LMFDB - Embedded newform 169.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,8,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.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:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-10.0000 q^{2} -73.0000 q^{3} -28.0000 q^{4} +295.000 q^{5} +730.000 q^{6} -1373.00 q^{7} +1560.00 q^{8} +3142.00 q^{9} +O(q^{10})$

\(q-10.0000 q^{2} -73.0000 q^{3} -28.0000 q^{4} +295.000 q^{5} +730.000 q^{6} -1373.00 q^{7} +1560.00 q^{8} +3142.00 q^{9} -2950.00 q^{10} +7646.00 q^{11} +2044.00 q^{12} +13730.0 q^{14} -21535.0 q^{15} -12016.0 q^{16} -4147.00 q^{17} -31420.0 q^{18} +3186.00 q^{19} -8260.00 q^{20} +100229. q^{21} -76460.0 q^{22} -17784.0 q^{23} -113880. q^{24} +8900.00 q^{25} -69715.0 q^{27} +38444.0 q^{28} -93322.0 q^{29} +215350. q^{30} +124484. q^{31} -79520.0 q^{32} -558158. q^{33} +41470.0 q^{34} -405035. q^{35} -87976.0 q^{36} -273661. q^{37} -31860.0 q^{38} +460200. q^{40} -585816. q^{41} -1.00229e6 q^{42} -533559. q^{43} -214088. q^{44} +926890. q^{45} +177840. q^{46} +530055. q^{47} +877168. q^{48} +1.06159e6 q^{49} -89000.0 q^{50} +302731. q^{51} -615288. q^{53} +697150. q^{54} +2.25557e6 q^{55} -2.14188e6 q^{56} -232578. q^{57} +933220. q^{58} +392514. q^{59} +602980. q^{60} +1.87806e6 q^{61} -1.24484e6 q^{62} -4.31397e6 q^{63} +2.33325e6 q^{64} +5.58158e6 q^{66} +3.97144e6 q^{67} +116116. q^{68} +1.29823e6 q^{69} +4.05035e6 q^{70} +3.74660e6 q^{71} +4.90152e6 q^{72} -2.48580e6 q^{73} +2.73661e6 q^{74} -649700. q^{75} -89208.0 q^{76} -1.04980e7 q^{77} -1.26446e6 q^{79} -3.54472e6 q^{80} -1.78236e6 q^{81} +5.85816e6 q^{82} -434308. q^{83} -2.80641e6 q^{84} -1.22336e6 q^{85} +5.33559e6 q^{86} +6.81251e6 q^{87} +1.19278e7 q^{88} -5.83081e6 q^{89} -9.26890e6 q^{90} +497952. q^{92} -9.08733e6 q^{93} -5.30055e6 q^{94} +939870. q^{95} +5.80496e6 q^{96} +2.04533e6 q^{97} -1.06159e7 q^{98} +2.40237e7 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−10.0000	−0.883883	−0.441942	−	0.897044i	$-0.645710\pi$
$2$	−10.0000	−0.883883	−0.441942	+	0.897044i	$0.645710\pi$
$3$	−73.0000	−1.56098	−0.780492	−	0.625166i	$-0.785031\pi$
$3$	−73.0000	−1.56098	−0.780492	+	0.625166i	$0.785031\pi$
$4$	−28.0000	−0.218750
$5$	295.000	1.05542	0.527712	−	0.849423i	$-0.323050\pi$
$5$	295.000	1.05542	0.527712	+	0.849423i	$0.323050\pi$
$6$	730.000	1.37973
$7$	−1373.00	−1.51296	−0.756480	−	0.654017i	$-0.773082\pi$
$7$	−1373.00	−1.51296	−0.756480	+	0.654017i	$0.773082\pi$
$8$	1560.00	1.07723
$9$	3142.00	1.43667
$10$	−2950.00	−0.932872
$11$	7646.00	1.73205	0.866024	−	0.500003i	$-0.166668\pi$
$11$	7646.00	1.73205	0.866024	+	0.500003i	$0.166668\pi$
$12$	2044.00	0.341465
$13$	0	0
$13$	0	0
$14$	13730.0	1.33728
$15$	−21535.0	−1.64750
$16$	−12016.0	−0.733398
$17$	−4147.00	−0.204721	−0.102361	−	0.994747i	$-0.532640\pi$
$17$	−4147.00	−0.204721	−0.102361	+	0.994747i	$0.532640\pi$
$18$	−31420.0	−1.26985
$19$	3186.00	0.106563	0.0532817	−	0.998580i	$-0.483032\pi$
$19$	3186.00	0.106563	0.0532817	+	0.998580i	$0.483032\pi$
$20$	−8260.00	−0.230874
$21$	100229.	2.36171
$22$	−76460.0	−1.53093
$23$	−17784.0	−0.304777	−0.152388	−	0.988321i	$-0.548696\pi$
$23$	−17784.0	−0.304777	−0.152388	+	0.988321i	$0.548696\pi$
$24$	−113880.	−1.68154
$25$	8900.00	0.113920
$26$	0	0
$27$	−69715.0	−0.681637
$28$	38444.0	0.330960
$29$	−93322.0	−0.710544	−0.355272	−	0.934763i	$-0.615612\pi$
$29$	−93322.0	−0.710544	−0.355272	+	0.934763i	$0.615612\pi$
$30$	215350.	1.45620
$31$	124484.	0.750495	0.375247	−	0.926925i	$-0.377558\pi$
$31$	124484.	0.750495	0.375247	+	0.926925i	$0.377558\pi$
$32$	−79520.0	−0.428994
$33$	−558158.	−2.70370
$34$	41470.0	0.180950
$35$	−405035.	−1.59681
$36$	−87976.0	−0.314272
$37$	−273661.	−0.888192	−0.444096	−	0.895979i	$-0.646475\pi$
$37$	−273661.	−0.888192	−0.444096	+	0.895979i	$0.646475\pi$
$38$	−31860.0	−0.0941897
$39$	0	0
$40$	460200.	1.13694
$41$	−585816.	−1.32745	−0.663724	−	0.747977i	$-0.731025\pi$
$41$	−585816.	−1.32745	−0.663724	+	0.747977i	$0.731025\pi$
$42$	−1.00229e6	−2.08747
$43$	−533559.	−1.02339	−0.511697	−	0.859166i	$-0.670983\pi$
$43$	−533559.	−1.02339	−0.511697	+	0.859166i	$0.670983\pi$
$44$	−214088.	−0.378885
$45$	926890.	1.51630
$46$	177840.	0.269387
$47$	530055.	0.744695	0.372347	−	0.928093i	$-0.378553\pi$
$47$	530055.	0.744695	0.372347	+	0.928093i	$0.378553\pi$
$48$	877168.	1.14482
$49$	1.06159e6	1.28905
$50$	−89000.0	−0.100692
$51$	302731.	0.319567
$52$	0	0
$53$	−615288.	−0.567692	−0.283846	−	0.958870i	$-0.591610\pi$
$53$	−615288.	−0.567692	−0.283846	+	0.958870i	$0.591610\pi$
$54$	697150.	0.602488
$55$	2.25557e6	1.82805
$56$	−2.14188e6	−1.62981
$57$	−232578.	−0.166344
$58$	933220.	0.628038
$59$	392514.	0.248813	0.124407	−	0.992231i	$-0.460297\pi$
$59$	392514.	0.248813	0.124407	+	0.992231i	$0.460297\pi$
$60$	602980.	0.360391
$61$	1.87806e6	1.05939	0.529695	−	0.848188i	$-0.322306\pi$
$61$	1.87806e6	1.05939	0.529695	+	0.848188i	$0.322306\pi$
$62$	−1.24484e6	−0.663350
$63$	−4.31397e6	−2.17363
$64$	2.33325e6	1.11258
$65$	0	0
$66$	5.58158e6	2.38976
$67$	3.97144e6	1.61319	0.806596	−	0.591103i	$-0.201307\pi$
$67$	3.97144e6	1.61319	0.806596	+	0.591103i	$0.201307\pi$
$68$	116116.	0.0447828
$69$	1.29823e6	0.475752
$70$	4.05035e6	1.41140
$71$	3.74660e6	1.24232	0.621160	−	0.783684i	$-0.286662\pi$
$71$	3.74660e6	1.24232	0.621160	+	0.783684i	$0.286662\pi$
$72$	4.90152e6	1.54763
$73$	−2.48580e6	−0.747888	−0.373944	−	0.927451i	$-0.621995\pi$
$73$	−2.48580e6	−0.747888	−0.373944	+	0.927451i	$0.621995\pi$
