LMFDB - Embedded newform 169.8.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [21,-31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.7930693068$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	\(q-19.6237 q^{2} -72.4635 q^{3} +257.091 q^{4} +413.760 q^{5} +1422.00 q^{6} +1005.88 q^{7} -2533.24 q^{8} +3063.96 q^{9} -8119.51 q^{10} +3482.63 q^{11} -18629.7 q^{12} -19739.1 q^{14} -29982.5 q^{15} +16804.1 q^{16} +6157.31 q^{17} -60126.4 q^{18} -41406.0 q^{19} +106374. q^{20} -72889.6 q^{21} -68342.2 q^{22} -63119.9 q^{23} +183568. q^{24} +93072.0 q^{25} -63547.8 q^{27} +258602. q^{28} +75193.0 q^{29} +588368. q^{30} -133868. q^{31} -5503.36 q^{32} -252364. q^{33} -120829. q^{34} +416192. q^{35} +787717. q^{36} +171939. q^{37} +812541. q^{38} -1.04815e6 q^{40} -747517. q^{41} +1.43037e6 q^{42} -187227. q^{43} +895352. q^{44} +1.26774e6 q^{45} +1.23865e6 q^{46} -226649. q^{47} -1.21768e6 q^{48} +188251. q^{49} -1.82642e6 q^{50} -446180. q^{51} -1.34632e6 q^{53} +1.24705e6 q^{54} +1.44097e6 q^{55} -2.54814e6 q^{56} +3.00043e6 q^{57} -1.47557e6 q^{58} -2.50878e6 q^{59} -7.70822e6 q^{60} -509752. q^{61} +2.62699e6 q^{62} +3.08198e6 q^{63} -2.04292e6 q^{64} +4.95232e6 q^{66} -4.07091e6 q^{67} +1.58299e6 q^{68} +4.57389e6 q^{69} -8.16725e6 q^{70} -256015. q^{71} -7.76177e6 q^{72} +5.38093e6 q^{73} -3.37408e6 q^{74} -6.74433e6 q^{75} -1.06451e7 q^{76} +3.50311e6 q^{77} +5.18777e6 q^{79} +6.95285e6 q^{80} -2.09599e6 q^{81} +1.46691e7 q^{82} +5.32418e6 q^{83} -1.87392e7 q^{84} +2.54765e6 q^{85} +3.67410e6 q^{86} -5.44875e6 q^{87} -8.82235e6 q^{88} -8.69805e6 q^{89} -2.48779e7 q^{90} -1.62276e7 q^{92} +9.70053e6 q^{93} +4.44770e6 q^{94} -1.71321e7 q^{95} +398793. q^{96} +3.76799e6 q^{97} -3.69419e6 q^{98} +1.06706e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 31 q^{2} - 26 q^{3} + 1409 q^{4} - 680 q^{5} - 1470 q^{6} - 2929 q^{7} - 4716 q^{8} + 15465 q^{9} - 5167 q^{10} - 14824 q^{11} + 21795 q^{12} - 179 q^{14} - 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots - 37605493 q^{99}+O(q^{100}) $ 21 * q - 31 * q^2 - 26 * q^3 + 1409 * q^4 - 680 * q^5 - 1470 * q^6 - 2929 * q^7 - 4716 * q^8 + 15465 * q^9 - 5167 * q^10 - 14824 * q^11 + 21795 * q^12 - 179 * q^14 - 36398 * q^15 + 113205 * q^16 + 45016 * q^17 - 72967 * q^18 - 119316 * q^19 - 131130 * q^20 - 94178 * q^21 - 274174 * q^22 - 128105 * q^23 - 316902 * q^24 + 366537 * q^25 - 106643 * q^27 - 170352 * q^28 - 62329 * q^29 + 9761 * q^30 - 467367 * q^31 - 677464 * q^32 - 587690 * q^33 - 578614 * q^34 + 848860 * q^35 - 274245 * q^36 - 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 - 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 + 263025 * q^44 + 1938345 * q^45 + 3170392 * q^46 - 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 - 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 - 