LMFDB - Embedded newform 490.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,8,Mod(1,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, 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("490.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	490.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:	$153.068662487$
Analytic rank:	$1$
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	$\chi$	$=$	490.1

$q$ -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} +30.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +240.000 q^{6} +512.000 q^{8} -1287.00 q^{9} -1000.00 q^{10} +3.00000 q^{11} +1920.00 q^{12} +1745.00 q^{13} -3750.00 q^{15} +4096.00 q^{16} -3786.00 q^{17} -10296.0 q^{18} -1945.00 q^{19} -8000.00 q^{20} +24.0000 q^{22} +79551.0 q^{23} +15360.0 q^{24} +15625.0 q^{25} +13960.0 q^{26} -104220. q^{27} -94926.0 q^{29} -30000.0 q^{30} +127628. q^{31} +32768.0 q^{32} +90.0000 q^{33} -30288.0 q^{34} -82368.0 q^{36} -128257. q^{37} -15560.0 q^{38} +52350.0 q^{39} -64000.0 q^{40} -298077. q^{41} -875626. q^{43} +192.000 q^{44} +160875. q^{45} +636408. q^{46} +611559. q^{47} +122880. q^{48} +125000. q^{50} -113580. q^{51} +111680. q^{52} -259137. q^{53} -833760. q^{54} -375.000 q^{55} -58350.0 q^{57} -759408. q^{58} -2.87734e6 q^{59} -240000. q^{60} -148564. q^{61} +1.02102e6 q^{62} +262144. q^{64} -218125. q^{65} +720.000 q^{66} -1.79088e6 q^{67} -242304. q^{68} +2.38653e6 q^{69} -493236. q^{71} -658944. q^{72} -2.05805e6 q^{73} -1.02606e6 q^{74} +468750. q^{75} -124480. q^{76} +418800. q^{78} -5.86707e6 q^{79} -512000. q^{80} -311931. q^{81} -2.38462e6 q^{82} -921132. q^{83} +473250. q^{85} -7.00501e6 q^{86} -2.84778e6 q^{87} +1536.00 q^{88} -5.12308e6 q^{89} +1.28700e6 q^{90} +5.09126e6 q^{92} +3.82884e6 q^{93} +4.89247e6 q^{94} +243125. q^{95} +983040. q^{96} -5.87831e6 q^{97} -3861.00 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
$3$	30.0000	0.641500	0.320750	−	0.947164i	$-0.396065\pi$
$3$	30.0000	0.641500	0.320750	+	0.947164i	$0.396065\pi$
$4$	64.0000	0.500000
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	240.000	0.453609
$7$	0	0
$7$	0	0
$8$	512.000	0.353553
$9$	−1287.00	−0.588477
$10$	−1000.00	−0.316228
$11$	3.00000	0.000679590 0	0.000339795	−	1.00000i	$-0.499892\pi$
$11$	3.00000	0.000679590 0	0.000339795		1.00000i	$0.499892\pi$
$12$	1920.00	0.320750
$13$	1745.00	0.220289	0.110145	−	0.993916i	$-0.464869\pi$
$13$	1745.00	0.220289	0.110145	+	0.993916i	$0.464869\pi$
$14$	0	0
$15$	−3750.00	−0.286888
$16$	4096.00	0.250000
$17$	−3786.00	−0.186900	−0.0934500	−	0.995624i	$-0.529790\pi$
$17$	−3786.00	−0.186900	−0.0934500	+	0.995624i	$0.529790\pi$
$18$	−10296.0	−0.416116
$19$	−1945.00	−0.0650552	−0.0325276	−	0.999471i	$-0.510356\pi$
$19$	−1945.00	−0.0650552	−0.0325276	+	0.999471i	$0.510356\pi$
$20$	−8000.00	−0.223607
$21$	0	0
$22$	24.0000	0.000480543 0
$23$	79551.0	1.36332	0.681661	−	0.731668i	$-0.261258\pi$
$23$	79551.0	1.36332	0.681661	+	0.731668i	$0.261258\pi$
$24$	15360.0	0.226805
$25$	15625.0	0.200000
$26$	13960.0	0.155768
$27$	−104220.	−1.01901
$28$	0	0
$29$	−94926.0	−0.722757	−0.361378	−	0.932419i	$-0.617694\pi$
$29$	−94926.0	−0.722757	−0.361378	+	0.932419i	$0.617694\pi$
$30$	−30000.0	−0.202860
$31$	127628.	0.769449	0.384725	−	0.923031i	$-0.374296\pi$
$31$	127628.	0.769449	0.384725	+	0.923031i	$0.374296\pi$
$32$	32768.0	0.176777
$33$	90.0000	0.000435957 0
$34$	−30288.0	−0.132158
$35$	0	0
$36$	−82368.0	−0.294239
$37$	−128257.	−0.416270	−0.208135	−	0.978100i	$-0.566739\pi$
$37$	−128257.	−0.416270	−0.208135	+	0.978100i	$0.566739\pi$
$38$	−15560.0	−0.0460010
$39$	52350.0	0.141316
$40$	−64000.0	−0.158114
$41$	−298077.	−0.675437	−0.337719	−	0.941247i	$-0.609655\pi$
$41$	−298077.	−0.675437	−0.337719	+	0.941247i	$0.609655\pi$
$42$	0	0
$43$	−875626.	−1.67950	−0.839748	−	0.542976i	$-0.817297\pi$
$43$	−875626.	−1.67950	−0.839748	+	0.542976i	$0.817297\pi$
$44$	192.000	0.000339795 0
$45$	160875.	0.263175
$46$	636408.	0.964014
$47$	611559.	0.859203	0.429602	−	0.903019i	$-0.358654\pi$
$47$	611559.	0.859203	0.429602	+	0.903019i	$0.358654\pi$
$48$	122880.	0.160375
$49$	0	0
$50$	125000.	0.141421
$51$	−113580.	−0.119896
$52$	111680.	0.110145
$53$	−259137.	−0.239091	−0.119546	−	0.992829i	$-0.538144\pi$
$53$	−259137.	−0.239091	−0.119546	+	0.992829i	$0.538144\pi$
$54$	−833760.	−0.720548
$55$	−375.000	−0.000303922 0
$56$	0	0
$57$	−58350.0	−0.0417329
$58$	−759408.	−0.511066
$59$	−2.87734e6	−1.82393	−0.911966	−	0.410266i	$-0.865436\pi$
$59$	−2.87734e6	−1.82393	−0.911966	+	0.410266i	$0.865436\pi$
$60$	−240000.	−0.143444
$61$	−148564.	−0.0838029	−0.0419015	−	0.999122i	$-0.513342\pi$
$61$	−148564.	−0.0838029	−0.0419015	+	0.999122i	$0.513342\pi$
$62$	1.02102e6	0.544083
$63$	0	0
$64$	262144.	0.125000
$65$	−218125.	−0.0985164
$66$	720.000	0.000308268 0
$67$	−1.79088e6	−0.727454	−0.363727	−	0.931506i	$-0.618496\pi$
$67$	−1.79088e6	−0.727454	−0.363727	+	0.931506i	$0.618496\pi$
$68$	−242304.	−0.0934500
$69$	2.38653e6	0.874571
$70$	0	0
$71$	−493236.	−0.163550	−0.0817750	−	0.996651i	$-0.526059\pi$
$71$	−493236.	−0.163550	−0.0817750	+	0.996651i	$0.526059\pi$
$72$	−658944.	−0.208058
$73$	−2.05805e6	−0.619193	−0.309597	−	0.950868i	$-0.600194\pi$
$73$	−2.05805e6	−0.619193	−0.309597	+	0.950868i	$0.600194\pi$
$74$	−1.02606e6	−0.294347
$75$	468750.	0.128300
$76$	−124480.	−0.0325276
$77$	0	0
$78$	418800.	0.0999253
