LMFDB - Embedded newform 338.8.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, 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("338.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	338.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:	$105.586138614$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6981x^{2} - 35424x + 7188480 $ x^4 - 6981x^2 - 35424x + 7188480
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$31.8716$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +31.8716 q^{3} +64.0000 q^{4} -54.4265 q^{5} -254.973 q^{6} -1112.71 q^{7} -512.000 q^{8} -1171.20 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +31.8716 q^{3} +64.0000 q^{4} -54.4265 q^{5} -254.973 q^{6} -1112.71 q^{7} -512.000 q^{8} -1171.20 q^{9} +435.412 q^{10} +7133.68 q^{11} +2039.79 q^{12} +8901.66 q^{14} -1734.66 q^{15} +4096.00 q^{16} -20505.6 q^{17} +9369.59 q^{18} +28071.7 q^{19} -3483.30 q^{20} -35463.8 q^{21} -57069.4 q^{22} +33922.5 q^{23} -16318.3 q^{24} -75162.8 q^{25} -107031. q^{27} -71213.3 q^{28} -175470. q^{29} +13877.3 q^{30} +15690.1 q^{31} -32768.0 q^{32} +227362. q^{33} +164045. q^{34} +60560.8 q^{35} -74956.7 q^{36} +1996.96 q^{37} -224573. q^{38} +27866.4 q^{40} +223640. q^{41} +283711. q^{42} -537931. q^{43} +456555. q^{44} +63744.2 q^{45} -271380. q^{46} +542249. q^{47} +130546. q^{48} +414575. q^{49} +601302. q^{50} -653549. q^{51} +1.85685e6 q^{53} +856250. q^{54} -388261. q^{55} +569706. q^{56} +894691. q^{57} +1.40376e6 q^{58} -1.33045e6 q^{59} -111018. q^{60} +2.86662e6 q^{61} -125521. q^{62} +1.30320e6 q^{63} +262144. q^{64} -1.81890e6 q^{66} -2.83157e6 q^{67} -1.31236e6 q^{68} +1.08117e6 q^{69} -484486. q^{70} -1.60186e6 q^{71} +599654. q^{72} +1.39213e6 q^{73} -15975.6 q^{74} -2.39556e6 q^{75} +1.79659e6 q^{76} -7.93770e6 q^{77} -2.33505e6 q^{79} -222931. q^{80} -849853. q^{81} -1.78912e6 q^{82} +2.37338e6 q^{83} -2.26968e6 q^{84} +1.11605e6 q^{85} +4.30345e6 q^{86} -5.59251e6 q^{87} -3.65244e6 q^{88} +7.05151e6 q^{89} -509954. q^{90} +2.17104e6 q^{92} +500068. q^{93} -4.33799e6 q^{94} -1.52784e6 q^{95} -1.04437e6 q^{96} +8.51370e6 q^{97} -3.31660e6 q^{98} -8.35495e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9	\( 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9} - 2224 q^{10} + 7392 q^{11} - 4384 q^{14} - 15528 q^{15} + 16384 q^{16} - 28316 q^{17} - 41712 q^{18} + 99888 q^{19} + 17792 q^{20} + 91074 q^{21} - 59136 q^{22} + 33388 q^{23} + 86878 q^{25} + 106272 q^{27} + 35072 q^{28} - 93140 q^{29} + 124224 q^{30} + 311160 q^{31} - 131072 q^{32} - 238638 q^{33} + 226528 q^{34} - 141544 q^{35} + 333696 q^{36} + 9636 q^{37} - 799104 q^{38} - 142336 q^{40} - 82892 q^{41} - 728592 q^{42} + 569264 q^{43} + 473088 q^{44} + 2303394 q^{45} - 267104 q^{46} - 574200 q^{47} + 717798 q^{49} - 695024 q^{50} - 2729928 q^{51} + 1235350 q^{53} - 850176 q^{54} + 1092512 q^{55} - 280576 q^{56} + 3528462 q^{57} + 745120 q^{58} - 231504 q^{59} - 993792 q^{60} - 685684 q^{61} - 2489280 q^{62} + 5951712 q^{63} + 1048576 q^{64} + 1909104 q^{66} - 3271056 q^{67} - 1812224 q^{68} - 5600034 q^{69} + 1132352 q^{70} + 175012 q^{71} - 2669568 q^{72} + 7137890 q^{73} - 77088 q^{74} - 22200960 q^{75} + 6392832 q^{76} - 13915206 q^{77} - 7053952 q^{79} + 1138688 q^{80} - 3758004 q^{81} + 663136 q^{82} + 657288 q^{83} + 5828736 q^{84} - 11814998 q^{85} - 4554112 q^{86} + 7182900 q^{87} - 3784704 q^{88} + 11452234 q^{89} - 18427152 q^{90} + 2136832 q^{92} - 2984688 q^{93} + 4593600 q^{94} + 23334088 q^{95} + 428002 q^{97} - 5742384 q^{98} - 10357656 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9 - 2224 * q^10 + 7392 * q^11 - 4384 * q^14 - 15528 * q^15 + 16384 * q^16 - 28316 * q^17 - 41712 * q^18 + 99888 * q^19 + 17792 * q^20 + 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 + 35072 * q^28 - 93140 * q^29 + 124224 * q^30 + 311160 * q^31 - 131072 * q^32 - 238638 * q^33 + 226528 * q^34 - 141544 * q^35 + 333696 * q^36 + 9636 * q^37 - 799104 * q^38 - 142336 * q^40 - 82892 * q^41 - 728592 * q^42 + 569264 * q^43 + 473088 * q^44 + 2303394 * q^45 - 267104 * q^46 - 574200 * q^47 + 717798 * q^49 - 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 - 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 + 3528462 * q^57 + 745120 * q^58 - 231504 * q^59 - 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 + 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 - 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 + 1132352 * q^70 + 175012 * q^71 - 2669568 * q^72 + 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 + 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 + 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 + 657288 * q^83 + 5828736 * q^84 - 11814998 * q^85 - 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 + 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 - 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 + 428002 * q^97 - 5742384 * q^98 - 10357656 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	31.8716	0.681522	0.340761	−	0.940150i	$-0.389315\pi$
$3$	31.8716	0.681522	0.340761	+	0.940150i	$0.389315\pi$
$4$	64.0000	0.500000
$5$	−54.4265	−0.194722	−0.0973611	−	0.995249i	$-0.531040\pi$
$5$	−54.4265	−0.194722	−0.0973611	+	0.995249i	$0.531040\pi$
$6$	−254.973	−0.481909
$7$	−1112.71	−1.22613	−0.613067	−	0.790031i	$-0.710064\pi$
$7$	−1112.71	−1.22613	−0.613067	+	0.790031i	$0.710064\pi$
$8$	−512.000	−0.353553
$9$	−1171.20	−0.535527
$10$	435.412	0.137689
$11$	7133.68	1.61599	0.807996	−	0.589189i	$-0.200553\pi$
$11$	7133.68	1.61599	0.807996	+	0.589189i	$0.200553\pi$
$12$	2039.79	0.340761
$13$	0	0
$13$	0	0
$14$	8901.66	0.867008
$15$	−1734.66	−0.132708
$16$	4096.00	0.250000
$17$	−20505.6	−1.01228	−0.506142	−	0.862450i	$-0.668929\pi$
$17$	−20505.6	−1.01228	−0.506142	+	0.862450i	$0.668929\pi$
$18$	9369.59	0.378675
$19$	28071.7	0.938925	0.469462	−	0.882952i	$-0.344448\pi$
$19$	28071.7	0.938925	0.469462	+	0.882952i	$0.344448\pi$
$20$	−3483.30	−0.0973611
$21$	−35463.8	−0.835638
$22$	−57069.4	−1.14268
$23$	33922.5	0.581354	0.290677	−	0.956821i	$-0.406120\pi$
$23$	33922.5	0.581354	0.290677	+	0.956821i	$0.406120\pi$
$24$	−16318.3	−0.240955
$25$	−75162.8	−0.962083
$26$	0	0
$27$	−107031.	−1.04650
$28$	−71213.3	−0.613067
$29$	−175470.	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$29$	−175470.	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$30$	13877.3	0.0938384
$31$	15690.1	0.0945930	0.0472965	−	0.998881i	$-0.484939\pi$