$74$	2.73661e6	0.785058
$75$	−649700.	−0.177827
$76$	−89208.0	−0.0233107
$77$	−1.04980e7	−2.62052
$78$	0	0
$79$	−1.26446e6	−0.288542	−0.144271	−	0.989538i	$-0.546084\pi$
$79$	−1.26446e6	−0.288542	−0.144271	+	0.989538i	$0.546084\pi$
$80$	−3.54472e6	−0.774046
$81$	−1.78236e6	−0.372647
$82$	5.85816e6	1.17331
$83$	−434308.	−0.0833728	−0.0416864	−	0.999131i	$-0.513273\pi$
$83$	−434308.	−0.0833728	−0.0416864	+	0.999131i	$0.513273\pi$
$84$	−2.80641e6	−0.516623
$85$	−1.22336e6	−0.216068
$86$	5.33559e6	0.904561
$87$	6.81251e6	1.10915
$88$	1.19278e7	1.86582
$89$	−5.83081e6	−0.876726	−0.438363	−	0.898798i	$-0.644442\pi$
$89$	−5.83081e6	−0.876726	−0.438363	+	0.898798i	$0.644442\pi$
$90$	−9.26890e6	−1.34023
$91$	0	0
$92$	497952.	0.0666699
$93$	−9.08733e6	−1.17151
$94$	−5.30055e6	−0.658224
$95$	939870.	0.112470
$96$	5.80496e6	0.669653
$97$	2.04533e6	0.227542	0.113771	−	0.993507i	$-0.463707\pi$
$97$	2.04533e6	0.227542	0.113771	+	0.993507i	$0.463707\pi$
$98$	−1.06159e7	−1.13937
$99$	2.40237e7	2.48838
$100$	−249200.	−0.0249200
$101$	−1.55142e6	−0.149832	−0.0749160	−	0.997190i	$-0.523869\pi$
$101$	−1.55142e6	−0.149832	−0.0749160	+	0.997190i	$0.523869\pi$
$102$	−3.02731e6	−0.282460
$103$	−1.68251e7	−1.51714	−0.758572	−	0.651590i	$-0.774102\pi$
$103$	−1.68251e7	−1.51714	−0.758572	+	0.651590i	$0.774102\pi$
$104$	0	0
$105$	2.95676e7	2.49260
$106$	6.15288e6	0.501774
$107$	2.19295e7	1.73055	0.865277	−	0.501294i	$-0.167142\pi$
$107$	2.19295e7	1.73055	0.865277	+	0.501294i	$0.167142\pi$
$108$	1.95202e6	0.149108
$109$	1.96595e7	1.45405	0.727024	−	0.686612i	$-0.240903\pi$
$109$	1.96595e7	1.45405	0.727024	+	0.686612i	$0.240903\pi$
$110$	−2.25557e7	−1.61578
$111$	1.99773e7	1.38645
$112$	1.64980e7	1.10960
$113$	−2.14963e7	−1.40149	−0.700744	−	0.713412i	$-0.747149\pi$
$113$	−2.14963e7	−1.40149	−0.700744	+	0.713412i	$0.747149\pi$
$114$	2.32578e6	0.147029
$115$	−5.24628e6	−0.321669
$116$	2.61302e6	0.155432
$117$	0	0
$118$	−3.92514e6	−0.219922
$119$	5.69383e6	0.309735
$120$	−3.35946e7	−1.77474
$121$	3.89741e7	1.99999
$122$	−1.87806e7	−0.936378
$123$	4.27646e7	2.07213
$124$	−3.48555e6	−0.164171
$125$	−2.04214e7	−0.935190
$126$	4.31397e7	1.92123
$127$	−1.77419e7	−0.768578	−0.384289	−	0.923213i	$-0.625553\pi$
$127$	−1.77419e7	−0.768578	−0.384289	+	0.923213i	$0.625553\pi$
$128$	−1.31539e7	−0.554396
$129$	3.89498e7	1.59750
$130$	0	0
$131$	−8.61184e6	−0.334693	−0.167346	−	0.985898i	$-0.553520\pi$
$131$	−8.61184e6	−0.334693	−0.167346	+	0.985898i	$0.553520\pi$
$132$	1.56284e7	0.591434
$133$	−4.37438e6	−0.161226
$134$	−3.97144e7	−1.42587
$135$	−2.05659e7	−0.719416
$136$	−6.46932e6	−0.220532
$137$	−6.30262e6	−0.209411	−0.104705	−	0.994503i	$-0.533390\pi$
$137$	−6.30262e6	−0.209411	−0.104705	+	0.994503i	$0.533390\pi$
$138$	−1.29823e7	−0.420509
$139$	1.34997e7	0.426355	0.213177	−	0.977014i	$-0.431619\pi$
$139$	1.34997e7	0.426355	0.213177	+	0.977014i	$0.431619\pi$
$140$	1.13410e7	0.349303
$141$	−3.86940e7	−1.16246
$142$	−3.74660e7	−1.09807
$143$	0	0
$144$	−3.77543e7	−1.05365
$145$	−2.75300e7	−0.749925
$146$	2.48580e7	0.661046
$147$	−7.74958e7	−2.01218
$148$	7.66251e6	0.194292
$149$	1.43791e7	0.356105	0.178053	−	0.984021i	$-0.443020\pi$
$149$	1.43791e7	0.356105	0.178053	+	0.984021i	$0.443020\pi$
$150$	6.49700e6	0.157179
$151$	−8.24764e7	−1.94944	−0.974721	−	0.223424i	$-0.928277\pi$
$151$	−8.24764e7	−1.94944	−0.974721	+	0.223424i	$0.928277\pi$
$152$	4.97016e6	0.114794
$153$	−1.30299e7	−0.294117
$154$	1.04980e8	2.31623
$155$	3.67228e7	0.792090
$156$	0	0
$157$	8.92107e6	0.183979	0.0919895	−	0.995760i	$-0.470677\pi$
$157$	8.92107e6	0.183979	0.0919895	+	0.995760i	$0.470677\pi$
$158$	1.26446e7	0.255037
$159$	4.49160e7	0.886158
$160$	−2.34584e7	−0.452771
$161$	2.44174e7	0.461115
$162$	1.78236e7	0.329377
$163$	−2.09065e7	−0.378116	−0.189058	−	0.981966i	$-0.560543\pi$
$163$	−2.09065e7	−0.378116	−0.189058	+	0.981966i	$0.560543\pi$
$164$	1.64028e7	0.290379
$165$	−1.64657e8	−2.85355
$166$	4.34308e6	0.0736919
$167$	1.88221e7	0.312724	0.156362	−	0.987700i	$-0.450023\pi$
$167$	1.88221e7	0.312724	0.156362	+	0.987700i	$0.450023\pi$
$168$	1.56357e8	2.54411
$169$	0	0
$170$	1.22336e7	0.190979
$171$	1.00104e7	0.153097
$172$	1.49397e7	0.223867
$173$	−4.78358e6	−0.0702412	−0.0351206	−	0.999383i	$-0.511182\pi$
$173$	−4.78358e6	−0.0702412	−0.0351206	+	0.999383i	$0.511182\pi$
$174$	−6.81251e7	−0.980358
$175$	−1.22197e7	−0.172356
$176$	−9.18743e7	−1.27028
$177$	−2.86535e7	−0.388393
$178$	5.83081e7	0.774924
$179$	9.09914e7	1.18581	0.592904	−	0.805273i	$-0.297981\pi$
$179$	9.09914e7	1.18581	0.592904	+	0.805273i	$0.297981\pi$
$180$	−2.59529e7	−0.331690
$181$	−1.72015e7	−0.215622	−0.107811	−	0.994171i	$-0.534384\pi$
$181$	−1.72015e7	−0.215622	−0.107811	+	0.994171i	$0.534384\pi$
$182$	0	0
$183$	−1.37099e8	−1.65369
$184$	−2.77430e7	−0.328316
$185$	−8.07300e7	−0.937419
$186$	9.08733e7	1.03548
$187$	−3.17080e7	−0.354587
$188$	−1.48415e7	−0.162902
$189$	9.57187e7	1.03129
$190$	−9.39870e6	−0.0994100
$191$	−6.68698e7	−0.694405	−0.347203	−	0.937790i	$-0.612868\pi$
$191$	−6.68698e7	−0.694405	−0.347203	+	0.937790i	$0.612868\pi$
$192$	−1.70327e8	−1.73672
$193$	4.86222e7	0.486838	0.243419	−	0.969921i	$-0.421731\pi$
$193$	4.86222e7	0.486838	0.243419	+	0.969921i	$0.421731\pi$
$194$	−2.04533e7	−0.201121
$195$	0	0
$196$	−2.97244e7	−0.281979
$197$	8.42682e7	0.785293	0.392646	−	0.919689i	$-0.371560\pi$
$197$	8.42682e7	0.785293	0.392646	+	0.919689i	$0.371560\pi$
$198$	−2.40237e8	−2.19944