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 - 5590305 * q^57 - 7510959 * q^58 - 10746233 * q^59 - 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 - 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 - 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 - 28670666 * q^70 - 23267025 * q^71 - 49774322 * q^72 - 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 - 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 - 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 - 21455789 * q^83 - 60768429 * q^84 - 26216249 * q^85 - 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 - 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 - 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 - 66667956 * q^96 - 10916609 * q^97 - 41738840 * q^98 - 37605493 * q^99

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 $

$2$	−19.6237	−1.73451	−0.867255	−	0.497865i	$-0.834118\pi$
$2$	−19.6237	−1.73451	−0.867255	+	0.497865i	$0.834118\pi$
$3$	−72.4635	−1.54951	−0.774756	−	0.632260i	$-0.782127\pi$
$3$	−72.4635	−1.54951	−0.774756	+	0.632260i	$0.782127\pi$
$4$	257.091	2.00852
$5$	413.760	1.48031	0.740156	−	0.672436i	$-0.234752\pi$
$5$	413.760	1.48031	0.740156	+	0.672436i	$0.234752\pi$
$6$	1422.00	2.68764
$7$	1005.88	1.10842	0.554208	−	0.832378i	$-0.313021\pi$
$7$	1005.88	1.10842	0.554208	+	0.832378i	$0.313021\pi$
$8$	−2533.24	−1.74929
$9$	3063.96	1.40099
$10$	−8119.51	−2.56761
$11$	3482.63	0.788920	0.394460	−	0.918913i	$-0.370932\pi$
$11$	3482.63	0.788920	0.394460	+	0.918913i	$0.370932\pi$
$12$	−18629.7	−3.11223
$13$	0	0
$13$	0	0
$14$	−19739.1	−1.92256
$15$	−29982.5	−2.29376
$16$	16804.1	1.02564
$17$	6157.31	0.303962	0.151981	−	0.988383i	$-0.451435\pi$
$17$	6157.31	0.303962	0.151981	+	0.988383i	$0.451435\pi$
$18$	−60126.4	−2.43003
$19$	−41406.0	−1.38492	−0.692462	−	0.721454i	$-0.743474\pi$
$19$	−41406.0	−1.38492	−0.692462	+	0.721454i	$0.743474\pi$
$20$	106374.	2.97324
$21$	−72889.6	−1.71751
$22$	−68342.2	−1.36839
$23$	−63119.9	−1.08173	−0.540865	−	0.841109i	$-0.681903\pi$
$23$	−63119.9	−1.08173	−0.540865	+	0.841109i	$0.681903\pi$
$24$	183568.	2.71055
$25$	93072.0	1.19132
$26$	0	0
$27$	−63547.8	−0.621337
$28$	258602.	2.22628
$29$	75193.0	0.572512	0.286256	−	0.958153i	$-0.407589\pi$
$29$	75193.0	0.572512	0.286256	+	0.958153i	$0.407589\pi$
$30$	588368.	3.97855
$31$	−133868.	−0.807068	−0.403534	−	0.914965i	$-0.632218\pi$
$31$	−133868.	−0.807068	−0.403534	+	0.914965i	$0.632218\pi$
$32$	−5503.36	−0.0296895
$33$	−252364.	−1.22244
$34$	−120829.	−0.527225
$35$	416192.	1.64080
$36$	787717.	2.81392
$37$	171939.	0.558043	0.279022	−	0.960285i	$-0.409990\pi$
$37$	171939.	0.558043	0.279022	+	0.960285i	$0.409990\pi$
$38$	812541.	2.40216
$39$	0	0
$40$	−1.04815e6	−2.58950
$41$	−747517.	−1.69386	−0.846930	−	0.531705i	$-0.821552\pi$
$41$	−747517.	−1.69386	−0.846930	+	0.531705i	$0.821552\pi$
$42$	1.43037e6	2.97903