$31$	15690.1	0.0945930	0.0472965	+	0.998881i	$0.484939\pi$
$32$	−32768.0	−0.176777
$33$	227362.	1.10133
$34$	164045.	0.715792
$35$	60560.8	0.238756
$36$	−74956.7	−0.267764
$37$	1996.96	0.00648130	0.00324065	−	0.999995i	$-0.498968\pi$
$37$	1996.96	0.00648130	0.00324065	+	0.999995i	$0.498968\pi$
$38$	−224573.	−0.663920
$39$	0	0
$40$	27866.4	0.0688447
$41$	223640.	0.506764	0.253382	−	0.967366i	$-0.418457\pi$
$41$	223640.	0.506764	0.253382	+	0.967366i	$0.418457\pi$
$42$	283711.	0.590885
$43$	−537931.	−1.03178	−0.515890	−	0.856655i	$-0.672539\pi$
$43$	−537931.	−1.03178	−0.515890	+	0.856655i	$0.672539\pi$
$44$	456555.	0.807996
$45$	63744.2	0.104279
$46$	−271380.	−0.411079
$47$	542249.	0.761826	0.380913	−	0.924611i	$-0.375610\pi$
$47$	542249.	0.761826	0.380913	+	0.924611i	$0.375610\pi$
$48$	130546.	0.170381
$49$	414575.	0.503405
$50$	601302.	0.680296
$51$	−653549.	−0.689894
$52$	0	0
$53$	1.85685e6	1.71322	0.856609	−	0.515967i	$-0.172567\pi$
$53$	1.85685e6	1.71322	0.856609	+	0.515967i	$0.172567\pi$
$54$	856250.	0.739985
$55$	−388261.	−0.314669
$56$	569706.	0.433504
$57$	894691.	0.639898
$58$	1.40376e6	0.944701
$59$	−1.33045e6	−0.843367	−0.421684	−	0.906743i	$-0.638561\pi$
$59$	−1.33045e6	−0.843367	−0.421684	+	0.906743i	$0.638561\pi$
$60$	−111018.	−0.0663538
$61$	2.86662e6	1.61702	0.808512	−	0.588480i	$-0.200273\pi$
$61$	2.86662e6	1.61702	0.808512	+	0.588480i	$0.200273\pi$
$62$	−125521.	−0.0668873
$63$	1.30320e6	0.656628
$64$	262144.	0.125000
$65$	0	0
$66$	−1.81890e6	−0.778761
$67$	−2.83157e6	−1.15018	−0.575091	−	0.818090i	$-0.695033\pi$
$67$	−2.83157e6	−1.15018	−0.575091	+	0.818090i	$0.695033\pi$
$68$	−1.31236e6	−0.506142
$69$	1.08117e6	0.396205
$70$	−484486.	−0.168826
$71$	−1.60186e6	−0.531153	−0.265576	−	0.964090i	$-0.585562\pi$
$71$	−1.60186e6	−0.531153	−0.265576	+	0.964090i	$0.585562\pi$
$72$	599654.	0.189337
$73$	1.39213e6	0.418841	0.209420	−	0.977826i	$-0.432842\pi$
$73$	1.39213e6	0.418841	0.209420	+	0.977826i	$0.432842\pi$
$74$	−15975.6	−0.00458297
$75$	−2.39556e6	−0.655681
$76$	1.79659e6	0.469462
$77$	−7.93770e6	−1.98142
$78$	0	0
$79$	−2.33505e6	−0.532847	−0.266423	−	0.963856i	$-0.585842\pi$
$79$	−2.33505e6	−0.532847	−0.266423	+	0.963856i	$0.585842\pi$
$80$	−222931.	−0.0486806
$81$	−849853.	−0.177683
$82$	−1.78912e6	−0.358336
$83$	2.37338e6	0.455610	0.227805	−	0.973707i	$-0.426845\pi$
$83$	2.37338e6	0.455610	0.227805	+	0.973707i	$0.426845\pi$
$84$	−2.26968e6	−0.417819
$85$	1.11605e6	0.197114
$86$	4.30345e6	0.729579
$87$	−5.59251e6	−0.910519
$88$	−3.65244e6	−0.571339
$89$	7.05151e6	1.06027	0.530135	−	0.847913i	$-0.322141\pi$
$89$	7.05151e6	1.06027	0.530135	+	0.847913i	$0.322141\pi$
$90$	−509954.	−0.0737364
$91$	0	0
$92$	2.17104e6	0.290677
$93$	500068.	0.0644672
$94$	−4.33799e6	−0.538693
$95$	−1.52784e6	−0.182830
$96$	−1.04437e6	−0.120477
$97$	8.51370e6	0.947146	0.473573	−	0.880754i	$-0.342964\pi$
$97$	8.51370e6	0.947146	0.473573	+	0.880754i	$0.342964\pi$
$98$	−3.31660e6	−0.355961
$99$	−8.35495e6	−0.865407
$100$	−4.81042e6	−0.481042
$101$	−1.05742e7	−1.02123	−0.510615	−	0.859810i	$-0.670582\pi$
$101$	−1.05742e7	−1.02123	−0.510615	+	0.859810i	$0.670582\pi$
$102$	5.22839e6	0.487829
$103$	3.01542e6	0.271905	0.135952	−	0.990715i	$-0.456591\pi$
$103$	3.01542e6	0.271905	0.135952	+	0.990715i	$0.456591\pi$
$104$	0	0
$105$	1.93017e6	0.162717
$106$	−1.48548e7	−1.21143
$107$	2.28239e7	1.80114	0.900568	−	0.434715i	$-0.143151\pi$
$107$	2.28239e7	1.80114	0.900568	+	0.434715i	$0.143151\pi$
$108$	−6.85000e6	−0.523248
$109$	4.55907e6	0.337197	0.168598	−	0.985685i	$-0.446076\pi$
$109$	4.55907e6	0.337197	0.168598	+	0.985685i	$0.446076\pi$
$110$	3.10609e6	0.222505
$111$	63646.2	0.00441715
$112$	−4.55765e6	−0.306534
$113$	2.07319e7	1.35165	0.675826	−	0.737061i	$-0.263787\pi$
$113$	2.07319e7	1.35165	0.675826	+	0.737061i	$0.263787\pi$
$114$	−7.15753e6	−0.452476
$115$	−1.84628e6	−0.113202
$116$	−1.12301e7	−0.668004
$117$	0	0
$118$	1.06436e7	0.596351
$119$	2.28168e7	1.24120
$120$	888147.	0.0469192
$121$	3.14022e7	1.61143
$122$	−2.29330e7	−1.14341
$123$	7.12776e6	0.345371
$124$	1.00416e6	0.0472965
$125$	8.34292e6	0.382061
$126$	−1.04256e7	−0.464306
$127$	−4.07871e7	−1.76689	−0.883445	−	0.468534i	$-0.844782\pi$
$127$	−4.07871e7	−1.76689	−0.883445	+	0.468534i	$0.844782\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.71448e7	−0.703182
$130$	0	0
$131$	4.11026e7	1.59742	0.798711	−	0.601715i	$-0.205516\pi$
$131$	4.11026e7	1.59742	0.798711	+	0.601715i	$0.205516\pi$
$132$	1.45512e7	0.550667
$133$	−3.12356e7	−1.15125
$134$	2.26526e7	0.813301
$135$	5.82534e6	0.203776
$136$	1.04989e7	0.357896
$137$	1.47207e7	0.489111	0.244555	−	0.969635i	$-0.421358\pi$
$137$	1.47207e7	0.489111	0.244555	+	0.969635i	$0.421358\pi$
$138$	−8.64933e6	−0.280160
$139$	−5.68489e7	−1.79544	−0.897719	−	0.440568i	$-0.854777\pi$
$139$	−5.68489e7	−1.79544	−0.897719	+	0.440568i	$0.854777\pi$
$140$	3.87589e6	0.119378
$141$	1.72824e7	0.519202
$142$	1.28149e7	0.375582
$143$	0	0
$144$	−4.79723e6	−0.133882
$145$	9.55020e6	0.260151
$146$	−1.11370e7	−0.296165
$147$	1.32132e7	0.343082
$148$	127805.	0.00324065
$149$	−3.85262e7	−0.954122	−0.477061	−	0.878870i	$-0.658298\pi$
$149$	−3.85262e7	−0.954122	−0.477061	+	0.878870i	$0.658298\pi$
$150$	1.91645e7	0.463637
$151$	2.94587e7	0.696297	0.348148	−	0.937439i	$-0.386811\pi$
$151$	2.94587e7	0.696297	0.348148	+	0.937439i	$0.386811\pi$
$152$	−1.43727e7	−0.331960
$153$	2.40162e7	0.542105
$154$	6.35016e7	1.40108
$155$	−853956.	−0.0184194
$156$	0	0
$157$	6.77251e7	1.39669	0.698346	−	0.715760i	$-0.253919\pi$
$157$	6.77251e7	1.39669	0.698346	+	0.715760i	$0.253919\pi$
$158$	1.86804e7	0.376779
$159$	5.91810e7	1.16760
$160$	1.78345e6	0.0344224
$161$	−3.77458e7	−0.712817
$162$	6.79882e6	0.125641
$163$	9.96365e7	1.80203	0.901015	−	0.433788i	$-0.142823\pi$
$163$	9.96365e7	1.80203	0.901015	+	0.433788i	$0.142823\pi$
$164$	1.43129e7	0.253382
$165$	−1.23745e7	−0.214454
$166$	−1.89870e7	−0.322165
$167$	3.31622e7	0.550979	0.275490	−	0.961304i	$-0.411160\pi$
$167$	3.31622e7	0.550979	0.275490	+	0.961304i	$0.411160\pi$
$168$	1.81575e7	0.295443
$169$	0	0
$170$	−8.92840e6	−0.139381
$171$	−3.28775e7	−0.502820
$172$	−3.44276e7	−0.515890
$173$	4.26162e7	0.625768	0.312884	−	0.949791i	$-0.398705\pi$
$173$	4.26162e7	0.625768	0.312884	+	0.949791i	$0.398705\pi$
$174$	4.47401e7	0.643834
$175$	8.36342e7	1.17964
$176$	2.92195e7	0.403998
$177$	−4.24037e7	−0.574774
$178$	−5.64120e7	−0.749725