$199$	1.39905e8	1.25849	0.629243	−	0.777208i	$-0.283365\pi$
$199$	1.39905e8	1.25849	0.629243	+	0.777208i	$0.283365\pi$
$200$	1.38840e7	0.122718
$201$	−2.89915e8	−2.51817
$202$	1.55142e7	0.132434
$203$	1.28131e8	1.07502
$204$	−8.47647e6	−0.0699052
$205$	−1.72816e8	−1.40102
$206$	1.68251e8	1.34098
$207$	−5.58773e7	−0.437864
$208$	0	0
$209$	2.43602e7	0.184573
$210$	−2.95676e8	−2.20317
$211$	2.26349e8	1.65878	0.829391	−	0.558669i	$-0.188688\pi$
$211$	2.26349e8	1.65878	0.829391	+	0.558669i	$0.188688\pi$
$212$	1.72281e7	0.124183
$213$	−2.73502e8	−1.93924
$214$	−2.19295e8	−1.52961
$215$	−1.57400e8	−1.08011
$216$	−1.08755e8	−0.734282
$217$	−1.70917e8	−1.13547
$218$	−1.96595e8	−1.28521
$219$	1.81464e8	1.16744
$220$	−6.31560e7	−0.399885
$221$	0	0
$222$	−1.99773e8	−1.22546
$223$	2.19897e8	1.32786	0.663929	−	0.747796i	$-0.268888\pi$
$223$	2.19897e8	1.32786	0.663929	+	0.747796i	$0.268888\pi$
$224$	1.09181e8	0.649051
$225$	2.79638e7	0.163666
$226$	2.14963e8	1.23875
$227$	2.30377e8	1.30722	0.653611	−	0.756831i	$-0.273253\pi$
$227$	2.30377e8	1.30722	0.653611	+	0.756831i	$0.273253\pi$
$228$	6.51218e6	0.0363877
$229$	5.41755e7	0.298111	0.149056	−	0.988829i	$-0.452377\pi$
$229$	5.41755e7	0.298111	0.149056	+	0.988829i	$0.452377\pi$
$230$	5.24628e7	0.284318
$231$	7.66351e8	4.09059
$232$	−1.45582e8	−0.765422
$233$	1.41580e8	0.733259	0.366629	−	0.930367i	$-0.380512\pi$
$233$	1.41580e8	0.733259	0.366629	+	0.930367i	$0.380512\pi$
$234$	0	0
$235$	1.56366e8	0.785969
$236$	−1.09904e7	−0.0544278
$237$	9.23053e7	0.450409
$238$	−5.69383e7	−0.273770
$239$	2.57365e8	1.21943	0.609715	−	0.792621i	$-0.291284\pi$
$239$	2.57365e8	1.21943	0.609715	+	0.792621i	$0.291284\pi$
$240$	2.58765e8	1.20827
$241$	2.46818e8	1.13584	0.567921	−	0.823083i	$-0.307748\pi$
$241$	2.46818e8	1.13584	0.567921	+	0.823083i	$0.307748\pi$
$242$	−3.89741e8	−1.76776
$243$	2.82579e8	1.26333
$244$	−5.25858e7	−0.231742
$245$	3.13168e8	1.36049
$246$	−4.27646e8	−1.83152
$247$	0	0
$248$	1.94195e8	0.808458
$249$	3.17045e7	0.130144
$250$	2.04214e8	0.826599
$251$	2.39628e7	0.0956490	0.0478245	−	0.998856i	$-0.484771\pi$
$251$	2.39628e7	0.0956490	0.0478245	+	0.998856i	$0.484771\pi$
$252$	1.20791e8	0.475481
$253$	−1.35976e8	−0.527888
$254$	1.77419e8	0.679333
$255$	8.93056e7	0.337278
$256$	−1.67117e8	−0.622558
$257$	−2.50050e8	−0.918885	−0.459443	−	0.888207i	$-0.651951\pi$
$257$	−2.50050e8	−0.918885	−0.459443	+	0.888207i	$0.651951\pi$
$258$	−3.89498e8	−1.41201
$259$	3.75737e8	1.34380
$260$	0	0
$261$	−2.93218e8	−1.02082
$262$	8.61184e7	0.295829
$263$	−2.09182e8	−0.709055	−0.354527	−	0.935046i	$-0.615358\pi$
$263$	−2.09182e8	−0.709055	−0.354527	+	0.935046i	$0.615358\pi$
$264$	−8.70726e8	−2.91251
$265$	−1.81510e8	−0.599156
$266$	4.37438e7	0.142505
$267$	4.25649e8	1.36856
$268$	−1.11200e8	−0.352886
$269$	3.71414e8	1.16339	0.581695	−	0.813407i	$-0.302390\pi$
$269$	3.71414e8	1.16339	0.581695	+	0.813407i	$0.302390\pi$
$270$	2.05659e8	0.635880
$271$	−3.30225e8	−1.00790	−0.503950	−	0.863733i	$-0.668121\pi$
$271$	−3.30225e8	−1.00790	−0.503950	+	0.863733i	$0.668121\pi$
$272$	4.98304e7	0.150142
$273$	0	0
$274$	6.30262e7	0.185095
$275$	6.80494e7	0.197315
$276$	−3.63505e7	−0.104071
$277$	5.06278e8	1.43123	0.715616	−	0.698494i	$-0.246146\pi$
$277$	5.06278e8	1.43123	0.715616	+	0.698494i	$0.246146\pi$
$278$	−1.34997e8	−0.376848
$279$	3.91129e8	1.07821
$280$	−6.31855e8	−1.72014
$281$	−1.06744e8	−0.286994	−0.143497	−	0.989651i	$-0.545835\pi$
$281$	−1.06744e8	−0.286994	−0.143497	+	0.989651i	$0.545835\pi$
$282$	3.86940e8	1.02748
$283$	−5.56521e8	−1.45958	−0.729792	−	0.683669i	$-0.760383\pi$
$283$	−5.56521e8	−1.45958	−0.729792	+	0.683669i	$0.760383\pi$
$284$	−1.04905e8	−0.271757
$285$	−6.86105e7	−0.175563
$286$	0	0
$287$	8.04325e8	2.00838
$288$	−2.49852e8	−0.616324
$289$	−3.93141e8	−0.958089
$290$	2.75300e8	0.662847
$291$	−1.49309e8	−0.355190
$292$	6.96025e7	0.163600
$293$	−2.23708e8	−0.519571	−0.259786	−	0.965666i	$-0.583652\pi$
$293$	−2.23708e8	−0.519571	−0.259786	+	0.965666i	$0.583652\pi$
$294$	7.74958e8	1.77853
$295$	1.15792e8	0.262603
$296$	−4.26911e8	−0.956790
$297$	−5.33041e8	−1.18063
$298$	−1.43791e8	−0.314756
$299$	0	0
$300$	1.81916e7	0.0388997
$301$	7.32577e8	1.54835
$302$	8.24764e8	1.72308
$303$	1.13254e8	0.233885
$304$	−3.82830e7	−0.0781534
$305$	5.54029e8	1.11811
$306$	1.30299e8	0.259965
$307$	9.91919e8	1.95655	0.978277	−	0.207300i	$-0.0664678\pi$
$307$	9.91919e8	1.95655	0.978277	+	0.207300i	$0.0664678\pi$
$308$	2.93943e8	0.573239
$309$	1.22823e9	2.36824
$310$	−3.67228e8	−0.700115
$311$	−2.48269e8	−0.468016	−0.234008	−	0.972235i	$-0.575184\pi$
$311$	−2.48269e8	−0.468016	−0.234008	+	0.972235i	$0.575184\pi$
$312$	0	0
$313$	−2.00737e8	−0.370018	−0.185009	−	0.982737i	$-0.559231\pi$
$313$	−2.00737e8	−0.370018	−0.185009	+	0.982737i	$0.559231\pi$
$314$	−8.92107e7	−0.162616
$315$	−1.27262e9	−2.29410
$316$	3.54048e7	0.0631185
$317$	1.02635e8	0.180962	0.0904808	−	0.995898i	$-0.471160\pi$
$317$	1.02635e8	0.180962	0.0904808	+	0.995898i	$0.471160\pi$
$318$	−4.49160e8	−0.783261
$319$	−7.13540e8	−1.23070
$320$	6.88308e8	1.17424
$321$	−1.60085e9	−2.70137
$322$	−2.44174e8	−0.407572
$323$	−1.32123e7	−0.0218158
$324$	4.99061e7	0.0815165
$325$	0	0
$326$	2.09065e8	0.334210
$327$	−1.43514e9	−2.26975
$328$	−9.13873e8	−1.42997
$329$	−7.27766e8	−1.12669
$330$	1.64657e9	2.52220
$331$	7.60053e8	1.15198	0.575991	−	0.817456i	$-0.304616\pi$
$331$	7.60053e8	1.15198	0.575991	+	0.817456i	$0.304616\pi$
