LMFDB - Embedded newform 350.8.a.g.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)

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")

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);

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,-3] 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 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} -3.00000 q^{3} +64.0000 q^{4} -24.0000 q^{6} +343.000 q^{7} +512.000 q^{8} -2178.00 q^{9} +2303.00 q^{11} -192.000 q^{12} -1381.00 q^{13} +2744.00 q^{14} +4096.00 q^{16} +4009.00 q^{17} -17424.0 q^{18} -7688.00 q^{19} -1029.00 q^{21} +18424.0 q^{22} -81810.0 q^{23} -1536.00 q^{24} -11048.0 q^{26} +13095.0 q^{27} +21952.0 q^{28} +157191. q^{29} -39834.0 q^{31} +32768.0 q^{32} -6909.00 q^{33} +32072.0 q^{34} -139392. q^{36} -125266. q^{37} -61504.0 q^{38} +4143.00 q^{39} -739014. q^{41} -8232.00 q^{42} -294604. q^{43} +147392. q^{44} -654480. q^{46} +655397. q^{47} -12288.0 q^{48} +117649. q^{49} -12027.0 q^{51} -88384.0 q^{52} -291934. q^{53} +104760. q^{54} +175616. q^{56} +23064.0 q^{57} +1.25753e6 q^{58} -2.54192e6 q^{59} +1.43728e6 q^{61} -318672. q^{62} -747054. q^{63} +262144. q^{64} -55272.0 q^{66} -3.15097e6 q^{67} +256576. q^{68} +245430. q^{69} +2.11758e6 q^{71} -1.11514e6 q^{72} -552310. q^{73} -1.00213e6 q^{74} -492032. q^{76} +789929. q^{77} +33144.0 q^{78} -2.33442e6 q^{79} +4.72400e6 q^{81} -5.91211e6 q^{82} -219508. q^{83} -65856.0 q^{84} -2.35683e6 q^{86} -471573. q^{87} +1.17914e6 q^{88} -3.15028e6 q^{89} -473683. q^{91} -5.23584e6 q^{92} +119502. q^{93} +5.24318e6 q^{94} -98304.0 q^{96} -1.21821e7 q^{97} +941192. q^{98} -5.01593e6 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
$3$	−3.00000	−0.0641500	−0.0320750	−	0.999485i	$-0.510212\pi$
$3$	−3.00000	−0.0641500	−0.0320750	+	0.999485i	$0.510212\pi$
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−24.0000	−0.0453609
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	512.000	0.353553
$9$	−2178.00	−0.995885
$10$	0	0
$11$	2303.00	0.521698	0.260849	−	0.965380i	$-0.415997\pi$
$11$	2303.00	0.521698	0.260849	+	0.965380i	$0.415997\pi$
$12$	−192.000	−0.0320750
$13$	−1381.00	−0.174338	−0.0871690	−	0.996194i	$-0.527782\pi$
$13$	−1381.00	−0.174338	−0.0871690	+	0.996194i	$0.527782\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	4009.00	0.197909	0.0989543	−	0.995092i	$-0.468450\pi$
$17$	4009.00	0.197909	0.0989543	+	0.995092i	$0.468450\pi$
$18$	−17424.0	−0.704197
$19$	−7688.00	−0.257144	−0.128572	−	0.991700i	$-0.541039\pi$
$19$	−7688.00	−0.257144	−0.128572	+	0.991700i	$0.541039\pi$
$20$	0	0
$21$	−1029.00	−0.0242464
$22$	18424.0	0.368897
$23$	−81810.0	−1.40204	−0.701018	−	0.713144i	$-0.747271\pi$
$23$	−81810.0	−1.40204	−0.701018	+	0.713144i	$0.747271\pi$
$24$	−1536.00	−0.0226805
$25$	0	0
$26$	−11048.0	−0.123276
$27$	13095.0	0.128036
$28$	21952.0	0.188982
$29$	157191.	1.19684	0.598418	−	0.801184i	$-0.295796\pi$
$29$	157191.	1.19684	0.598418	+	0.801184i	$0.295796\pi$
$30$	0	0
$31$	−39834.0	−0.240153	−0.120076	−	0.992765i	$-0.538314\pi$
$31$	−39834.0	−0.240153	−0.120076	+	0.992765i	$0.538314\pi$
$32$	32768.0	0.176777
$33$	−6909.00	−0.0334670
$34$	32072.0	0.139943
$35$	0	0
$36$	−139392.	−0.497942
$37$	−125266.	−0.406562	−0.203281	−	0.979120i	$-0.565161\pi$
$37$	−125266.	−0.406562	−0.203281	+	0.979120i	$0.565161\pi$
$38$	−61504.0	−0.181828
$39$	4143.00	0.0111838
$40$	0	0
$41$	−739014.	−1.67459	−0.837296	−	0.546749i	$-0.815865\pi$
$41$	−739014.	−1.67459	−0.837296	+	0.546749i	$0.815865\pi$
$42$	−8232.00	−0.0171448
$43$	−294604.	−0.565066	−0.282533	−	0.959258i	$-0.591175\pi$
$43$	−294604.	−0.565066	−0.282533	+	0.959258i	$0.591175\pi$
$44$	147392.	0.260849
$45$	0	0
$46$	−654480.	−0.991389
$47$	655397.	0.920793	0.460396	−	0.887713i	$-0.347707\pi$
$47$	655397.	0.920793	0.460396	+	0.887713i	$0.347707\pi$
$48$	−12288.0	−0.0160375
$49$	117649.	0.142857
$50$	0	0
$51$	−12027.0	−0.0126958
$52$	−88384.0	−0.0871690
$53$	−291934.	−0.269351	−0.134676	−	0.990890i	$-0.542999\pi$
$53$	−291934.	−0.269351	−0.134676	+	0.990890i	$0.542999\pi$
$54$	104760.	0.0905352
$55$	0	0
$56$	175616.	0.133631
$57$	23064.0	0.0164958
$58$	1.25753e6	0.846291
$59$	−2.54192e6	−1.61131	−0.805657	−	0.592382i	$-0.798188\pi$
$59$	−2.54192e6	−1.61131	−0.805657	+	0.592382i	$0.798188\pi$
$60$	0	0
$61$	1.43728e6	0.810750	0.405375	−	0.914150i	$-0.367141\pi$
$61$	1.43728e6	0.810750	0.405375	+	0.914150i	$0.367141\pi$
$62$	−318672.	−0.169814
$63$	−747054.	−0.376409
$64$	262144.	0.125000
$65$	0	0
$66$	−55272.0	−0.0236647
$67$	−3.15097e6	−1.27992	−0.639959	−	0.768409i	$-0.721049\pi$
$67$	−3.15097e6	−1.27992	−0.639959	+	0.768409i	$0.721049\pi$
$68$	256576.	0.0989543
$69$	245430.	0.0899406
$70$	0	0
$71$	2.11758e6	0.702158	0.351079	−	0.936346i	$-0.385815\pi$
$71$	2.11758e6	0.702158	0.351079	+	0.936346i	$0.385815\pi$
$72$	−1.11514e6	−0.352098
$73$	−552310.	−0.166170	−0.0830851	−	0.996542i	$-0.526477\pi$
$73$	−552310.	−0.166170	−0.0830851	+	0.996542i	$0.526477\pi$
$74$	−1.00213e6	−0.287483
$75$	0	0
$76$	−492032.	−0.128572
$77$	789929.	0.197183
$78$	33144.0	0.00790813
$79$	−2.33442e6	−0.532702	−0.266351	−	0.963876i	$-0.585818\pi$
$79$	−2.33442e6	−0.532702	−0.266351	+	0.963876i	$0.585818\pi$
$80$	0	0
$81$	4.72400e6	0.987671
