.713248	+	0.700911i	$0.752777\pi$
$450$	0	0
$451$	1.27394e7	0.00653932
$452$	6.54641e8	0.333441
$453$	4.28288e8	0.216467
$454$	−1.54901e8	−0.0776890
$455$	0	0
$456$	1.56193e9	0.771411
$457$	−2.43053e9	−1.19123	−0.595614	−	0.803271i	$-0.703091\pi$
$457$	−2.43053e9	−1.19123	−0.595614	+	0.803271i	$0.703091\pi$
$458$	−2.46525e9	−1.19904
$459$	4.33644e7	0.0209309
$460$	0	0
$461$	3.94884e9	1.87723	0.938613	−	0.344971i	$-0.112111\pi$
$461$	3.94884e9	1.87723	0.938613	+	0.344971i	$0.112111\pi$
$462$	−7.24416e6	−0.00341776
$463$	−2.57453e9	−1.20549	−0.602746	−	0.797933i	$-0.705927\pi$
$463$	−2.57453e9	−1.20549	−0.602746	+	0.797933i	$0.705927\pi$
$464$	−5.17464e8	−0.240474
$465$	0	0
$466$	2.84638e8	0.130299
$467$	−2.98482e8	−0.135616	−0.0678078	−	0.997698i	$-0.521600\pi$
$467$	−2.98482e8	−0.135616	−0.0678078	+	0.997698i	$0.521600\pi$
$468$	6.18009e8	0.278698
$469$	3.41034e8	0.152649
$470$	0	0
$471$	4.16335e9	1.83599
$472$	6.00010e8	0.262640
$473$	−3.10976e6	−0.00135118
$474$	1.25462e9	0.541112
$475$	0	0
$476$	−8.01292e8	−0.340539
$477$	−3.47751e9	−1.46708
$478$	1.84045e9	0.770772
$479$	2.62000e9	1.08925	0.544625	−	0.838680i	$-0.316672\pi$
$479$	2.62000e9	1.08925	0.544625	+	0.838680i	$0.316672\pi$
$480$	0	0
$481$	−9.32872e7	−0.0382221
$482$	−4.02796e7	−0.0163840
$483$	2.38152e9	0.961698
$484$	−1.24708e9	−0.499959
$485$	0	0
$486$	2.52524e9	0.997873
$487$	4.16662e9	1.63468	0.817339	−	0.576157i	$-0.195448\pi$
$487$	4.16662e9	1.63468	0.817339	+	0.576157i	$0.195448\pi$
$488$	1.05926e9	0.412605
$489$	−5.49299e9	−2.12436
$490$	0	0
$491$	2.41300e9	0.919967	0.459983	−	0.887928i	$-0.347855\pi$
$491$	2.41300e9	0.919967	0.459983	+	0.887928i	$0.347855\pi$
$492$	1.34528e9	0.509256
$493$	4.61144e9	1.73330
$494$	1.64624e9	0.614398
$495$	0	0
$496$	−7.00269e8	−0.257679
$497$	1.14150e7	0.00417090
$498$	−1.12061e9	−0.406584
$499$	−1.04092e9	−0.375029	−0.187515	−	0.982262i	$-0.560043\pi$
$499$	−1.04092e9	−0.375029	−0.187515	+	0.982262i	$0.560043\pi$
$500$	0	0
$501$	−2.02564e9	−0.719666
$502$	8.26261e8	0.291510
$503$	1.17273e9	0.410876	0.205438	−	0.978670i	$-0.434138\pi$
$503$	1.17273e9	0.410876	0.205438	+	0.978670i	$0.434138\pi$
$504$	−3.80911e8	−0.132531
$505$	0	0
$506$	−3.36640e7	−0.0115515
$507$	−2.83326e9	−0.965515
$508$	−3.84464e8	−0.130117
$509$	−8.13818e7	−0.0273536	−0.0136768	−	0.999906i	$-0.504354\pi$
$509$	−8.13818e7	−0.0273536	−0.0136768	+	0.999906i	$0.504354\pi$
$510$	0	0
$511$	1.01921e9	0.337901
$512$	−1.34218e8	−0.0441942
$513$	5.49117e7	0.0179579
$514$	−1.85826e9	−0.603580
$515$	0	0
$516$	−3.28391e8	−0.105224
$517$	−2.81486e7	−0.00895861
$518$	5.74978e7	0.0181759
$519$	3.48043e9	1.09282
$520$	0	0
$521$	−2.77458e9	−0.859540	−0.429770	−	0.902939i	$-0.641405\pi$