$332$	1.21606e7	0.0182378
$333$	−8.59843e8	−1.27604
$334$	−1.88221e8	−0.276411
$335$	1.17157e9	1.70260
$336$	−1.20435e9	−1.73207
$337$	−4.36659e7	−0.0621495	−0.0310748	−	0.999517i	$-0.509893\pi$
$337$	−4.36659e7	−0.0621495	−0.0310748	+	0.999517i	$0.509893\pi$
$338$	0	0
$339$	1.56923e9	2.18770
$340$	3.42542e7	0.0472648
$341$	9.51805e8	1.29989
$342$	−1.00104e8	−0.135320
$343$	−3.26833e8	−0.437317
$344$	−8.32352e8	−1.10243
$345$	3.82978e8	0.502120
$346$	4.78358e7	0.0620851
$347$	−1.82053e8	−0.233907	−0.116954	−	0.993137i	$-0.537313\pi$
$347$	−1.82053e8	−0.233907	−0.116954	+	0.993137i	$0.537313\pi$
$348$	−1.90750e8	−0.242626
$349$	5.80955e8	0.731566	0.365783	−	0.930700i	$-0.380801\pi$
$349$	5.80955e8	0.731566	0.365783	+	0.930700i	$0.380801\pi$
$350$	1.22197e8	0.152343
$351$	0	0
$352$	−6.08010e8	−0.743039
$353$	−5.45624e8	−0.660210	−0.330105	−	0.943944i	$-0.607084\pi$
$353$	−5.45624e8	−0.660210	−0.330105	+	0.943944i	$0.607084\pi$
$354$	2.86535e8	0.343294
$355$	1.10525e9	1.31117
$356$	1.63263e8	0.191784
$357$	−4.15650e8	−0.483491
$358$	−9.09914e8	−1.04812
$359$	1.05196e9	1.19996	0.599981	−	0.800014i	$-0.295175\pi$
$359$	1.05196e9	1.19996	0.599981	+	0.800014i	$0.295175\pi$
$360$	1.44595e9	1.63341
$361$	−8.83721e8	−0.988644
$362$	1.72015e8	0.190584
$363$	−2.84511e9	−3.12195
$364$	0	0
$365$	−7.33312e8	−0.789339
$366$	1.37099e9	1.46167
$367$	7.29203e8	0.770047	0.385024	−	0.922907i	$-0.374193\pi$
$367$	7.29203e8	0.770047	0.385024	+	0.922907i	$0.374193\pi$
$368$	2.13693e8	0.223523
$369$	−1.84063e9	−1.90711
$370$	8.07300e8	0.828569
$371$	8.44790e8	0.858895
$372$	2.54445e8	0.256268
$373$	1.32385e9	1.32087	0.660434	−	0.750884i	$-0.270372\pi$
$373$	1.32385e9	1.32087	0.660434	+	0.750884i	$0.270372\pi$
$374$	3.17080e8	0.313414
$375$	1.49076e9	1.45982
$376$	8.26886e8	0.802210
$377$	0	0
$378$	−9.57187e8	−0.911539
$379$	1.08474e8	0.102350	0.0511752	−	0.998690i	$-0.483703\pi$
$379$	1.08474e8	0.102350	0.0511752	+	0.998690i	$0.483703\pi$
$380$	−2.63164e7	−0.0246027
$381$	1.29516e9	1.19974
$382$	6.68698e8	0.613773
$383$	−2.84754e8	−0.258985	−0.129492	−	0.991580i	$-0.541335\pi$
$383$	−2.84754e8	−0.258985	−0.129492	+	0.991580i	$0.541335\pi$
$384$	9.60236e8	0.865404
$385$	−3.09690e9	−2.76576
$386$	−4.86222e8	−0.430308
$387$	−1.67644e9	−1.47028
$388$	−5.72692e7	−0.0497749
$389$	3.22741e8	0.277991	0.138996	−	0.990293i	$-0.455613\pi$
$389$	3.22741e8	0.277991	0.138996	+	0.990293i	$0.455613\pi$
$390$	0	0
$391$	7.37502e7	0.0623943
$392$	1.65607e9	1.38860
$393$	6.28664e8	0.522450
$394$	−8.42682e8	−0.694108
$395$	−3.73015e8	−0.304534
$396$	−6.72664e8	−0.544334
$397$	8.64634e8	0.693531	0.346765	−	0.937952i	$-0.387280\pi$
$397$	8.64634e8	0.693531	0.346765	+	0.937952i	$0.387280\pi$
$398$	−1.39905e9	−1.11236
$399$	3.19330e8	0.251671
$400$	−1.06942e8	−0.0835488
$401$	1.31166e9	1.01582	0.507909	−	0.861411i	$-0.330419\pi$
$401$	1.31166e9	1.01582	0.507909	+	0.861411i	$0.330419\pi$
$402$	2.89915e9	2.22577
$403$	0	0
$404$	4.34398e7	0.0327758
$405$	−5.25796e8	−0.393301
$406$	−1.28131e9	−0.950197
$407$	−2.09241e9	−1.53839
$408$	4.72260e8	0.344248
$409$	4.00024e7	0.0289104	0.0144552	−	0.999896i	$-0.495399\pi$
$409$	4.00024e7	0.0289104	0.0144552	+	0.999896i	$0.495399\pi$
$410$	1.72816e9	1.23834
$411$	4.60091e8	0.326887
$412$	4.71102e8	0.331875
$413$	−5.38922e8	−0.376444
$414$	5.58773e8	0.387021
$415$	−1.28121e8	−0.0879937
$416$	0	0
$417$	−9.85475e8	−0.665533
$418$	−2.43602e8	−0.163141
$419$	2.64978e9	1.75979	0.879896	−	0.475166i	$-0.157612\pi$
$419$	2.64978e9	1.75979	0.879896	+	0.475166i	$0.157612\pi$
$420$	−8.27892e8	−0.545257
$421$	−8.99741e8	−0.587665	−0.293833	−	0.955857i	$-0.594931\pi$
$421$	−8.99741e8	−0.587665	−0.293833	+	0.955857i	$0.594931\pi$
$422$	−2.26349e9	−1.46617
$423$	1.66543e9	1.06988
$424$	−9.59849e8	−0.611537
$425$	−3.69083e7	−0.0233218
$426$	2.73502e9	1.71406
$427$	−2.57858e9	−1.60281
$428$	−6.14026e8	−0.378559
$429$	0	0
$430$	1.57400e9	0.954695
$431$	−3.69212e8	−0.222129	−0.111065	−	0.993813i	$-0.535426\pi$
$431$	−3.69212e8	−0.222129	−0.111065	+	0.993813i	$0.535426\pi$
$432$	8.37695e8	0.499911
$433$	2.63280e9	1.55851	0.779255	−	0.626707i	$-0.215598\pi$
$433$	2.63280e9	1.55851	0.779255	+	0.626707i	$0.215598\pi$
$434$	1.70917e9	1.00362
$435$	2.00969e9	1.17062
$436$	−5.50465e8	−0.318073
$437$	−5.66598e7	−0.0324781
$438$	−1.81464e9	−1.03188
$439$	3.44814e8	0.194518	0.0972588	−	0.995259i	$-0.468993\pi$
$439$	3.44814e8	0.194518	0.0972588	+	0.995259i	$0.468993\pi$
$440$	3.51869e9	1.96923
$441$	3.33550e9	1.85194
$442$	0	0
$443$	−1.68347e9	−0.920012	−0.460006	−	0.887916i	$-0.652153\pi$
$443$	−1.68347e9	−0.920012	−0.460006	+	0.887916i	$0.652153\pi$
$444$	−5.59363e8	−0.303287
$445$	−1.72009e9	−0.925318
$446$	−2.19897e9	−1.17367
$447$	−1.04967e9	−0.555875
$448$	−3.20355e9	−1.68329
$449$	−3.20869e9	−1.67288	−0.836440	−	0.548058i	$-0.815367\pi$
$449$	−3.20869e9	−1.67288	−0.836440	+	0.548058i	$0.815367\pi$
$450$	−2.79638e8	−0.144661
$451$	−4.47915e9	−2.29920
$452$	6.01897e8	0.306576
$453$	6.02078e9	3.04305
$454$	−2.30377e9	−1.15543
$455$	0	0
$456$	−3.62822e8	−0.179191
$457$	8.53834e8	0.418473	0.209236	−	0.977865i	$-0.432902\pi$
$457$	8.53834e8	0.418473	0.209236	+	0.977865i	$0.432902\pi$
$458$	−5.41755e8	−0.263496
$459$	2.89108e8	0.139546
$460$	1.46896e8	0.0703651
$461$	3.91799e9	1.86256	0.931279	−	0.364308i	$-0.118694\pi$
$461$	3.91799e9	1.86256	0.931279	+	0.364308i	$0.118694\pi$
$462$	−7.66351e9	−3.61560
$463$	9.00831e8	0.421803	0.210902	−	0.977507i	$-0.432360\pi$