$82$	−5.91211e6	−1.18412
$83$	−219508.	−0.0421383	−0.0210692	−	0.999778i	$-0.506707\pi$
$83$	−219508.	−0.0421383	−0.0210692	+	0.999778i	$0.506707\pi$
$84$	−65856.0	−0.0121232
$85$	0	0
$86$	−2.35683e6	−0.399562
$87$	−471573.	−0.0767771
$88$	1.17914e6	0.184448
$89$	−3.15028e6	−0.473679	−0.236840	−	0.971549i	$-0.576112\pi$
$89$	−3.15028e6	−0.473679	−0.236840	+	0.971549i	$0.576112\pi$
$90$	0	0
$91$	−473683.	−0.0658936
$92$	−5.23584e6	−0.701018
$93$	119502.	0.0154058
$94$	5.24318e6	0.651099
$95$	0	0
$96$	−98304.0	−0.0113402
$97$	−1.21821e7	−1.35526	−0.677630	−	0.735403i	$-0.736993\pi$
$97$	−1.21821e7	−1.35526	−0.677630	+	0.735403i	$0.736993\pi$
$98$	941192.	0.101015
$99$	−5.01593e6	−0.519552
$100$	0	0
$101$	−8.78918e6	−0.848836	−0.424418	−	0.905466i	$-0.639521\pi$
$101$	−8.78918e6	−0.848836	−0.424418	+	0.905466i	$0.639521\pi$
$102$	−96216.0	−0.00897732
$103$	6.08668e6	0.548846	0.274423	−	0.961609i	$-0.411513\pi$
$103$	6.08668e6	0.548846	0.274423	+	0.961609i	$0.411513\pi$
$104$	−707072.	−0.0616378
$105$	0	0
$106$	−2.33547e6	−0.190460
$107$	3.17738e6	0.250742	0.125371	−	0.992110i	$-0.459988\pi$
$107$	3.17738e6	0.250742	0.125371	+	0.992110i	$0.459988\pi$
$108$	838080.	0.0640180
$109$	−1.50644e7	−1.11419	−0.557093	−	0.830450i	$-0.688083\pi$
$109$	−1.50644e7	−1.11419	−0.557093	+	0.830450i	$0.688083\pi$
$110$	0	0
$111$	375798.	0.0260810
$112$	1.40493e6	0.0944911
$113$	−1.74841e7	−1.13990	−0.569952	−	0.821678i	$-0.693038\pi$
$113$	−1.74841e7	−1.13990	−0.569952	+	0.821678i	$0.693038\pi$
$114$	184512.	0.0116643
$115$	0	0
$116$	1.00602e7	0.598418
$117$	3.00782e6	0.173621
$118$	−2.03354e7	−1.13937
$119$	1.37509e6	0.0748024
$120$	0	0
$121$	−1.41834e7	−0.727831
$122$	1.14982e7	0.573287
$123$	2.21704e6	0.107425
$124$	−2.54938e6	−0.120076
$125$	0	0
$126$	−5.97643e6	−0.266161
$127$	3.54207e7	1.53442	0.767210	−	0.641396i	$-0.221644\pi$
$127$	3.54207e7	1.53442	0.767210	+	0.641396i	$0.221644\pi$
$128$	2.09715e6	0.0883883
$129$	883812.	0.0362490
$130$	0	0
$131$	−8.64463e6	−0.335967	−0.167984	−	0.985790i	$-0.553726\pi$
$131$	−8.64463e6	−0.335967	−0.167984	+	0.985790i	$0.553726\pi$
$132$	−442176.	−0.0167335
$133$	−2.63698e6	−0.0971912
$134$	−2.52077e7	−0.905038
$135$	0	0
$136$	2.05261e6	0.0699713
$137$	−4.91067e7	−1.63162	−0.815809	−	0.578321i	$-0.803708\pi$
$137$	−4.91067e7	−1.63162	−0.815809	+	0.578321i	$0.803708\pi$
$138$	1.96344e6	0.0635976
$139$	5.04104e6	0.159209	0.0796047	−	0.996827i	$-0.474634\pi$
$139$	5.04104e6	0.159209	0.0796047	+	0.996827i	$0.474634\pi$
$140$	0	0
$141$	−1.96619e6	−0.0590689
$142$	1.69406e7	0.496501
$143$	−3.18044e6	−0.0909518
$144$	−8.92109e6	−0.248971
$145$	0	0
$146$	−4.41848e6	−0.117500
$147$	−352947.	−0.00916429
$148$	−8.01702e6	−0.203281
$149$	−3.39189e7	−0.840021	−0.420011	−	0.907519i	$-0.637974\pi$
$149$	−3.39189e7	−0.840021	−0.420011	+	0.907519i	$0.637974\pi$
$150$	0	0
$151$	−3.61357e7	−0.854116	−0.427058	−	0.904224i	$-0.640450\pi$
$151$	−3.61357e7	−0.854116	−0.427058	+	0.904224i	$0.640450\pi$
$152$	−3.93626e6	−0.0909140
$153$	−8.73160e6	−0.197094
$154$	6.31943e6	0.139430
$155$	0	0
$156$	265152.	0.00559189
$157$	1.67447e7	0.345326	0.172663	−	0.984981i	$-0.444763\pi$
$157$	1.67447e7	0.345326	0.172663	+	0.984981i	$0.444763\pi$
$158$	−1.86754e7	−0.376677
$159$	875802.	0.0172789
$160$	0	0
$161$	−2.80608e7	−0.529920
$162$	3.77920e7	0.698389
$163$	5.94510e7	1.07523	0.537616	−	0.843190i	$-0.319325\pi$
$163$	5.94510e7	1.07523	0.537616	+	0.843190i	$0.319325\pi$
$164$	−4.72969e7	−0.837296
$165$	0	0
$166$	−1.75606e6	−0.0297963
$167$	−5.24249e7	−0.871024	−0.435512	−	0.900183i	$-0.643433\pi$
$167$	−5.24249e7	−0.871024	−0.435512	+	0.900183i	$0.643433\pi$
$168$	−526848.	−0.00857241
$169$	−6.08414e7	−0.969606
$170$	0	0
$171$	1.67445e7	0.256085
$172$	−1.88547e7	−0.282533
$173$	9.61135e7	1.41131	0.705656	−	0.708554i	$-0.250652\pi$
$173$	9.61135e7	1.41131	0.705656	+	0.708554i	$0.250652\pi$
$174$	−3.77258e6	−0.0542896
$175$	0	0
$176$	9.43309e6	0.130425
$177$	7.62577e6	0.103366
$178$	−2.52022e7	−0.334942
$179$	−1.28054e8	−1.66882	−0.834409	−	0.551146i	$-0.814191\pi$
$179$	−1.28054e8	−1.66882	−0.834409	+	0.551146i	$0.814191\pi$
$180$	0	0
$181$	−1.38094e8	−1.73101	−0.865504	−	0.500901i	$-0.833002\pi$
$181$	−1.38094e8	−1.73101	−0.865504	+	0.500901i	$0.833002\pi$
$182$	−3.78946e6	−0.0465938
$183$	−4.31184e6	−0.0520096
$184$	−4.18867e7	−0.495694
$185$	0	0
$186$	956016.	0.0108936
$187$	9.23273e6	0.103249
$188$	4.19454e7	0.460396
$189$	4.49158e6	0.0483931
$190$	0	0
$191$	2.12961e7	0.221148	0.110574	−	0.993868i	$-0.464731\pi$
$191$	2.12961e7	0.221148	0.110574	+	0.993868i	$0.464731\pi$
$192$	−786432.	−0.00801875
$193$	1.65697e7	0.165907	0.0829535	−	0.996553i	$-0.473565\pi$
$193$	1.65697e7	0.165907	0.0829535	+	0.996553i	$0.473565\pi$
$194$	−9.74571e7	−0.958313
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	4.23674e7	0.394821	0.197411	−	0.980321i	$-0.436747\pi$
$197$	4.23674e7	0.394821	0.197411	+	0.980321i	$0.436747\pi$
$198$	−4.01275e7	−0.367378
$199$	−1.73127e8	−1.55732	−0.778661	−	0.627445i	$-0.784101\pi$
$199$	−1.73127e8	−1.55732	−0.778661	+	0.627445i	$0.784101\pi$
$200$	0	0
$201$	9.45290e6	0.0821067
$202$	−7.03134e7	−0.600217
$203$	5.39165e7	0.452362
$204$	−769728.	−0.00634792
$205$	0	0
$206$	4.86935e7	0.388093