$521$	−2.77458e9	−0.859540	−0.429770	+	0.902939i	$0.641405\pi$
$522$	2.19215e9	0.674564
$523$	−4.99213e9	−1.52591	−0.762957	−	0.646449i	$-0.776253\pi$
$523$	−4.99213e9	−1.52591	−0.762957	+	0.646449i	$0.776253\pi$
$524$	1.32094e9	0.401072
$525$	0	0
$526$	3.33399e9	0.998883
$527$	6.24053e9	1.85731
$528$	1.08134e7	0.00319702
$529$	7.66221e9	2.25040
$530$	0	0
$531$	−2.54184e9	−0.736745
$532$	−1.01467e9	−0.292168
$533$	1.41790e9	0.405602
$534$	−3.65394e9	−1.03841
$535$	0	0
$536$	−5.09065e8	−0.142790
$537$	5.56207e9	1.54998
$538$	2.51743e9	0.696978
$539$	4.70596e6	0.00129446
$540$	0	0
$541$	1.63095e9	0.442844	0.221422	−	0.975178i	$-0.428930\pi$
$541$	1.63095e9	0.442844	0.221422	+	0.975178i	$0.428930\pi$
$542$	−1.53710e9	−0.414671
$543$	−6.86112e9	−1.83906
$544$	1.19610e9	0.318545
$545$	0	0
$546$	−8.06275e8	−0.211987
$547$	−2.00950e9	−0.524967	−0.262484	−	0.964936i	$-0.584542\pi$
$547$	−2.00950e9	−0.524967	−0.262484	+	0.964936i	$0.584542\pi$
$548$	−3.04767e8	−0.0791109
$549$	−4.48738e9	−1.15742
$550$	0	0
$551$	5.83941e9	1.48709
$552$	−3.55492e9	−0.899586
$553$	−8.15026e8	−0.204943
$554$	−3.52315e9	−0.880332
$555$	0	0
$556$	−3.23663e8	−0.0798604
$557$	4.47959e9	1.09836	0.549180	−	0.835704i	$-0.314940\pi$
$557$	4.47959e9	1.09836	0.549180	+	0.835704i	$0.314940\pi$
$558$	2.96657e9	0.722827
$559$	−3.46116e8	−0.0838071
$560$	0	0
$561$	−9.63653e7	−0.0230436
$562$	−2.87388e9	−0.682954
$563$	1.50730e9	0.355976	0.177988	−	0.984033i	$-0.443041\pi$
$563$	1.50730e9	0.355976	0.177988	+	0.984033i	$0.443041\pi$
$564$	−2.97250e9	−0.697661
$565$	0	0
$566$	−6.49174e8	−0.150489
$567$	−1.65395e9	−0.381050
$568$	−1.70394e7	−0.00390152
$569$	2.33088e9	0.530428	0.265214	−	0.964190i	$-0.414557\pi$
$569$	2.33088e9	0.530428	0.265214	+	0.964190i	$0.414557\pi$
$570$	0	0
$571$	−2.91101e9	−0.654362	−0.327181	−	0.944962i	$-0.606099\pi$
$571$	−2.91101e9	−0.654362	−0.327181	+	0.944962i	$0.606099\pi$
$572$	1.13971e7	0.00254630
$573$	−8.23517e9	−1.82865
$574$	−8.73926e8	−0.192878
$575$	0	0
$576$	5.68590e8	0.123971
$577$	8.64805e9	1.87414	0.937072	−	0.349137i	$-0.113525\pi$
$577$	8.64805e9	1.87414	0.937072	+	0.349137i	$0.113525\pi$
$578$	−7.37646e9	−1.58891
$579$	−9.74086e9	−2.08556
$580$	0	0
$581$	7.27969e8	0.153991
$582$	2.61503e9	0.549853
$583$	−6.41311e7	−0.0134038
$584$	−1.52138e9	−0.316078
$585$	0	0
$586$	2.02733e9	0.416181
$587$	6.33513e9	1.29277	0.646387	−	0.763010i	$-0.276279\pi$
$587$	6.33513e9	1.29277	0.646387	+	0.763010i	$0.276279\pi$
$588$	4.96949e8	0.100807
$589$	7.90230e9	1.59349
$590$	0	0
$591$	−1.02836e10	−2.04921
$592$	−8.58276e7	−0.0170020
$593$	1.70162e9	0.335098	0.167549	−	0.985864i	$-0.446415\pi$
$593$	1.70162e9	0.335098	0.167549	+	0.985864i	$0.446415\pi$
$594$	380160.	7.44241e−5 0
$595$	0	0
$596$	−1.74551e9	−0.337723