$463$	9.00831e8	0.421803	0.210902	+	0.977507i	$0.432360\pi$
$464$	1.12136e9	0.521112
$465$	−2.68076e9	−1.23644
$466$	−1.41580e9	−0.648115
$467$	−8.14889e7	−0.0370245	−0.0185123	−	0.999829i	$-0.505893\pi$
$467$	−8.14889e7	−0.0370245	−0.0185123	+	0.999829i	$0.505893\pi$
$468$	0	0
$469$	−5.45278e9	−2.44069
$470$	−1.56366e9	−0.694705
$471$	−6.51238e8	−0.287188
$472$	6.12322e8	0.268030
$473$	−4.07959e9	−1.77257
$474$	−9.23053e8	−0.398109
$475$	2.83554e7	0.0121397
$476$	−1.59427e8	−0.0677545
$477$	−1.93323e9	−0.815587
$478$	−2.57365e9	−1.07783
$479$	−4.28234e9	−1.78036	−0.890178	−	0.455612i	$-0.849420\pi$
$479$	−4.28234e9	−1.78036	−0.890178	+	0.455612i	$0.849420\pi$
$480$	1.71246e9	0.706768
$481$	0	0
$482$	−2.46818e9	−1.00395
$483$	−1.78247e9	−0.719793
$484$	−1.09128e9	−0.437498
$485$	6.03372e8	0.240154
$486$	−2.82579e9	−1.11664
$487$	−4.24063e9	−1.66371	−0.831857	−	0.554990i	$-0.812722\pi$
$487$	−4.24063e9	−1.66371	−0.831857	+	0.554990i	$0.812722\pi$
$488$	2.92978e9	1.14121
$489$	1.52617e9	0.590233
$490$	−3.13168e9	−1.20252
$491$	2.21387e9	0.844046	0.422023	−	0.906585i	$-0.361320\pi$
$491$	2.21387e9	0.844046	0.422023	+	0.906585i	$0.361320\pi$
$492$	−1.19741e9	−0.453278
$493$	3.87006e8	0.145463
$494$	0	0
$495$	7.08700e9	2.62630
$496$	−1.49580e9	−0.550412
$497$	−5.14408e9	−1.87958
$498$	−3.17045e8	−0.115032
$499$	2.45975e9	0.886215	0.443108	−	0.896468i	$-0.353876\pi$
$499$	2.45975e9	0.886215	0.443108	+	0.896468i	$0.353876\pi$
$500$	5.71798e8	0.204573
$501$	−1.37401e9	−0.488156
$502$	−2.39628e8	−0.0845426
$503$	3.72798e9	1.30613	0.653063	−	0.757303i	$-0.273484\pi$
$503$	3.72798e9	1.30613	0.653063	+	0.757303i	$0.273484\pi$
$504$	−6.72979e9	−2.34150
$505$	−4.57669e8	−0.158136
$506$	1.35976e9	0.466592
$507$	0	0
$508$	4.96774e8	0.168126
$509$	−1.48553e9	−0.499308	−0.249654	−	0.968335i	$-0.580317\pi$
$509$	−1.48553e9	−0.499308	−0.249654	+	0.968335i	$0.580317\pi$
$510$	−8.93056e8	−0.298115
$511$	3.41301e9	1.13152
$512$	3.35487e9	1.10466
$513$	−2.22112e8	−0.0726376
$514$	2.50050e9	0.812187
$515$	−4.96340e9	−1.60123
$516$	−1.09059e9	−0.349453
$517$	4.05280e9	1.28985
$518$	−3.75737e9	−1.18776
$519$	3.49202e8	0.109645
$520$	0	0
$521$	1.06857e9	0.331031	0.165516	−	0.986207i	$-0.447071\pi$
$521$	1.06857e9	0.331031	0.165516	+	0.986207i	$0.447071\pi$
$522$	2.93218e9	0.902284
$523$	−3.85266e8	−0.117762	−0.0588809	−	0.998265i	$-0.518753\pi$
$523$	−3.85266e8	−0.117762	−0.0588809	+	0.998265i	$0.518753\pi$
$524$	2.41131e8	0.0732140
$525$	8.92038e8	0.269046
$526$	2.09182e9	0.626722
$527$	−5.16235e8	−0.153642
$528$	6.70683e9	1.98289
$529$	−3.08855e9	−0.907111
$530$	1.81510e9	0.529584
$531$	1.23328e9	0.357463
$532$	1.22483e8	0.0352682
$533$	0	0
$534$	−4.25649e9	−1.20964
$535$	6.46920e9	1.82647
$536$	6.19544e9	1.73778
$537$	−6.64237e9	−1.85103
$538$	−3.71414e9	−1.02830
$539$	8.11689e9	2.23269
$540$	5.75846e8	0.157372
$541$	2.81334e9	0.763891	0.381946	−	0.924185i	$-0.375254\pi$
$541$	2.81334e9	0.763891	0.381946	+	0.924185i	$0.375254\pi$
$542$	3.30225e9	0.890867
$543$	1.25571e9	0.336582
$544$	3.29769e8	0.0878242
$545$	5.79954e9	1.53464
$546$	0	0
$547$	1.67344e9	0.437174	0.218587	−	0.975818i	$-0.429855\pi$
$547$	1.67344e9	0.437174	0.218587	+	0.975818i	$0.429855\pi$
$548$	1.76473e8	0.0458086
$549$	5.90088e9	1.52200
$550$	−6.80494e8	−0.174403
$551$	−2.97324e8	−0.0757180
$552$	2.02524e9	0.512496
$553$	1.73610e9	0.436552
$554$	−5.06278e9	−1.26504
$555$	5.89329e9	1.46330
$556$	−3.77991e8	−0.0932651
$557$	4.46631e9	1.09511	0.547553	−	0.836771i	$-0.315559\pi$
$557$	4.46631e9	1.09511	0.547553	+	0.836771i	$0.315559\pi$
$558$	−3.91129e9	−0.953016
$559$	0	0
$560$	4.86690e9	1.17110
$561$	2.31468e9	0.553505
$562$	1.06744e9	0.253669
$563$	−5.06446e9	−1.19606	−0.598031	−	0.801473i	$-0.704050\pi$
$563$	−5.06446e9	−1.19606	−0.598031	+	0.801473i	$0.704050\pi$
$564$	1.08343e9	0.254287
$565$	−6.34142e9	−1.47917
$566$	5.56521e9	1.29010
$567$	2.44718e9	0.563800
$568$	5.84470e9	1.33827
$569$	4.07861e9	0.928152	0.464076	−	0.885795i	$-0.346386\pi$
$569$	4.07861e9	0.928152	0.464076	+	0.885795i	$0.346386\pi$
$570$	6.86105e8	0.155177
$571$	−7.82983e9	−1.76005	−0.880027	−	0.474923i	$-0.842476\pi$
$571$	−7.82983e9	−1.76005	−0.880027	+	0.474923i	$0.842476\pi$
$572$	0	0
$573$	4.88149e9	1.08396
$574$	−8.04325e9	−1.77517
$575$	−1.58278e8	−0.0347202
$576$	7.33107e9	1.59841
$577$	7.94179e9	1.72109	0.860544	−	0.509376i	$-0.170124\pi$
$577$	7.94179e9	1.72109	0.860544	+	0.509376i	$0.170124\pi$
$578$	3.93141e9	0.846839
$579$	−3.54942e9	−0.759946
$580$	7.70840e8	0.164046
$581$	5.96305e8	0.126140
$582$	1.49309e9	0.313947
$583$	−4.70449e9	−0.983270
$584$	−3.87785e9	−0.805650
$585$	0	0
$586$	2.23708e9	0.459241
$587$	2.02009e9	0.412227	0.206114	−	0.978528i	$-0.433918\pi$
$587$	2.02009e9	0.412227	0.206114	+	0.978528i	$0.433918\pi$
$588$	2.16988e9	0.440165
$589$	3.96606e8	0.0799753
$590$	−1.15792e9	−0.232111
$591$	−6.15158e9	−1.22583
$592$	3.28831e9	0.651399
$593$	−5.19728e9	−1.02349	−0.511746	−	0.859137i	$-0.671001\pi$
$593$	−5.19728e9	−1.02349	−0.511746	+	0.859137i	$0.671001\pi$
$594$	5.33041e9	1.04354
$595$	1.67968e9	0.326902
$596$	−4.02614e8	−0.0778980
$597$	−1.02131e10	−1.96448
$598$	0	0
$599$	−3.92347e9	−0.745893	−0.372946	−	0.927853i	$-0.621652\pi$
$599$	−3.92347e9	−0.745893	−0.372946	+	0.927853i	$0.621652\pi$
$600$	−1.01353e9	−0.191561
$601$	−9.51281e8	−0.178751	−0.0893754	−	0.995998i	$-0.528487\pi$
$601$	−9.51281e8	−0.178751	−0.0893754	+	0.995998i	$0.528487\pi$