$207$	1.78182e8	1.39627
$208$	−5.65658e6	−0.0435845
$209$	−1.77055e7	−0.134151
$210$	0	0
$211$	1.94275e8	1.42373	0.711866	−	0.702315i	$-0.247850\pi$
$211$	1.94275e8	1.42373	0.711866	+	0.702315i	$0.247850\pi$
$212$	−1.86838e7	−0.134676
$213$	−6.35273e6	−0.0450435
$214$	2.54191e7	0.177301
$215$	0	0
$216$	6.70464e6	0.0452676
$217$	−1.36631e7	−0.0907693
$218$	−1.20515e8	−0.787849
$219$	1.65693e6	0.0106598
$220$	0	0
$221$	−5.53643e6	−0.0345030
$222$	3.00638e6	0.0184420
$223$	8.21095e7	0.495822	0.247911	−	0.968783i	$-0.420256\pi$
$223$	8.21095e7	0.495822	0.247911	+	0.968783i	$0.420256\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	−1.39873e8	−0.806034
$227$	3.15611e8	1.79086	0.895430	−	0.445202i	$-0.146868\pi$
$227$	3.15611e8	1.79086	0.895430	+	0.445202i	$0.146868\pi$
$228$	1.47610e6	0.00824789
$229$	2.36109e8	1.29924	0.649619	−	0.760260i	$-0.274928\pi$
$229$	2.36109e8	1.29924	0.649619	+	0.760260i	$0.274928\pi$
$230$	0	0
$231$	−2.36979e6	−0.0126493
$232$	8.04818e7	0.423145
$233$	1.60204e8	0.829711	0.414856	−	0.909887i	$-0.363832\pi$
$233$	1.60204e8	0.829711	0.414856	+	0.909887i	$0.363832\pi$
$234$	2.40625e7	0.122768
$235$	0	0
$236$	−1.62683e8	−0.805657
$237$	7.00326e6	0.0341728
$238$	1.10007e7	0.0528933
$239$	−3.55108e8	−1.68255	−0.841274	−	0.540609i	$-0.818194\pi$
$239$	−3.55108e8	−1.68255	−0.841274	+	0.540609i	$0.818194\pi$
$240$	0	0
$241$	−9.64026e7	−0.443638	−0.221819	−	0.975088i	$-0.571199\pi$
$241$	−9.64026e7	−0.443638	−0.221819	+	0.975088i	$0.571199\pi$
$242$	−1.13467e8	−0.514654
$243$	−4.28108e7	−0.191395
$244$	9.19859e7	0.405375
$245$	0	0
$246$	1.77363e7	0.0759611
$247$	1.06171e7	0.0448299
$248$	−2.03950e7	−0.0849069
$249$	658524.	0.00270317
$250$	0	0
$251$	1.57871e8	0.630151	0.315076	−	0.949067i	$-0.397970\pi$
$251$	1.57871e8	0.630151	0.315076	+	0.949067i	$0.397970\pi$
$252$	−4.78115e7	−0.188205
$253$	−1.88408e8	−0.731440
$254$	2.83366e8	1.08500
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−1.86351e8	−0.684803	−0.342402	−	0.939554i	$-0.611240\pi$
$257$	−1.86351e8	−0.684803	−0.342402	+	0.939554i	$0.611240\pi$
$258$	7.07050e6	0.0256319
$259$	−4.29662e7	−0.153666
$260$	0	0
$261$	−3.42362e8	−1.19191
$262$	−6.91570e7	−0.237565
$263$	−2.33579e8	−0.791751	−0.395875	−	0.918304i	$-0.629559\pi$
$263$	−2.33579e8	−0.791751	−0.395875	+	0.918304i	$0.629559\pi$
$264$	−3.53741e6	−0.0118324
$265$	0	0
$266$	−2.10959e7	−0.0687245
$267$	9.45084e6	0.0303865
$268$	−2.01662e8	−0.639959
$269$	−8.11207e7	−0.254097	−0.127048	−	0.991897i	$-0.540550\pi$
$269$	−8.11207e7	−0.254097	−0.127048	+	0.991897i	$0.540550\pi$
$270$	0	0
$271$	−4.27197e8	−1.30387	−0.651937	−	0.758273i	$-0.726043\pi$
$271$	−4.27197e8	−1.30387	−0.651937	+	0.758273i	$0.726043\pi$
$272$	1.64209e7	0.0494772
$273$	1.42105e6	0.00422707
$274$	−3.92854e8	−1.15373
$275$	0	0
$276$	1.57075e7	0.0449703
$277$	2.93503e8	0.829724	0.414862	−	0.909884i	$-0.363830\pi$
$277$	2.93503e8	0.829724	0.414862	+	0.909884i	$0.363830\pi$
$278$	4.03283e7	0.112578
$279$	8.67585e7	0.239165
$280$	0	0
$281$	4.33253e8	1.16485	0.582424	−	0.812885i	$-0.302104\pi$
$281$	4.33253e8	1.16485	0.582424	+	0.812885i	$0.302104\pi$
$282$	−1.57295e7	−0.0417680
$283$	−2.49936e8	−0.655505	−0.327752	−	0.944764i	$-0.606291\pi$
$283$	−2.49936e8	−0.655505	−0.327752	+	0.944764i	$0.606291\pi$
$284$	1.35525e8	0.351079
$285$	0	0
$286$	−2.54435e7	−0.0643127
$287$	−2.53482e8	−0.632937
$288$	−7.13687e7	−0.176049
$289$	−3.94267e8	−0.960832
$290$	0	0
$291$	3.65464e7	0.0869399
$292$	−3.53478e7	−0.0830851
$293$	8.02521e8	1.86389	0.931943	−	0.362604i	$-0.118112\pi$
$293$	8.02521e8	1.86389	0.931943	+	0.362604i	$0.118112\pi$
$294$	−2.82358e6	−0.00648013
$295$	0	0
$296$	−6.41362e7	−0.143742
$297$	3.01578e7	0.0667962
$298$	−2.71352e8	−0.593985
$299$	1.12980e8	0.244428
$300$	0	0
$301$	−1.01049e8	−0.213575
$302$	−2.89085e8	−0.603951
$303$	2.63675e7	0.0544528
$304$	−3.14900e7	−0.0642859
$305$	0	0
$306$	−6.98528e7	−0.139367
$307$	−3.02427e8	−0.596536	−0.298268	−	0.954482i	$-0.596409\pi$
$307$	−3.02427e8	−0.596536	−0.298268	+	0.954482i	$0.596409\pi$
$308$	5.05555e7	0.0985917
$309$	−1.82601e7	−0.0352085
$310$	0	0
$311$	5.70669e8	1.07578	0.537889	−	0.843015i	$-0.319222\pi$
$311$	5.70669e8	1.07578	0.537889	+	0.843015i	$0.319222\pi$
$312$	2.12122e6	0.00395407
$313$	4.23943e8	0.781452	0.390726	−	0.920507i	$-0.372224\pi$
$313$	4.23943e8	0.781452	0.390726	+	0.920507i	$0.372224\pi$
$314$	1.33958e8	0.244182
$315$	0	0
$316$	−1.49403e8	−0.266351
$317$	−7.40954e8	−1.30642	−0.653212	−	0.757175i	$-0.726579\pi$
$317$	−7.40954e8	−1.30642	−0.653212	+	0.757175i	$0.726579\pi$
$318$	7.00642e6	0.0122180
$319$	3.62011e8	0.624388
$320$	0	0
$321$	−9.53215e6	−0.0160851
$322$	−2.24487e8	−0.374710
$323$	−3.08212e7	−0.0508910
$324$	3.02336e8	0.493836
$325$	0	0
$326$	4.75608e8	0.760304
$327$	4.51931e7	0.0714751
$328$	−3.78375e8	−0.592058
$329$	2.24801e8	0.348027
$330$	0	0
$331$	6.85452e8	1.03891	0.519457	−	0.854497i	$-0.326134\pi$
$331$	6.85452e8	1.03891	0.519457	+	0.854497i	$0.326134\pi$
$332$	−1.40485e7	−0.0210692
$333$	2.72829e8	0.404889
$334$	−4.19399e8	−0.615907
$335$	0	0
$336$	−4.21478e6	−0.00606161
$337$	7.06213e7	0.100515	0.0502576	−	0.998736i	$-0.483996\pi$
$337$	7.06213e7	0.100515	0.0502576	+	0.998736i	$0.483996\pi$
$338$	−4.86731e8	−0.685615