$597$	−8.79071e8	−0.169088
$598$	−3.74680e9	−0.716484
$599$	3.01977e9	0.574090	0.287045	−	0.957917i	$-0.407327\pi$
$599$	3.01977e9	0.574090	0.287045	+	0.957917i	$0.407327\pi$
$600$	0	0
$601$	−5.92708e9	−1.11373	−0.556865	−	0.830603i	$-0.687996\pi$
$601$	−5.92708e9	−1.11373	−0.556865	+	0.830603i	$0.687996\pi$
$602$	2.13330e8	0.0398532
$603$	2.15657e9	0.400546
$604$	4.15309e8	0.0766906
$605$	0	0
$606$	−1.70544e9	−0.311302
$607$	1.45649e9	0.264331	0.132165	−	0.991228i	$-0.457807\pi$
$607$	1.45649e9	0.264331	0.132165	+	0.991228i	$0.457807\pi$
$608$	1.51460e9	0.273298
$609$	−2.85995e9	−0.513095
$610$	0	0
$611$	−3.13294e9	−0.555659
$612$	−5.06706e9	−0.893565
$613$	6.71607e9	1.17762	0.588808	−	0.808273i	$-0.299597\pi$
$613$	6.71607e9	1.17762	0.588808	+	0.808273i	$0.299597\pi$
$614$	6.98164e9	1.21722
$615$	0	0
$616$	−7.02464e6	−0.00121085
$617$	−7.02027e9	−1.20325	−0.601625	−	0.798779i	$-0.705480\pi$
$617$	−7.02027e9	−1.20325	−0.601625	+	0.798779i	$0.705480\pi$
$618$	−9.49921e8	−0.161893
$619$	5.14352e9	0.871652	0.435826	−	0.900031i	$-0.356456\pi$
$619$	5.14352e9	0.871652	0.435826	+	0.900031i	$0.356456\pi$
$620$	0	0
$621$	−1.24978e8	−0.0209417
$622$	5.37289e9	0.895245
$623$	2.37368e9	0.393291
$624$	1.20354e9	0.198296
$625$	0	0
$626$	−1.53773e9	−0.250536
$627$	−1.22026e8	−0.0197704
$628$	4.03719e9	0.650459
$629$	7.64863e8	0.122548
$630$	0	0
$631$	−4.41574e9	−0.699681	−0.349841	−	0.936809i	$-0.613764\pi$
$631$	−4.41574e9	−0.699681	−0.349841	+	0.936809i	$0.613764\pi$
$632$	1.21660e9	0.191707
$633$	−1.35260e10	−2.11962
$634$	−1.07070e9	−0.166861
$635$	0	0
$636$	−6.77225e9	−1.04384
$637$	5.23773e8	0.0802889
$638$	4.04269e7	0.00616308
$639$	7.21843e7	0.0109443
$640$	0	0
$641$	6.94176e8	0.104104	0.0520519	−	0.998644i	$-0.483424\pi$
$641$	6.94176e8	0.104104	0.0520519	+	0.998644i	$0.483424\pi$
$642$	−8.25944e9	−1.23191
$643$	−9.50809e9	−1.41044	−0.705220	−	0.708988i	$-0.749152\pi$
$643$	−9.50809e9	−1.41044	−0.705220	+	0.708988i	$0.749152\pi$
$644$	2.30935e9	0.340713
$645$	0	0
$646$	−1.34976e10	−1.96989
$647$	7.73215e9	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$647$	7.73215e9	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$648$	2.46887e9	0.356439
$649$	−4.68758e7	−0.00673119
$650$	0	0
$651$	−3.87028e9	−0.549806
$652$	−5.32654e9	−0.752624
$653$	5.06321e9	0.711590	0.355795	−	0.934564i	$-0.384210\pi$
$653$	5.06321e9	0.711590	0.355795	+	0.934564i	$0.384210\pi$
$654$	3.33638e9	0.466395
$655$	0	0
$656$	1.30452e9	0.180421
$657$	6.44508e9	0.886645
$658$	1.93100e9	0.264235
$659$	8.08113e9	1.09995	0.549975	−	0.835181i	$-0.314637\pi$
$659$	8.08113e9	1.09995	0.549975	+	0.835181i	$0.314637\pi$
$660$	0	0
$661$	6.30089e9	0.848588	0.424294	−	0.905524i	$-0.360522\pi$