$602$	−7.32577e9	−1.36856
$603$	1.24783e10	2.31763
$604$	2.30934e9	0.426441
$605$	1.14974e10	2.11084
$606$	−1.13254e9	−0.206727
$607$	8.27679e9	1.50211	0.751055	−	0.660240i	$-0.229545\pi$
$607$	8.27679e9	1.50211	0.751055	+	0.660240i	$0.229545\pi$
$608$	−2.53351e8	−0.0457151
$609$	−9.35357e9	−1.67810
$610$	−5.54029e9	−0.988275
$611$	0	0
$612$	3.64836e8	0.0643381
$613$	−2.92674e9	−0.513183	−0.256591	−	0.966520i	$-0.582599\pi$
$613$	−2.92674e9	−0.513183	−0.256591	+	0.966520i	$0.582599\pi$
$614$	−9.91919e9	−1.72937
$615$	1.26155e10	2.18697
$616$	−1.63768e10	−2.82291
$617$	−8.88587e9	−1.52301	−0.761503	−	0.648161i	$-0.775538\pi$
$617$	−8.88587e9	−1.52301	−0.761503	+	0.648161i	$0.775538\pi$
$618$	−1.22823e10	−2.09324
$619$	4.16163e9	0.705255	0.352627	−	0.935764i	$-0.385288\pi$
$619$	4.16163e9	0.705255	0.352627	+	0.935764i	$0.385288\pi$
$620$	−1.02824e9	−0.173270
$621$	1.23981e9	0.207747
$622$	2.48269e9	0.413671
$623$	8.00570e9	1.32645
$624$	0	0
$625$	−6.71962e9	−1.10094
$626$	2.00737e9	0.327052
$627$	−1.77829e9	−0.288115
$628$	−2.49790e8	−0.0402454
$629$	1.13487e9	0.181832
$630$	1.27262e10	2.02771
$631$	7.30070e9	1.15681	0.578405	−	0.815750i	$-0.303676\pi$
$631$	7.30070e9	1.15681	0.578405	+	0.815750i	$0.303676\pi$
$632$	−1.97255e9	−0.310827
$633$	−1.65234e10	−2.58933
$634$	−1.02635e9	−0.159949
$635$	−5.23387e9	−0.811176
$636$	−1.25765e9	−0.193847
$637$	0	0
$638$	7.13540e9	1.08779
$639$	1.17718e10	1.78480
$640$	−3.88041e9	−0.585123
$641$	3.46867e9	0.520187	0.260094	−	0.965583i	$-0.416247\pi$
$641$	3.46867e9	0.520187	0.260094	+	0.965583i	$0.416247\pi$
$642$	1.60085e10	2.38769
$643$	3.72175e9	0.552089	0.276045	−	0.961145i	$-0.410976\pi$
$643$	3.72175e9	0.552089	0.276045	+	0.961145i	$0.410976\pi$
$644$	−6.83688e8	−0.100869
$645$	1.14902e10	1.68604
$646$	1.32123e8	0.0192826
$647$	−8.03095e9	−1.16574	−0.582870	−	0.812565i	$-0.698070\pi$
$647$	−8.03095e9	−1.16574	−0.582870	+	0.812565i	$0.698070\pi$
$648$	−2.78048e9	−0.401428
$649$	3.00116e9	0.430956
$650$	0	0
$651$	1.24769e10	1.77245
$652$	5.85382e8	0.0827128
$653$	8.22151e9	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$653$	8.22151e9	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$654$	1.43514e10	2.00619
$655$	−2.54049e9	−0.353243
$656$	7.03917e9	0.973549
$657$	−7.81039e9	−1.07447
$658$	7.27766e9	0.995866
$659$	6.47061e9	0.880737	0.440369	−	0.897817i	$-0.354848\pi$
$659$	6.47061e9	0.880737	0.440369	+	0.897817i	$0.354848\pi$
$660$	4.61039e9	0.624214
$661$	3.64380e9	0.490738	0.245369	−	0.969430i	$-0.421091\pi$
$661$	3.64380e9	0.490738	0.245369	+	0.969430i	$0.421091\pi$
$662$	−7.60053e9	−1.01822
$663$	0	0
$664$	−6.77520e8	−0.0898120
$665$	−1.29044e9	−0.170162
$666$	8.59843e9	1.12787
$667$	1.65964e9	0.216557
$668$	−5.27019e8	−0.0684083
$669$	−1.60524e10	−2.07276
$670$	−1.17157e10	−1.50490
$671$	1.43597e10	1.83491
$672$	−7.97021e9	−1.01316
$673$	2.32463e9	0.293968	0.146984	−	0.989139i	$-0.453043\pi$
$673$	2.32463e9	0.293968	0.146984	+	0.989139i	$0.453043\pi$
$674$	4.36659e8	0.0549330
$675$	−6.20464e8	−0.0776521
$676$	0	0
$677$	2.19098e9	0.271380	0.135690	−	0.990751i	$-0.456675\pi$
$677$	2.19098e9	0.271380	0.135690	+	0.990751i	$0.456675\pi$
$678$	−1.56923e10	−1.93367
$679$	−2.80824e9	−0.344262
$680$	−1.90845e9	−0.232755
$681$	−1.68175e10	−2.04055
$682$	−9.51805e9	−1.14895
$683$	1.70757e9	0.205072	0.102536	−	0.994729i	$-0.467304\pi$
$683$	1.70757e9	0.205072	0.102536	+	0.994729i	$0.467304\pi$
$684$	−2.80292e8	−0.0334899
$685$	−1.85927e9	−0.221017
$686$	3.26833e9	0.386537
$687$	−3.95481e9	−0.465347
$688$	6.41124e9	0.750556
$689$	0	0
$690$	−3.82978e9	−0.443816
$691$	−1.48657e10	−1.71400	−0.857000	−	0.515316i	$-0.827675\pi$
$691$	−1.48657e10	−1.71400	−0.857000	+	0.515316i	$0.827675\pi$
$692$	1.33940e8	0.0153653
$693$	−3.29846e10	−3.76482
$694$	1.82053e9	0.206747
$695$	3.98240e9	0.449985
$696$	1.06275e10	1.19481
$697$	2.42938e9	0.271757
$698$	−5.80955e9	−0.646619
$699$	−1.03354e10	−1.14460
$700$	3.42152e8	0.0377030
$701$	8.96793e9	0.983284	0.491642	−	0.870797i	$-0.336397\pi$
$701$	8.96793e9	0.983284	0.491642	+	0.870797i	$0.336397\pi$
$702$	0	0
$703$	−8.71884e8	−0.0946488
$704$	1.78400e10	1.92704
$705$	−1.14147e10	−1.22689
$706$	5.45624e9	0.583549
$707$	2.13010e9	0.226690
$708$	8.02299e8	0.0849610
$709$	−1.31197e10	−1.38249	−0.691247	−	0.722619i	$-0.742938\pi$
$709$	−1.31197e10	−1.38249	−0.691247	+	0.722619i	$0.742938\pi$
$710$	−1.10525e10	−1.15892
$711$	−3.97292e9	−0.414540
$712$	−9.09606e9	−0.944438
$713$	−2.21382e9	−0.228733
$714$	4.15650e9	0.427350
$715$	0	0
$716$	−2.54776e9	−0.259396
$717$	−1.87876e10	−1.90351
$718$	−1.05196e10	−1.06063
$719$	−1.16702e10	−1.17092	−0.585462	−	0.810700i	$-0.699087\pi$
$719$	−1.16702e10	−1.17092	−0.585462	+	0.810700i	$0.699087\pi$
$720$	−1.11375e10	−1.11205
$721$	2.31008e10	2.29538
$722$	8.83721e9	0.873846
$723$	−1.80177e10	−1.77303
$724$	4.81643e8	0.0471673
$725$	−8.30566e8	−0.0809452
$726$	2.84511e10	2.75944
$727$	1.03092e10	0.995068	0.497534	−	0.867444i	$-0.334239\pi$
$727$	1.03092e10	0.995068	0.497534	+	0.867444i	$0.334239\pi$
$728$	0	0
$729$	−1.67302e10	−1.59940
$730$	7.33312e9	0.697684
$731$	2.21267e9	0.209510
$732$	3.83876e9	0.361745
$733$	1.00112e10	0.938910	0.469455	−	0.882956i	$-0.344450\pi$
$733$	1.00112e10	0.938910	0.469455	+	0.882956i	$0.344450\pi$
$734$	−7.29203e9	−0.680632
$735$	−2.28613e10	−2.12371
$736$	1.41418e9	0.130748
$737$	3.03656e10	2.79413
$738$	1.84063e10	1.68566
$739$	−3.30781e9	−0.301498	−0.150749	−	0.988572i	$-0.548169\pi$