$79$	−5.86707e6	−1.33883	−0.669417	−	0.742887i	$-0.733456\pi$
$79$	−5.86707e6	−1.33883	−0.669417	+	0.742887i	$0.733456\pi$
$80$	−512000.	−0.111803
$81$	−311931.	−0.0652170
$82$	−2.38462e6	−0.477606
$83$	−921132.	−0.176827	−0.0884135	−	0.996084i	$-0.528180\pi$
$83$	−921132.	−0.176827	−0.0884135	+	0.996084i	$0.528180\pi$
$84$	0	0
$85$	473250.	0.0835842
$86$	−7.00501e6	−1.18758
$87$	−2.84778e6	−0.463649
$88$	1536.00	0.000240271 0
$89$	−5.12308e6	−0.770311	−0.385156	−	0.922852i	$-0.625852\pi$
$89$	−5.12308e6	−0.770311	−0.385156	+	0.922852i	$0.625852\pi$
$90$	1.28700e6	0.186093
$91$	0	0
$92$	5.09126e6	0.681661
$93$	3.82884e6	0.493602
$94$	4.89247e6	0.607548
$95$	243125.	0.0290936
$96$	983040.	0.113402
$97$	−5.87831e6	−0.653960	−0.326980	−	0.945031i	$-0.606031\pi$
$97$	−5.87831e6	−0.653960	−0.326980	+	0.945031i	$0.606031\pi$
$98$	0	0
$99$	−3861.00	−0.000399923 0
$100$	1.00000e6	0.100000
$101$	2.06994e6	0.199909	0.0999546	−	0.994992i	$-0.468130\pi$
$101$	2.06994e6	0.199909	0.0999546	+	0.994992i	$0.468130\pi$
$102$	−908640.	−0.0847796
$103$	1.88625e7	1.70086	0.850430	−	0.526088i	$-0.176342\pi$
$103$	1.88625e7	1.70086	0.850430	+	0.526088i	$0.176342\pi$
$104$	893440.	0.0778841
$105$	0	0
$106$	−2.07310e6	−0.169063
$107$	−7.91987e6	−0.624992	−0.312496	−	0.949919i	$-0.601165\pi$
$107$	−7.91987e6	−0.624992	−0.312496	+	0.949919i	$0.601165\pi$
$108$	−6.67008e6	−0.509504
$109$	−463012.	−0.0342452	−0.0171226	−	0.999853i	$-0.505451\pi$
$109$	−463012.	−0.0342452	−0.0171226	+	0.999853i	$0.505451\pi$
$110$	−3000.00	−0.000214905 0
$111$	−3.84771e6	−0.267037
$112$	0	0
$113$	−9.89485e6	−0.645111	−0.322556	−	0.946550i	$-0.604542\pi$
$113$	−9.89485e6	−0.645111	−0.322556	+	0.946550i	$0.604542\pi$
$114$	−466800.	−0.0295096
$115$	−9.94388e6	−0.609696
$116$	−6.07526e6	−0.361378
$117$	−2.24582e6	−0.129635
$118$	−2.30187e7	−1.28971
$119$	0	0
$120$	−1.92000e6	−0.101430
$121$	−1.94872e7	−1.00000
$122$	−1.18851e6	−0.0592576
$123$	−8.94231e6	−0.433293
$124$	8.16819e6	0.384725
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.93788e7	−1.70588	−0.852942	−	0.522006i	$-0.825184\pi$
$127$	−3.93788e7	−1.70588	−0.852942	+	0.522006i	$0.825184\pi$
$128$	2.09715e6	0.0883883
$129$	−2.62688e7	−1.07740
$130$	−1.74500e6	−0.0696616
$131$	2.90434e6	0.112875	0.0564375	−	0.998406i	$-0.482026\pi$
$131$	2.90434e6	0.112875	0.0564375	+	0.998406i	$0.482026\pi$
$132$	5760.00	0.000217979 0
$133$	0	0
$134$	−1.43271e7	−0.514388
$135$	1.30275e7	0.455715
$136$	−1.93843e6	−0.0660791
$137$	2.12920e7	0.707448	0.353724	−	0.935350i	$-0.384915\pi$
$137$	2.12920e7	0.707448	0.353724	+	0.935350i	$0.384915\pi$
$138$	1.90922e7	0.618415
$139$	−1.03342e7	−0.326382	−0.163191	−	0.986594i	$-0.552179\pi$
$139$	−1.03342e7	−0.326382	−0.163191	+	0.986594i	$0.552179\pi$
$140$	0	0
$141$	1.83468e7	0.551179
$142$	−3.94589e6	−0.115647
$143$	5235.00	0.000149706 0
$144$	−5.27155e6	−0.147119
$145$	1.18657e7	0.323227
$146$	−1.64644e7	−0.437836
$147$	0	0
$148$	−8.20845e6	−0.208135
$149$	−5.87706e7	−1.45549	−0.727743	−	0.685850i	$-0.759431\pi$
$149$	−5.87706e7	−1.45549	−0.727743	+	0.685850i	$0.759431\pi$
$150$	3.75000e6	0.0907218
$151$	2.78922e7	0.659271	0.329635	−	0.944108i	$-0.393074\pi$
$151$	2.78922e7	0.659271	0.329635	+	0.944108i	$0.393074\pi$
$152$	−995840.	−0.0230005
$153$	4.87258e6	0.109986
$154$	0	0
$155$	−1.59535e7	−0.344108
$156$	3.35040e6	0.0706579
$157$	7.22654e7	1.49033	0.745164	−	0.666881i	$-0.232371\pi$
$157$	7.22654e7	1.49033	0.745164	+	0.666881i	$0.232371\pi$
$158$	−4.69366e7	−0.946699
$159$	−7.77411e6	−0.153377
$160$	−4.09600e6	−0.0790569
$161$	0	0
$162$	−2.49545e6	−0.0461154
$163$	1.84531e7	0.333744	0.166872	−	0.985979i	$-0.446633\pi$
$163$	1.84531e7	0.333744	0.166872	+	0.985979i	$0.446633\pi$
$164$	−1.90769e7	−0.337719
$165$	−11250.0	−0.000194966 0
$166$	−7.36906e6	−0.125036
$167$	3.66176e7	0.608390	0.304195	−	0.952610i	$-0.401613\pi$
$167$	3.66176e7	0.608390	0.304195	+	0.952610i	$0.401613\pi$
$168$	0	0
$169$	−5.97035e7	−0.951473
$170$	3.78600e6	0.0591030
$171$	2.50322e6	0.0382835
$172$	−5.60401e7	−0.839748
$173$	7.35761e7	1.08038	0.540189	−	0.841544i	$-0.318353\pi$
$173$	7.35761e7	1.08038	0.540189	+	0.841544i	$0.318353\pi$
$174$	−2.27822e7	−0.327849
$175$	0	0
$176$	12288.0	0.000169897 0
$177$	−8.63201e7	−1.17005
$178$	−4.09847e7	−0.544692
$179$	3.42619e7	0.446505	0.223252	−	0.974761i	$-0.428333\pi$
$179$	3.42619e7	0.446505	0.223252	+	0.974761i	$0.428333\pi$
$180$	1.02960e7	0.131588
$181$	6.81694e7	0.854505	0.427252	−	0.904132i	$-0.359482\pi$
$181$	6.81694e7	0.854505	0.427252	+	0.904132i	$0.359482\pi$
$182$	0	0
$183$	−4.45692e6	−0.0537596
$184$	4.07301e7	0.482007
$185$	1.60321e7	0.186162
$186$	3.06307e7	0.349029
$187$	−11358.0	−0.000127015 0
$188$	3.91398e7	0.429602
$189$	0	0
$190$	1.94500e6	0.0205723
$191$	−1.85214e8	−1.92334	−0.961672	−	0.274203i	$-0.911586\pi$
$191$	−1.85214e8	−1.92334	−0.961672	+	0.274203i	$0.911586\pi$
$192$	7.86432e6	0.0801875
$193$	−1.82070e8	−1.82301	−0.911504	−	0.411291i	$-0.865078\pi$
$193$	−1.82070e8	−1.82301	−0.911504	+	0.411291i	$0.865078\pi$
$194$	−4.70264e7	−0.462419
$195$	−6.54375e6	−0.0631983
$196$	0	0
$197$	1.28308e8	1.19570	0.597850	−	0.801608i	$-0.296022\pi$
$197$	1.28308e8	1.19570	0.597850	+	0.801608i	$0.296022\pi$
$198$	−30888.0	−0.000282788 0
$199$	1.18055e8	1.06194	0.530968	−	0.847392i	$-0.321828\pi$
$199$	1.18055e8	1.06194	0.530968	+	0.847392i	$0.321828\pi$