$43$	−187227.	−0.359112	−0.179556	−	0.983748i	$-0.557466\pi$
$43$	−187227.	−0.359112	−0.179556	+	0.983748i	$0.557466\pi$
$44$	895352.	1.58456
$45$	1.26774e6	2.07390
$46$	1.23865e6	1.87627
$47$	−226649.	−0.318428	−0.159214	−	0.987244i	$-0.550896\pi$
$47$	−226649.	−0.318428	−0.159214	+	0.987244i	$0.550896\pi$
$48$	−1.21768e6	−1.58924
$49$	188251.	0.228587
$50$	−1.82642e6	−2.06636
$51$	−446180.	−0.470993
$52$	0	0
$53$	−1.34632e6	−1.24218	−0.621088	−	0.783740i	$-0.713309\pi$
$53$	−1.34632e6	−1.24218	−0.621088	+	0.783740i	$0.713309\pi$
$54$	1.24705e6	1.07772
$55$	1.44097e6	1.16785
$56$	−2.54814e6	−1.93894
$57$	3.00043e6	2.14596
$58$	−1.47557e6	−0.993027
$59$	−2.50878e6	−1.59031	−0.795153	−	0.606409i	$-0.792610\pi$
$59$	−2.50878e6	−1.59031	−0.795153	+	0.606409i	$0.792610\pi$
$60$	−7.70822e6	−4.60707
$61$	−509752.	−0.287544	−0.143772	−	0.989611i	$-0.545923\pi$
$61$	−509752.	−0.287544	−0.143772	+	0.989611i	$0.545923\pi$
$62$	2.62699e6	1.39987
$63$	3.08198e6	1.55288
$64$	−2.04292e6	−0.974142
$65$	0	0
$66$	4.95232e6	2.12034
$67$	−4.07091e6	−1.65360	−0.826799	−	0.562498i	$-0.809840\pi$
$67$	−4.07091e6	−1.65360	−0.826799	+	0.562498i	$0.809840\pi$
$68$	1.58299e6	0.610515
$69$	4.57389e6	1.67616
$70$	−8.16725e6	−2.84599
$71$	−256015.	−0.0848910	−0.0424455	−	0.999099i	$-0.513515\pi$
$71$	−256015.	−0.0848910	−0.0424455	+	0.999099i	$0.513515\pi$
$72$	−7.76177e6	−2.45074
$73$	5.38093e6	1.61893	0.809463	−	0.587171i	$-0.199758\pi$
$73$	5.38093e6	1.61893	0.809463	+	0.587171i	$0.199758\pi$
$74$	−3.37408e6	−0.967931
$75$	−6.74433e6	−1.84597
$76$	−1.06451e7	−2.78165
$77$	3.50311e6	0.874452
$78$	0	0
$79$	5.18777e6	1.18382	0.591911	−	0.806004i	$-0.298374\pi$
$79$	5.18777e6	1.18382	0.591911	+	0.806004i	$0.298374\pi$
$80$	6.95285e6	1.51827
$81$	−2.09599e6	−0.438219
$82$	1.46691e7	2.93801
$83$	5.32418e6	1.02207	0.511034	−	0.859561i	$-0.329263\pi$
$83$	5.32418e6	1.02207	0.511034	+	0.859561i	$0.329263\pi$
$84$	−1.87392e7	−3.44965
$85$	2.54765e6	0.449959
$86$	3.67410e6	0.622883
$87$	−5.44875e6	−0.887114
$88$	−8.82235e6	−1.38005
$89$	−8.69805e6	−1.30785	−0.653923	−	0.756561i	$-0.726878\pi$
$89$	−8.69805e6	−1.30785	−0.653923	+	0.756561i	$0.726878\pi$
$90$	−2.48779e7	−3.59720
$91$	0	0
$92$	−1.62276e7	−2.17268
$93$	9.70053e6	1.25056
$94$	4.44770e6	0.552317
$95$	−1.71321e7	−2.05012
$96$	398793.	0.0460043
$97$	3.76799e6	0.419187	0.209594	−	0.977789i	$-0.432786\pi$
$97$	3.76799e6	0.419187	0.209594	+	0.977789i	$0.432786\pi$
$98$	−3.69419e6	−0.396486
$99$	1.06706e7	1.10527
$100$	2.39280e7	2.39280
$101$	1.07132e7	1.03465	0.517325	−	0.855789i	$-0.326928\pi$
$101$	1.07132e7	1.03465	0.517325	+	0.855789i	$0.326928\pi$
$102$	8.75572e6	0.816942
$103$	−8.08002e6	−0.728588	−0.364294	−	0.931284i	$-0.618690\pi$
$103$	−8.08002e6	−0.728588	−0.364294	+	0.931284i	$0.618690\pi$
$104$	0	0
$105$	−3.01588e7	−2.54244
$106$	2.64199e7	2.15457