$739$	−3.30781e9	−0.301498	−0.150749	+	0.988572i	$0.548169\pi$
$740$	2.26044e9	0.205060
$741$	0	0
$742$	−8.44790e9	−0.759163
$743$	2.17089e10	1.94168	0.970838	−	0.239735i	$-0.0770607\pi$
$743$	2.17089e10	1.94168	0.970838	+	0.239735i	$0.0770607\pi$
$744$	−1.41762e10	−1.26199
$745$	4.24182e9	0.375842
$746$	−1.32385e10	−1.16749
$747$	−1.36460e9	−0.119779
$748$	8.87823e8	0.0775659
$749$	−3.01092e10	−2.61826
$750$	−1.49076e10	−1.29031
$751$	9.19095e9	0.791809	0.395904	−	0.918292i	$-0.370431\pi$
$751$	9.19095e9	0.791809	0.395904	+	0.918292i	$0.370431\pi$
$752$	−6.36914e9	−0.546158
$753$	−1.74929e9	−0.149307
$754$	0	0
$755$	−2.43305e10	−2.05749
$756$	−2.68012e9	−0.225595
$757$	9.78965e9	0.820222	0.410111	−	0.912036i	$-0.365490\pi$
$757$	9.78965e9	0.820222	0.410111	+	0.912036i	$0.365490\pi$
$758$	−1.08474e9	−0.0904658
$759$	9.92628e9	0.824025
$760$	1.46620e9	0.121156
$761$	2.03733e10	1.67577	0.837886	−	0.545845i	$-0.183791\pi$
$761$	2.03733e10	1.67577	0.837886	+	0.545845i	$0.183791\pi$
$762$	−1.29516e10	−1.06043
$763$	−2.69924e10	−2.19992
$764$	1.87235e9	0.151901
$765$	−3.84381e9	−0.310418
$766$	2.84754e9	0.228913
$767$	0	0
$768$	1.21995e10	0.971803
$769$	8.96000e9	0.710503	0.355251	−	0.934771i	$-0.384395\pi$
$769$	8.96000e9	0.710503	0.355251	+	0.934771i	$0.384395\pi$
$770$	3.09690e10	2.44461
$771$	1.82537e10	1.43436
$772$	−1.36142e9	−0.106496
$773$	−7.61579e9	−0.593044	−0.296522	−	0.955026i	$-0.595827\pi$
$773$	−7.61579e9	−0.593044	−0.296522	+	0.955026i	$0.595827\pi$
$774$	1.67644e10	1.29956
$775$	1.10791e9	0.0854964
$776$	3.19071e9	0.245116
$777$	−2.74288e10	−2.09765
$778$	−3.22741e9	−0.245712
$779$	−1.86641e9	−0.141457
$780$	0	0
$781$	2.86465e10	2.15176
$782$	−7.37502e8	−0.0551493
$783$	6.50594e9	0.484333
$784$	−1.27560e10	−0.945385
$785$	2.63172e9	0.194176
$786$	−6.28664e9	−0.461785
$787$	−2.15840e10	−1.57841	−0.789205	−	0.614130i	$-0.789507\pi$
$787$	−2.15840e10	−1.57841	−0.789205	+	0.614130i	$0.789507\pi$
$788$	−2.35951e9	−0.171783
$789$	1.52703e10	1.10682
$790$	3.73015e9	0.269173
$791$	2.95145e10	2.12040
$792$	3.74770e10	2.68057
$793$	0	0
$794$	−8.64634e9	−0.613000
$795$	1.32502e10	0.935273
$796$	−3.91735e9	−0.275294
$797$	−1.58880e10	−1.11164	−0.555822	−	0.831301i	$-0.687597\pi$
$797$	−1.58880e10	−1.11164	−0.555822	+	0.831301i	$0.687597\pi$
$798$	−3.19330e9	−0.222448
$799$	−2.19814e9	−0.152455
$800$	−7.07728e8	−0.0488710
$801$	−1.83204e10	−1.25957
$802$	−1.31166e10	−0.897865
$803$	−1.90064e10	−1.29538
$804$	8.11762e9	0.550849
$805$	7.20314e9	0.486672
$806$	0	0
$807$	−2.71132e10	−1.81603
$808$	−2.42022e9	−0.161404
$809$	8.28899e9	0.550404	0.275202	−	0.961386i	$-0.411255\pi$
$809$	8.28899e9	0.550404	0.275202	+	0.961386i	$0.411255\pi$
$810$	5.25796e9	0.347632
$811$	6.46851e9	0.425825	0.212913	−	0.977071i	$-0.431705\pi$
$811$	6.46851e9	0.425825	0.212913	+	0.977071i	$0.431705\pi$
$812$	−3.58767e9	−0.235162
$813$	2.41064e10	1.57332
$814$	2.09241e10	1.35976
$815$	−6.16742e9	−0.399072
$816$	−3.63762e9	−0.234370
$817$	−1.69992e9	−0.109056
$818$	−4.00024e8	−0.0255534
$819$	0	0
$820$	4.83884e9	0.306473
$821$	−1.65268e10	−1.04228	−0.521142	−	0.853470i	$-0.674494\pi$
$821$	−1.65268e10	−1.04228	−0.521142	+	0.853470i	$0.674494\pi$
$822$	−4.60091e9	−0.288930
$823$	−2.05119e10	−1.28264	−0.641322	−	0.767272i	$-0.721614\pi$
$823$	−2.05119e10	−1.28264	−0.641322	+	0.767272i	$0.721614\pi$
$824$	−2.62471e10	−1.63432
$825$	−4.96761e9	−0.308005
$826$	5.38922e9	0.332733
$827$	−8.63679e9	−0.530986	−0.265493	−	0.964113i	$-0.585535\pi$
$827$	−8.63679e9	−0.530986	−0.265493	+	0.964113i	$0.585535\pi$
$828$	1.56457e9	0.0957828
$829$	8.81187e9	0.537189	0.268595	−	0.963253i	$-0.413441\pi$
$829$	8.81187e9	0.537189	0.268595	+	0.963253i	$0.413441\pi$
$830$	1.28121e9	0.0777762
$831$	−3.69583e10	−2.23413
$832$	0	0
$833$	−4.40240e9	−0.263895
$834$	9.85475e9	0.588254
$835$	5.55252e9	0.330056
$836$	−6.82084e8	−0.0403753
$837$	−8.67840e9	−0.511565
$838$	−2.64978e10	−1.55545
$839$	−1.73321e10	−1.01317	−0.506586	−	0.862189i	$-0.669093\pi$
$839$	−1.73321e10	−1.01317	−0.506586	+	0.862189i	$0.669093\pi$
$840$	4.61254e10	2.68511
$841$	−8.54088e9	−0.495127
$842$	8.99741e9	0.519428
$843$	7.79234e9	0.447993
$844$	−6.33776e9	−0.362859
$845$	0	0
$846$	−1.66543e10	−0.945651
$847$	−5.35115e10	−3.02590
$848$	7.39330e9	0.416345
$849$	4.06260e10	2.27839
$850$	3.69083e8	0.0206138
$851$	4.86679e9	0.270700
$852$	7.65805e9	0.424209
$853$	1.62475e10	0.896322	0.448161	−	0.893953i	$-0.352079\pi$
$853$	1.62475e10	0.896322	0.448161	+	0.893953i	$0.352079\pi$
$854$	2.57858e10	1.41670
$855$	2.95307e9	0.161582
$856$	3.42100e10	1.86421
$857$	−2.45158e10	−1.33049	−0.665247	−	0.746623i	$-0.731674\pi$
$857$	−2.45158e10	−1.33049	−0.665247	+	0.746623i	$0.731674\pi$
$858$	0	0
$859$	9.28369e9	0.499741	0.249870	−	0.968279i	$-0.419612\pi$
$859$	9.28369e9	0.499741	0.249870	+	0.968279i	$0.419612\pi$
$860$	4.40720e9	0.236275
$861$	−5.87158e10	−3.13504
$862$	3.69212e9	0.196336
$863$	6.17565e9	0.327073	0.163536	−	0.986537i	$-0.447710\pi$
$863$	6.17565e9	0.327073	0.163536	+	0.986537i	$0.447710\pi$
$864$	5.54374e9	0.292418
$865$	−1.41116e9	−0.0741343
$866$	−2.63280e10	−1.37754
$867$	2.86993e10	1.49556
$868$	4.78566e9	0.248384
$869$	−9.66803e9	−0.499768
$870$	−2.00969e10	−1.03469
$871$	0	0
$872$	3.06688e10	1.56635
$873$	6.42643e9	0.326904
$874$	5.66598e8	0.0287068
$875$	2.80385e10	1.41491
$876$	−5.08098e9	−0.255378
$877$	1.34392e10	0.672782	0.336391	−	0.941722i	$-0.390794\pi$