$339$	5.24522e7	0.0731249
$340$	0	0
$341$	−9.17377e7	−0.125287
$342$	1.33956e8	0.181080
$343$	4.03536e7	0.0539949
$344$	−1.50837e8	−0.199781
$345$	0	0
$346$	7.68908e8	0.997949
$347$	−5.95905e8	−0.765639	−0.382819	−	0.923823i	$-0.625047\pi$
$347$	−5.95905e8	−0.765639	−0.382819	+	0.923823i	$0.625047\pi$
$348$	−3.01807e7	−0.0383885
$349$	1.11603e9	1.40536	0.702680	−	0.711506i	$-0.251987\pi$
$349$	1.11603e9	1.40536	0.702680	+	0.711506i	$0.251987\pi$
$350$	0	0
$351$	−1.80842e7	−0.0223215
$352$	7.54647e7	0.0922241
$353$	1.26050e9	1.52522	0.762609	−	0.646860i	$-0.223918\pi$
$353$	1.26050e9	1.52522	0.762609	+	0.646860i	$0.223918\pi$
$354$	6.10061e7	0.0730907
$355$	0	0
$356$	−2.01618e8	−0.236840
$357$	−4.12526e6	−0.00479858
$358$	−1.02444e9	−1.18003
$359$	1.44717e7	0.0165078	0.00825390	−	0.999966i	$-0.497373\pi$
$359$	1.44717e7	0.0165078	0.00825390	+	0.999966i	$0.497373\pi$
$360$	0	0
$361$	−8.34766e8	−0.933877
$362$	−1.10475e9	−1.22401
$363$	4.25501e7	0.0466904
$364$	−3.03157e7	−0.0329468
$365$	0	0
$366$	−3.44947e7	−0.0367764
$367$	1.35964e9	1.43580	0.717900	−	0.696146i	$-0.245103\pi$
$367$	1.35964e9	1.43580	0.717900	+	0.696146i	$0.245103\pi$
$368$	−3.35094e8	−0.350509
$369$	1.60957e9	1.66770
$370$	0	0
$371$	−1.00133e8	−0.101805
$372$	7.64813e6	0.00770291
$373$	8.65452e8	0.863499	0.431750	−	0.901993i	$-0.357896\pi$
$373$	8.65452e8	0.863499	0.431750	+	0.901993i	$0.357896\pi$
$374$	7.38618e7	0.0730078
$375$	0	0
$376$	3.35563e8	0.325549
$377$	−2.17081e8	−0.208654
$378$	3.59327e7	0.0342191
$379$	2.06822e8	0.195146	0.0975731	−	0.995228i	$-0.468892\pi$
$379$	2.06822e8	0.195146	0.0975731	+	0.995228i	$0.468892\pi$
$380$	0	0
$381$	−1.06262e8	−0.0984331
$382$	1.70368e8	0.156375
$383$	3.21422e8	0.292335	0.146167	−	0.989260i	$-0.453306\pi$
$383$	3.21422e8	0.292335	0.146167	+	0.989260i	$0.453306\pi$
$384$	−6.29146e6	−0.00567012
$385$	0	0
$386$	1.32558e8	0.117314
$387$	6.41648e8	0.562740
$388$	−7.79657e8	−0.677630
$389$	5.80546e8	0.500050	0.250025	−	0.968239i	$-0.419561\pi$
$389$	5.80546e8	0.500050	0.250025	+	0.968239i	$0.419561\pi$
$390$	0	0
$391$	−3.27976e8	−0.277475
$392$	6.02363e7	0.0505076
$393$	2.59339e7	0.0215523
$394$	3.38939e8	0.279181
$395$	0	0
$396$	−3.21020e8	−0.259776
$397$	1.02652e9	0.823380	0.411690	−	0.911324i	$-0.364939\pi$
$397$	1.02652e9	0.823380	0.411690	+	0.911324i	$0.364939\pi$
$398$	−1.38501e9	−1.10119
$399$	7.91095e6	0.00623482
$400$	0	0
$401$	4.12720e8	0.319632	0.159816	−	0.987147i	$-0.448910\pi$
$401$	4.12720e8	0.319632	0.159816	+	0.987147i	$0.448910\pi$
$402$	7.56232e7	0.0580582
$403$	5.50108e7	0.0418678
$404$	−5.62508e8	−0.424418
$405$	0	0
$406$	4.31332e8	0.319868
$407$	−2.88488e8	−0.212103
$408$	−6.15782e6	−0.00448866
$409$	1.84599e9	1.33413	0.667064	−	0.745001i	$-0.267551\pi$
$409$	1.84599e9	1.33413	0.667064	+	0.745001i	$0.267551\pi$
$410$	0	0
$411$	1.47320e8	0.104668
$412$	3.89548e8	0.274423
$413$	−8.71879e8	−0.609019
$414$	1.42546e9	0.987309
$415$	0	0
$416$	−4.52526e7	−0.0308189
$417$	−1.51231e7	−0.0102133
$418$	−1.41644e8	−0.0948594
$419$	1.27567e9	0.847205	0.423603	−	0.905848i	$-0.360765\pi$
$419$	1.27567e9	0.847205	0.423603	+	0.905848i	$0.360765\pi$
$420$	0	0
$421$	1.30451e8	0.0852041	0.0426020	−	0.999092i	$-0.486435\pi$
$421$	1.30451e8	0.0852041	0.0426020	+	0.999092i	$0.486435\pi$
$422$	1.55420e9	1.00673
$423$	−1.42745e9	−0.917004
$424$	−1.49470e8	−0.0952301
$425$	0	0
$426$	−5.08218e7	−0.0318505
$427$	4.92987e8	0.306435
$428$	2.03353e8	0.125371
$429$	9.54133e6	0.00583456
$430$	0	0
$431$	2.60852e9	1.56937	0.784683	−	0.619897i	$-0.212826\pi$
$431$	2.60852e9	1.56937	0.784683	+	0.619897i	$0.212826\pi$
$432$	5.36371e7	0.0320090
$433$	6.77620e8	0.401124	0.200562	−	0.979681i	$-0.435723\pi$
$433$	6.77620e8	0.401124	0.200562	+	0.979681i	$0.435723\pi$
$434$	−1.09304e8	−0.0641836
$435$	0	0
$436$	−9.64119e8	−0.557093
$437$	6.28955e8	0.360524
$438$	1.32554e7	0.00753763
$439$	−7.90402e8	−0.445884	−0.222942	−	0.974832i	$-0.571566\pi$
$439$	−7.90402e8	−0.445884	−0.222942	+	0.974832i	$0.571566\pi$
$440$	0	0
$441$	−2.56240e8	−0.142269
$442$	−4.42914e7	−0.0243973
$443$	2.13809e9	1.16846	0.584228	−	0.811589i	$-0.301397\pi$
$443$	2.13809e9	1.16846	0.584228	+	0.811589i	$0.301397\pi$
$444$	2.40511e7	0.0130405
$445$	0	0
$446$	6.56876e8	0.350599
$447$	1.01757e8	0.0538874
$448$	8.99154e7	0.0472456
$449$	2.81523e7	0.0146775	0.00733875	−	0.999973i	$-0.497664\pi$
$449$	2.81523e7	0.0146775	0.00733875	+	0.999973i	$0.497664\pi$
$450$	0	0
$451$	−1.70195e9	−0.873632
$452$	−1.11898e9	−0.569952
$453$	1.08407e8	0.0547916
$454$	2.52489e9	1.26633
$455$	0	0
$456$	1.18088e7	0.00583214
$457$	−7.59179e8	−0.372081	−0.186041	−	0.982542i	$-0.559566\pi$
$457$	−7.59179e8	−0.372081	−0.186041	+	0.982542i	$0.559566\pi$
$458$	1.88887e9	0.918700
$459$	5.24979e7	0.0253394
$460$	0	0
$461$	3.30359e9	1.57048	0.785240	−	0.619191i	$-0.212539\pi$
$461$	3.30359e9	1.57048	0.785240	+	0.619191i	$0.212539\pi$
$462$	−1.89583e7	−0.00894442
$463$	3.42142e9	1.60204	0.801020	−	0.598638i	$-0.204291\pi$
$463$	3.42142e9	1.60204	0.801020	+	0.598638i	$0.204291\pi$
$464$	6.43854e8	0.299209
$465$	0	0
$466$	1.28163e9	0.586695
$467$	2.96740e9	1.34824	0.674120	−	0.738622i	$-0.264523\pi$
$467$	2.96740e9	1.34824	0.674120	+	0.738622i	$0.264523\pi$