$179$	−2.02658e7	−0.264105	−0.132053	−	0.991243i	$-0.542157\pi$
$179$	−2.02658e7	−0.264105	−0.132053	+	0.991243i	$0.542157\pi$
$180$	4.07963e6	0.0521395
$181$	−3.09055e7	−0.387401	−0.193700	−	0.981061i	$-0.562049\pi$
$181$	−3.09055e7	−0.387401	−0.193700	+	0.981061i	$0.562049\pi$
$182$	0	0
$183$	9.13640e7	1.10204
$184$	−1.73683e7	−0.205540
$185$	−108687.	−0.00126205
$186$	−4.00055e6	−0.0455852
$187$	−1.46281e8	−1.63584
$188$	3.47039e7	0.380913
$189$	1.19095e8	1.28314
$190$	1.22228e7	0.129280
$191$	8.07132e7	0.838162	0.419081	−	0.907949i	$-0.362352\pi$
$191$	8.07132e7	0.838162	0.419081	+	0.907949i	$0.362352\pi$
$192$	8.35496e6	0.0851903
$193$	1.38387e8	1.38562	0.692810	−	0.721120i	$-0.256373\pi$
$193$	1.38387e8	1.38562	0.692810	+	0.721120i	$0.256373\pi$
$194$	−6.81096e7	−0.669734
$195$	0	0
$196$	2.65328e7	0.251702
$197$	1.73847e8	1.62008	0.810038	−	0.586377i	$-0.199446\pi$
$197$	1.73847e8	1.62008	0.810038	+	0.586377i	$0.199446\pi$
$198$	6.68396e7	0.611935
$199$	2.39009e7	0.214995	0.107498	−	0.994205i	$-0.465716\pi$
$199$	2.39009e7	0.214995	0.107498	+	0.994205i	$0.465716\pi$
$200$	3.84833e7	0.340148
$201$	−9.02470e7	−0.783874
$202$	8.45938e7	0.722119
$203$	1.95246e8	1.63813
$204$	−4.18271e7	−0.344947
$205$	−1.21719e7	−0.0986781
$206$	−2.41233e7	−0.192266
$207$	−3.97300e7	−0.311331
$208$	0	0
$209$	2.00254e8	1.51729
$210$	−1.54414e7	−0.115058
$211$	−5.47761e7	−0.401423	−0.200711	−	0.979650i	$-0.564325\pi$
$211$	−5.47761e7	−0.401423	−0.200711	+	0.979650i	$0.564325\pi$
$212$	1.18839e8	0.856609
$213$	−5.10538e7	−0.361993
$214$	−1.82591e8	−1.27360
$215$	2.92777e7	0.200911
$216$	5.48000e7	0.369992
$217$	−1.74585e7	−0.115984
$218$	−3.64726e7	−0.238434
$219$	4.43694e7	0.285449
$220$	−2.48487e7	−0.157335
$221$	0	0
$222$	−509170.	−0.00312340
$223$	2.72218e8	1.64380	0.821901	−	0.569631i	$-0.192914\pi$
$223$	2.72218e8	1.64380	0.821901	+	0.569631i	$0.192914\pi$
$224$	3.64612e7	0.216752
$225$	8.80305e7	0.515222
$226$	−1.65855e8	−0.955762
$227$	−4.49554e7	−0.255089	−0.127544	−	0.991833i	$-0.540710\pi$
$227$	−4.49554e7	−0.255089	−0.127544	+	0.991833i	$0.540710\pi$
$228$	5.72602e7	0.319949
$229$	1.23460e8	0.679362	0.339681	−	0.940541i	$-0.389681\pi$
$229$	1.23460e8	0.679362	0.339681	+	0.940541i	$0.389681\pi$
$230$	1.47703e7	0.0800462
$231$	−2.52987e8	−1.35038
$232$	8.98405e7	0.472350
$233$	1.26541e8	0.655371	0.327685	−	0.944787i	$-0.393731\pi$
$233$	1.26541e8	0.655371	0.327685	+	0.944787i	$0.393731\pi$
$234$	0	0
$235$	−2.95127e7	−0.148345
$236$	−8.51488e7	−0.421684
$237$	−7.44220e7	−0.363147
$238$	−1.82534e8	−0.877658
$239$	8.04943e6	0.0381393	0.0190696	−	0.999818i	$-0.493930\pi$
$239$	8.04943e6	0.0381393	0.0190696	+	0.999818i	$0.493930\pi$
$240$	−7.10518e6	−0.0331769
$241$	−3.87577e8	−1.78360	−0.891801	−	0.452428i	$-0.850558\pi$
$241$	−3.87577e8	−1.78360	−0.891801	+	0.452428i	$0.850558\pi$
$242$	−2.51217e8	−1.13945
$243$	2.06991e8	0.925401
$244$	1.83464e8	0.808512
$245$	−2.25639e7	−0.0980241
$246$	−5.70221e7	−0.244214
$247$	0	0
$248$	−8.03331e6	−0.0334437
$249$	7.56434e7	0.310509
$250$	−6.67433e7	−0.270158
$251$	7.40469e7	0.295562	0.147781	−	0.989020i	$-0.452787\pi$
$251$	7.40469e7	0.295562	0.147781	+	0.989020i	$0.452787\pi$
$252$	8.34049e7	0.328314
$253$	2.41992e8	0.939462
$254$	3.26297e8	1.24938
$255$	3.55704e7	0.134338
$256$	1.67772e7	0.0625000
$257$	−1.49301e8	−0.548651	−0.274326	−	0.961637i	$-0.588455\pi$
$257$	−1.49301e8	−0.548651	−0.274326	+	0.961637i	$0.588455\pi$
$258$	1.37158e8	0.497224
$259$	−2.22203e6	−0.00794695
$260$	0	0
$261$	2.05510e8	0.715469
$262$	−3.28821e8	−1.12955
$263$	1.73457e8	0.587958	0.293979	−	0.955812i	$-0.405020\pi$
$263$	1.73457e8	0.587958	0.293979	+	0.955812i	$0.405020\pi$
$264$	−1.16409e8	−0.389380
$265$	−1.01062e8	−0.333601
$266$	2.49885e8	0.814055
$267$	2.24743e8	0.722598
$268$	−1.81221e8	−0.575091
$269$	−4.30910e8	−1.34975	−0.674875	−	0.737932i	$-0.735802\pi$
$269$	−4.30910e8	−1.34975	−0.674875	+	0.737932i	$0.735802\pi$
$270$	−4.66027e7	−0.144091
$271$	3.10232e8	0.946878	0.473439	−	0.880827i	$-0.343012\pi$
$271$	3.10232e8	0.946878	0.473439	+	0.880827i	$0.343012\pi$
$272$	−8.39911e7	−0.253071
$273$	0	0
$274$	−1.17766e8	−0.345853
$275$	−5.36187e8	−1.55472
$276$	6.91946e7	0.198103
$277$	3.90038e8	1.10263	0.551313	−	0.834299i	$-0.314127\pi$
$277$	3.90038e8	1.10263	0.551313	+	0.834299i	$0.314127\pi$
$278$	4.54792e8	1.26957
$279$	−1.83762e7	−0.0506571
$280$	−3.10071e7	−0.0844128
$281$	−2.24392e8	−0.603303	−0.301652	−	0.953418i	$-0.597538\pi$
$281$	−2.24392e8	−0.603303	−0.301652	+	0.953418i	$0.597538\pi$
$282$	−1.38259e8	−0.367131
$283$	−1.25189e7	−0.0328332	−0.0164166	−	0.999865i	$-0.505226\pi$
$283$	−1.25189e7	−0.0328332	−0.0164166	+	0.999865i	$0.505226\pi$
$284$	−1.02519e8	−0.265576
$285$	−4.86949e7	−0.124602
$286$	0	0
$287$	−2.48846e8	−0.621360
$288$	3.83778e7	0.0946687
$289$	1.01427e7	0.0247178
$290$	−7.64016e7	−0.183954
$291$	2.71346e8	0.645501
$292$	8.90961e7	0.209420
$293$	−4.26673e8	−0.990966	−0.495483	−	0.868618i	$-0.665009\pi$
$293$	−4.26673e8	−0.990966	−0.495483	+	0.868618i	$0.665009\pi$
$294$	−1.05706e8	−0.242595
$295$	7.24118e7	0.164222
$296$	−1.02244e6	−0.00229149
$297$	−7.63527e8	−1.69113
$298$	3.08209e8	0.674666
$299$	0	0
$300$	−1.53316e8	−0.327841
$301$	5.98561e8	1.26510
$302$	−2.35670e8	−0.492356
$303$	−3.37018e8	−0.695991
$304$	1.14982e8	0.234731
$305$	−1.56020e8	−0.314870
$306$	−1.92129e8	−0.383326
$307$	2.83423e8	0.559051	0.279525	−	0.960138i	$-0.409823\pi$
$307$	2.83423e8	0.559051	0.279525	+	0.960138i	$0.409823\pi$
$308$	−5.08013e8	−0.990711
$309$	9.61063e7	0.185309
$310$	6.83165e6	0.0130244
$311$	−7.14238e8	−1.34642	−0.673212	−	0.739450i	$-0.735086\pi$
$311$	−7.14238e8	−1.34642	−0.673212	+	0.739450i	$0.735086\pi$
$312$	0	0
$313$	−3.51984e8	−0.648811	−0.324406	−	0.945918i	$-0.605164\pi$
$313$	−3.51984e8	−0.648811	−0.324406	+	0.945918i	$0.605164\pi$
$314$	−5.41801e8	−0.987611
$315$	−7.09287e7	−0.127860
$316$	−1.49443e8	−0.266423
$317$	−4.55628e8	−0.803346	−0.401673	−	0.915783i	$-0.631571\pi$
$317$	−4.55628e8	−0.803346	−0.401673	+	0.915783i	$0.631571\pi$
$318$	−4.73448e8	−0.825615
$319$	−1.25174e9	−2.15898
$320$	−1.42676e7	−0.0243403
$321$	7.27435e8	1.22751
$322$	3.01967e8	0.504038
$323$	−5.75628e8	−0.950458
$324$	−5.43906e7	−0.0888416
$325$	0	0
$326$	−7.97092e8	−1.27423
$327$	1.45305e8	0.229807
$328$	−1.14504e8	−0.179168
$329$	−6.03364e8	−0.934101
$330$	9.89962e7	0.151642
$331$	6.65210e8	1.00823	0.504117	−	0.863636i	$-0.331818\pi$