$661$	6.30089e9	0.848588	0.424294	+	0.905524i	$0.360522\pi$
$662$	−3.40239e9	−0.455806
$663$	−1.07255e10	−1.42928
$664$	−1.08665e9	−0.144046
$665$	0	0
$666$	3.63594e8	0.0476932
$667$	−1.32903e10	−1.73419
$668$	−1.96426e9	−0.254965
$669$	4.52006e9	0.583651
$670$	0	0
$671$	−8.27549e7	−0.0105746
$672$	−7.41802e8	−0.0942965
$673$	9.62624e9	1.21732	0.608659	−	0.793432i	$-0.291708\pi$
$673$	9.62624e9	1.21732	0.608659	+	0.793432i	$0.291708\pi$
$674$	8.61620e9	1.08394
$675$	0	0
$676$	−2.74741e9	−0.342066
$677$	9.45429e9	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$677$	9.45429e9	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$678$	−5.40079e9	−0.665508
$679$	−1.69878e9	−0.208254
$680$	0	0
$681$	1.27794e9	0.155058
$682$	5.47085e7	0.00660403
$683$	−2.48879e8	−0.0298893	−0.0149447	−	0.999888i	$-0.504757\pi$
$683$	−2.48879e8	−0.0298893	−0.0149447	+	0.999888i	$0.504757\pi$
$684$	−6.41635e9	−0.766641
$685$	0	0
$686$	−3.22829e8	−0.0381802
$687$	2.03383e10	2.39313
$688$	−3.18439e8	−0.0372793
$689$	−7.13779e9	−0.831375
$690$	0	0
$691$	−3.46412e9	−0.399411	−0.199705	−	0.979856i	$-0.563998\pi$
$691$	−3.46412e9	−0.399411	−0.199705	+	0.979856i	$0.563998\pi$
$692$	3.37497e9	0.387167
$693$	2.97587e7	0.00339662
$694$	−5.79011e9	−0.657550
$695$	0	0
$696$	4.26908e9	0.479956
$697$	−1.16254e10	−1.30045
$698$	3.58506e9	0.399027
$699$	−2.34826e9	−0.260062
$700$	0	0
$701$	5.56322e9	0.609976	0.304988	−	0.952356i	$-0.401347\pi$
$701$	5.56322e9	0.609976	0.304988	+	0.952356i	$0.401347\pi$
$702$	4.23118e7	0.00461617
$703$	9.68536e8	0.105141
$704$	1.04858e7	0.00113265
$705$	0	0
$706$	−1.19956e10	−1.28294
$707$	1.10789e9	0.117904
$708$	−4.95008e9	−0.524199
$709$	8.23697e9	0.867971	0.433986	−	0.900920i	$-0.357107\pi$
$709$	8.23697e9	0.867971	0.433986	+	0.900920i	$0.357107\pi$
$710$	0	0
$711$	−5.15391e9	−0.537766
$712$	−3.54322e9	−0.367890
$713$	−1.79854e10	−1.85826
$714$	6.61066e9	0.679674
$715$	0	0
$716$	5.39353e9	0.549133
$717$	−1.51837e10	−1.53837
$718$	2.05512e9	0.207206
$719$	5.85212e9	0.587168	0.293584	−	0.955933i	$-0.405152\pi$
$719$	5.85212e9	0.587168	0.293584	+	0.955933i	$0.405152\pi$
$720$	0	0
$721$	6.17089e8	0.0613160
$722$	−9.94081e9	−0.982973
$723$	3.32307e8	0.0327005
$724$	−6.65320e9	−0.651547
$725$	0	0
$726$	1.02884e10	0.997858
$727$	−1.51706e10	−1.46431	−0.732154	−	0.681139i	$-0.761485\pi$
$727$	−1.51706e10	−1.46431	−0.732154	+	0.681139i	$0.761485\pi$
$728$	−7.81842e8	−0.0751034
$729$	−1.02875e10	−0.983472
$730$	0	0
$731$	2.83781e9	0.268703
$732$	−8.73892e9	−0.823510
$733$	1.55969e10	1.46277	0.731383	−	0.681967i	$-0.238875\pi$
$733$	1.55969e10	1.46277	0.731383	+	0.681967i	$0.238875\pi$
$734$	5.20339e9	0.485680
$735$	0	0
$736$	−3.44719e9	−0.318708
$737$	3.97707e7	0.00365955
$738$	−5.52637e9	−0.506107