$200$	8.00000e6	0.0707107
$201$	−5.37265e7	−0.466662
$202$	1.65595e7	0.141357
$203$	0	0
$204$	−7.26912e6	−0.0599482
$205$	3.72596e7	0.302065
$206$	1.50900e8	1.20269
$207$	−1.02382e8	−0.802284
$208$	7.14752e6	0.0550724
$209$	−5835.00	−4.42108e−5 0
$210$	0	0
$211$	3.23323e7	0.236945	0.118473	−	0.992957i	$-0.462200\pi$
$211$	3.23323e7	0.236945	0.118473	+	0.992957i	$0.462200\pi$
$212$	−1.65848e7	−0.119546
$213$	−1.47971e7	−0.104917
$214$	−6.33589e7	−0.441936
$215$	1.09453e8	0.751094
$216$	−5.33606e7	−0.360274
$217$	0	0
$218$	−3.70410e6	−0.0242150
$219$	−6.17416e7	−0.397213
$220$	−24000.0	−0.000151961 0
$221$	−6.60657e6	−0.0411721
$222$	−3.07817e7	−0.188824
$223$	−3.00852e8	−1.81671	−0.908355	−	0.418200i	$-0.862661\pi$
$223$	−3.00852e8	−1.81671	−0.908355	+	0.418200i	$0.862661\pi$
$224$	0	0
$225$	−2.01094e7	−0.117695
$226$	−7.91588e7	−0.456163
$227$	1.80589e8	1.02471	0.512356	−	0.858773i	$-0.328773\pi$
$227$	1.80589e8	1.02471	0.512356	+	0.858773i	$0.328773\pi$
$228$	−3.73440e6	−0.0208665
$229$	2.37722e8	1.30811	0.654057	−	0.756445i	$-0.273066\pi$
$229$	2.37722e8	1.30811	0.654057	+	0.756445i	$0.273066\pi$
$230$	−7.95510e7	−0.431120
$231$	0	0
$232$	−4.86021e7	−0.255533
$233$	−3.86316e7	−0.200077	−0.100038	−	0.994984i	$-0.531897\pi$
$233$	−3.86316e7	−0.200077	−0.100038	+	0.994984i	$0.531897\pi$
$234$	−1.79665e7	−0.0916660
$235$	−7.64449e7	−0.384247
$236$	−1.84150e8	−0.911966
$237$	−1.76012e8	−0.858862
$238$	0	0
$239$	−2.37392e8	−1.12479	−0.562397	−	0.826867i	$-0.690121\pi$
$239$	−2.37392e8	−1.12479	−0.562397	+	0.826867i	$0.690121\pi$
$240$	−1.53600e7	−0.0717219
$241$	−2.29529e6	−0.0105628	−0.00528138	−	0.999986i	$-0.501681\pi$
$241$	−2.29529e6	−0.0105628	−0.00528138	+	0.999986i	$0.501681\pi$
$242$	−1.55897e8	−0.707106
$243$	2.18571e8	0.977172
$244$	−9.50810e6	−0.0419015
$245$	0	0
$246$	−7.15385e7	−0.306385
$247$	−3.39402e6	−0.0143310
$248$	6.53455e7	0.272041
$249$	−2.76340e7	−0.113435
$250$	−1.56250e7	−0.0632456
$251$	−3.71573e8	−1.48315	−0.741577	−	0.670868i	$-0.765922\pi$
$251$	−3.71573e8	−1.48315	−0.741577	+	0.670868i	$0.765922\pi$
$252$	0	0
$253$	238653.	0.000926499 0
$254$	−3.15030e8	−1.20624
$255$	1.41975e7	0.0536193
$256$	1.67772e7	0.0625000
$257$	−3.15777e8	−1.16042	−0.580209	−	0.814468i	$-0.697029\pi$
$257$	−3.15777e8	−1.16042	−0.580209	+	0.814468i	$0.697029\pi$
$258$	−2.10150e8	−0.761835
$259$	0	0
$260$	−1.39600e7	−0.0492582
$261$	1.22170e8	0.425326
$262$	2.32347e7	0.0798147
$263$	−2.91438e7	−0.0987873	−0.0493936	−	0.998779i	$-0.515729\pi$
$263$	−2.91438e7	−0.0987873	−0.0493936	+	0.998779i	$0.515729\pi$
$264$	46080.0	0.000154134 0
$265$	3.23921e7	0.106925
$266$	0	0
$267$	−1.53692e8	−0.494155
$268$	−1.14617e8	−0.363727
$269$	1.56638e8	0.490642	0.245321	−	0.969442i	$-0.421107\pi$
$269$	1.56638e8	0.490642	0.245321	+	0.969442i	$0.421107\pi$
$270$	1.04220e8	0.322239
$271$	2.36298e8	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$271$	2.36298e8	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$272$	−1.55075e7	−0.0467250
$273$	0	0
$274$	1.70336e8	0.500241
$275$	46875.0	0.000135918 0
$276$	1.52738e8	0.437286
$277$	1.67160e8	0.472555	0.236278	−	0.971686i	$-0.424072\pi$
$277$	1.67160e8	0.472555	0.236278	+	0.971686i	$0.424072\pi$
$278$	−8.26739e7	−0.230787
$279$	−1.64257e8	−0.452804
$280$	0	0
$281$	4.24035e8	1.14007	0.570033	−	0.821622i	$-0.306931\pi$
$281$	4.24035e8	1.14007	0.570033	+	0.821622i	$0.306931\pi$
$282$	1.46774e8	0.389742
$283$	5.29481e7	0.138867	0.0694333	−	0.997587i	$-0.477881\pi$
$283$	5.29481e7	0.138867	0.0694333	+	0.997587i	$0.477881\pi$
$284$	−3.15671e7	−0.0817750
$285$	7.29375e6	0.0186635
$286$	41880.0	0.000105858 0
$287$	0	0
$288$	−4.21724e7	−0.104029
$289$	−3.96005e8	−0.965068
$290$	9.49260e7	0.228556
$291$	−1.76349e8	−0.419515
$292$	−1.31715e8	−0.309597
$293$	−8.24478e6	−0.0191488	−0.00957442	−	0.999954i	$-0.503048\pi$
$293$	−8.24478e6	−0.0191488	−0.00957442	+	0.999954i	$0.503048\pi$
$294$	0	0
$295$	3.59667e8	0.815687
$296$	−6.56676e7	−0.147174
$297$	−312660.	−0.000692508 0
$298$	−4.70165e8	−1.02918
$299$	1.38816e8	0.300325
$300$	3.00000e7	0.0641500
$301$	0	0
$302$	2.23138e8	0.466175
$303$	6.20982e7	0.128242
$304$	−7.96672e6	−0.0162638
$305$	1.85705e7	0.0374778
$306$	3.89807e7	0.0777722
$307$	−9.83032e8	−1.93902	−0.969512	−	0.245044i	$-0.921198\pi$
$307$	−9.83032e8	−1.93902	−0.969512	+	0.245044i	$0.921198\pi$
$308$	0	0
$309$	5.65875e8	1.09110
$310$	−1.27628e8	−0.243321
$311$	−6.46188e8	−1.21814	−0.609071	−	0.793116i	$-0.708457\pi$
$311$	−6.46188e8	−1.21814	−0.609071	+	0.793116i	$0.708457\pi$
$312$	2.68032e7	0.0499627
$313$	−2.28436e8	−0.421074	−0.210537	−	0.977586i	$-0.567521\pi$
$313$	−2.28436e8	−0.421074	−0.210537	+	0.977586i	$0.567521\pi$
$314$	5.78123e8	1.05382
$315$	0	0
$316$	−3.75493e8	−0.669417
$317$	−5.90783e8	−1.04165	−0.520823	−	0.853665i	$-0.674375\pi$
$317$	−5.90783e8	−1.04165	−0.520823	+	0.853665i	$0.674375\pi$
$318$	−6.21929e7	−0.108454
$319$	−284778.	−0.000491178 0
$320$	−3.27680e7	−0.0559017
$321$	−2.37596e8	−0.400933
$322$	0	0
$323$	7.36377e6	0.0121588
$324$	−1.99636e7	−0.0326085
$325$	2.72656e7	0.0440579
$326$	1.47625e8	0.235992
$327$	−1.38904e7	−0.0219683
$328$	−1.52615e8	−0.238803
$329$	0	0
$330$	−90000.0	−0.000137862 0
$331$	1.57789e8	0.239154	0.119577	−	0.992825i	$-0.461846\pi$
$331$	1.57789e8	0.239154	0.119577	+	0.992825i	$0.461846\pi$
$332$	−5.89524e7	−0.0884135