$331$	6.65210e8	1.00823	0.504117	+	0.863636i	$0.331818\pi$
$332$	1.51896e8	0.227805
$333$	−2.33883e6	−0.00347091
$334$	−2.65297e8	−0.389601
$335$	1.54113e8	0.223966
$336$	−1.45260e8	−0.208909
$337$	9.20291e8	1.30985	0.654924	−	0.755695i	$-0.272701\pi$
$337$	9.20291e8	1.30985	0.654924	+	0.755695i	$0.272701\pi$
$338$	0	0
$339$	6.60760e8	0.921181
$340$	7.14272e7	0.0985570
$341$	1.11928e8	0.152861
$342$	2.63020e8	0.355547
$343$	4.55061e8	0.608892
$344$	2.75421e8	0.364790
$345$	−5.88441e7	−0.0771500
$346$	−3.40930e8	−0.442485
$347$	1.36499e9	1.75378	0.876890	−	0.480691i	$-0.159614\pi$
$347$	1.36499e9	1.75378	0.876890	+	0.480691i	$0.159614\pi$
$348$	−3.57921e8	−0.455260
$349$	−8.97223e8	−1.12983	−0.564913	−	0.825151i	$-0.691090\pi$
$349$	−8.97223e8	−1.12983	−0.564913	+	0.825151i	$0.691090\pi$
$350$	−6.69073e8	−0.834134
$351$	0	0
$352$	−2.33756e8	−0.285670
$353$	−2.69605e8	−0.326225	−0.163112	−	0.986608i	$-0.552153\pi$
$353$	−2.69605e8	−0.326225	−0.163112	+	0.986608i	$0.552153\pi$
$354$	3.39229e8	0.406426
$355$	8.71835e7	0.103427
$356$	4.51296e8	0.530135
$357$	7.27208e8	0.845902
$358$	1.62126e8	0.186751
$359$	−2.23928e8	−0.255433	−0.127717	−	0.991811i	$-0.540765\pi$
$359$	−2.23928e8	−0.255433	−0.127717	+	0.991811i	$0.540765\pi$
$360$	−3.26371e7	−0.0368682
$361$	−1.05852e8	−0.118420
$362$	2.47244e8	0.273934
$363$	1.00084e9	1.09822
$364$	0	0
$365$	−7.57686e7	−0.0815576
$366$	−7.30912e8	−0.779258
$367$	−2.80857e8	−0.296588	−0.148294	−	0.988943i	$-0.547378\pi$
$367$	−2.80857e8	−0.296588	−0.148294	+	0.988943i	$0.547378\pi$
$368$	1.38947e8	0.145338
$369$	−2.61926e8	−0.271386
$370$	869499.	0.000892407 0
$371$	−2.06614e9	−2.10063
$372$	3.20044e7	0.0322336
$373$	1.75524e9	1.75128	0.875642	−	0.482961i	$-0.160439\pi$
$373$	1.75524e9	1.75128	0.875642	+	0.482961i	$0.160439\pi$
$374$	1.17024e9	1.15671
$375$	2.65903e8	0.260383
$376$	−2.77631e8	−0.269346
$377$	0	0
$378$	−9.52756e8	−0.907320
$379$	3.59233e8	0.338953	0.169477	−	0.985534i	$-0.445792\pi$
$379$	3.59233e8	0.338953	0.169477	+	0.985534i	$0.445792\pi$
$380$	−9.77820e7	−0.0914148
$381$	−1.29995e9	−1.20418
$382$	−6.45706e8	−0.592670
$383$	−1.87869e9	−1.70868	−0.854338	−	0.519718i	$-0.826037\pi$
$383$	−1.87869e9	−1.70868	−0.854338	+	0.519718i	$0.826037\pi$
$384$	−6.68397e7	−0.0602386
$385$	4.32021e8	0.385827
$386$	−1.10709e9	−0.979781
$387$	6.30024e8	0.552547
$388$	5.44877e8	0.473573
$389$	4.34490e8	0.374245	0.187122	−	0.982337i	$-0.440084\pi$
$389$	4.34490e8	0.374245	0.187122	+	0.982337i	$0.440084\pi$
$390$	0	0
$391$	−6.95603e8	−0.588495
$392$	−2.12263e8	−0.177980
$393$	1.31001e9	1.08868
$394$	−1.39078e9	−1.14557
$395$	1.27089e8	0.103757
$396$	−5.34717e8	−0.432704
$397$	2.03158e9	1.62954	0.814772	−	0.579781i	$-0.196862\pi$
$397$	2.03158e9	1.62954	0.814772	+	0.579781i	$0.196862\pi$
$398$	−1.91207e8	−0.152025
$399$	−9.95529e8	−0.784601
$400$	−3.07867e8	−0.240521
$401$	−8.03077e8	−0.621945	−0.310973	−	0.950419i	$-0.600655\pi$
$401$	−8.03077e8	−0.621945	−0.310973	+	0.950419i	$0.600655\pi$
$402$	7.21976e8	0.554283
$403$	0	0
$404$	−6.76750e8	−0.510615
$405$	4.62545e7	0.0345989
$406$	−1.56197e9	−1.15833
$407$	1.42456e7	0.0104737
$408$	3.34617e8	0.243914
$409$	−2.53740e9	−1.83382	−0.916912	−	0.399089i	$-0.869326\pi$
$409$	−2.53740e9	−1.83382	−0.916912	+	0.399089i	$0.869326\pi$
$410$	9.73754e7	0.0697760
$411$	4.69174e8	0.333340
$412$	1.92987e8	0.135952
$413$	1.48040e9	1.03408
$414$	3.17840e8	0.220144
$415$	−1.29175e8	−0.0887174
$416$	0	0
$417$	−1.81187e9	−1.22363
$418$	−1.60203e9	−1.07289
$419$	−1.78594e8	−0.118609	−0.0593044	−	0.998240i	$-0.518888\pi$
$419$	−1.78594e8	−0.118609	−0.0593044	+	0.998240i	$0.518888\pi$
$420$	1.23531e8	0.0813586
$421$	2.07782e9	1.35713	0.678563	−	0.734542i	$-0.262603\pi$
$421$	2.07782e9	1.35713	0.678563	+	0.734542i	$0.262603\pi$
$422$	4.38208e8	0.283849
$423$	−6.35081e8	−0.407979
$424$	−9.50710e8	−0.605714
$425$	1.54126e9	0.973901
$426$	4.08431e8	0.255967
$427$	−3.18971e9	−1.98269
$428$	1.46073e9	0.900568
$429$	0	0
$430$	−2.34222e8	−0.142065
$431$	1.58186e9	0.951693	0.475846	−	0.879528i	$-0.342142\pi$
$431$	1.58186e9	0.951693	0.475846	+	0.879528i	$0.342142\pi$
$432$	−4.38400e8	−0.261624
$433$	1.33652e9	0.791168	0.395584	−	0.918430i	$-0.370542\pi$
$433$	1.33652e9	0.791168	0.395584	+	0.918430i	$0.370542\pi$
$434$	1.39668e8	0.0820128
$435$	3.04381e8	0.177298
$436$	2.91781e8	0.168598
$437$	9.52262e8	0.545847
$438$	−3.54955e8	−0.201843
$439$	1.99532e9	1.12561	0.562805	−	0.826590i	$-0.309722\pi$
$439$	1.99532e9	1.12561	0.562805	+	0.826590i	$0.309722\pi$
$440$	1.98790e8	0.111252
$441$	−4.85550e8	−0.269587
$442$	0	0
$443$	1.51862e8	0.0829921	0.0414960	−	0.999139i	$-0.486788\pi$
$443$	1.51862e8	0.0829921	0.0414960	+	0.999139i	$0.486788\pi$
$444$	4.07336e6	0.00220858
$445$	−3.83789e8	−0.206458
$446$	−2.17774e9	−1.16234
$447$	−1.22789e9	−0.650255
$448$	−2.91690e8	−0.153267
$449$	−1.60651e8	−0.0837571	−0.0418786	−	0.999123i	$-0.513334\pi$
$449$	−1.60651e8	−0.0837571	−0.0418786	+	0.999123i	$0.513334\pi$
$450$	−7.04244e8	−0.364317
$451$	1.59537e9	0.818925
$452$	1.32684e9	0.675826
$453$	9.38897e8	0.474542
$454$	3.59643e8	0.180375
$455$	0	0
$456$	−4.58082e8	−0.226238
$457$	1.02168e9	0.500735	0.250367	−	0.968151i	$-0.419449\pi$
$457$	1.02168e9	0.500735	0.250367	+	0.968151i	$0.419449\pi$
$458$	−9.87678e8	−0.480381
$459$	2.19475e9	1.05935
$460$	−1.18162e8	−0.0566012
$461$	2.70289e9	1.28492	0.642458	−	0.766321i	$-0.277915\pi$
$461$	2.70289e9	1.28492	0.642458	+	0.766321i	$0.277915\pi$
$462$	2.02390e9	0.954865
$463$	−2.68582e9	−1.25760	−0.628801	−	0.777566i	$-0.716454\pi$
$463$	−2.68582e9	−1.25760	−0.628801	+	0.777566i	$0.716454\pi$
$464$	−7.18724e8	−0.334002
$465$	−2.72170e7	−0.0125532
$466$	−1.01233e9	−0.463417
$467$	−4.54161e8	−0.206348	−0.103174	−	0.994663i	$-0.532900\pi$
$467$	−4.54161e8	−0.206348	−0.103174	+	0.994663i	$0.532900\pi$
$468$	0	0
$469$	3.15072e9	1.41028
$470$	2.36102e8	0.104895
$471$	2.15851e9	0.951877
$472$	6.81191e8	0.298175
$473$	−3.83743e9	−1.66735
$474$	5.95376e8	0.256784
$475$	−2.10995e9	−0.903324
$476$	1.46027e9	0.620598
$477$	−2.17475e9	−0.917475
$478$	−6.43954e7	−0.0269685
$479$	−1.99643e9	−0.830003	−0.415001	−	0.909821i	$-0.636219\pi$
$479$	−1.99643e9	−0.830003	−0.415001	+	0.909821i	$0.636219\pi$
$480$	5.68414e7	0.0234596
$481$	0	0
$482$	3.10061e9	1.26120
$483$	−1.20302e9	−0.485801
$484$	2.00974e9	0.805714
$485$	−4.63371e8	−0.184430
$486$	−1.65593e9	−0.654357
$487$	1.61212e9	0.632479	0.316240	−	0.948679i	$-0.397580\pi$