$739$	−6.95573e9	−0.633997	−0.316998	−	0.948426i	$-0.602675\pi$
$739$	−6.95573e9	−0.633997	−0.316998	+	0.948426i	$0.602675\pi$
$740$	0	0
$741$	−1.35815e10	−1.22626
$742$	4.39939e9	0.395348
$743$	1.17803e10	1.05365	0.526824	−	0.849975i	$-0.323383\pi$
$743$	1.17803e10	1.05365	0.526824	+	0.849975i	$0.323383\pi$
$744$	5.77722e9	0.514296
$745$	0	0
$746$	−3.72902e9	−0.328858
$747$	4.60339e9	0.404070
$748$	−9.34451e7	−0.00816396
$749$	5.36551e9	0.466578
$750$	0	0
$751$	−6.96200e9	−0.599783	−0.299892	−	0.953973i	$-0.596951\pi$
$751$	−6.96200e9	−0.599783	−0.299892	+	0.953973i	$0.596951\pi$
$752$	−2.88242e9	−0.247170
$753$	−6.81665e9	−0.581820
$754$	4.49951e9	0.382266
$755$	0	0
$756$	−2.60790e7	−0.00219515
$757$	2.07114e10	1.73530	0.867648	−	0.497179i	$-0.165631\pi$
$757$	2.07114e10	1.73530	0.867648	+	0.497179i	$0.165631\pi$
$758$	−2.28688e9	−0.190722
$759$	2.77728e8	0.0230554
$760$	0	0
$761$	1.65392e10	1.36041	0.680204	−	0.733023i	$-0.261891\pi$
$761$	1.65392e10	1.36041	0.680204	+	0.733023i	$0.261891\pi$
$762$	3.17183e9	0.259697
$763$	−2.16738e9	−0.176644
$764$	−7.98562e9	−0.647861
$765$	0	0
$766$	−1.32060e10	−1.06162
$767$	−5.21727e9	−0.417503
$768$	1.10730e9	0.0882063
$769$	−2.33650e10	−1.85278	−0.926391	−	0.376562i	$-0.877106\pi$
$769$	−2.33650e10	−1.85278	−0.926391	+	0.376562i	$0.877106\pi$
$770$	0	0
$771$	1.53306e10	1.20467
$772$	−9.44568e9	−0.738878
$773$	−7.09263e9	−0.552305	−0.276152	−	0.961114i	$-0.589059\pi$
$773$	−7.09263e9	−0.552305	−0.276152	+	0.961114i	$0.589059\pi$
$774$	1.34901e9	0.104574
$775$	0	0
$776$	2.53579e9	0.194804
$777$	−4.74357e8	−0.0362770
$778$	−1.21043e10	−0.921536
$779$	−1.47211e10	−1.11573
$780$	0	0
$781$	1.33120e6	9.99919e−5 0
$782$	3.07201e10	2.29720
$783$	1.50085e8	0.0111730
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	−1.08977e10	−0.800491
$787$	1.23030e10	0.899703	0.449851	−	0.893103i	$-0.351477\pi$
$787$	1.23030e10	0.899703	0.449851	+	0.893103i	$0.351477\pi$
$788$	−9.97194e9	−0.726002
$789$	−2.75054e10	−1.99365
$790$	0	0
$791$	3.50847e9	0.252057
$792$	−4.44211e7	−0.00317725
$793$	−9.21062e9	−0.655892
$794$	−6.91036e9	−0.489924
$795$	0	0
$796$	−8.52433e8	−0.0599052
$797$	3.66650e9	0.256535	0.128268	−	0.991740i	$-0.459058\pi$
$797$	3.66650e9	0.256535	0.128268	+	0.991740i	$0.459058\pi$
$798$	8.37099e9	0.583132
$799$	2.56870e10	1.78156
$800$	0	0
$801$	1.50102e10	1.03199
$802$	−9.12334e8	−0.0624516
$803$	1.18858e8	0.00810074
$804$	4.19979e9	0.284991
$805$	0	0
$806$	6.08905e9	0.409616
$807$	−2.07688e10	−1.39109
$808$	−1.65376e9	−0.110289
$809$	−2.96609e10	−1.96954	−0.984770	−	0.173861i	$-0.944376\pi$
$809$	−2.96609e10	−1.96954	−0.984770	+	0.173861i	$0.944376\pi$
$810$	0	0
$811$	2.51278e10	1.65417	0.827087	−	0.562073i	$-0.189996\pi$