$107$	−2.35427e6	−0.185786	−0.0928931	−	0.995676i	$-0.529611\pi$
$107$	−2.35427e6	−0.185786	−0.0928931	+	0.995676i	$0.529611\pi$
$108$	−1.63376e7	−1.24797
$109$	289028.	0.0213770	0.0106885	−	0.999943i	$-0.496598\pi$
$109$	289028.	0.0213770	0.0106885	+	0.999943i	$0.496598\pi$
$110$	−2.82772e7	−2.02564
$111$	−1.24593e7	−0.864695
$112$	1.69029e7	1.13684
$113$	4.19514e6	0.273509	0.136755	−	0.990605i	$-0.456333\pi$
$113$	4.19514e6	0.273509	0.136755	+	0.990605i	$0.456333\pi$
$114$	−5.88796e7	−3.72218
$115$	−2.61165e7	−1.60130
$116$	1.93314e7	1.14990
$117$	0	0
$118$	4.92317e7	2.75840
$119$	6.19351e6	0.336917
$120$	7.59529e7	4.01246
$121$	−7.35846e6	−0.377605
$122$	1.00032e7	0.498748
$123$	5.41677e7	2.62466
$124$	−3.44162e7	−1.62101
$125$	6.18447e6	0.283216
$126$	−6.04799e7	−2.69348
$127$	3.35328e7	1.45264	0.726318	−	0.687358i	$-0.241230\pi$
$127$	3.35328e7	1.45264	0.726318	+	0.687358i	$0.241230\pi$
$128$	4.07942e7	1.71935
$129$	1.35672e7	0.556449
$130$	0	0
$131$	−2.99754e7	−1.16497	−0.582487	−	0.812840i	$-0.697920\pi$
$131$	−2.99754e7	−1.16497	−0.582487	+	0.812840i	$0.697920\pi$
$132$	−6.48804e7	−2.45530
$133$	−4.16495e7	−1.53507
$134$	7.98865e7	2.86818
$135$	−2.62935e7	−0.919773
$136$	−1.55980e7	−0.531719
$137$	−1.77421e7	−0.589501	−0.294750	−	0.955574i	$-0.595236\pi$
$137$	−1.77421e7	−0.589501	−0.294750	+	0.955574i	$0.595236\pi$
$138$	−8.97569e7	−2.90731
$139$	−2.24539e7	−0.709152	−0.354576	−	0.935027i	$-0.615375\pi$
$139$	−2.24539e7	−0.709152	−0.354576	+	0.935027i	$0.615375\pi$
$140$	1.06999e8	3.29559
$141$	1.64238e7	0.493409
$142$	5.02398e6	0.147244
$143$	0	0
$144$	5.14871e7	1.43691
$145$	3.11118e7	0.847495
$146$	−1.05594e8	−2.80804
$147$	−1.36413e7	−0.354198
$148$	4.42039e7	1.12084
$149$	−1.63242e7	−0.404278	−0.202139	−	0.979357i	$-0.564789\pi$
$149$	−1.63242e7	−0.404278	−0.202139	+	0.979357i	$0.564789\pi$
$150$	1.32349e8	3.20185
$151$	−3.54101e7	−0.836965	−0.418483	−	0.908225i	$-0.637438\pi$
$151$	−3.54101e7	−0.836965	−0.418483	+	0.908225i	$0.637438\pi$
$152$	1.04892e8	2.42264
$153$	1.88658e7	0.425848
$154$	−6.87440e7	−1.51674
$155$	−5.53891e7	−1.19471
$156$	0	0
$157$	−1.64297e7	−0.338830	−0.169415	−	0.985545i	$-0.554188\pi$
$157$	−1.64297e7	−0.338830	−0.169415	+	0.985545i	$0.554188\pi$
$158$	−1.01803e8	−2.05335
$159$	9.75593e7	1.92477
$160$	−2.27707e6	−0.0439497
$161$	−6.34911e7	−1.19901
$162$	4.11311e7	0.760095
$163$	7.24842e7	1.31095	0.655475	−	0.755216i	$-0.272468\pi$
$163$	7.24842e7	1.31095	0.655475	+	0.755216i	$0.272468\pi$
$164$	−1.92180e8	−3.40215
$165$	−1.04418e8	−1.80959
$166$	−1.04480e8	−1.77279
$167$	−9.79620e7	−1.62761	−0.813804	−	0.581139i	$-0.802607\pi$
$167$	−9.79620e7	−1.62761	−0.813804	+	0.581139i	$0.802607\pi$
$168$	1.84647e8	3.00442
$169$	0	0
$170$	−4.99943e7	−0.780458
$171$	−1.26867e8	−1.94026
$172$	−4.81345e7	−0.721284
$173$	5.57283e6	0.0818303	0.0409151	−	0.999163i	$-0.486973\pi$