$468$	1.92500e8	0.0868103
$469$	−1.08078e9	−0.483763
$470$	0	0
$471$	−5.02342e7	−0.0221527
$472$	−1.30146e9	−0.569686
$473$	−6.78473e8	−0.294794
$474$	5.60261e7	0.0241638
$475$	0	0
$476$	8.80056e7	0.0374012
$477$	6.35832e8	0.268243
$478$	−2.84086e9	−1.18974
$479$	3.49512e9	1.45308	0.726538	−	0.687127i	$-0.241128\pi$
$479$	3.49512e9	1.45308	0.726538	+	0.687127i	$0.241128\pi$
$480$	0	0
$481$	1.72992e8	0.0708793
$482$	−7.71221e8	−0.313700
$483$	8.41825e7	0.0339944
$484$	−9.07735e8	−0.363915
$485$	0	0
$486$	−3.42486e8	−0.135337
$487$	−2.32240e9	−0.911141	−0.455571	−	0.890200i	$-0.650565\pi$
$487$	−2.32240e9	−0.911141	−0.455571	+	0.890200i	$0.650565\pi$
$488$	7.35887e8	0.286643
$489$	−1.78353e8	−0.0689762
$490$	0	0
$491$	−3.54551e9	−1.35174	−0.675871	−	0.737020i	$-0.736232\pi$
$491$	−3.54551e9	−1.35174	−0.675871	+	0.737020i	$0.736232\pi$
$492$	1.41891e8	0.0537126
$493$	6.30179e8	0.236864
$494$	8.49370e7	0.0316995
$495$	0	0
$496$	−1.63160e8	−0.0600382
$497$	7.26329e8	0.265391
$498$	5.26819e6	0.00191143
$499$	−5.30900e9	−1.91276	−0.956381	−	0.292123i	$-0.905638\pi$
$499$	−5.30900e9	−1.91276	−0.956381	+	0.292123i	$0.905638\pi$
$500$	0	0
$501$	1.57275e8	0.0558762
$502$	1.26297e9	0.445584
$503$	−2.07831e9	−0.728151	−0.364076	−	0.931369i	$-0.618615\pi$
$503$	−2.07831e9	−0.728151	−0.364076	+	0.931369i	$0.618615\pi$
$504$	−3.82492e8	−0.133081
$505$	0	0
$506$	−1.50727e9	−0.517206
$507$	1.82524e8	0.0622003
$508$	2.26693e9	0.767210
$509$	−5.30242e9	−1.78222	−0.891112	−	0.453784i	$-0.850074\pi$
$509$	−5.30242e9	−1.78222	−0.891112	+	0.453784i	$0.850074\pi$
$510$	0	0
$511$	−1.89442e8	−0.0628064
$512$	1.34218e8	0.0441942
$513$	−1.00674e8	−0.0329237
$514$	−1.49081e9	−0.484229
$515$	0	0
$516$	5.65640e7	0.0181245
$517$	1.50938e9	0.480376
$518$	−3.43730e8	−0.108658
$519$	−2.88341e8	−0.0905357
$520$	0	0
$521$	−1.30449e9	−0.404118	−0.202059	−	0.979373i	$-0.564763\pi$
$521$	−1.30449e9	−0.404118	−0.202059	+	0.979373i	$0.564763\pi$
$522$	−2.73890e9	−0.842808
$523$	2.26437e9	0.692136	0.346068	−	0.938210i	$-0.387517\pi$
$523$	2.26437e9	0.692136	0.346068	+	0.938210i	$0.387517\pi$
$524$	−5.53256e8	−0.167984
$525$	0	0
$526$	−1.86863e9	−0.559852
$527$	−1.59695e8	−0.0475284
$528$	−2.82993e7	−0.00836674
$529$	3.28805e9	0.965703
$530$	0	0
$531$	5.53631e9	1.60468
$532$	−1.68767e8	−0.0485956
$533$	1.02058e9	0.291945
$534$	7.56067e7	0.0214865
$535$	0	0
$536$	−1.61329e9	−0.452519
$537$	3.84163e8	0.107055
$538$	−6.48966e8	−0.179673
$539$	2.70946e8	0.0745283
$540$	0	0
$541$	2.94233e9	0.798915	0.399458	−	0.916752i	$-0.369198\pi$
$541$	2.94233e9	0.798915	0.399458	+	0.916752i	$0.369198\pi$
$542$	−3.41757e9	−0.921978
$543$	4.14281e8	0.111044
$544$	1.31367e8	0.0349856
$545$	0	0
$546$	1.13684e7	0.00298899
$547$	−2.04414e9	−0.534016	−0.267008	−	0.963694i	$-0.586035\pi$
$547$	−2.04414e9	−0.534016	−0.267008	+	0.963694i	$0.586035\pi$
$548$	−3.14283e9	−0.815809
$549$	−3.13040e9	−0.807414
$550$	0	0
$551$	−1.20848e9	−0.307759
$552$	1.25660e8	0.0317988
$553$	−8.00706e8	−0.201342
$554$	2.34803e9	0.586704
$555$	0	0
$556$	3.22627e8	0.0796047
$557$	5.47095e9	1.34144	0.670718	−	0.741713i	$-0.265986\pi$
$557$	5.47095e9	1.34144	0.670718	+	0.741713i	$0.265986\pi$
$558$	6.94068e8	0.169115
$559$	4.06848e8	0.0985124
$560$	0	0
$561$	−2.76982e7	−0.00662340
$562$	3.46602e9	0.823672
$563$	−3.36364e9	−0.794384	−0.397192	−	0.917735i	$-0.630015\pi$
$563$	−3.36364e9	−0.794384	−0.397192	+	0.917735i	$0.630015\pi$
$564$	−1.25836e8	−0.0295344
$565$	0	0
$566$	−1.99949e9	−0.463512
$567$	1.62033e9	0.373305
$568$	1.08420e9	0.248250
$569$	−4.28774e9	−0.975744	−0.487872	−	0.872915i	$-0.662227\pi$
$569$	−4.28774e9	−0.975744	−0.487872	+	0.872915i	$0.662227\pi$
$570$	0	0
$571$	−8.17590e9	−1.83785	−0.918923	−	0.394437i	$-0.870940\pi$
$571$	−8.17590e9	−1.83785	−0.918923	+	0.394437i	$0.870940\pi$
$572$	−2.03548e8	−0.0454759
$573$	−6.38882e7	−0.0141866
$574$	−2.02785e9	−0.447554
$575$	0	0
$576$	−5.70950e8	−0.124486
$577$	−2.91623e9	−0.631986	−0.315993	−	0.948762i	$-0.602338\pi$
$577$	−2.91623e9	−0.631986	−0.315993	+	0.948762i	$0.602338\pi$
$578$	−3.15413e9	−0.679411
$579$	−4.97092e7	−0.0106429
$580$	0	0
$581$	−7.52912e7	−0.0159268
$582$	2.92371e8	0.0614758
$583$	−6.72324e8	−0.140520
$584$	−2.82783e8	−0.0587500
$585$	0	0
$586$	6.42016e9	1.31797
$587$	−2.55546e9	−0.521477	−0.260739	−	0.965409i	$-0.583966\pi$
$587$	−2.55546e9	−0.521477	−0.260739	+	0.965409i	$0.583966\pi$
$588$	−2.25886e7	−0.00458214
$589$	3.06244e8	0.0617538
$590$	0	0
$591$	−1.27102e8	−0.0253278
$592$	−5.13090e8	−0.101641
$593$	−7.96504e9	−1.56854	−0.784272	−	0.620417i	$-0.786963\pi$
$593$	−7.96504e9	−1.56854	−0.784272	+	0.620417i	$0.786963\pi$
$594$	2.41262e8	0.0472321
$595$	0	0
$596$	−2.17081e9	−0.420011
$597$	5.19380e8	0.0999023
$598$	9.03837e8	0.172837
$599$	5.10160e9	0.969869	0.484934	−	0.874551i	$-0.338844\pi$
$599$	5.10160e9	0.969869	0.484934	+	0.874551i	$0.338844\pi$
$600$	0	0
$601$	−2.27123e9	−0.426776	−0.213388	−	0.976967i	$-0.568450\pi$
$601$	−2.27123e9	−0.426776	−0.213388	+	0.976967i	$0.568450\pi$
$602$	−8.08393e8	−0.151020
$603$	6.86280e9	1.27465
$604$	−2.31268e9	−0.427058
$605$	0	0
$606$	2.10940e8	0.0385040
$607$	−3.21591e9	−0.583638	−0.291819	−	0.956474i	$-0.594260\pi$
$607$	−3.21591e9	−0.583638	−0.291819	+	0.956474i	$0.594260\pi$