$811$	2.51278e10	1.65417	0.827087	+	0.562073i	$0.189996\pi$
$812$	−2.77328e9	−0.181781
$813$	1.26810e10	0.827633
$814$	6.70528e6	0.000435744 0
$815$	0	0
$816$	−9.86780e9	−0.635777
$817$	3.59348e9	0.230536
$818$	9.46623e9	0.604700
$819$	3.31214e9	0.210676
$820$	0	0
$821$	4.57772e9	0.288701	0.144350	−	0.989527i	$-0.453891\pi$
$821$	4.57772e9	0.288701	0.144350	+	0.989527i	$0.453891\pi$
$822$	2.51433e9	0.157896
$823$	−1.93133e9	−0.120769	−0.0603846	−	0.998175i	$-0.519233\pi$
$823$	−1.93133e9	−0.120769	−0.0603846	+	0.998175i	$0.519233\pi$
$824$	−9.21135e8	−0.0573559
$825$	0	0
$826$	3.21568e9	0.198537
$827$	−1.58094e10	−0.971958	−0.485979	−	0.873971i	$-0.661537\pi$
$827$	−1.58094e10	−0.971958	−0.485979	+	0.873971i	$0.661537\pi$
$828$	1.46034e10	0.894024
$829$	−2.46536e9	−0.150293	−0.0751465	−	0.997173i	$-0.523942\pi$
$829$	−2.46536e9	−0.150293	−0.0751465	+	0.997173i	$0.523942\pi$
$830$	0	0
$831$	2.90660e10	1.75704
$832$	1.16707e9	0.0702528
$833$	−4.29442e9	−0.257423
$834$	2.67022e9	0.159392
$835$	0	0
$836$	−1.18328e8	−0.00700433
$837$	2.03105e8	0.0119724
$838$	−1.82368e9	−0.107052
$839$	2.51861e10	1.47229	0.736147	−	0.676822i	$-0.236643\pi$
$839$	2.51861e10	1.47229	0.736147	+	0.676822i	$0.236643\pi$
$840$	0	0
$841$	−1.28960e9	−0.0747598
$842$	3.12560e8	0.0180444
$843$	2.37095e10	1.36309
$844$	−1.31162e10	−0.750945
$845$	0	0
$846$	1.22109e10	0.693347
$847$	−6.68355e9	−0.377933
$848$	−6.56703e9	−0.369814
$849$	5.35568e9	0.300357
$850$	0	0
$851$	−2.20436e9	−0.122611
$852$	1.40575e8	0.00778697
$853$	−1.07306e10	−0.591972	−0.295986	−	0.955192i	$-0.595648\pi$
$853$	−1.07306e10	−0.591972	−0.295986	+	0.955192i	$0.595648\pi$
$854$	5.67698e9	0.311900
$855$	0	0
$856$	−8.00915e9	−0.436444
$857$	2.79332e10	1.51596	0.757979	−	0.652279i	$-0.226187\pi$
$857$	2.79332e10	1.51596	0.757979	+	0.652279i	$0.226187\pi$
$858$	−9.40262e7	−0.00508210
$859$	−1.94983e10	−1.04959	−0.524795	−	0.851229i	$-0.675858\pi$
$859$	−1.94983e10	−1.04959	−0.524795	+	0.851229i	$0.675858\pi$
$860$	0	0
$861$	7.20989e9	0.384962
$862$	2.06896e10	1.10021
$863$	1.63551e10	0.866193	0.433096	−	0.901348i	$-0.357421\pi$
$863$	1.63551e10	0.866193	0.433096	+	0.901348i	$0.357421\pi$
$864$	3.89284e7	0.00205338
$865$	0	0
$866$	−1.43114e10	−0.748807
$867$	6.08558e10	3.17128
$868$	−3.75300e9	−0.194787
$869$	−9.50467e7	−0.00491324
$870$	0	0
$871$	4.42648e9	0.226984
$872$	3.23528e9	0.165236
$873$	−1.07424e10	−0.546453
$874$	3.89004e10	1.97090
$875$	0	0
$876$	1.25514e10	0.630854
$877$	−2.68874e10	−1.34601	−0.673007	−	0.739636i	$-0.734998\pi$
$877$	−2.68874e10	−1.34601	−0.673007	+	0.739636i	$0.734998\pi$
$878$	3.67124e9	0.183055
$879$	−1.67255e10	−0.830648
$880$	0	0
$881$	1.08918e10	0.536644	0.268322	−	0.963329i	$-0.413531\pi$