$333$	1.65067e8	0.244965
$334$	2.92941e8	0.430196
$335$	2.23860e8	0.325327
$336$	0	0
$337$	6.78321e8	0.965451	0.482726	−	0.875772i	$-0.339647\pi$
$337$	6.78321e8	0.965451	0.482726	+	0.875772i	$0.339647\pi$
$338$	−4.77628e8	−0.672793
$339$	−2.96846e8	−0.413839
$340$	3.02880e7	0.0417921
$341$	382884.	0.000522910 0
$342$	2.00257e7	0.0270705
$343$	0	0
$344$	−4.48321e8	−0.593792
$345$	−2.98316e8	−0.391120
$346$	5.88609e8	0.763942
$347$	1.01563e9	1.30492	0.652460	−	0.757823i	$-0.273737\pi$
$347$	1.01563e9	1.30492	0.652460	+	0.757823i	$0.273737\pi$
$348$	−1.82258e8	−0.231824
$349$	3.29134e8	0.414462	0.207231	−	0.978292i	$-0.433555\pi$
$349$	3.29134e8	0.414462	0.207231	+	0.978292i	$0.433555\pi$
$350$	0	0
$351$	−1.81864e8	−0.224477
$352$	98304.0	0.000120136 0
$353$	1.09376e9	1.32347	0.661733	−	0.749740i	$-0.269821\pi$
$353$	1.09376e9	1.32347	0.661733	+	0.749740i	$0.269821\pi$
$354$	−6.90561e8	−0.827352
$355$	6.16545e7	0.0731418
$356$	−3.27877e8	−0.385156
$357$	0	0
$358$	2.74095e8	0.315727
$359$	−9.05883e8	−1.03334	−0.516668	−	0.856186i	$-0.672828\pi$
$359$	−9.05883e8	−1.03334	−0.516668	+	0.856186i	$0.672828\pi$
$360$	8.23680e7	0.0930464
$361$	−8.90089e8	−0.995768
$362$	5.45355e8	0.604226
$363$	−5.84615e8	−0.641500
$364$	0	0
$365$	2.57256e8	0.276912
$366$	−3.56554e7	−0.0380138
$367$	5.52154e8	0.583082	0.291541	−	0.956558i	$-0.405832\pi$
$367$	5.52154e8	0.583082	0.291541	+	0.956558i	$0.405832\pi$
$368$	3.25841e8	0.340830
$369$	3.83625e8	0.397479
$370$	1.28257e8	0.131636
$371$	0	0
$372$	2.45046e8	0.246801
$373$	−1.03969e9	−1.03735	−0.518673	−	0.854973i	$-0.673574\pi$
$373$	−1.03969e9	−1.03735	−0.518673	+	0.854973i	$0.673574\pi$
$374$	−90864.0	−8.98134e−5 0
$375$	−5.85938e7	−0.0573775
$376$	3.13118e8	0.303774
$377$	−1.65646e8	−0.159216
$378$	0	0
$379$	−1.29056e9	−1.21770	−0.608850	−	0.793285i	$-0.708369\pi$
$379$	−1.29056e9	−1.21770	−0.608850	+	0.793285i	$0.708369\pi$
$380$	1.55600e7	0.0145468
$381$	−1.18136e9	−1.09432
$382$	−1.48171e9	−1.36001
$383$	−4.06216e8	−0.369454	−0.184727	−	0.982790i	$-0.559140\pi$
$383$	−4.06216e8	−0.369454	−0.184727	+	0.982790i	$0.559140\pi$
$384$	6.29146e7	0.0567012
$385$	0	0
$386$	−1.45656e9	−1.28906
$387$	1.12693e9	0.988346
$388$	−3.76212e8	−0.326980
$389$	1.11374e9	0.959315	0.479657	−	0.877456i	$-0.340761\pi$
$389$	1.11374e9	0.959315	0.479657	+	0.877456i	$0.340761\pi$
$390$	−5.23500e7	−0.0446880
$391$	−3.01180e8	−0.254805
$392$	0	0
$393$	8.71302e7	0.0724093
$394$	1.02646e9	0.845487
$395$	7.33384e8	0.598745
$396$	−247104.	−0.000199962 0
$397$	1.22618e9	0.983527	0.491764	−	0.870729i	$-0.336352\pi$
$397$	1.22618e9	0.983527	0.491764	+	0.870729i	$0.336352\pi$
$398$	9.44439e8	0.750902
$399$	0	0
$400$	6.40000e7	0.0500000
$401$	1.30987e9	1.01443	0.507217	−	0.861818i	$-0.330674\pi$
$401$	1.30987e9	1.01443	0.507217	+	0.861818i	$0.330674\pi$
$402$	−4.29812e8	−0.329980
$403$	2.22711e8	0.169502
$404$	1.32476e8	0.0999546
$405$	3.89914e7	0.0291659
$406$	0	0
$407$	−384771.	−0.000282893 0
$408$	−5.81530e7	−0.0423898
$409$	−9.92387e7	−0.0717215	−0.0358608	−	0.999357i	$-0.511417\pi$
$409$	−9.92387e7	−0.0717215	−0.0358608	+	0.999357i	$0.511417\pi$
$410$	2.98077e8	0.213592
$411$	6.38760e8	0.453828
$412$	1.20720e9	0.850430
$413$	0	0
$414$	−8.19057e8	−0.567300
$415$	1.15142e8	0.0790794
$416$	5.71802e7	0.0389420
$417$	−3.10027e8	−0.209374
$418$	−46680.0	−3.12618e−5 0
$419$	−6.61810e8	−0.439526	−0.219763	−	0.975553i	$-0.570528\pi$
$419$	−6.61810e8	−0.439526	−0.219763	+	0.975553i	$0.570528\pi$
$420$	0	0
$421$	−1.91861e9	−1.25314	−0.626569	−	0.779366i	$-0.715541\pi$
$421$	−1.91861e9	−1.25314	−0.626569	+	0.779366i	$0.715541\pi$
$422$	2.58658e8	0.167545
$423$	−7.87076e8	−0.505622
$424$	−1.32678e8	−0.0845316
$425$	−5.91562e7	−0.0373800
$426$	−1.18377e8	−0.0741878
$427$	0	0
$428$	−5.06872e8	−0.312496
$429$	157050.	9.60367e−5 0
$430$	8.75626e8	0.531103
$431$	−9.10212e8	−0.547611	−0.273805	−	0.961785i	$-0.588282\pi$
$431$	−9.10212e8	−0.547611	−0.273805	+	0.961785i	$0.588282\pi$
$432$	−4.26885e8	−0.254752
$433$	1.70048e9	1.00661	0.503307	−	0.864108i	$-0.332117\pi$
$433$	1.70048e9	1.00661	0.503307	+	0.864108i	$0.332117\pi$
$434$	0	0
$435$	3.55972e8	0.207350
$436$	−2.96328e7	−0.0171226
$437$	−1.54727e8	−0.0886911
$438$	−4.93932e8	−0.280872
$439$	8.26867e8	0.466455	0.233227	−	0.972422i	$-0.425071\pi$
$439$	8.26867e8	0.466455	0.233227	+	0.972422i	$0.425071\pi$
$440$	−192000.	−0.000107453 0
$441$	0	0
$442$	−5.28526e7	−0.0291131
$443$	−2.46337e8	−0.134622	−0.0673111	−	0.997732i	$-0.521442\pi$
$443$	−2.46337e8	−0.134622	−0.0673111	+	0.997732i	$0.521442\pi$
$444$	−2.46253e8	−0.133519
$445$	6.40385e8	0.344494
$446$	−2.40681e9	−1.28461
$447$	−1.76312e9	−0.933695
$448$	0	0
$449$	−1.84688e9	−0.962890	−0.481445	−	0.876476i	$-0.659888\pi$
$449$	−1.84688e9	−0.962890	−0.481445	+	0.876476i	$0.659888\pi$
$450$	−1.60875e8	−0.0832233
$451$	−894231.	−0.000459020 0
$452$	−6.33271e8	−0.322556
$453$	8.36766e8	0.422922
$454$	1.44471e9	0.724580
$455$	0	0
$456$	−2.98752e7	−0.0147548
$457$	2.48561e9	1.21822	0.609110	−	0.793086i	$-0.291527\pi$
$457$	2.48561e9	1.21822	0.609110	+	0.793086i	$0.291527\pi$
$458$	1.90178e9	0.924977
$459$	3.94577e8	0.190453
$460$	−6.36408e8	−0.304848
$461$	2.20516e9	1.04830	0.524152	−	0.851625i	$-0.324382\pi$
$461$	2.20516e9	1.04830	0.524152	+	0.851625i	$0.324382\pi$
$462$	0	0
$463$	4.01804e9	1.88140	0.940699	−	0.339241i	$-0.110170\pi$