$173$	5.57283e6	0.0818303	0.0409151	+	0.999163i	$0.486973\pi$
$174$	1.06925e8	1.53871
$175$	9.36193e7	1.32048
$176$	5.85224e7	0.809147
$177$	1.81795e8	2.46420
$178$	1.70688e8	2.26847
$179$	7.61885e7	0.992896	0.496448	−	0.868067i	$-0.334637\pi$
$179$	7.61885e7	0.992896	0.496448	+	0.868067i	$0.334637\pi$
$180$	3.25925e8	4.16547
$181$	5.13200e7	0.643297	0.321648	−	0.946859i	$-0.395763\pi$
$181$	5.13200e7	0.643297	0.321648	+	0.946859i	$0.395763\pi$
$182$	0	0
$183$	3.69384e7	0.445553
$184$	1.59898e8	1.89226
$185$	7.11413e7	0.826078
$186$	−1.90361e8	−2.16911
$187$	2.14436e7	0.239802
$188$	−5.82694e7	−0.639570
$189$	−6.39215e7	−0.688701
$190$	3.36197e8	3.55595
$191$	1.19362e7	0.123951	0.0619753	−	0.998078i	$-0.480260\pi$
$191$	1.19362e7	0.123951	0.0619753	+	0.998078i	$0.480260\pi$
$192$	1.48038e8	1.50945
$193$	1.37269e8	1.37443	0.687216	−	0.726453i	$-0.258833\pi$
$193$	1.37269e8	1.37443	0.687216	+	0.726453i	$0.258833\pi$
$194$	−7.39419e7	−0.727084
$195$	0	0
$196$	4.83976e7	0.459122
$197$	−9.34188e7	−0.870567	−0.435284	−	0.900293i	$-0.643352\pi$
$197$	−9.34188e7	−0.870567	−0.435284	+	0.900293i	$0.643352\pi$
$198$	−2.09398e8	−1.91710
$199$	1.28639e8	1.15714	0.578571	−	0.815632i	$-0.303611\pi$
$199$	1.28639e8	1.15714	0.578571	+	0.815632i	$0.303611\pi$
$200$	−2.35774e8	−2.08397
$201$	2.94993e8	2.56227
$202$	−2.10232e8	−1.79461
$203$	7.56351e7	0.634581
$204$	−1.14709e8	−0.946001
$205$	−3.09292e8	−2.50744
$206$	1.58560e8	1.26374
$207$	−1.93397e8	−1.51549
$208$	0	0
$209$	−1.44202e8	−1.09259
$210$	5.91828e8	4.40989
$211$	−1.53936e8	−1.12811	−0.564054	−	0.825738i	$-0.690759\pi$
$211$	−1.53936e8	−1.12811	−0.564054	+	0.825738i	$0.690759\pi$
$212$	−3.46127e8	−2.49494
$213$	1.85518e7	0.131540
$214$	4.61996e7	0.322248
$215$	−7.74672e7	−0.531598
$216$	1.60982e8	1.08690
$217$	−1.34655e8	−0.894568
$218$	−5.67181e6	−0.0370786
$219$	−3.89921e8	−2.50855
$220$	3.70461e8	2.34565
$221$	0	0
$222$	2.44498e8	1.49982
$223$	−1.30054e8	−0.785336	−0.392668	−	0.919680i	$-0.628448\pi$
$223$	−1.30054e8	−0.785336	−0.392668	+	0.919680i	$0.628448\pi$
$224$	−5.53572e6	−0.0329083
$225$	2.85169e8	1.66903
$226$	−8.23244e7	−0.474405
$227$	−3.46925e8	−1.96855	−0.984273	−	0.176657i	$-0.943472\pi$
$227$	−3.46925e8	−1.96855	−0.984273	+	0.176657i	$0.943472\pi$
$228$	7.71382e8	4.31020
$229$	−8.27429e7	−0.455309	−0.227655	−	0.973742i	$-0.573106\pi$
$229$	−8.27429e7	−0.455309	−0.227655	+	0.973742i	$0.573106\pi$
$230$	5.12503e8	2.77747
$231$	−2.53848e8	−1.35497
$232$	−1.90482e8	−1.00149
$233$	−1.51866e7	−0.0786530	−0.0393265	−	0.999226i	$-0.512521\pi$
$233$	−1.51866e7	−0.0786530	−0.0393265	+	0.999226i	$0.512521\pi$
$234$	0	0
$235$	−9.37783e7	−0.471373
$236$	−6.44985e8	−3.19417
$237$	−3.75924e8	−1.83435
$238$	−1.21540e8	−0.584385
$239$	−1.61515e8	−0.765280	−0.382640	−	0.923898i	$-0.624985\pi$
$239$	−1.61515e8	−0.765280	−0.382640	+	0.923898i	$0.624985\pi$