$881$	1.08918e10	0.536644	0.268322	+	0.963329i	$0.413531\pi$
$882$	−2.04145e9	−0.100184
$883$	3.99542e10	1.95299	0.976495	−	0.215542i	$-0.0691517\pi$
$883$	3.99542e10	1.95299	0.976495	+	0.215542i	$0.0691517\pi$
$884$	−1.04004e10	−0.506371
$885$	0	0
$886$	1.10767e10	0.535048
$887$	−1.30306e10	−0.626946	−0.313473	−	0.949597i	$-0.601493\pi$
$887$	−1.30306e10	−0.626946	−0.313473	+	0.949597i	$0.601493\pi$
$888$	7.08078e8	0.0339340
$889$	−2.06049e9	−0.0983589
$890$	0	0
$891$	−1.92880e8	−0.00913516
$892$	4.38309e9	0.206778
$893$	3.25272e10	1.52850
$894$	1.44005e10	0.674056
$895$	0	0
$896$	−7.19323e8	−0.0334077
$897$	3.09111e10	1.43002
$898$	2.18889e10	1.00869
$899$	2.15986e10	0.991439
$900$	0	0
$901$	5.85229e10	2.66556
$902$	−1.01916e8	−0.00462400
$903$	−1.75997e9	−0.0795422
$904$	−5.23713e9	−0.235778
$905$	0	0
$906$	−3.42630e9	−0.153065
$907$	−4.04015e10	−1.79792	−0.898962	−	0.438026i	$-0.855678\pi$
$907$	−4.04015e10	−1.79792	−0.898962	+	0.438026i	$0.855678\pi$
$908$	1.23921e9	0.0549344
$909$	7.00587e9	0.309377
$910$	0	0
$911$	1.98919e10	0.871690	0.435845	−	0.900022i	$-0.356450\pi$
$911$	1.98919e10	0.871690	0.435845	+	0.900022i	$0.356450\pi$
$912$	−1.24955e10	−0.545470
$913$	8.48943e7	0.00369174
$914$	1.94443e10	0.842325
$915$	0	0
$916$	1.97220e10	0.847847
$917$	7.07939e9	0.303182
$918$	−3.46915e8	−0.0148004
$919$	−4.10990e10	−1.74674	−0.873368	−	0.487061i	$-0.838069\pi$
$919$	−4.10990e10	−1.74674	−0.873368	+	0.487061i	$0.838069\pi$
$920$	0	0
$921$	−5.75986e10	−2.42942
$922$	−3.15908e10	−1.32740
$923$	1.48163e8	0.00620201
$924$	5.79533e7	0.00241672
$925$	0	0
$926$	2.05962e10	0.852412
$927$	3.90223e9	0.160892
$928$	4.13971e9	0.170040
$929$	−1.90374e10	−0.779027	−0.389513	−	0.921021i	$-0.627357\pi$
$929$	−1.90374e10	−0.779027	−0.389513	+	0.921021i	$0.627357\pi$
$930$	0	0
$931$	−5.43797e9	−0.220858
$932$	−2.27710e9	−0.0921355
$933$	−4.43264e10	−1.78680
$934$	2.38786e9	0.0958947
$935$	0	0
$936$	−4.94407e9	−0.197069
$937$	−3.93830e10	−1.56394	−0.781971	−	0.623315i	$-0.785786\pi$
$937$	−3.93830e10	−1.56394	−0.781971	+	0.623315i	$0.785786\pi$
$938$	−2.72827e9	−0.107939
$939$	1.26863e10	0.500040
$940$	0	0
$941$	−1.59186e10	−0.622791	−0.311395	−	0.950280i	$-0.600796\pi$
$941$	−1.59186e10	−0.622791	−0.311395	+	0.950280i	$0.600796\pi$
$942$	−3.33068e10	−1.29824
$943$	3.35047e10	1.30111
$944$	−4.80008e9	−0.185715
$945$	0	0
$946$	2.48781e7	0.000955428 0
$947$	−3.57237e9	−0.136688	−0.0683442	−	0.997662i	$-0.521772\pi$
$947$	−3.57237e9	−0.136688	−0.0683442	+	0.997662i	$0.521772\pi$
$948$	−1.00369e10	−0.382624
$949$	1.32289e10	0.502450
$950$	0	0
$951$	8.83326e9	0.333034
$952$	6.41034e9	0.240797
$953$	−4.05101e9	−0.151614	−0.0758068	−	0.997123i	$-0.524153\pi$
$953$	−4.05101e9	−0.151614	−0.0758068	+	0.997123i	$0.524153\pi$