$463$	4.01804e9	1.88140	0.940699	+	0.339241i	$0.110170\pi$
$464$	−3.88817e8	−0.180689
$465$	−4.78605e8	−0.220746
$466$	−3.09053e8	−0.141476
$467$	−2.95028e9	−1.34046	−0.670231	−	0.742153i	$-0.733805\pi$
$467$	−2.95028e9	−1.34046	−0.670231	+	0.742153i	$0.733805\pi$
$468$	−1.43732e8	−0.0648177
$469$	0	0
$470$	−6.11559e8	−0.271704
$471$	2.16796e9	0.956046
$472$	−1.47320e9	−0.644857
$473$	−2.62688e6	−0.00114137
$474$	−1.40810e9	−0.607307
$475$	−3.03906e7	−0.0130110
$476$	0	0
$477$	3.33509e8	0.140700
$478$	−1.89914e9	−0.795350
$479$	−3.33419e9	−1.38617	−0.693084	−	0.720857i	$-0.743749\pi$
$479$	−3.33419e9	−1.38617	−0.693084	+	0.720857i	$0.743749\pi$
$480$	−1.22880e8	−0.0507151
$481$	−2.23808e8	−0.0916999
$482$	−1.83623e7	−0.00746900
$483$	0	0
$484$	−1.24718e9	−0.500000
$485$	7.34788e8	0.292460
$486$	1.74857e9	0.690965
$487$	4.40620e9	1.72867	0.864337	−	0.502913i	$-0.167738\pi$
$487$	4.40620e9	1.72867	0.864337	+	0.502913i	$0.167738\pi$
$488$	−7.60648e7	−0.0296288
$489$	5.53593e8	0.214097
$490$	0	0
$491$	9.56781e8	0.364777	0.182389	−	0.983227i	$-0.441617\pi$
$491$	9.56781e8	0.364777	0.182389	+	0.983227i	$0.441617\pi$
$492$	−5.72308e8	−0.216647
$493$	3.59390e8	0.135083
$494$	−2.71522e7	−0.0101335
$495$	482625.	0.000178851 0
$496$	5.22764e8	0.192362
$497$	0	0
$498$	−2.21072e8	−0.0802104
$499$	1.02883e9	0.370675	0.185337	−	0.982675i	$-0.440662\pi$
$499$	1.02883e9	0.370675	0.185337	+	0.982675i	$0.440662\pi$
$500$	−1.25000e8	−0.0447214
$501$	1.09853e9	0.390282
$502$	−2.97258e9	−1.04875
$503$	3.54264e9	1.24119	0.620596	−	0.784130i	$-0.286891\pi$
$503$	3.54264e9	1.24119	0.620596	+	0.784130i	$0.286891\pi$
$504$	0	0
$505$	−2.58743e8	−0.0894021
$506$	1.90922e6	0.000655134 0
$507$	−1.79110e9	−0.610370
$508$	−2.52024e9	−0.852942
$509$	2.60762e9	0.876460	0.438230	−	0.898863i	$-0.355606\pi$
$509$	2.60762e9	0.876460	0.438230	+	0.898863i	$0.355606\pi$
$510$	1.13580e8	0.0379146
$511$	0	0
$512$	1.34218e8	0.0441942
$513$	2.02708e8	0.0662918
$514$	−2.52622e9	−0.820540
$515$	−2.35781e9	−0.760648
$516$	−1.68120e9	−0.538699
$517$	1.83468e6	0.000583906 0
$518$	0	0
$519$	2.20728e9	0.693063
$520$	−1.11680e8	−0.0348308
$521$	−5.41367e9	−1.67710	−0.838552	−	0.544821i	$-0.816597\pi$
$521$	−5.41367e9	−1.67710	−0.838552	+	0.544821i	$0.816597\pi$
$522$	9.77358e8	0.300751
$523$	4.66750e9	1.42669	0.713343	−	0.700815i	$-0.247180\pi$
$523$	4.66750e9	1.42669	0.713343	+	0.700815i	$0.247180\pi$
$524$	1.85878e8	0.0564375
$525$	0	0
$526$	−2.33150e8	−0.0698532
$527$	−4.83200e8	−0.143810
$528$	368640.	0.000108989 0
$529$	2.92354e9	0.858645
$530$	2.59137e8	0.0756073
$531$	3.70313e9	1.07334
$532$	0	0
$533$	−5.20144e8	−0.148792
$534$	−1.22954e9	−0.349420
$535$	9.89983e8	0.279505
$536$	−9.16933e8	−0.257194
$537$	1.02786e9	0.286433
$538$	1.25311e9	0.346937
$539$	0	0
$540$	8.33760e8	0.227857
$541$	2.57833e9	0.700082	0.350041	−	0.936734i	$-0.386168\pi$
$541$	2.57833e9	0.700082	0.350041	+	0.936734i	$0.386168\pi$
$542$	1.89038e9	0.509978
$543$	2.04508e9	0.548165
$544$	−1.24060e8	−0.0330396
$545$	5.78765e7	0.0153149
$546$	0	0
$547$	3.05920e9	0.799195	0.399598	−	0.916691i	$-0.369150\pi$
$547$	3.05920e9	0.799195	0.399598	+	0.916691i	$0.369150\pi$
$548$	1.36269e9	0.353724
$549$	1.91202e8	0.0493161
$550$	375000.	9.61085e−5 0
$551$	1.84631e8	0.0470191
$552$	1.22190e9	0.309208
$553$	0	0
$554$	1.33728e9	0.334147
$555$	4.80964e8	0.119423
$556$	−6.61391e8	−0.163191
$557$	4.32915e9	1.06147	0.530737	−	0.847537i	$-0.321915\pi$
$557$	4.32915e9	1.06147	0.530737	+	0.847537i	$0.321915\pi$
$558$	−1.31406e9	−0.320180
$559$	−1.52797e9	−0.369975
$560$	0	0
$561$	−340740.	−8.14804e−5 0
$562$	3.39228e9	0.806148
$563$	−4.80781e9	−1.13545	−0.567725	−	0.823218i	$-0.692176\pi$
$563$	−4.80781e9	−1.13545	−0.567725	+	0.823218i	$0.692176\pi$
$564$	1.17419e9	0.275590
$565$	1.23686e9	0.288503
$566$	4.23584e8	0.0981934
$567$	0	0
$568$	−2.52537e8	−0.0578237
$569$	5.92596e8	0.134855	0.0674273	−	0.997724i	$-0.478521\pi$
$569$	5.92596e8	0.134855	0.0674273	+	0.997724i	$0.478521\pi$
$570$	5.83500e7	0.0131971
$571$	6.79980e9	1.52852	0.764258	−	0.644911i	$-0.223105\pi$
$571$	6.79980e9	1.52852	0.764258	+	0.644911i	$0.223105\pi$
$572$	335040.	7.48532e−5 0
$573$	−5.55642e9	−1.23383
$574$	0	0
$575$	1.24298e9	0.272664
$576$	−3.37379e8	−0.0735597
$577$	3.38667e9	0.733936	0.366968	−	0.930234i	$-0.380396\pi$
$577$	3.38667e9	0.733936	0.366968	+	0.930234i	$0.380396\pi$
$578$	−3.16804e9	−0.682406
$579$	−5.46211e9	−1.16946
$580$	7.59408e8	0.161613
$581$	0	0
$582$	−1.41079e9	−0.296642
$583$	−777411.	−0.000162484 0
$584$	−1.05372e9	−0.218918
$585$	2.80727e8	0.0579747
$586$	−6.59583e7	−0.0135403
$587$	2.26024e9	0.461233	0.230616	−	0.973045i	$-0.425926\pi$
$587$	2.26024e9	0.461233	0.230616	+	0.973045i	$0.425926\pi$
$588$	0	0
$589$	−2.48236e8	−0.0500567
$590$	2.87734e9	0.576778
$591$	3.84924e9	0.767042
$592$	−5.25341e8	−0.104067
$593$	9.56957e9	1.88452	0.942260	−	0.334881i	$-0.108696\pi$
$593$	9.56957e9	1.88452	0.942260	+	0.334881i	$0.108696\pi$
$594$	−2.50128e6	−0.000489677 0
$595$	0	0
$596$	−3.76132e9	−0.727743
$597$	3.54165e9	0.681232
$598$	1.11053e9	0.212362
$599$	4.56293e9	0.867461	0.433730	−	0.901043i	$-0.357197\pi$
$599$	4.56293e9	0.867461	0.433730	+	0.901043i	$0.357197\pi$
$600$	2.40000e8	0.0453609
$601$	−8.24377e7	−0.0154905	−0.00774525	−	0.999970i	$-0.502465\pi$
$601$	−8.24377e7	−0.0154905	−0.00774525	+	0.999970i	$0.502465\pi$