$240$	−5.03828e8	−2.35257
$241$	−1.29574e8	−0.596292	−0.298146	−	0.954520i	$-0.596368\pi$
$241$	−1.29574e8	−0.596292	−0.298146	+	0.954520i	$0.596368\pi$
$242$	1.44400e8	0.654960
$243$	2.90862e8	1.30036
$244$	−1.31053e8	−0.577538
$245$	7.78907e7	0.338380
$246$	−1.06297e9	−4.55249
$247$	0	0
$248$	3.39120e8	1.41180
$249$	−3.85809e8	−1.58371
$250$	−1.21362e8	−0.491241
$251$	−1.44103e8	−0.575196	−0.287598	−	0.957751i	$-0.592857\pi$
$251$	−1.44103e8	−0.575196	−0.287598	+	0.957751i	$0.592857\pi$
$252$	7.92348e8	3.11899
$253$	−2.19823e8	−0.853399
$254$	−6.58039e8	−2.51961
$255$	−1.84611e8	−0.697217
$256$	−5.39041e8	−2.00808
$257$	2.48876e8	0.914572	0.457286	−	0.889320i	$-0.348822\pi$
$257$	2.48876e8	0.914572	0.457286	+	0.889320i	$0.348822\pi$
$258$	−2.66238e8	−0.965165
$259$	1.72950e8	0.618544
$260$	0	0
$261$	2.30388e8	0.802082
$262$	5.88230e8	2.02066
$263$	1.91513e8	0.649163	0.324582	−	0.945858i	$-0.394777\pi$
$263$	1.91513e8	0.649163	0.324582	+	0.945858i	$0.394777\pi$
$264$	6.39299e8	2.13841
$265$	−5.57054e8	−1.83881
$266$	8.17318e8	2.66260
$267$	6.30291e8	2.02652
$268$	−1.04659e9	−3.32129
$269$	−2.90813e8	−0.910922	−0.455461	−	0.890256i	$-0.650526\pi$
$269$	−2.90813e8	−0.910922	−0.455461	+	0.890256i	$0.650526\pi$
$270$	5.15977e8	1.59535
$271$	6.34456e7	0.193646	0.0968232	−	0.995302i	$-0.469132\pi$
$271$	6.34456e7	0.193646	0.0968232	+	0.995302i	$0.469132\pi$
$272$	1.03468e8	0.311756
$273$	0	0
$274$	3.48167e8	1.02249
$275$	3.24135e8	0.939858
$276$	1.17591e9	3.36660
$277$	2.25716e8	0.638090	0.319045	−	0.947739i	$-0.396638\pi$
$277$	2.25716e8	0.638090	0.319045	+	0.947739i	$0.396638\pi$
$278$	4.40629e8	1.23003
$279$	−4.10166e8	−1.13069
$280$	−1.05432e9	−2.87024
$281$	−6.67278e6	−0.0179405	−0.00897025	−	0.999960i	$-0.502855\pi$
$281$	−6.67278e6	−0.0179405	−0.00897025	+	0.999960i	$0.502855\pi$
$282$	−3.22296e8	−0.855822
$283$	1.62048e8	0.425002	0.212501	−	0.977161i	$-0.431839\pi$
$283$	1.62048e8	0.425002	0.212501	+	0.977161i	$0.431839\pi$
$284$	−6.58192e7	−0.170505
$285$	1.24146e9	3.17669
$286$	0	0
$287$	−7.51912e8	−1.87750
$288$	−1.68621e7	−0.0415947
$289$	−3.72426e8	−0.907607
$290$	−6.10530e8	−1.46999
$291$	−2.73042e8	−0.649536
$292$	1.38339e9	3.25165
$293$	−3.53069e8	−0.820017	−0.410009	−	0.912082i	$-0.634474\pi$
$293$	−3.53069e8	−0.820017	−0.410009	+	0.912082i	$0.634474\pi$
$294$	2.67694e8	0.614360
$295$	−1.03803e9	−2.35415
$296$	−4.35563e8	−0.976180
$297$	−2.21314e8	−0.490186
$298$	3.20342e8	0.701224
$299$	0	0
$300$	−1.73390e9	−3.70767
$301$	−1.88328e8	−0.398046
$302$	6.94878e8	1.45172
$303$	−7.76314e8	−1.60320
$304$	−6.95790e8	−1.42043
$305$	−2.10915e8	−0.425655
$306$	−3.70217e8	−0.738637
$307$	−6.79916e7	−0.134113	−0.0670565	−	0.997749i	$-0.521361\pi$
$307$	−6.79916e7	−0.134113	−0.0670565	+	0.997749i	$0.521361\pi$
$308$	9.00617e8	1.75636
$309$	5.85507e8	1.12896
$310$	1.08694e9	2.07224