$608$	−2.51920e8	−0.0454570
$609$	−1.61750e8	−0.0290190
$610$	0	0
$611$	−9.05103e8	−0.160529
$612$	−5.58823e8	−0.0985471
$613$	−2.39288e9	−0.419574	−0.209787	−	0.977747i	$-0.567277\pi$
$613$	−2.39288e9	−0.419574	−0.209787	+	0.977747i	$0.567277\pi$
$614$	−2.41942e9	−0.421815
$615$	0	0
$616$	4.04444e8	0.0697149
$617$	−6.09839e9	−1.04524	−0.522621	−	0.852565i	$-0.675046\pi$
$617$	−6.09839e9	−1.04524	−0.522621	+	0.852565i	$0.675046\pi$
$618$	−1.46080e8	−0.0248962
$619$	4.49455e9	0.761674	0.380837	−	0.924642i	$-0.375636\pi$
$619$	4.49455e9	0.761674	0.380837	+	0.924642i	$0.375636\pi$
$620$	0	0
$621$	−1.07130e9	−0.179511
$622$	4.56535e9	0.760690
$623$	−1.08055e9	−0.179034
$624$	1.69697e7	0.00279595
$625$	0	0
$626$	3.39154e9	0.552570
$627$	5.31164e7	0.00860582
$628$	1.07166e9	0.172663
$629$	−5.02191e8	−0.0804622
$630$	0	0
$631$	−5.29152e9	−0.838451	−0.419225	−	0.907882i	$-0.637698\pi$
$631$	−5.29152e9	−0.838451	−0.419225	+	0.907882i	$0.637698\pi$
$632$	−1.19522e9	−0.188338
$633$	−5.82825e8	−0.0913324
$634$	−5.92763e9	−0.923781
$635$	0	0
$636$	5.60513e7	0.00863945
$637$	−1.62473e8	−0.0249054
$638$	2.89609e9	0.441509
$639$	−4.61208e9	−0.699268
$640$	0	0
$641$	1.94450e9	0.291612	0.145806	−	0.989313i	$-0.453422\pi$
$641$	1.94450e9	0.291612	0.145806	+	0.989313i	$0.453422\pi$
$642$	−7.62572e7	−0.0113739
$643$	5.84495e9	0.867047	0.433523	−	0.901142i	$-0.357270\pi$
$643$	5.84495e9	0.867047	0.433523	+	0.901142i	$0.357270\pi$
$644$	−1.79589e9	−0.264960
$645$	0	0
$646$	−2.46570e8	−0.0359853
$647$	−8.35753e9	−1.21315	−0.606573	−	0.795028i	$-0.707456\pi$
$647$	−8.35753e9	−1.21315	−0.606573	+	0.795028i	$0.707456\pi$
$648$	2.41869e9	0.349195
$649$	−5.85405e9	−0.840620
$650$	0	0
$651$	4.09892e7	0.00582285
$652$	3.80486e9	0.537616
$653$	−9.47179e9	−1.33118	−0.665588	−	0.746319i	$-0.731819\pi$
$653$	−9.47179e9	−1.33118	−0.665588	+	0.746319i	$0.731819\pi$
$654$	3.61545e8	0.0505405
$655$	0	0
$656$	−3.02700e9	−0.418648
$657$	1.20293e9	0.165486
$658$	1.79841e9	0.246092
$659$	−9.59528e9	−1.30605	−0.653024	−	0.757338i	$-0.726500\pi$
$659$	−9.59528e9	−1.30605	−0.653024	+	0.757338i	$0.726500\pi$
$660$	0	0
$661$	−3.05680e9	−0.411683	−0.205841	−	0.978585i	$-0.565993\pi$
$661$	−3.05680e9	−0.411683	−0.205841	+	0.978585i	$0.565993\pi$
$662$	5.48362e9	0.734622
$663$	1.66093e7	0.00221337
$664$	−1.12388e8	−0.0148981
$665$	0	0
$666$	2.18263e9	0.286300
$667$	−1.28598e10	−1.67801
$668$	−3.35519e9	−0.435512
$669$	−2.46328e8	−0.0318070
$670$	0	0
$671$	3.31006e9	0.422967
$672$	−3.37183e7	−0.00428620
$673$	2.91477e9	0.368597	0.184299	−	0.982870i	$-0.440999\pi$
$673$	2.91477e9	0.368597	0.184299	+	0.982870i	$0.440999\pi$
$674$	5.64971e8	0.0710749
$675$	0	0
$676$	−3.89385e9	−0.484803
$677$	1.04043e10	1.28870	0.644348	−	0.764732i	$-0.277129\pi$
$677$	1.04043e10	1.28870	0.644348	+	0.764732i	$0.277129\pi$
$678$	4.19618e8	0.0517071
$679$	−4.17847e9	−0.512240
$680$	0	0
$681$	−9.46833e8	−0.114884
$682$	−7.33902e8	−0.0885916
$683$	−9.43735e9	−1.13339	−0.566693	−	0.823929i	$-0.691777\pi$
$683$	−9.43735e9	−1.13339	−0.566693	+	0.823929i	$0.691777\pi$
$684$	1.07165e9	0.128043
$685$	0	0
$686$	3.22829e8	0.0381802
$687$	−7.08328e8	−0.0833462
$688$	−1.20670e9	−0.141266
$689$	4.03161e8	0.0469582
$690$	0	0
$691$	−2.32820e9	−0.268440	−0.134220	−	0.990952i	$-0.542853\pi$
$691$	−2.32820e9	−0.268440	−0.134220	+	0.990952i	$0.542853\pi$
$692$	6.15127e9	0.705656
$693$	−1.72047e9	−0.196372
$694$	−4.76724e9	−0.541388
$695$	0	0
$696$	−2.41445e8	−0.0271448
$697$	−2.96271e9	−0.331416
$698$	8.92825e9	0.993739
$699$	−4.80611e8	−0.0532260
$700$	0	0
$701$	1.49770e9	0.164215	0.0821075	−	0.996623i	$-0.473835\pi$
$701$	1.49770e9	0.164215	0.0821075	+	0.996623i	$0.473835\pi$
$702$	−1.44674e8	−0.0157837
$703$	9.63045e8	0.104545
$704$	6.03718e8	0.0652123
$705$	0	0
$706$	1.00840e10	1.07849
$707$	−3.01469e9	−0.320830
$708$	4.88049e8	0.0516829
$709$	−7.48971e9	−0.789229	−0.394615	−	0.918847i	$-0.629122\pi$
$709$	−7.48971e9	−0.789229	−0.394615	+	0.918847i	$0.629122\pi$
$710$	0	0
$711$	5.08436e9	0.530509
$712$	−1.61294e9	−0.167471
$713$	3.25882e9	0.336703
$714$	−3.30021e7	−0.00339311
$715$	0	0
$716$	−8.19548e9	−0.834409
$717$	1.06532e9	0.107935
$718$	1.15774e8	0.0116728
$719$	−4.25246e9	−0.426667	−0.213333	−	0.976979i	$-0.568432\pi$
$719$	−4.25246e9	−0.426667	−0.213333	+	0.976979i	$0.568432\pi$
$720$	0	0
$721$	2.08773e9	0.207444
$722$	−6.67813e9	−0.660351
$723$	2.89208e8	0.0284594
$724$	−8.83800e9	−0.865504
$725$	0	0
$726$	3.40401e8	0.0330151
$727$	−3.55527e9	−0.343165	−0.171582	−	0.985170i	$-0.554888\pi$
$727$	−3.55527e9	−0.343165	−0.171582	+	0.985170i	$0.554888\pi$
$728$	−2.42526e8	−0.0232969
$729$	−1.02030e10	−0.975393
$730$	0	0
$731$	−1.18107e9	−0.111831
$732$	−2.75958e8	−0.0260048
$733$	1.97546e10	1.85270	0.926350	−	0.376664i	$-0.122929\pi$
$733$	1.97546e10	1.85270	0.926350	+	0.376664i	$0.122929\pi$
$734$	1.08772e10	1.01526
$735$	0	0
$736$	−2.68075e9	−0.247847
$737$	−7.25667e9	−0.667731
$738$	1.28766e10	1.17924
$739$	3.56697e9	0.325121	0.162560	−	0.986699i	$-0.448025\pi$
$739$	3.56697e9	0.325121	0.162560	+	0.986699i	$0.448025\pi$
$740$	0	0
$741$	−3.18514e7	−0.00287584
$742$	−8.01067e8	−0.0719872
$743$	−8.93144e9	−0.798842	−0.399421	−	0.916768i	$-0.630789\pi$