$602$	0	0
$603$	2.30487e9	0.428090
$604$	1.78510e9	0.329635
$605$	2.43590e9	0.447213
$606$	4.96786e8	0.0906807
$607$	−3.35909e8	−0.0609623	−0.0304812	−	0.999535i	$-0.509704\pi$
$607$	−3.35909e8	−0.0609623	−0.0304812	+	0.999535i	$0.509704\pi$
$608$	−6.37338e7	−0.0115002
$609$	0	0
$610$	1.48564e8	0.0265008
$611$	1.06717e9	0.189273
$612$	3.11845e8	0.0549932
$613$	7.29431e9	1.27901	0.639503	−	0.768789i	$-0.279140\pi$
$613$	7.29431e9	1.27901	0.639503	+	0.768789i	$0.279140\pi$
$614$	−7.86425e9	−1.37110
$615$	1.11779e9	0.193775
$616$	0	0
$617$	3.33942e9	0.572365	0.286183	−	0.958175i	$-0.407614\pi$
$617$	3.33942e9	0.572365	0.286183	+	0.958175i	$0.407614\pi$
$618$	4.52700e9	0.771526
$619$	−2.83754e9	−0.480867	−0.240434	−	0.970666i	$-0.577290\pi$
$619$	−2.83754e9	−0.480867	−0.240434	+	0.970666i	$0.577290\pi$
$620$	−1.02102e9	−0.172054
$621$	−8.29081e9	−1.38924
$622$	−5.16951e9	−0.861356
$623$	0	0
$624$	2.14426e8	0.0353289
$625$	2.44141e8	0.0400000
$626$	−1.82748e9	−0.297744
$627$	−175050.	−2.83613e−5 0
$628$	4.62499e9	0.745164
$629$	4.85581e8	0.0778009
$630$	0	0
$631$	2.68392e8	0.0425272	0.0212636	−	0.999774i	$-0.493231\pi$
$631$	2.68392e8	0.0425272	0.0212636	+	0.999774i	$0.493231\pi$
$632$	−3.00394e9	−0.473349
$633$	9.69968e8	0.152000
$634$	−4.72626e9	−0.736555
$635$	4.92235e9	0.762894
$636$	−4.97543e8	−0.0766886
$637$	0	0
$638$	−2.27822e6	−0.000347315 0
$639$	6.34795e8	0.0962455
$640$	−2.62144e8	−0.0395285
$641$	8.11172e9	1.21649	0.608247	−	0.793748i	$-0.291873\pi$
$641$	8.11172e9	1.21649	0.608247	+	0.793748i	$0.291873\pi$
$642$	−1.90077e9	−0.283502
$643$	7.28529e8	0.108071	0.0540354	−	0.998539i	$-0.482792\pi$
$643$	7.28529e8	0.108071	0.0540354	+	0.998539i	$0.482792\pi$
$644$	0	0
$645$	3.28360e9	0.481827
$646$	5.89102e7	0.00859758
$647$	−1.01278e10	−1.47011	−0.735056	−	0.678006i	$-0.762844\pi$
$647$	−1.01278e10	−1.47011	−0.735056	+	0.678006i	$0.762844\pi$
$648$	−1.59709e8	−0.0230577
$649$	−8.63201e6	−0.00123953
$650$	2.18125e8	0.0311536
$651$	0	0
$652$	1.18100e9	0.166872
$653$	1.11058e10	1.56082	0.780412	−	0.625265i	$-0.215009\pi$
$653$	1.11058e10	1.56082	0.780412	+	0.625265i	$0.215009\pi$
$654$	−1.11123e8	−0.0155339
$655$	−3.63042e8	−0.0504792
$656$	−1.22092e9	−0.168859
$657$	2.64871e9	0.364381
$658$	0	0
$659$	3.44065e9	0.468319	0.234159	−	0.972198i	$-0.424766\pi$
$659$	3.44065e9	0.468319	0.234159	+	0.972198i	$0.424766\pi$
$660$	−720000.	−9.74830e−5 0
$661$	1.34018e10	1.80492	0.902462	−	0.430769i	$-0.141758\pi$
$661$	1.34018e10	1.80492	0.902462	+	0.430769i	$0.141758\pi$
$662$	1.26231e9	0.169108
$663$	−1.98197e8	−0.0264119
$664$	−4.71620e8	−0.0625178
$665$	0	0
$666$	1.32053e9	0.173217
$667$	−7.55146e9	−0.985350
$668$	2.34352e9	0.304195
$669$	−9.02555e9	−1.16542
$670$	1.79088e9	0.230041
$671$	−445692.	−5.69516e−5 0
$672$	0	0
$673$	5.62819e9	0.711732	0.355866	−	0.934537i	$-0.384186\pi$
$673$	5.62819e9	0.711732	0.355866	+	0.934537i	$0.384186\pi$
$674$	5.42656e9	0.682677
$675$	−1.62844e9	−0.203802
$676$	−3.82102e9	−0.475736
$677$	−1.00469e10	−1.24443	−0.622217	−	0.782845i	$-0.713768\pi$
$677$	−1.00469e10	−1.24443	−0.622217	+	0.782845i	$0.713768\pi$
$678$	−2.37476e9	−0.292628
$679$	0	0
$680$	2.42304e8	0.0295515
$681$	5.41768e9	0.657352
$682$	3.06307e6	0.000369753 0
$683$	−4.49436e9	−0.539754	−0.269877	−	0.962895i	$-0.586983\pi$
$683$	−4.49436e9	−0.539754	−0.269877	+	0.962895i	$0.586983\pi$
$684$	1.60206e8	0.0191418
$685$	−2.66150e9	−0.316380
$686$	0	0
$687$	7.13167e9	0.839156
$688$	−3.58656e9	−0.419874
$689$	−4.52194e8	−0.0526693
$690$	−2.38653e9	−0.276564
$691$	−1.60700e10	−1.85286	−0.926428	−	0.376472i	$-0.877137\pi$
$691$	−1.60700e10	−1.85286	−0.926428	+	0.376472i	$0.877137\pi$
$692$	4.70887e9	0.540189
$693$	0	0
$694$	8.12507e9	0.922718
$695$	1.29178e9	0.145963
$696$	−1.45806e9	−0.163925
$697$	1.12852e9	0.126239
$698$	2.63308e9	0.293069
$699$	−1.15895e9	−0.128349
$700$	0	0
$701$	−1.03290e10	−1.13252	−0.566262	−	0.824226i	$-0.691611\pi$
$701$	−1.03290e10	−1.13252	−0.566262	+	0.824226i	$0.691611\pi$
$702$	−1.45491e9	−0.158729
$703$	2.49460e8	0.0270805
$704$	786432.	8.49487e−5 0
$705$	−2.29335e9	−0.246495
$706$	8.75012e9	0.935831
$707$	0	0
$708$	−5.52449e9	−0.585026
$709$	−1.18240e10	−1.24596	−0.622978	−	0.782239i	$-0.714077\pi$
$709$	−1.18240e10	−1.24596	−0.622978	+	0.782239i	$0.714077\pi$
$710$	4.93236e8	0.0517191
$711$	7.55092e9	0.787874
$712$	−2.62302e9	−0.272346
$713$	1.01529e10	1.04901
$714$	0	0
$715$	−654375.	−6.69508e−5 0
$716$	2.19276e9	0.223252
$717$	−7.12176e9	−0.721556
$718$	−7.24706e9	−0.730679
$719$	3.97665e9	0.398994	0.199497	−	0.979898i	$-0.436069\pi$
$719$	3.97665e9	0.398994	0.199497	+	0.979898i	$0.436069\pi$
$720$	6.58944e8	0.0657938
$721$	0	0
$722$	−7.12071e9	−0.704114
$723$	−6.88587e7	−0.00677602
$724$	4.36284e9	0.427252
$725$	−1.48322e9	−0.144551
$726$	−4.67692e9	−0.453609
$727$	−1.64415e9	−0.158698	−0.0793489	−	0.996847i	$-0.525284\pi$
$727$	−1.64415e9	−0.158698	−0.0793489	+	0.996847i	$0.525284\pi$
$728$	0	0
$729$	7.23933e9	0.692073
$730$	2.05805e9	0.195806
$731$	3.31512e9	0.313898
$732$	−2.85243e8	−0.0268798
$733$	4.93406e9	0.462743	0.231372	−	0.972865i	$-0.425679\pi$
$733$	4.93406e9	0.462743	0.231372	+	0.972865i	$0.425679\pi$
$734$	4.41724e9	0.412301
$735$	0	0
$736$	2.60673e9	0.241003
$737$	−5.37265e6	−0.000494370 0
$738$	3.06900e9	0.281060
$739$	−5.87025e9	−0.535059	−0.267529	−	0.963550i	$-0.586207\pi$