$311$	−6.98524e8	−1.31680	−0.658401	−	0.752668i	$-0.728767\pi$
$311$	−6.98524e8	−1.31680	−0.658401	+	0.752668i	$0.728767\pi$
$312$	0	0
$313$	8.41764e8	1.55162	0.775810	−	0.630967i	$-0.217342\pi$
$313$	8.41764e8	1.55162	0.775810	+	0.630967i	$0.217342\pi$
$314$	3.22412e8	0.587703
$315$	1.27520e9	2.29874
$316$	1.33373e9	2.37773
$317$	9.91402e8	1.74800	0.874002	−	0.485922i	$-0.161516\pi$
$317$	9.91402e8	1.74800	0.874002	+	0.485922i	$0.161516\pi$
$318$	−1.91448e9	−3.33853
$319$	2.61869e8	0.451666
$320$	−8.45280e8	−1.44203
$321$	1.70599e8	0.287878
$322$	1.24593e9	2.07969
$323$	−2.54950e8	−0.420965
$324$	−5.38859e8	−0.880172
$325$	0	0
$326$	−1.42241e9	−2.27386
$327$	−2.09440e7	−0.0331240
$328$	1.89364e9	2.96305
$329$	−2.27982e8	−0.352951
$330$	2.04907e9	3.13876
$331$	1.46324e8	0.221777	0.110889	−	0.993833i	$-0.464630\pi$
$331$	1.46324e8	0.221777	0.110889	+	0.993833i	$0.464630\pi$
$332$	1.36880e9	2.05284
$333$	5.26814e8	0.781812
$334$	1.92238e9	2.82310
$335$	−1.68438e9	−2.44784
$336$	−1.22484e9	−1.76154
$337$	5.22422e8	0.743561	0.371780	−	0.928321i	$-0.378747\pi$
$337$	5.22422e8	0.743561	0.371780	+	0.928321i	$0.378747\pi$
$338$	0	0
$339$	−3.03995e8	−0.423806
$340$	6.54976e8	0.903752
$341$	−4.66212e8	−0.636712
$342$	2.48960e9	3.36541
$343$	−6.39027e8	−0.855047
$344$	4.74293e8	0.628191
$345$	1.89249e9	2.48123
$346$	−1.09360e8	−0.141935
$347$	6.74172e7	0.0866199	0.0433100	−	0.999062i	$-0.486210\pi$
$347$	6.74172e7	0.0866199	0.0433100	+	0.999062i	$0.486210\pi$
$348$	−1.40082e9	−1.78179
$349$	7.24565e8	0.912406	0.456203	−	0.889876i	$-0.349209\pi$
$349$	7.24565e8	0.912406	0.456203	+	0.889876i	$0.349209\pi$
$350$	−1.83716e9	−2.29039
$351$	0	0
$352$	−1.91662e7	−0.0234227
$353$	4.32002e8	0.522726	0.261363	−	0.965241i	$-0.415828\pi$
$353$	4.32002e8	0.522726	0.261363	+	0.965241i	$0.415828\pi$
$354$	−3.56750e9	−4.27418
$355$	−1.05929e8	−0.125665
$356$	−2.23619e9	−2.62684
$357$	−4.48804e8	−0.522057
$358$	−1.49510e9	−1.72219
$359$	−1.10242e9	−1.25752	−0.628761	−	0.777598i	$-0.716438\pi$
$359$	−1.10242e9	−1.25752	−0.628761	+	0.777598i	$0.716438\pi$
$360$	−3.21151e9	−3.62785
$361$	8.20588e8	0.918015
$362$	−1.00709e9	−1.11580
$363$	5.33220e8	0.585104
$364$	0	0
$365$	2.22641e9	2.39652
$366$	−7.24869e8	−0.772816
$367$	1.31669e9	1.39044	0.695222	−	0.718796i	$-0.255306\pi$
$367$	1.31669e9	1.39044	0.695222	+	0.718796i	$0.255306\pi$
$368$	−1.06067e9	−1.10947
$369$	−2.29036e9	−2.37308
$370$	−1.39606e9	−1.43284
$371$	−1.35424e9	−1.37685
$372$	2.49392e9	2.51178
$373$	1.44085e9	1.43760	0.718799	−	0.695218i	$-0.244692\pi$
$373$	1.44085e9	1.43760	0.718799	+	0.695218i	$0.244692\pi$
$374$	−4.20804e8	−0.415939
$375$	−4.48149e8	−0.438847
$376$	5.74158e8	0.557024
$377$	0	0
$378$	1.25438e9	1.19456
$379$	−1.01968e8	−0.0962111	−0.0481056	−	0.998842i	$-0.515318\pi$
$379$	−1.01968e8	−0.0962111	−0.0481056	+	0.998842i	$0.515318\pi$