$487$	1.61212e9	0.632479	0.316240	+	0.948679i	$0.397580\pi$
$488$	−1.46771e9	−0.571704
$489$	3.17558e9	1.22812
$490$	1.80511e8	0.0693135
$491$	−5.18143e9	−1.97544	−0.987722	−	0.156221i	$-0.950069\pi$
$491$	−5.18143e9	−1.97544	−0.987722	+	0.156221i	$0.950069\pi$
$492$	4.56177e8	0.172685
$493$	3.59812e9	1.35242
$494$	0	0
$495$	4.54731e8	0.168514
$496$	6.42665e7	0.0236482
$497$	1.78240e9	0.651265
$498$	−6.05148e8	−0.219563
$499$	2.11935e9	0.763574	0.381787	−	0.924250i	$-0.375309\pi$
$499$	2.11935e9	0.763574	0.381787	+	0.924250i	$0.375309\pi$
$500$	5.33947e8	0.191031
$501$	1.05693e9	0.375505
$502$	−5.92376e8	−0.208994
$503$	−1.90758e8	−0.0668336	−0.0334168	−	0.999442i	$-0.510639\pi$
$503$	−1.90758e8	−0.0668336	−0.0334168	+	0.999442i	$0.510639\pi$
$504$	−6.67239e8	−0.232153
$505$	5.75518e8	0.198856
$506$	−1.93594e9	−0.664300
$507$	0	0
$508$	−2.61037e9	−0.883445
$509$	5.14477e9	1.72923	0.864617	−	0.502431i	$-0.167561\pi$
$509$	5.14477e9	1.72923	0.864617	+	0.502431i	$0.167561\pi$
$510$	−2.84563e8	−0.0949911
$511$	−1.54903e9	−0.513555
$512$	−1.34218e8	−0.0441942
$513$	−3.00455e9	−0.982581
$514$	1.19441e9	0.387955
$515$	−1.64119e8	−0.0529459
$516$	−1.09726e9	−0.351591
$517$	3.86823e9	1.23110
$518$	1.77762e7	0.00561934
$519$	1.35825e9	0.426475
$520$	0	0
$521$	1.41370e9	0.437950	0.218975	−	0.975730i	$-0.429729\pi$
$521$	1.41370e9	0.437950	0.218975	+	0.975730i	$0.429729\pi$
$522$	−1.64408e9	−0.505913
$523$	−3.46950e9	−1.06050	−0.530251	−	0.847841i	$-0.677902\pi$
$523$	−3.46950e9	−1.06050	−0.530251	+	0.847841i	$0.677902\pi$
$524$	2.63057e9	0.798711
$525$	2.66556e9	0.803953
$526$	−1.38766e9	−0.415749
$527$	−3.21735e8	−0.0957549
$528$	9.31275e8	0.275333
$529$	−2.25409e9	−0.662028
$530$	8.08497e8	0.235892
$531$	1.55822e9	0.451646
$532$	−1.99908e9	−0.575624
$533$	0	0
$534$	−1.79794e9	−0.510954
$535$	−1.24223e9	−0.350721
$536$	1.44977e9	0.406650
$537$	−6.45903e8	−0.179994
$538$	3.44728e9	0.954417
$539$	2.95745e9	0.813498
$540$	3.72822e8	0.101888
$541$	−2.01148e9	−0.546166	−0.273083	−	0.961990i	$-0.588043\pi$
$541$	−2.01148e9	−0.546166	−0.273083	+	0.961990i	$0.588043\pi$
$542$	−2.48186e9	−0.669544
$543$	−9.85008e8	−0.264022
$544$	6.71929e8	0.178948
$545$	−2.48134e8	−0.0656597
$546$	0	0
$547$	−1.25989e8	−0.0329137	−0.0164569	−	0.999865i	$-0.505239\pi$
$547$	−1.25989e8	−0.0329137	−0.0164569	+	0.999865i	$0.505239\pi$
$548$	9.42126e8	0.244555
$549$	−3.35739e9	−0.865960
$550$	4.28949e9	1.09935
$551$	−4.92573e9	−1.25441
$552$	−5.53557e8	−0.140080
$553$	2.59823e9	0.653341
$554$	−3.12031e9	−0.779674
$555$	−3.46404e6	−0.000860118 0
$556$	−3.63833e9	−0.897719
$557$	−2.26177e7	−0.00554569	−0.00277284	−	0.999996i	$-0.500883\pi$
$557$	−2.26177e7	−0.00554569	−0.00277284	+	0.999996i	$0.500883\pi$
$558$	1.47009e8	0.0358200
$559$	0	0
$560$	2.48057e8	0.0596889
$561$	−4.66220e9	−1.11486
$562$	1.79514e9	0.426600
$563$	3.40008e9	0.802988	0.401494	−	0.915862i	$-0.368491\pi$
$563$	3.40008e9	0.802988	0.401494	+	0.915862i	$0.368491\pi$
$564$	1.10607e9	0.259601
$565$	−1.12837e9	−0.263197
$566$	1.00151e8	0.0232166
$567$	9.45638e8	0.217863
$568$	8.20151e8	0.187791
$569$	−8.44190e9	−1.92109	−0.960544	−	0.278128i	$-0.910286\pi$
$569$	−8.44190e9	−1.92109	−0.960544	+	0.278128i	$0.910286\pi$
$570$	3.89559e8	0.0881072
$571$	−8.49495e9	−1.90956	−0.954782	−	0.297307i	$-0.903912\pi$
$571$	−8.49495e9	−1.90956	−0.954782	+	0.297307i	$0.903912\pi$
$572$	0	0
$573$	2.57246e9	0.571226
$574$	1.99076e9	0.439368
$575$	−2.54971e9	−0.559311
$576$	−3.07023e8	−0.0669409
$577$	−3.24606e9	−0.703462	−0.351731	−	0.936101i	$-0.614407\pi$
$577$	−3.24606e9	−0.703462	−0.351731	+	0.936101i	$0.614407\pi$
$578$	−8.11412e7	−0.0174781
$579$	4.41061e9	0.944331
$580$	6.11213e8	0.130075
$581$	−2.64088e9	−0.558639
$582$	−2.17076e9	−0.456438
$583$	1.32462e10	2.76854
$584$	−7.12769e8	−0.148082
$585$	0	0
$586$	3.41338e9	0.700718
$587$	−5.74253e9	−1.17184	−0.585922	−	0.810367i	$-0.699268\pi$
$587$	−5.74253e9	−1.17184	−0.585922	+	0.810367i	$0.699268\pi$
$588$	8.45645e8	0.171541
$589$	4.40447e8	0.0888157
$590$	−5.79294e8	−0.116123
$591$	5.54079e9	1.10412
$592$	8.17953e6	0.00162033
$593$	−5.53884e9	−1.09076	−0.545378	−	0.838190i	$-0.683614\pi$
$593$	−5.53884e9	−1.09076	−0.545378	+	0.838190i	$0.683614\pi$
$594$	6.10821e9	1.19581
$595$	−1.24184e9	−0.241688
$596$	−2.46567e9	−0.477061
$597$	7.61761e8	0.146524
$598$	0	0
$599$	−5.75425e8	−0.109394	−0.0546971	−	0.998503i	$-0.517419\pi$
$599$	−5.75425e8	−0.109394	−0.0546971	+	0.998503i	$0.517419\pi$
$600$	1.22653e9	0.231818
$601$	3.63524e9	0.683082	0.341541	−	0.939867i	$-0.389051\pi$
$601$	3.63524e9	0.683082	0.341541	+	0.939867i	$0.389051\pi$
$602$	−4.78848e9	−0.894562
$603$	3.31634e9	0.615953
$604$	1.88536e9	0.348148
$605$	−1.70911e9	−0.313781
$606$	2.69614e9	0.492140
$607$	8.99932e8	0.163324	0.0816619	−	0.996660i	$-0.473977\pi$
$607$	8.99932e8	0.163324	0.0816619	+	0.996660i	$0.473977\pi$
$608$	−9.19853e8	−0.165980
$609$	6.22283e9	1.11642
$610$	1.24816e9	0.222647
$611$	0	0
$612$	1.53703e9	0.271053
$613$	1.94949e9	0.341829	0.170914	−	0.985286i	$-0.445328\pi$
$613$	1.94949e9	0.341829	0.170914	+	0.985286i	$0.445328\pi$
$614$	−2.26739e9	−0.395308
$615$	−3.87939e8	−0.0672513
$616$	4.06410e9	0.700538
$617$	−5.89463e9	−1.01032	−0.505160	−	0.863026i	$-0.668566\pi$
$617$	−5.89463e9	−1.01032	−0.505160	+	0.863026i	$0.668566\pi$
$618$	−7.68851e8	−0.131033
$619$	5.43625e9	0.921259	0.460630	−	0.887592i	$-0.347624\pi$
$619$	5.43625e9	0.921259	0.460630	+	0.887592i	$0.347624\pi$
$620$	−5.46532e7	−0.00920968
$621$	−3.63077e9	−0.608384
$622$	5.71390e9	0.952065
$623$	−7.84626e9	−1.30003
$624$	0	0
$625$	5.41801e9	0.887687
$626$	2.81588e9	0.458779
$627$	6.38243e9	1.03407
$628$	4.33441e9	0.698346
$629$	−4.09488e7	−0.00656092
$630$	5.67430e8	0.0904108
$631$	−1.16333e9	−0.184332	−0.0921660	−	0.995744i	$-0.529379\pi$
$631$	−1.16333e9	−0.184332	−0.0921660	+	0.995744i	$0.529379\pi$
$632$	1.19555e9	0.188390
$633$	−1.74580e9	−0.273579
$634$	3.64502e9	0.568052
$635$	2.21990e9	0.344053
$636$	3.78759e9	0.583798
$637$	0	0
$638$	1.00140e10	1.52663
$639$	1.87609e9	0.284447
$640$	1.14141e8	0.0172112
$641$	2.62975e9	0.394377	0.197189	−	0.980366i	$-0.436819\pi$
$641$	2.62975e9	0.394377	0.197189	+	0.980366i	$0.436819\pi$
$642$	−5.81948e9	−0.867984
$643$	−2.92590e9	−0.434032	−0.217016	−	0.976168i	$-0.569632\pi$
$643$	−2.92590e9	−0.434032	−0.217016	+	0.976168i	$0.569632\pi$
$644$	−2.41573e9	−0.356409
$645$	9.33130e8	0.136925
$646$	4.60502e9	0.672075