$877$	1.34392e10	0.672782	0.336391	+	0.941722i	$0.390794\pi$
$878$	−3.44814e9	−0.171931
$879$	1.63307e10	0.811043
$880$	−2.71029e10	−1.34069
$881$	3.85616e9	0.189994	0.0949968	−	0.995478i	$-0.469716\pi$
$881$	3.85616e9	0.189994	0.0949968	+	0.995478i	$0.469716\pi$
$882$	−3.33550e10	−1.63690
$883$	2.27185e10	1.11050	0.555248	−	0.831685i	$-0.312623\pi$
$883$	2.27185e10	1.11050	0.555248	+	0.831685i	$0.312623\pi$
$884$	0	0
$885$	−8.45279e9	−0.409920
$886$	1.68347e10	0.813183
$887$	2.21671e10	1.06654	0.533270	−	0.845945i	$-0.320963\pi$
$887$	2.21671e10	1.06654	0.533270	+	0.845945i	$0.320963\pi$
$888$	3.11645e10	1.49353
$889$	2.43597e10	1.16283
$890$	1.72009e10	0.817873
$891$	−1.36279e10	−0.645442
$892$	−6.15710e9	−0.290469
$893$	1.68876e9	0.0793572
$894$	1.04967e10	0.491328
$895$	2.68425e10	1.25153
$896$	1.80603e10	0.838779
$897$	0	0
$898$	3.20869e10	1.47863
$899$	−1.16171e10	−0.533260
$900$	−7.82986e8	−0.0358018
$901$	2.55160e9	0.116219
$902$	4.47915e10	2.03223
$903$	−5.34781e10	−2.41696
$904$	−3.35343e10	−1.50973
$905$	−5.07446e9	−0.227572
$906$	−6.02078e10	−2.68970
$907$	−3.55279e10	−1.58104	−0.790522	−	0.612434i	$-0.790191\pi$
$907$	−3.55279e10	−1.58104	−0.790522	+	0.612434i	$0.790191\pi$
$908$	−6.45056e9	−0.285955
$909$	−4.87456e9	−0.215259
$910$	0	0
$911$	−4.10088e9	−0.179706	−0.0898530	−	0.995955i	$-0.528640\pi$
$911$	−4.10088e9	−0.179706	−0.0898530	+	0.995955i	$0.528640\pi$
$912$	2.79466e9	0.121996
$913$	−3.32072e9	−0.144406
$914$	−8.53834e9	−0.369881
$915$	−4.04441e10	−1.74535
$916$	−1.51691e9	−0.0652119
$917$	1.18240e10	0.506377
$918$	−2.89108e9	−0.123342
$919$	9.41768e8	0.0400258	0.0200129	−	0.999800i	$-0.493629\pi$
$919$	9.41768e8	0.0400258	0.0200129	+	0.999800i	$0.493629\pi$
$920$	−8.18420e9	−0.346512
$921$	−7.24101e10	−3.05415
$922$	−3.91799e10	−1.64628
$923$	0	0
$924$	−2.14578e10	−0.894816
$925$	−2.43558e9	−0.101183
$926$	−9.00831e9	−0.372825
$927$	−5.28644e10	−2.17964
$928$	7.42097e9	0.304819
$929$	1.39001e10	0.568803	0.284402	−	0.958705i	$-0.408205\pi$
$929$	1.39001e10	0.568803	0.284402	+	0.958705i	$0.408205\pi$
$930$	2.68076e10	1.09287
$931$	3.38221e9	0.137365
$932$	−3.96425e9	−0.160400
$933$	1.81236e10	0.730565
$934$	8.14889e8	0.0327254
$935$	−9.35385e9	−0.374240
$936$	0	0
$937$	−1.57057e10	−0.623689	−0.311844	−	0.950133i	$-0.600947\pi$
$937$	−1.57057e10	−0.623689	−0.311844	+	0.950133i	$0.600947\pi$
$938$	5.45278e10	2.15729
$939$	1.46538e10	0.577592
$940$	−4.37825e9	−0.171931
$941$	3.84758e10	1.50530	0.752651	−	0.658419i	$-0.228775\pi$
$941$	3.84758e10	1.50530	0.752651	+	0.658419i	$0.228775\pi$
$942$	6.51238e9	0.253841
$943$	1.04182e10	0.404576
$944$	−4.71645e9	−0.182479
$945$	2.82370e10	1.08845
$946$	4.07959e10	1.56674
$947$	1.18417e10	0.453095	0.226548	−	0.974000i	$-0.427256\pi$
$947$	1.18417e10	0.453095	0.226548	+	0.974000i	$0.427256\pi$
$948$	−2.58455e9	−0.0985270
$949$	0	0
$950$	−2.83554e8	−0.0107301
$951$	−7.49233e9	−0.282478
$952$	8.88238e9	0.333657
$953$	8.06299e9	0.301767	0.150883	−	0.988552i	$-0.451788\pi$
$953$	8.06299e9	0.301767	0.150883	+	0.988552i	$0.451788\pi$
$954$	1.93323e10	0.720884
$955$	−1.97266e10	−0.732892
$956$	−7.20622e9	−0.266750
$957$	5.20884e10	1.92110
$958$	4.28234e10	1.57363
$959$	8.65349e9	0.316830
$960$	−5.02465e10	−1.83297
$961$	−1.20163e10	−0.436758
$962$	0	0
$963$	6.89024e10	2.48624
$964$	−6.91091e9	−0.248465
$965$	1.43436e10	0.513820
$966$	1.78247e10	0.636214
$967$	−2.18672e10	−0.777679	−0.388840	−	0.921305i	$-0.627124\pi$
$967$	−2.18672e10	−0.777679	−0.388840	+	0.921305i	$0.627124\pi$
$968$	6.07997e10	2.15446
$969$	9.64501e8	0.0340541
$970$	−6.03372e9	−0.212268
$971$	1.48206e10	0.519516	0.259758	−	0.965674i	$-0.416357\pi$
$971$	1.48206e10	0.519516	0.259758	+	0.965674i	$0.416357\pi$
$972$	−7.91221e9	−0.276354
$973$	−1.85350e10	−0.645058
$974$	4.24063e10	1.47053
$975$	0	0
$976$	−2.25668e10	−0.776955
$977$	4.40385e10	1.51078	0.755391	−	0.655274i	$-0.227447\pi$
$977$	4.40385e10	1.51078	0.755391	+	0.655274i	$0.227447\pi$
$978$	−1.52617e10	−0.521697
$979$	−4.45824e10	−1.51853
$980$	−8.76870e9	−0.297608
$981$	6.17700e10	2.08899
$982$	−2.21387e10	−0.746038
$983$	−2.32688e10	−0.781333	−0.390666	−	0.920532i	$-0.627755\pi$
$983$	−2.32688e10	−0.781333	−0.390666	+	0.920532i	$0.627755\pi$
$984$	6.67127e10	2.23216
$985$	2.48591e10	0.828817
$986$	−3.87006e9	−0.128573
$987$	5.31269e10	1.75875
$988$	0	0
$989$	9.48881e9	0.311907
$990$	−7.08700e10	−2.32134
$991$	1.25560e10	0.409821	0.204911	−	0.978781i	$-0.434310\pi$
$991$	1.25560e10	0.409821	0.204911	+	0.978781i	$0.434310\pi$
$992$	−9.89897e9	−0.321958
$993$	−5.54839e10	−1.79823
$994$	5.14408e10	1.66133
$995$	4.12721e10	1.32824
$996$	−8.87726e8	−0.0284689
$997$	3.50176e10	1.11906	0.559530	−	0.828810i	$-0.310982\pi$
$997$	3.50176e10	1.11906	0.559530	+	0.828810i	$0.310982\pi$
$998$	−2.45975e10	−0.783311
$999$	1.90783e10	0.605424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.a.1.1		1
13.5	odd	4		169.8.b.a.168.2		2
13.8	odd	4		169.8.b.a.168.1		2
13.12	even	2		13.8.a.a.1.1	✓	1
39.38	odd	2		117.8.a.a.1.1		1
52.51	odd	2		208.8.a.d.1.1		1
65.64	even	2		325.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.a.a.1.1	✓	1	13.12	even	2
117.8.a.a.1.1		1	39.38	odd	2
169.8.a.a.1.1		1	1.1	even	1	trivial
169.8.b.a.168.1		2	13.8	odd	4
169.8.b.a.168.2		2	13.5	odd	4
208.8.a.d.1.1		1	52.51	odd	2
325.8.a.a.1.1		1	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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