$743$	−8.93144e9	−0.798842	−0.399421	+	0.916768i	$0.630789\pi$
$744$	6.11850e7	0.00544678
$745$	0	0
$746$	6.92362e9	0.610586
$747$	4.78088e8	0.0419649
$748$	5.90895e8	0.0516243
$749$	1.08984e9	0.0947714
$750$	0	0
$751$	−8.26094e9	−0.711688	−0.355844	−	0.934545i	$-0.615807\pi$
$751$	−8.26094e9	−0.711688	−0.355844	+	0.934545i	$0.615807\pi$
$752$	2.68451e9	0.230198
$753$	−4.73613e8	−0.0404242
$754$	−1.73665e9	−0.147541
$755$	0	0
$756$	2.87461e8	0.0241965
$757$	1.65500e10	1.38664	0.693318	−	0.720632i	$-0.256148\pi$
$757$	1.65500e10	1.38664	0.693318	+	0.720632i	$0.256148\pi$
$758$	1.65458e9	0.137989
$759$	5.65225e8	0.0469219
$760$	0	0
$761$	−7.87845e9	−0.648029	−0.324014	−	0.946052i	$-0.605033\pi$
$761$	−7.87845e9	−0.648029	−0.324014	+	0.946052i	$0.605033\pi$
$762$	−8.50097e8	−0.0696027
$763$	−5.16708e9	−0.421123
$764$	1.36295e9	0.110574
$765$	0	0
$766$	2.57138e9	0.206712
$767$	3.51039e9	0.280913
$768$	−5.03316e7	−0.00400938
$769$	9.00174e9	0.713813	0.356907	−	0.934140i	$-0.383831\pi$
$769$	9.00174e9	0.713813	0.356907	+	0.934140i	$0.383831\pi$
$770$	0	0
$771$	5.59053e8	0.0439301
$772$	1.06046e9	0.0829535
$773$	−2.39995e10	−1.86885	−0.934424	−	0.356163i	$-0.884085\pi$
$773$	−2.39995e10	−1.86885	−0.934424	+	0.356163i	$0.884085\pi$
$774$	5.13318e9	0.397918
$775$	0	0
$776$	−6.23725e9	−0.479156
$777$	1.28899e8	0.00985769
$778$	4.64437e9	0.353588
$779$	5.68154e9	0.430611
$780$	0	0
$781$	4.87678e9	0.366315
$782$	−2.62381e9	−0.196204
$783$	2.05842e9	0.153238
$784$	4.81890e8	0.0357143
$785$	0	0
$786$	2.07471e8	0.0152398
$787$	1.58050e9	0.115580	0.0577901	−	0.998329i	$-0.481595\pi$
$787$	1.58050e9	0.115580	0.0577901	+	0.998329i	$0.481595\pi$
$788$	2.71152e9	0.197411
$789$	7.00737e8	0.0507908
$790$	0	0
$791$	−5.99704e9	−0.430843
$792$	−2.56816e9	−0.183689
$793$	−1.98488e9	−0.141344
$794$	8.21216e9	0.582218
$795$	0	0
$796$	−1.10801e10	−0.778661
$797$	1.88835e10	1.32123	0.660613	−	0.750726i	$-0.270296\pi$
$797$	1.88835e10	1.32123	0.660613	+	0.750726i	$0.270296\pi$
$798$	6.32876e7	0.00440868
$799$	2.62749e9	0.182233
$800$	0	0
$801$	6.86131e9	0.471730
$802$	3.30176e9	0.226014
$803$	−1.27197e9	−0.0866907
$804$	6.04985e8	0.0410534
$805$	0	0
$806$	4.40086e8	0.0296050
$807$	2.43362e8	0.0163003
$808$	−4.50006e9	−0.300109
$809$	5.61160e8	0.0372621	0.0186310	−	0.999826i	$-0.494069\pi$
$809$	5.61160e8	0.0372621	0.0186310	+	0.999826i	$0.494069\pi$
$810$	0	0
$811$	6.94523e9	0.457208	0.228604	−	0.973520i	$-0.426584\pi$
$811$	6.94523e9	0.457208	0.228604	+	0.973520i	$0.426584\pi$
$812$	3.45066e9	0.226181
$813$	1.28159e9	0.0836435
$814$	−2.30790e9	−0.149979
$815$	0	0
$816$	−4.92626e7	−0.00317396
$817$	2.26492e9	0.145303
$818$	1.47679e10	0.943370
$819$	1.03168e9	0.0656224
$820$	0	0
$821$	8.87330e9	0.559608	0.279804	−	0.960057i	$-0.409731\pi$
$821$	8.87330e9	0.559608	0.279804	+	0.960057i	$0.409731\pi$
$822$	1.17856e9	0.0740117
$823$	−1.87347e10	−1.17152	−0.585758	−	0.810486i	$-0.699203\pi$
$823$	−1.87347e10	−1.17152	−0.585758	+	0.810486i	$0.699203\pi$
$824$	3.11638e9	0.194046
$825$	0	0
$826$	−6.97503e9	−0.430642
$827$	1.69343e10	1.04111	0.520556	−	0.853828i	$-0.325725\pi$
$827$	1.69343e10	1.04111	0.520556	+	0.853828i	$0.325725\pi$
$828$	1.14037e10	0.698133
$829$	−3.44175e8	−0.0209816	−0.0104908	−	0.999945i	$-0.503339\pi$
$829$	−3.44175e8	−0.0209816	−0.0104908	+	0.999945i	$0.503339\pi$
$830$	0	0
$831$	−8.80510e8	−0.0532268
$832$	−3.62021e8	−0.0217922
$833$	4.71655e8	0.0282727
$834$	−1.20985e8	−0.00722188
$835$	0	0
$836$	−1.13315e9	−0.0670757
$837$	−5.21626e8	−0.0307482
$838$	1.02053e10	0.599065
$839$	2.98584e10	1.74542	0.872709	−	0.488241i	$-0.162361\pi$
$839$	2.98584e10	1.74542	0.872709	+	0.488241i	$0.162361\pi$
$840$	0	0
$841$	7.45913e9	0.432417
$842$	1.04361e9	0.0602484
$843$	−1.29976e9	−0.0747250
$844$	1.24336e10	0.711866
$845$	0	0
$846$	−1.14196e10	−0.648419
$847$	−4.86489e9	−0.275094
$848$	−1.19576e9	−0.0673378
$849$	7.49807e8	0.0420506
$850$	0	0
$851$	1.02480e10	0.570015
$852$	−4.06575e8	−0.0225217
$853$	−9.16319e9	−0.505504	−0.252752	−	0.967531i	$-0.581336\pi$
$853$	−9.16319e9	−0.505504	−0.252752	+	0.967531i	$0.581336\pi$
$854$	3.94390e9	0.216682
$855$	0	0
$856$	1.62682e9	0.0886506
$857$	1.12085e10	0.608297	0.304148	−	0.952625i	$-0.401628\pi$
$857$	1.12085e10	0.608297	0.304148	+	0.952625i	$0.401628\pi$
$858$	7.63306e7	0.00412566
$859$	7.88838e9	0.424631	0.212315	−	0.977201i	$-0.431900\pi$
$859$	7.88838e9	0.424631	0.212315	+	0.977201i	$0.431900\pi$
$860$	0	0
$861$	7.60445e8	0.0406029
$862$	2.08682e10	1.10971
$863$	−2.69012e10	−1.42473	−0.712367	−	0.701807i	$-0.752377\pi$
$863$	−2.69012e10	−1.42473	−0.712367	+	0.701807i	$0.752377\pi$
$864$	4.29097e8	0.0226338
$865$	0	0
$866$	5.42096e9	0.283637
$867$	1.18280e9	0.0616374
$868$	−8.74436e8	−0.0453846
$869$	−5.37617e9	−0.277910
$870$	0	0
$871$	4.35148e9	0.223138
$872$	−7.71295e9	−0.393924
$873$	2.65327e10	1.34968
$874$	5.03164e9	0.254929
$875$	0	0
$876$	1.06044e8	0.00532991
$877$	1.00443e10	0.502830	0.251415	−	0.967879i	$-0.419104\pi$
$877$	1.00443e10	0.502830	0.251415	+	0.967879i	$0.419104\pi$
$878$	−6.32322e9	−0.315288
$879$	−2.40756e9	−0.119568
$880$	0	0
$881$	−2.81294e10	−1.38594	−0.692971	−	0.720965i	$-0.743699\pi$
$881$	−2.81294e10	−1.38594	−0.692971	+	0.720965i	$0.743699\pi$
$882$	−2.04992e9	−0.100600