$647$	−1.37294e9	−0.199291	−0.0996455	−	0.995023i	$-0.531771\pi$
$647$	−1.37294e9	−0.199291	−0.0996455	+	0.995023i	$0.531771\pi$
$648$	4.35125e8	0.0628205
$649$	−9.49100e9	−1.36287
$650$	0	0
$651$	−5.56430e8	−0.0790455
$652$	6.37674e9	0.901015
$653$	−4.51279e9	−0.634233	−0.317116	−	0.948387i	$-0.602715\pi$
$653$	−4.51279e9	−0.634233	−0.317116	+	0.948387i	$0.602715\pi$
$654$	−1.16244e9	−0.162498
$655$	−2.23707e9	−0.311054
$656$	9.16028e8	0.126691
$657$	−1.63046e9	−0.224301
$658$	4.82691e9	0.660509
$659$	−5.41645e9	−0.737252	−0.368626	−	0.929578i	$-0.620172\pi$
$659$	−5.41645e9	−0.737252	−0.368626	+	0.929578i	$0.620172\pi$
$660$	−7.91969e8	−0.107227
$661$	−1.41912e9	−0.191124	−0.0955618	−	0.995423i	$-0.530465\pi$
$661$	−1.41912e9	−0.191124	−0.0955618	+	0.995423i	$0.530465\pi$
$662$	−5.32168e9	−0.712929
$663$	0	0
$664$	−1.21517e9	−0.161083
$665$	1.70004e9	0.224174
$666$	1.87106e7	0.00245431
$667$	−5.95237e9	−0.776693
$668$	2.12238e9	0.275490
$669$	8.67603e9	1.12029
$670$	−1.23290e9	−0.158368
$671$	2.04496e10	2.61310
$672$	1.16208e9	0.147721
$673$	−5.50060e9	−0.695597	−0.347799	−	0.937569i	$-0.613071\pi$
$673$	−5.50060e9	−0.695597	−0.347799	+	0.937569i	$0.613071\pi$
$674$	−7.36233e9	−0.926202
$675$	8.04477e9	1.00682
$676$	0	0
$677$	−8.33416e9	−1.03229	−0.516145	−	0.856501i	$-0.672633\pi$
$677$	−8.33416e9	−1.03229	−0.516145	+	0.856501i	$0.672633\pi$
$678$	−5.28608e9	−0.651373
$679$	−9.47326e9	−1.16133
$680$	−5.71418e8	−0.0696904
$681$	−1.43280e9	−0.173849
$682$	−8.95423e8	−0.108089
$683$	1.49205e10	1.79189	0.895943	−	0.444169i	$-0.146501\pi$
$683$	1.49205e10	1.79189	0.895943	+	0.444169i	$0.146501\pi$
$684$	−2.10416e9	−0.251410
$685$	−8.01198e8	−0.0952407
$686$	−3.64049e9	−0.430552
$687$	3.93486e9	0.463000
$688$	−2.20337e9	−0.257945
$689$	0	0
$690$	4.70753e8	0.0545533
$691$	4.89733e9	0.564659	0.282329	−	0.959318i	$-0.408893\pi$
$691$	4.89733e9	0.564659	0.282329	+	0.959318i	$0.408893\pi$
$692$	2.72744e9	0.312884
$693$	9.29662e9	1.06111
$694$	−1.09199e10	−1.24011
$695$	3.09409e9	0.349612
$696$	2.86336e9	0.321917
$697$	−4.58588e9	−0.512988
$698$	7.17779e9	0.798908
$699$	4.03308e9	0.446650
$700$	5.35259e9	0.589822
$701$	−2.20310e9	−0.241558	−0.120779	−	0.992679i	$-0.538539\pi$
$701$	−2.20310e9	−0.241558	−0.120779	+	0.992679i	$0.538539\pi$
$702$	0	0
$703$	5.60579e7	0.00608546
$704$	1.87005e9	0.201999
$705$	−9.40619e8	−0.101100
$706$	2.15684e9	0.230676
$707$	1.17660e10	1.25216
$708$	−2.71383e9	−0.287387
$709$	8.69875e9	0.916632	0.458316	−	0.888789i	$-0.348453\pi$
$709$	8.69875e9	0.916632	0.458316	+	0.888789i	$0.348453\pi$
$710$	−6.97468e8	−0.0731341
$711$	2.73481e9	0.285354
$712$	−3.61037e9	−0.374862
$713$	5.32246e8	0.0549920
$714$	−5.81767e9	−0.598143
$715$	0	0
$716$	−1.29701e9	−0.132053
$717$	2.56548e8	0.0259928
$718$	1.79142e9	0.180619
$719$	−1.09362e9	−0.109728	−0.0548640	−	0.998494i	$-0.517473\pi$
$719$	−1.09362e9	−0.109728	−0.0548640	+	0.998494i	$0.517473\pi$
$720$	2.61096e8	0.0260698
$721$	−3.35528e9	−0.333392
$722$	8.46819e8	0.0837356
$723$	−1.23527e10	−1.21556
$724$	−1.97795e9	−0.193700
$725$	1.31888e10	1.28535
$726$	−8.00671e9	−0.776561
$727$	9.47790e9	0.914832	0.457416	−	0.889253i	$-0.348775\pi$
$727$	9.47790e9	0.914832	0.457416	+	0.889253i	$0.348775\pi$
$728$	0	0
$729$	8.45578e9	0.808365
$730$	6.06149e8	0.0576699
$731$	1.10306e10	1.04445
$732$	5.84730e9	0.551019
$733$	5.75895e9	0.540107	0.270053	−	0.962845i	$-0.412959\pi$
$733$	5.75895e9	0.540107	0.270053	+	0.962845i	$0.412959\pi$
$734$	2.24686e9	0.209720
$735$	−7.19148e8	−0.0668056
$736$	−1.11157e9	−0.102770
$737$	−2.01995e10	−1.85868
$738$	2.09541e9	0.191899
$739$	7.10506e9	0.647608	0.323804	−	0.946124i	$-0.395038\pi$
$739$	7.10506e9	0.647608	0.323804	+	0.946124i	$0.395038\pi$
$740$	−6.95599e6	−0.000631027 0
$741$	0	0
$742$	1.65291e10	1.48537
$743$	−9.84108e9	−0.880202	−0.440101	−	0.897948i	$-0.645057\pi$
$743$	−9.84108e9	−0.880202	−0.440101	+	0.897948i	$0.645057\pi$
$744$	−2.56035e8	−0.0227926
$745$	2.09684e9	0.185789
$746$	−1.40420e10	−1.23834
$747$	−2.77970e9	−0.243992
$748$	−9.36196e9	−0.817921
$749$	−2.53963e10	−2.20843
$750$	−2.12722e9	−0.184119
$751$	1.29230e10	1.11333	0.556663	−	0.830738i	$-0.312081\pi$
$751$	1.29230e10	1.11333	0.556663	+	0.830738i	$0.312081\pi$
$752$	2.22105e9	0.190457
$753$	2.36000e9	0.201432
$754$	0	0
$755$	−1.60333e9	−0.135584
$756$	7.62205e9	0.641572
$757$	−1.51951e10	−1.27311	−0.636556	−	0.771230i	$-0.719642\pi$
$757$	−1.51951e10	−1.27311	−0.636556	+	0.771230i	$0.719642\pi$
$758$	−2.87387e9	−0.239676
$759$	7.71269e9	0.640265
$760$	7.82256e8	0.0646400
$761$	8.68678e9	0.714517	0.357259	−	0.934006i	$-0.383711\pi$
$761$	8.68678e9	0.714517	0.357259	+	0.934006i	$0.383711\pi$
$762$	1.03996e10	0.851481
$763$	−5.07291e9	−0.413449
$764$	5.16565e9	0.419081
$765$	−1.30712e9	−0.105560
$766$	1.50295e10	1.20822
$767$	0	0
$768$	5.34717e8	0.0425951
$769$	−2.15004e10	−1.70492	−0.852461	−	0.522792i	$-0.824891\pi$
$769$	−2.15004e10	−1.70492	−0.852461	+	0.522792i	$0.824891\pi$
$770$	−3.45617e9	−0.272821
$771$	−4.75846e9	−0.373918
$772$	8.85675e9	0.692810
$773$	5.01385e9	0.390430	0.195215	−	0.980760i	$-0.437460\pi$
$773$	5.01385e9	0.390430	0.195215	+	0.980760i	$0.437460\pi$
$774$	−5.04020e9	−0.390710
$775$	−1.17931e9	−0.0910063
$776$	−4.35901e9	−0.334867
$777$	−7.08197e7	−0.00541602
$778$	−3.47592e9	−0.264631
$779$	6.27794e9	0.475813
$780$	0	0
$781$	−1.14271e10	−0.858338
$782$	5.56482e9	0.416129
$783$	1.87808e10	1.39813
$784$	1.69810e9	0.125851
$785$	−3.68604e9	−0.271967
$786$	−1.04801e10	−0.769812
$787$	−9.99315e9	−0.730787	−0.365393	−	0.930853i	$-0.619066\pi$
$787$	−9.99315e9	−0.730787	−0.365393	+	0.930853i	$0.619066\pi$
$788$	1.11262e10	0.810038
$789$	5.52836e9	0.400707
$790$	−1.01671e9	−0.0733673
$791$	−2.30686e10	−1.65731
$792$	4.27773e9	0.305968
$793$	0	0
$794$	−1.62526e10	−1.15226
$795$	−3.22102e9	−0.227357
$796$	1.52966e9	0.107498
$797$	−2.78073e10	−1.94561	−0.972803	−	0.231634i	$-0.925593\pi$
$797$	−2.78073e10	−1.94561	−0.972803	+	0.231634i	$0.925593\pi$
$798$	7.96423e9	0.554797
$799$	−1.11192e10	−0.771184
$800$	2.46293e9	0.170074
$801$	−8.25871e9	−0.567804
$802$	6.42462e9	0.439782
$803$	9.93098e9	0.676843
$804$	−5.77580e9	−0.391937
$805$	2.05437e9	0.138801
$806$	0	0
$807$	−1.37338e10	−0.919884
$808$	5.41400e9	0.361059
$809$	2.83343e9	0.188145	0.0940724	−	0.995565i	$-0.470011\pi$
$809$	2.83343e9	0.188145	0.0940724	+	0.995565i	$0.470011\pi$
$810$	−3.70036e8	−0.0244651