$739$	−5.87025e9	−0.535059	−0.267529	+	0.963550i	$0.586207\pi$
$740$	1.02606e9	0.0930808
$741$	−1.01821e8	−0.00919332
$742$	0	0
$743$	5.52257e8	0.0493947	0.0246974	−	0.999695i	$-0.492138\pi$
$743$	5.52257e8	0.0493947	0.0246974	+	0.999695i	$0.492138\pi$
$744$	1.96037e9	0.174515
$745$	7.34632e9	0.650913
$746$	−8.31754e9	−0.733515
$747$	1.18550e9	0.104059
$748$	−726912.	−6.35077e−5 0
$749$	0	0
$750$	−4.68750e8	−0.0405720
$751$	−2.19410e9	−0.189024	−0.0945118	−	0.995524i	$-0.530129\pi$
$751$	−2.19410e9	−0.189024	−0.0945118	+	0.995524i	$0.530129\pi$
$752$	2.50495e9	0.214801
$753$	−1.11472e10	−0.951443
$754$	−1.32517e9	−0.112583
$755$	−3.48653e9	−0.294835
$756$	0	0
$757$	−1.90630e10	−1.59719	−0.798593	−	0.601871i	$-0.794422\pi$
$757$	−1.90630e10	−1.59719	−0.798593	+	0.601871i	$0.794422\pi$
$758$	−1.03245e10	−0.861044
$759$	7.15959e6	0.000594350 0
$760$	1.24480e8	0.0102861
$761$	2.17378e10	1.78801	0.894003	−	0.448062i	$-0.147885\pi$
$761$	2.17378e10	1.78801	0.894003	+	0.448062i	$0.147885\pi$
$762$	−9.45091e9	−0.773805
$763$	0	0
$764$	−1.18537e10	−0.961672
$765$	−6.09073e8	−0.0491874
$766$	−3.24972e9	−0.261244
$767$	−5.02095e9	−0.401793
$768$	5.03316e8	0.0400938
$769$	−1.63154e10	−1.29376	−0.646882	−	0.762590i	$-0.723927\pi$
$769$	−1.63154e10	−1.29376	−0.646882	+	0.762590i	$0.723927\pi$
$770$	0	0
$771$	−9.47331e9	−0.744409
$772$	−1.16525e10	−0.911504
$773$	5.85889e9	0.456233	0.228117	−	0.973634i	$-0.426743\pi$
$773$	5.85889e9	0.456233	0.228117	+	0.973634i	$0.426743\pi$
$774$	9.01545e9	0.698866
$775$	1.99419e9	0.153890
$776$	−3.00969e9	−0.231210
$777$	0	0
$778$	8.90994e9	0.678338
$779$	5.79760e8	0.0439407
$780$	−4.18800e8	−0.0315992
$781$	−1.47971e6	−0.000111147 0
$782$	−2.40944e9	−0.180174
$783$	9.89319e9	0.736495
$784$	0	0
$785$	−9.03318e9	−0.666495
$786$	6.97041e8	0.0512011
$787$	−6.24055e9	−0.456364	−0.228182	−	0.973619i	$-0.573278\pi$
$787$	−6.24055e9	−0.456364	−0.228182	+	0.973619i	$0.573278\pi$
$788$	8.21172e9	0.597850
$789$	−8.74314e8	−0.0633721
$790$	5.86707e9	0.423376
$791$	0	0
$792$	−1.97683e6	−0.000141394 0
$793$	−2.59244e8	−0.0184609
$794$	9.80942e9	0.695459
$795$	9.71764e8	0.0685924
$796$	7.55551e9	0.530968
$797$	−1.35158e10	−0.945664	−0.472832	−	0.881153i	$-0.656768\pi$
$797$	−1.35158e10	−0.945664	−0.472832	+	0.881153i	$0.656768\pi$
$798$	0	0
$799$	−2.31536e9	−0.160585
$800$	5.12000e8	0.0353553
$801$	6.59341e9	0.453311
$802$	1.04790e10	0.717313
$803$	−6.17416e6	−0.000420798 0
$804$	−3.43850e9	−0.233331
$805$	0	0
$806$	1.78169e9	0.119856
$807$	4.69915e9	0.314747
$808$	1.05981e9	0.0706786
$809$	−7.67388e9	−0.509560	−0.254780	−	0.966999i	$-0.582003\pi$
$809$	−7.67388e9	−0.509560	−0.254780	+	0.966999i	$0.582003\pi$
$810$	3.11931e8	0.0206234
$811$	6.26932e9	0.412713	0.206356	−	0.978477i	$-0.433839\pi$
$811$	6.26932e9	0.412713	0.206356	+	0.978477i	$0.433839\pi$
$812$	0	0
$813$	7.08893e9	0.462662
$814$	−3.07817e6	−0.000200035 0
$815$	−2.30664e9	−0.149255
$816$	−4.65224e8	−0.0299741
$817$	1.70309e9	0.109260
$818$	−7.93910e8	−0.0507148
$819$	0	0
$820$	2.38462e9	0.151032
$821$	2.15871e10	1.36143	0.680713	−	0.732550i	$-0.261670\pi$
$821$	2.15871e10	1.36143	0.680713	+	0.732550i	$0.261670\pi$
$822$	5.11008e9	0.320905
$823$	8.71475e9	0.544949	0.272474	−	0.962163i	$-0.412158\pi$
$823$	8.71475e9	0.544949	0.272474	+	0.962163i	$0.412158\pi$
$824$	9.65759e9	0.601345
$825$	1.40625e6	8.71914e−5 0
$826$	0	0
$827$	1.63977e9	0.100812	0.0504062	−	0.998729i	$-0.483948\pi$
$827$	1.63977e9	0.100812	0.0504062	+	0.998729i	$0.483948\pi$
$828$	−6.55246e9	−0.401142
$829$	1.10484e10	0.673534	0.336767	−	0.941588i	$-0.390667\pi$
$829$	1.10484e10	0.673534	0.336767	+	0.941588i	$0.390667\pi$
$830$	9.21132e8	0.0559176
$831$	5.01480e9	0.303144
$832$	4.57441e8	0.0275362
$833$	0	0
$834$	−2.48022e9	−0.148050
$835$	−4.57720e9	−0.272080
$836$	−373440.	−2.21054e−5 0
$837$	−1.33014e10	−0.784076
$838$	−5.29448e9	−0.310792
$839$	−1.71398e10	−1.00193	−0.500967	−	0.865467i	$-0.667022\pi$
$839$	−1.71398e10	−1.00193	−0.500967	+	0.865467i	$0.667022\pi$
$840$	0	0
$841$	−8.23893e9	−0.477623
$842$	−1.53489e10	−0.886102
$843$	1.27211e10	0.731352
$844$	2.06927e9	0.118473
$845$	7.46294e9	0.425511
$846$	−6.29661e9	−0.357528
$847$	0	0
$848$	−1.06143e9	−0.0597728
$849$	1.58844e9	0.0890829
$850$	−4.73250e8	−0.0264317
$851$	−1.02030e10	−0.567510
$852$	−9.47013e8	−0.0524587
$853$	−1.08837e10	−0.600420	−0.300210	−	0.953873i	$-0.597057\pi$
$853$	−1.08837e10	−0.600420	−0.300210	+	0.953873i	$0.597057\pi$
$854$	0	0
$855$	−3.12902e8	−0.0171209
$856$	−4.05497e9	−0.220968
$857$	−3.46402e10	−1.87996	−0.939978	−	0.341235i	$-0.889155\pi$
$857$	−3.46402e10	−1.87996	−0.939978	+	0.341235i	$0.889155\pi$
$858$	1.25640e6	6.79082e−5 0
$859$	1.88741e10	1.01599	0.507995	−	0.861360i	$-0.330387\pi$
$859$	1.88741e10	1.01599	0.507995	+	0.861360i	$0.330387\pi$
$860$	7.00501e9	0.375547
$861$	0	0
$862$	−7.28169e9	−0.387219
$863$	−3.52861e10	−1.86881	−0.934407	−	0.356208i	$-0.884069\pi$
$863$	−3.52861e10	−1.86881	−0.934407	+	0.356208i	$0.884069\pi$
$864$	−3.41508e9	−0.180137
$865$	−9.19702e9	−0.483160
$866$	1.36038e10	0.711783
$867$	−1.18801e10	−0.619092
$868$	0	0
$869$	−1.76012e7	−0.000909858 0
$870$	2.84778e9	0.146619
$871$	−3.12509e9	−0.160250
$872$	−2.37062e8	−0.0121075
$873$	7.56538e9	0.384841
$874$	−1.23781e9	−0.0627141
$875$	0	0
$876$	−3.95146e9	−0.198606
$877$	−3.86940e10	−1.93707	−0.968534	−	0.248880i	$-0.919937\pi$