$883$	−2.91109e10	−1.42296	−0.711480	−	0.702706i	$-0.751975\pi$
$883$	−2.91109e10	−1.42296	−0.711480	+	0.702706i	$0.751975\pi$
$884$	−3.54331e8	−0.0172515
$885$	0	0
$886$	1.71047e10	0.826224
$887$	−2.06382e10	−0.992976	−0.496488	−	0.868044i	$-0.665377\pi$
$887$	−2.06382e10	−0.992976	−0.496488	+	0.868044i	$0.665377\pi$
$888$	1.92409e8	0.00922102
$889$	1.21493e10	0.579956
$890$	0	0
$891$	1.08794e10	0.515267
$892$	5.25500e9	0.247911
$893$	−5.03869e9	−0.236776
$894$	8.14055e8	0.0381041
$895$	0	0
$896$	7.19323e8	0.0334077
$897$	−3.38939e8	−0.0156801
$898$	2.25219e8	0.0103786
$899$	−6.26155e9	−0.287424
$900$	0	0
$901$	−1.17036e9	−0.0533070
$902$	−1.36156e10	−0.617751
$903$	3.03148e8	0.0137008
$904$	−8.95185e9	−0.403017
$905$	0	0
$906$	8.67256e8	0.0387435
$907$	−1.03208e10	−0.459289	−0.229645	−	0.973275i	$-0.573756\pi$
$907$	−1.03208e10	−0.459289	−0.229645	+	0.973275i	$0.573756\pi$
$908$	2.01991e10	0.895430
$909$	1.91428e10	0.845342
$910$	0	0
$911$	2.93871e10	1.28778	0.643890	−	0.765118i	$-0.277319\pi$
$911$	2.93871e10	1.28778	0.643890	+	0.765118i	$0.277319\pi$
$912$	9.44701e7	0.00412394
$913$	−5.05527e8	−0.0219835
$914$	−6.07343e9	−0.263101
$915$	0	0
$916$	1.51110e10	0.649619
$917$	−2.96511e9	−0.126984
$918$	4.19983e8	0.0179177
$919$	1.36213e10	0.578914	0.289457	−	0.957191i	$-0.406525\pi$
$919$	1.36213e10	0.578914	0.289457	+	0.957191i	$0.406525\pi$
$920$	0	0
$921$	9.07282e8	0.0382678
$922$	2.64287e10	1.11050
$923$	−2.92437e9	−0.122413
$924$	−1.51666e8	−0.00632466
$925$	0	0
$926$	2.73714e10	1.13281
$927$	−1.32568e10	−0.546587
$928$	5.15083e9	0.211573
$929$	3.82858e10	1.56669	0.783344	−	0.621588i	$-0.213512\pi$
$929$	3.82858e10	1.56669	0.783344	+	0.621588i	$0.213512\pi$
$930$	0	0
$931$	−9.04486e8	−0.0367348
$932$	1.02530e10	0.414856
$933$	−1.71201e9	−0.0690112
$934$	2.37392e10	0.953350
$935$	0	0
$936$	1.54000e9	0.0613841
$937$	−4.25939e10	−1.69145	−0.845724	−	0.533621i	$-0.820831\pi$
$937$	−4.25939e10	−1.69145	−0.845724	+	0.533621i	$0.820831\pi$
$938$	−8.64625e9	−0.342072
$939$	−1.27183e9	−0.0501301
$940$	0	0
$941$	8.79813e9	0.344213	0.172106	−	0.985078i	$-0.444943\pi$
$941$	8.79813e9	0.344213	0.172106	+	0.985078i	$0.444943\pi$
$942$	−4.01874e8	−0.0156643
$943$	6.04587e10	2.34784
$944$	−1.04117e10	−0.402828
$945$	0	0
$946$	−5.42778e9	−0.208451
$947$	−3.15926e10	−1.20882	−0.604408	−	0.796675i	$-0.706590\pi$
$947$	−3.15926e10	−1.20882	−0.604408	+	0.796675i	$0.706590\pi$
$948$	4.48208e8	0.0170864
$949$	7.62740e8	0.0289698
$950$	0	0
$951$	2.22286e9	0.0838071
$952$	7.04045e8	0.0264467
$953$	4.47398e10	1.67444	0.837218	−	0.546869i	$-0.184181\pi$
$953$	4.47398e10	1.67444	0.837218	+	0.546869i	$0.184181\pi$
$954$	5.08666e9	0.189676
$955$	0	0
$956$	−2.27269e10	−0.841274
$957$	−1.08603e9	−0.0400545
$958$	2.79610e10	1.02748
$959$	−1.68436e10	−0.616694
$960$	0	0
$961$	−2.59259e10	−0.942327
$962$	1.38394e9	0.0501192
$963$	−6.92034e9	−0.249710
$964$	−6.16977e9	−0.221819
$965$	0	0
$966$	6.73460e8	0.0240376
$967$	1.69742e10	0.603666	0.301833	−	0.953361i	$-0.402402\pi$
$967$	1.69742e10	0.603666	0.301833	+	0.953361i	$0.402402\pi$
$968$	−7.26188e9	−0.257327
$969$	9.24636e7	0.00326466
$970$	0	0
$971$	2.06446e10	0.723667	0.361834	−	0.932243i	$-0.382151\pi$
$971$	2.06446e10	0.723667	0.361834	+	0.932243i	$0.382151\pi$
$972$	−2.73989e9	−0.0956976
$973$	1.72908e9	0.0601755
$974$	−1.85792e10	−0.644274
$975$	0	0
$976$	5.88710e9	0.202688
$977$	−4.79396e10	−1.64461	−0.822307	−	0.569045i	$-0.807313\pi$
$977$	−4.79396e10	−1.64461	−0.822307	+	0.569045i	$0.807313\pi$
$978$	−1.42682e9	−0.0487735
$979$	−7.25509e9	−0.247118
$980$	0	0
$981$	3.28102e10	1.10960
$982$	−2.83641e10	−0.955826
$983$	−3.01960e10	−1.01394	−0.506970	−	0.861964i	$-0.669234\pi$
$983$	−3.01960e10	−1.01394	−0.506970	+	0.861964i	$0.669234\pi$
$984$	1.13513e9	0.0379805
$985$	0	0
$986$	5.04143e9	0.167488
$987$	−6.74404e8	−0.0223259
$988$	6.79496e8	0.0224149
$989$	2.41016e10	0.792242
$990$	0	0
$991$	1.83662e10	0.599463	0.299731	−	0.954024i	$-0.403103\pi$
$991$	1.83662e10	0.599463	0.299731	+	0.954024i	$0.403103\pi$
$992$	−1.30528e9	−0.0424535
$993$	−2.05636e9	−0.0666463
$994$	5.81063e9	0.187660
$995$	0	0
$996$	4.21455e7	0.00135159
$997$	6.24126e9	0.199452	0.0997262	−	0.995015i	$-0.468203\pi$
$997$	6.24126e9	0.199452	0.0997262	+	0.995015i	$0.468203\pi$
$998$	−4.24720e10	−1.35253
$999$	−1.64036e9	−0.0520546

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a_n

instead) (See

a_n

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.a.g.1.1		1
5.2	odd	4		70.8.c.b.29.2	yes	2
5.3	odd	4		70.8.c.b.29.1	✓	2
5.4	even	2		350.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.8.c.b.29.1	✓	2	5.3	odd	4
70.8.c.b.29.2	yes	2	5.2	odd	4
350.8.a.b.1.1		1	5.4	even	2
350.8.a.g.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 $\F_{q}$
Belyi maps

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Fields

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Groups

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Properties

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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	350.8.a.g.1.1

Level	$350$
Weight	$8$
Character	350.1
Self dual	yes
Analytic conductor	$109.335$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$