$811$	−2.48489e10	−1.63581	−0.817906	−	0.575352i	$-0.804865\pi$
$811$	−2.48489e10	−1.63581	−0.817906	+	0.575352i	$0.804865\pi$
$812$	1.24958e10	0.819063
$813$	9.88761e9	0.645319
$814$	−1.13965e8	−0.00740604
$815$	−5.42287e9	−0.350895
$816$	−2.67693e9	−0.172473
$817$	−1.51006e10	−0.968765
$818$	2.02992e10	1.29671
$819$	0	0
$820$	−7.79003e8	−0.0493391
$821$	2.96860e10	1.87219	0.936096	−	0.351745i	$-0.114412\pi$
$821$	2.96860e10	1.87219	0.936096	+	0.351745i	$0.114412\pi$
$822$	−3.75339e9	−0.235707
$823$	2.35999e10	1.47575	0.737873	−	0.674940i	$-0.235830\pi$
$823$	2.35999e10	1.47575	0.737873	+	0.674940i	$0.235830\pi$
$824$	−1.54389e9	−0.0961329
$825$	−1.70892e10	−1.05957
$826$	−1.18432e10	−0.731206
$827$	−1.10498e10	−0.679335	−0.339668	−	0.940546i	$-0.610315\pi$
$827$	−1.10498e10	−0.679335	−0.339668	+	0.940546i	$0.610315\pi$
$828$	−2.54272e9	−0.155665
$829$	−1.36783e10	−0.833858	−0.416929	−	0.908939i	$-0.636894\pi$
$829$	−1.36783e10	−0.833858	−0.416929	+	0.908939i	$0.636894\pi$
$830$	1.03340e9	0.0627327
$831$	1.24312e10	0.751464
$832$	0	0
$833$	−8.50113e9	−0.509588
$834$	1.44950e10	0.865238
$835$	−1.80490e9	−0.107288
$836$	1.28163e10	0.758647
$837$	−1.67933e9	−0.0989912
$838$	1.42875e9	0.0838690
$839$	−2.90433e9	−0.169777	−0.0848886	−	0.996390i	$-0.527053\pi$
$839$	−2.90433e9	−0.169777	−0.0848886	+	0.996390i	$0.527053\pi$
$840$	−9.88248e8	−0.0575292
$841$	1.35397e10	0.784918
$842$	−1.66226e10	−0.959634
$843$	−7.15175e9	−0.411165
$844$	−3.50567e9	−0.200711
$845$	0	0
$846$	5.08065e9	0.288485
$847$	−3.49414e10	−1.97583
$848$	7.60568e9	0.428304
$849$	−3.98997e8	−0.0223765
$850$	−1.23301e10	−0.688652
$851$	6.77417e7	0.00376793
$852$	−3.26744e9	−0.180996
$853$	2.26605e10	1.25011	0.625054	−	0.780581i	$-0.285077\pi$
$853$	2.26605e10	1.25011	0.625054	+	0.780581i	$0.285077\pi$
$854$	2.55177e10	1.40197
$855$	1.78941e9	0.0979102
$856$	−1.16858e10	−0.636798
$857$	−2.06344e10	−1.11985	−0.559925	−	0.828544i	$-0.689170\pi$
$857$	−2.06344e10	−1.11985	−0.559925	+	0.828544i	$0.689170\pi$
$858$	0	0
$859$	−1.43580e10	−0.772891	−0.386445	−	0.922312i	$-0.626297\pi$
$859$	−1.43580e10	−0.772891	−0.386445	+	0.922312i	$0.626297\pi$
$860$	1.87378e9	0.100455
$861$	−7.93112e9	−0.423471
$862$	−1.26549e10	−0.672948
$863$	−5.30271e9	−0.280841	−0.140420	−	0.990092i	$-0.544845\pi$
$863$	−5.30271e9	−0.280841	−0.140420	+	0.990092i	$0.544845\pi$
$864$	3.50720e9	0.184996
$865$	−2.31945e9	−0.121851
$866$	−1.06922e10	−0.559440
$867$	3.23263e8	0.0168457
$868$	−1.11734e9	−0.0579918
$869$	−1.66575e10	−0.861075
$870$	−2.43505e9	−0.125369
$871$	0	0
$872$	−2.33424e9	−0.119217
$873$	−9.97123e9	−0.507223
$874$	−7.61809e9	−0.385972
$875$	−9.28323e9	−0.468458
$876$	2.83964e9	0.142725
$877$	−1.04975e10	−0.525519	−0.262759	−	0.964861i	$-0.584633\pi$
$877$	−1.04975e10	−0.525519	−0.262759	+	0.964861i	$0.584633\pi$
$878$	−1.59626e10	−0.795926
$879$	−1.35988e10	−0.675365
$880$	−1.59032e9	−0.0786673
$881$	2.56783e10	1.26517	0.632587	−	0.774490i	$-0.281993\pi$
$881$	2.56783e10	1.26517	0.632587	+	0.774490i	$0.281993\pi$
$882$	3.88440e9	0.190627
$883$	1.36057e10	0.665058	0.332529	−	0.943093i	$-0.392098\pi$
$883$	1.36057e10	0.665058	0.332529	+	0.943093i	$0.392098\pi$
$884$	0	0
$885$	2.30788e9	0.111921
$886$	−1.21490e9	−0.0586843
$887$	−5.79488e9	−0.278812	−0.139406	−	0.990235i	$-0.544519\pi$
$887$	−5.79488e9	−0.278812	−0.139406	+	0.990235i	$0.544519\pi$
$888$	−3.25869e7	−0.00156170
$889$	4.53841e10	2.16645
$890$	3.07031e9	0.145988
$891$	−6.06258e9	−0.287134
$892$	1.74219e10	0.821901
$893$	1.52218e10	0.715298
$894$	9.82314e9	0.459800
$895$	1.10299e9	0.0514272
$896$	2.33352e9	0.108376
$897$	0	0
$898$	1.28521e9	0.0592252
$899$	−2.75313e9	−0.126377
$900$	5.63395e9	0.257611
$901$	−3.80760e10	−1.73426
$902$	−1.27630e10	−0.579068
$903$	1.90771e10	0.862195
$904$	−1.06147e10	−0.477881
$905$	1.68208e9	0.0754355
$906$	−7.51118e9	−0.335552
$907$	3.99131e10	1.77619	0.888096	−	0.459659i	$-0.152028\pi$
$907$	3.99131e10	1.77619	0.888096	+	0.459659i	$0.152028\pi$
$908$	−2.87715e9	−0.127544
$909$	1.23845e10	0.546896
$910$	0	0
$911$	−3.59021e9	−0.157328	−0.0786639	−	0.996901i	$-0.525065\pi$
$911$	−3.59021e9	−0.157328	−0.0786639	+	0.996901i	$0.525065\pi$
$912$	3.66465e9	0.159975
$913$	1.69309e10	0.736262
$914$	−8.17343e9	−0.354073
$915$	−4.97263e9	−0.214591
$916$	7.90142e9	0.339681
$917$	−4.57352e10	−1.95865
$918$	−1.75580e10	−0.749074
$919$	−6.24523e9	−0.265427	−0.132713	−	0.991154i	$-0.542369\pi$
$919$	−6.24523e9	−0.265427	−0.132713	+	0.991154i	$0.542369\pi$
$920$	9.45297e8	0.0400231
$921$	9.03316e9	0.381005
$922$	−2.16231e10	−0.908573
$923$	0	0
$924$	−1.61912e10	−0.675192
$925$	−1.50097e8	−0.00623555
$926$	2.14866e10	0.889260
$927$	−3.53165e9	−0.145613
$928$	5.74979e9	0.236175
$929$	3.69092e9	0.151036	0.0755179	−	0.997144i	$-0.475939\pi$
$929$	3.69092e9	0.151036	0.0755179	+	0.997144i	$0.475939\pi$
$930$	2.17736e8	0.00887645
$931$	1.16378e10	0.472659
$932$	8.09865e9	0.327685
$933$	−2.27639e10	−0.917618
$934$	3.63329e9	0.145910
$935$	7.96154e9	0.318535
$936$	0	0
$937$	−3.30027e10	−1.31057	−0.655286	−	0.755381i	$-0.727452\pi$
$937$	−3.30027e10	−1.31057	−0.655286	+	0.755381i	$0.727452\pi$
$938$	−2.52057e10	−0.997216
$939$	−1.12183e10	−0.442179
$940$	−1.88881e9	−0.0741723
$941$	2.73399e10	1.06963	0.534814	−	0.844970i	$-0.320382\pi$
$941$	2.73399e10	1.06963	0.534814	+	0.844970i	$0.320382\pi$
$942$	−1.72681e10	−0.673079
$943$	7.58642e9	0.294609
$944$	−5.44953e9	−0.210842
$945$	−6.48190e9	−0.249857
$946$	3.06994e10	1.17899
$947$	−2.86589e10	−1.09657	−0.548283	−	0.836293i	$-0.684718\pi$
$947$	−2.86589e10	−1.09657	−0.548283	+	0.836293i	$0.684718\pi$
$948$	−4.76301e9	−0.181573
$949$	0	0
$950$	1.68796e10	0.638746
$951$	−1.45216e10	−0.547498
$952$	−1.16822e10	−0.438829
$953$	−3.66898e10	−1.37316	−0.686578	−	0.727056i	$-0.740888\pi$
$953$	−3.66898e10	−1.37316	−0.686578	+	0.727056i	$0.740888\pi$
$954$	1.73980e10	0.648753
$955$	−4.39294e9	−0.163209
$956$	5.15163e8	0.0190696
$957$	−3.98951e10	−1.47139
$958$	1.59714e10	0.586900
$959$	−1.63799e10	−0.599715
$960$	−4.54731e8	−0.0165884
$961$	−2.72664e10	−0.991052
$962$	0	0
$963$	−2.67313e10	−0.964558
$964$	−2.48049e10	−0.891801
$965$	−7.53191e9	−0.269811
$966$	9.62417e9	0.343513
$967$	2.06429e10	0.734139	0.367069	−	0.930194i	$-0.380361\pi$
$967$	2.06429e10	0.734139	0.367069	+	0.930194i	$0.380361\pi$
$968$	−1.60779e10	−0.569726
$969$	−1.83462e10	−0.647758
$970$	3.70697e9	0.130412
$971$	−1.62503e9	−0.0569631	−0.0284815	−	0.999594i	$-0.509067\pi$