$877$	−3.86940e10	−1.93707	−0.968534	+	0.248880i	$0.919937\pi$
$878$	6.61493e9	0.329833
$879$	−2.47343e8	−0.0122840
$880$	−1.53600e6	−7.59805e−5 0
$881$	3.55977e9	0.175390	0.0876952	−	0.996147i	$-0.472050\pi$
$881$	3.55977e9	0.175390	0.0876952	+	0.996147i	$0.472050\pi$
$882$	0	0
$883$	1.74416e9	0.0852556	0.0426278	−	0.999091i	$-0.486427\pi$
$883$	1.74416e9	0.0852556	0.0426278	+	0.999091i	$0.486427\pi$
$884$	−4.22820e8	−0.0205861
$885$	1.07900e10	0.523263
$886$	−1.97070e9	−0.0951923
$887$	1.86696e10	0.898262	0.449131	−	0.893466i	$-0.351734\pi$
$887$	1.86696e10	0.898262	0.449131	+	0.893466i	$0.351734\pi$
$888$	−1.97003e9	−0.0944119
$889$	0	0
$890$	5.12308e9	0.243594
$891$	−935793.	−4.43208e−5 0
$892$	−1.92545e10	−0.908355
$893$	−1.18948e9	−0.0558956
$894$	−1.41049e10	−0.660222
$895$	−4.28274e9	−0.199683
$896$	0	0
$897$	4.16449e9	0.192659
$898$	−1.47750e10	−0.680866
$899$	−1.21152e10	−0.556125
$900$	−1.28700e9	−0.0588477
$901$	9.81093e8	0.0446862
$902$	−7.15385e6	−0.000324576 0
$903$	0	0
$904$	−5.06616e9	−0.228081
$905$	−8.52117e9	−0.382146
$906$	6.69413e9	0.299051
$907$	−7.48063e9	−0.332899	−0.166449	−	0.986050i	$-0.553230\pi$
$907$	−7.48063e9	−0.332899	−0.166449	+	0.986050i	$0.553230\pi$
$908$	1.15577e10	0.512356
$909$	−2.66401e9	−0.117642
$910$	0	0
$911$	−2.66809e10	−1.16919	−0.584597	−	0.811324i	$-0.698747\pi$
$911$	−2.66809e10	−1.16919	−0.584597	+	0.811324i	$0.698747\pi$
$912$	−2.39002e8	−0.0104332
$913$	−2.76340e6	−0.000120170 0
$914$	1.98848e10	0.861411
$915$	5.57115e8	0.0240420
$916$	1.52142e10	0.654057
$917$	0	0
$918$	3.15662e9	0.134670
$919$	−8.40441e9	−0.357193	−0.178596	−	0.983922i	$-0.557156\pi$
$919$	−8.40441e9	−0.357193	−0.178596	+	0.983922i	$0.557156\pi$
$920$	−5.09126e9	−0.215560
$921$	−2.94909e10	−1.24388
$922$	1.76413e10	0.741263
$923$	−8.60697e8	−0.0360283
$924$	0	0
$925$	−2.00402e9	−0.0832540
$926$	3.21443e10	1.33035
$927$	−2.42760e10	−1.00092
$928$	−3.11054e9	−0.127767
$929$	−4.47848e10	−1.83263	−0.916317	−	0.400453i	$-0.868853\pi$
$929$	−4.47848e10	−1.83263	−0.916317	+	0.400453i	$0.868853\pi$
$930$	−3.82884e9	−0.156091
$931$	0	0
$932$	−2.47242e9	−0.100038
$933$	−1.93856e10	−0.781438
$934$	−2.36023e10	−0.947850
$935$	1.41975e6	5.68030e−5 0
$936$	−1.14986e9	−0.0458330
$937$	−1.65517e10	−0.657284	−0.328642	−	0.944455i	$-0.606591\pi$
$937$	−1.65517e10	−0.657284	−0.328642	+	0.944455i	$0.606591\pi$
$938$	0	0
$939$	−6.85307e9	−0.270119
$940$	−4.89247e9	−0.192124
$941$	3.17238e10	1.24114	0.620571	−	0.784150i	$-0.286901\pi$
$941$	3.17238e10	1.24114	0.620571	+	0.784150i	$0.286901\pi$
$942$	1.73437e10	0.676026
$943$	−2.37123e10	−0.920838
$944$	−1.17856e10	−0.455983
$945$	0	0
$946$	−2.10150e7	−0.000807069 0
$947$	−3.79796e10	−1.45320	−0.726599	−	0.687062i	$-0.758900\pi$
$947$	−3.79796e10	−1.45320	−0.726599	+	0.687062i	$0.758900\pi$
$948$	−1.12648e10	−0.429431
$949$	−3.59130e9	−0.136402
$950$	−2.43125e8	−0.00920019
$951$	−1.77235e10	−0.668216
$952$	0	0
$953$	2.44707e10	0.915845	0.457922	−	0.888992i	$-0.348594\pi$
$953$	2.44707e10	0.915845	0.457922	+	0.888992i	$0.348594\pi$
$954$	2.66807e9	0.0994898
$955$	2.31517e10	0.860145
$956$	−1.51931e10	−0.562397
$957$	−8.54334e6	−0.000315091 0
$958$	−2.66735e10	−0.980169
$959$	0	0
$960$	−9.83040e8	−0.0358610
$961$	−1.12237e10	−0.407948
$962$	−1.79047e9	−0.0648416
$963$	1.01929e10	0.367794
$964$	−1.46898e8	−0.00528138
$965$	2.27588e10	0.815274
$966$	0	0
$967$	−1.06887e10	−0.380131	−0.190065	−	0.981771i	$-0.560870\pi$
$967$	−1.06887e10	−0.380131	−0.190065	+	0.981771i	$0.560870\pi$
$968$	−9.97743e9	−0.353553
$969$	2.20913e8	0.00779989
$970$	5.87831e9	0.206800
$971$	−2.32485e10	−0.814944	−0.407472	−	0.913218i	$-0.633590\pi$
$971$	−2.32485e10	−0.814944	−0.407472	+	0.913218i	$0.633590\pi$
$972$	1.39886e10	0.488586
$973$	0	0
$974$	3.52496e10	1.22236
$975$	8.17969e8	0.0282631
$976$	−6.08518e8	−0.0209507
$977$	−2.55772e10	−0.877448	−0.438724	−	0.898622i	$-0.644569\pi$
$977$	−2.55772e10	−0.877448	−0.438724	+	0.898622i	$0.644569\pi$
$978$	4.42875e9	0.151389
$979$	−1.53692e7	−0.000523496 0
$980$	0	0
$981$	5.95896e8	0.0201525
$982$	7.65425e9	0.257936
$983$	1.48993e10	0.500298	0.250149	−	0.968207i	$-0.419520\pi$
$983$	1.48993e10	0.500298	0.250149	+	0.968207i	$0.419520\pi$
$984$	−4.57846e9	−0.153192
$985$	−1.60385e10	−0.534733
$986$	2.87512e9	0.0955183
$987$	0	0
$988$	−2.17218e8	−0.00716549
$989$	−6.96569e10	−2.28969
$990$	3.86100e6	0.000126467 0
$991$	−1.53863e10	−0.502199	−0.251100	−	0.967961i	$-0.580792\pi$
$991$	−1.53863e10	−0.502199	−0.251100	+	0.967961i	$0.580792\pi$
$992$	4.18211e9	0.136021
$993$	4.73366e9	0.153418
$994$	0	0
$995$	−1.47569e10	−0.474912
$996$	−1.76857e9	−0.0567173
$997$	−2.07687e10	−0.663706	−0.331853	−	0.943331i	$-0.607674\pi$
$997$	−2.07687e10	−0.663706	−0.331853	+	0.943331i	$0.607674\pi$
$998$	8.23066e9	0.262106
$999$	1.33669e10	0.424183

Display

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.8.a.d.1.1		1
7.2	even	3		70.8.e.a.11.1	✓	2
7.4	even	3		70.8.e.a.51.1	yes	2
7.6	odd	2		490.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.8.e.a.11.1	✓	2	7.2	even	3
70.8.e.a.51.1	yes	2	7.4	even	3
490.8.a.a.1.1		1	7.6	odd	2
490.8.a.d.1.1		1	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	490.8.a.d.1.1

Level	$490$
Weight	$8$
Character	490.1
Self dual	yes
Analytic conductor	$153.069$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$