$954$	2.78201e10	1.03738
$955$	0	0
$956$	−1.47236e10	−0.545018
$957$	−3.33522e8	−0.0123008
$958$	−2.09600e10	−0.770216
$959$	−1.63336e9	−0.0598022
$960$	0	0
$961$	1.71608e9	0.0623741
$962$	7.46298e8	0.0270271
$963$	3.39294e10	1.22429
$964$	3.22237e8	0.0115853
$965$	0	0
$966$	−1.90521e10	−0.680023
$967$	−2.55791e10	−0.909689	−0.454844	−	0.890571i	$-0.650305\pi$
$967$	−2.55791e10	−0.909689	−0.454844	+	0.890571i	$0.650305\pi$
$968$	9.97661e9	0.353524
$969$	1.11355e11	3.93166
$970$	0	0
$971$	4.10323e10	1.43833	0.719165	−	0.694840i	$-0.244525\pi$
$971$	4.10323e10	1.43833	0.719165	+	0.694840i	$0.244525\pi$
$972$	−2.02019e10	−0.705603
$973$	−1.73463e9	−0.0603688
$974$	−3.33330e10	−1.15589
$975$	0	0
$976$	−8.47410e9	−0.291756
$977$	−4.87277e10	−1.67165	−0.835824	−	0.548998i	$-0.815010\pi$
$977$	−4.87277e10	−1.67165	−0.835824	+	0.548998i	$0.815010\pi$
$978$	4.39439e10	1.50215
$979$	2.76814e8	0.00942863
$980$	0	0
$981$	−1.37057e10	−0.463511
$982$	−1.93040e10	−0.650515
$983$	6.94762e9	0.233291	0.116646	−	0.993174i	$-0.462786\pi$
$983$	6.94762e9	0.233291	0.116646	+	0.993174i	$0.462786\pi$
$984$	−1.07623e10	−0.360099
$985$	0	0
$986$	−3.68915e10	−1.22563
$987$	−1.59307e10	−0.527382
$988$	−1.31699e10	−0.434445
$989$	−8.17867e9	−0.268841
$990$	0	0
$991$	−1.83565e10	−0.599144	−0.299572	−	0.954074i	$-0.596844\pi$
$991$	−1.83565e10	−0.599144	−0.299572	+	0.954074i	$0.596844\pi$
$992$	5.60215e9	0.182206
$993$	2.80697e10	0.909735
$994$	−9.13203e7	−0.00294927
$995$	0	0
$996$	8.96484e9	0.287498
$997$	−3.44954e10	−1.10237	−0.551185	−	0.834383i	$-0.685824\pi$
$997$	−3.44954e10	−1.10237	−0.551185	+	0.834383i	$0.685824\pi$
$998$	8.32735e9	0.265186
$999$	2.48934e7	0.000789958 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.d.1.1		1
5.2	odd	4		350.8.c.b.99.1		2
5.3	odd	4		350.8.c.b.99.2		2
5.4	even	2		14.8.a.b.1.1	✓	1
15.14	odd	2		126.8.a.c.1.1		1
20.19	odd	2		112.8.a.d.1.1		1
35.4	even	6		98.8.c.b.79.1		2
35.9	even	6		98.8.c.b.67.1		2
35.19	odd	6		98.8.c.a.67.1		2
35.24	odd	6		98.8.c.a.79.1		2
35.34	odd	2		98.8.a.c.1.1		1
40.19	odd	2		448.8.a.b.1.1		1
40.29	even	2		448.8.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.b.1.1	✓	1	5.4	even	2
98.8.a.c.1.1		1	35.34	odd	2
98.8.c.a.67.1		2	35.19	odd	6
98.8.c.a.79.1		2	35.24	odd	6
98.8.c.b.67.1		2	35.9	even	6
98.8.c.b.79.1		2	35.4	even	6
112.8.a.d.1.1		1	20.19	odd	2
126.8.a.c.1.1		1	15.14	odd	2
350.8.a.d.1.1		1	1.1	even	1	trivial
350.8.c.b.99.1		2	5.2	odd	4
350.8.c.b.99.2		2	5.3	odd	4
448.8.a.b.1.1		1	40.19	odd	2
448.8.a.i.1.1		1	40.29	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 $\F_{q}$
Belyi maps

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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