$971$	−1.62503e9	−0.0569631	−0.0284815	+	0.999594i	$0.509067\pi$
$972$	1.32474e10	0.462701
$973$	6.32563e10	2.20145
$974$	−1.28970e10	−0.447230
$975$	0	0
$976$	1.17417e10	0.404256
$977$	−9.36392e9	−0.321238	−0.160619	−	0.987016i	$-0.551349\pi$
$977$	−9.36392e9	−0.321238	−0.160619	+	0.987016i	$0.551349\pi$
$978$	−2.54046e10	−0.868414
$979$	5.03031e10	1.71339
$980$	−1.44409e9	−0.0490120
$981$	−5.33958e9	−0.180578
$982$	4.14515e10	1.39685
$983$	3.28194e10	1.10203	0.551016	−	0.834495i	$-0.314240\pi$
$983$	3.28194e10	1.10203	0.551016	+	0.834495i	$0.314240\pi$
$984$	−3.64942e9	−0.122107
$985$	−9.46189e9	−0.315465
$986$	−2.87850e10	−0.956305
$987$	−1.92302e10	−0.636611
$988$	0	0
$989$	−1.82480e10	−0.599829
$990$	−3.63785e9	−0.119157
$991$	−3.66148e10	−1.19508	−0.597542	−	0.801838i	$-0.703856\pi$
$991$	−3.66148e10	−1.19508	−0.597542	+	0.801838i	$0.703856\pi$
$992$	−5.14132e8	−0.0167218
$993$	2.12013e10	0.687134
$994$	−1.42592e10	−0.460514
$995$	−1.30084e9	−0.0418643
$996$	4.84118e9	0.155254
$997$	4.33458e10	1.38520	0.692602	−	0.721320i	$-0.256464\pi$
$997$	4.33458e10	1.38520	0.692602	+	0.721320i	$0.256464\pi$
$998$	−1.69548e10	−0.539928
$999$	−2.13737e8	−0.00678266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.a.i.1.3		4
13.3	even	3		26.8.c.b.9.2	yes	8
13.5	odd	4		338.8.b.h.337.7		8
13.8	odd	4		338.8.b.h.337.3		8
13.9	even	3		26.8.c.b.3.2	✓	8
13.12	even	2		338.8.a.j.1.3		4
39.29	odd	6		234.8.h.b.217.3		8
39.35	odd	6		234.8.h.b.55.3		8
52.3	odd	6		208.8.i.b.113.3		8
52.35	odd	6		208.8.i.b.81.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.c.b.3.2	✓	8	13.9	even	3
26.8.c.b.9.2	yes	8	13.3	even	3
208.8.i.b.81.3		8	52.35	odd	6
208.8.i.b.113.3		8	52.3	odd	6
234.8.h.b.55.3		8	39.35	odd	6
234.8.h.b.217.3		8	39.29	odd	6
338.8.a.i.1.3		4	1.1	even	1	trivial
338.8.a.j.1.3		4	13.12	even	2
338.8.b.h.337.3		8	13.8	odd	4
338.8.b.h.337.7		8	13.5	odd	4

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +31.8716 q^{3} +64.0000 q^{4} -54.4265 q^{5} -254.973 q^{6} -1112.71 q^{7} -512.000 q^{8} -1171.20 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +31.8716 q^{3} +64.0000 q^{4} -54.4265 q^{5} -254.973 q^{6} -1112.71 q^{7} -512.000 q^{8} -1171.20 q^{9} +435.412 q^{10} +7133.68 q^{11} +2039.79 q^{12} +8901.66 q^{14} -1734.66 q^{15} +4096.00 q^{16} -20505.6 q^{17} +9369.59 q^{18} +28071.7 q^{19} -3483.30 q^{20} -35463.8 q^{21} -57069.4 q^{22} +33922.5 q^{23} -16318.3 q^{24} -75162.8 q^{25} -107031. q^{27} -71213.3 q^{28} -175470. q^{29} +13877.3 q^{30} +15690.1 q^{31} -32768.0 q^{32} +227362. q^{33} +164045. q^{34} +60560.8 q^{35} -74956.7 q^{36} +1996.96 q^{37} -224573. q^{38} +27866.4 q^{40} +223640. q^{41} +283711. q^{42} -537931. q^{43} +456555. q^{44} +63744.2 q^{45} -271380. q^{46} +542249. q^{47} +130546. q^{48} +414575. q^{49} +601302. q^{50} -653549. q^{51} +1.85685e6 q^{53} +856250. q^{54} -388261. q^{55} +569706. q^{56} +894691. q^{57} +1.40376e6 q^{58} -1.33045e6 q^{59} -111018. q^{60} +2.86662e6 q^{61} -125521. q^{62} +1.30320e6 q^{63} +262144. q^{64} -1.81890e6 q^{66} -2.83157e6 q^{67} -1.31236e6 q^{68} +1.08117e6 q^{69} -484486. q^{70} -1.60186e6 q^{71} +599654. q^{72} +1.39213e6 q^{73} -15975.6 q^{74} -2.39556e6 q^{75} +1.79659e6 q^{76} -7.93770e6 q^{77} -2.33505e6 q^{79} -222931. q^{80} -849853. q^{81} -1.78912e6 q^{82} +2.37338e6 q^{83} -2.26968e6 q^{84} +1.11605e6 q^{85} +4.30345e6 q^{86} -5.59251e6 q^{87} -3.65244e6 q^{88} +7.05151e6 q^{89} -509954. q^{90} +2.17104e6 q^{92} +500068. q^{93} -4.33799e6 q^{94} -1.52784e6 q^{95} -1.04437e6 q^{96} +8.51370e6 q^{97} -3.31660e6 q^{98} -8.35495e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9	\( 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9} - 2224 q^{10} + 7392 q^{11} - 4384 q^{14} - 15528 q^{15} + 16384 q^{16} - 28316 q^{17} - 41712 q^{18} + 99888 q^{19} + 17792 q^{20} + 91074 q^{21} - 59136 q^{22} + 33388 q^{23} + 86878 q^{25} + 106272 q^{27} + 35072 q^{28} - 93140 q^{29} + 124224 q^{30} + 311160 q^{31} - 131072 q^{32} - 238638 q^{33} + 226528 q^{34} - 141544 q^{35} + 333696 q^{36} + 9636 q^{37} - 799104 q^{38} - 142336 q^{40} - 82892 q^{41} - 728592 q^{42} + 569264 q^{43} + 473088 q^{44} + 2303394 q^{45} - 267104 q^{46} - 574200 q^{47} + 717798 q^{49} - 695024 q^{50} - 2729928 q^{51} + 1235350 q^{53} - 850176 q^{54} + 1092512 q^{55} - 280576 q^{56} + 3528462 q^{57} + 745120 q^{58} - 231504 q^{59} - 993792 q^{60} - 685684 q^{61} - 2489280 q^{62} + 5951712 q^{63} + 1048576 q^{64} + 1909104 q^{66} - 3271056 q^{67} - 1812224 q^{68} - 5600034 q^{69} + 1132352 q^{70} + 175012 q^{71} - 2669568 q^{72} + 7137890 q^{73} - 77088 q^{74} - 22200960 q^{75} + 6392832 q^{76} - 13915206 q^{77} - 7053952 q^{79} + 1138688 q^{80} - 3758004 q^{81} + 663136 q^{82} + 657288 q^{83} + 5828736 q^{84} - 11814998 q^{85} - 4554112 q^{86} + 7182900 q^{87} - 3784704 q^{88} + 11452234 q^{89} - 18427152 q^{90} + 2136832 q^{92} - 2984688 q^{93} + 4593600 q^{94} + 23334088 q^{95} + 428002 q^{97} - 5742384 q^{98} - 10357656 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9 - 2224 * q^10 + 7392 * q^11 - 4384 * q^14 - 15528 * q^15 + 16384 * q^16 - 28316 * q^17 - 41712 * q^18 + 99888 * q^19 + 17792 * q^20 + 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 + 35072 * q^28 - 93140 * q^29 + 124224 * q^30 + 311160 * q^31 - 131072 * q^32 - 238638 * q^33 + 226528 * q^34 - 141544 * q^35 + 333696 * q^36 + 9636 * q^37 - 799104 * q^38 - 142336 * q^40 - 82892 * q^41 - 728592 * q^42 + 569264 * q^43 + 473088 * q^44 + 2303394 * q^45 - 267104 * q^46 - 574200 * q^47 + 717798 * q^49 - 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 - 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 + 3528462 * q^57 + 745120 * q^58 - 231504 * q^59 - 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 + 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 - 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 + 1132352 * q^70 + 175012 * q^71 - 2669568 * q^72 + 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 + 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 + 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 + 657288 * q^83 + 5828736 * q^84 - 11814998 * q^85 - 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 + 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 - 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 + 428002 * q^97 - 5742384 * q^98 - 10357656 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more