$380$	−4.40452e9	−4.11771
$381$	−2.42991e9	−2.25088
$382$	−2.34232e8	−0.214993
$383$	−1.94208e9	−1.76633	−0.883164	−	0.469065i	$-0.844591\pi$
$383$	−1.94208e9	−1.76633	−0.883164	+	0.469065i	$0.844591\pi$
$384$	−2.95609e9	−2.66415
$385$	1.44944e9	1.29446
$386$	−2.69374e9	−2.38397
$387$	−5.73658e8	−0.503112
$388$	9.68715e8	0.841947
$389$	1.75657e9	1.51301	0.756506	−	0.653987i	$-0.226905\pi$
$389$	1.75657e9	1.51301	0.756506	+	0.653987i	$0.226905\pi$
$390$	0	0
$391$	−3.88649e8	−0.328805
$392$	−4.76886e8	−0.399865
$393$	2.17213e9	1.80514
$394$	1.83322e9	1.51001
$395$	2.14649e9	1.75242
$396$	2.74333e9	2.21996
$397$	−2.22292e9	−1.78302	−0.891511	−	0.452998i	$-0.850354\pi$
$397$	−2.22292e9	−1.78302	−0.891511	+	0.452998i	$0.850354\pi$
$398$	−2.52437e9	−2.00707
$399$	3.01807e9	2.37861
$400$	1.56399e9	1.22187
$401$	−1.61957e9	−1.25428	−0.627139	−	0.778908i	$-0.715774\pi$
$401$	−1.61957e9	−1.25428	−0.627139	+	0.778908i	$0.715774\pi$
$402$	−5.78885e9	−4.44428
$403$	0	0
$404$	2.75426e9	2.07812
$405$	−8.67235e8	−0.648700
$406$	−1.48424e9	−1.10069
$407$	5.98799e8	0.440251
$408$	1.13028e9	0.823905
$409$	2.57824e9	1.86334	0.931670	−	0.363307i	$-0.118352\pi$
$409$	2.57824e9	1.86334	0.931670	+	0.363307i	$0.118352\pi$
$410$	6.06947e9	4.34918
$411$	1.28566e9	0.913438
$412$	−2.07730e9	−1.46339
$413$	−2.52353e9	−1.76272
$414$	3.79517e9	2.62864
$415$	2.20293e9	1.51298
$416$	0	0
$417$	1.62709e9	1.09884
$418$	2.82978e9	1.89512
$419$	1.55385e9	1.03196	0.515978	−	0.856602i	$-0.327429\pi$
$419$	1.55385e9	1.03196	0.515978	+	0.856602i	$0.327429\pi$
$420$	−7.75354e9	−5.10655
$421$	1.34533e9	0.878699	0.439350	−	0.898316i	$-0.355209\pi$
$421$	1.34533e9	0.878699	0.439350	+	0.898316i	$0.355209\pi$
$422$	3.02079e9	1.95671
$423$	−6.94445e8	−0.446115
$424$	3.41056e9	2.17293
$425$	5.73073e8	0.362117
$426$	−3.64055e8	−0.228157
$427$	−5.12749e8	−0.318718
$428$	−6.05262e8	−0.373156
$429$	0	0
$430$	1.52019e9	0.922061
$431$	−1.60812e9	−0.967495	−0.483747	−	0.875208i	$-0.660725\pi$
$431$	−1.60812e9	−0.967495	−0.483747	+	0.875208i	$0.660725\pi$
$432$	−1.06786e9	−0.637268
$433$	−1.89392e9	−1.12112	−0.560562	−	0.828113i	$-0.689415\pi$
$433$	−1.89392e9	−1.12112	−0.560562	+	0.828113i	$0.689415\pi$
$434$	2.64243e9	1.55164
$435$	−2.25447e9	−1.31320
$436$	7.43065e7	0.0429362
$437$	2.61355e9	1.49812
$438$	7.65171e9	4.35110
$439$	1.08620e9	0.612749	0.306375	−	0.951911i	$-0.400884\pi$
$439$	1.08620e9	0.612749	0.306375	+	0.951911i	$0.400884\pi$
$440$	−3.65033e9	−2.04290
$441$	5.76794e8	0.320248
$442$	0	0
$443$	−2.67935e9	−1.46425	−0.732127	−	0.681168i	$-0.761472\pi$
$443$	−2.67935e9	−1.46425	−0.732127	+	0.681168i	$0.761472\pi$
$444$	−3.20317e9	−1.73676
$445$	−3.59890e9	−1.93602
$446$	2.55214e9	1.36217
$447$	1.18291e9	0.626434
$448$	−2.05494e9	−1.07976
$449$	2.77150e9	1.44495	0.722474	−	0.691398i	$-0.243005\pi$
$449$	2.77150e9	1.44495	0.722474	+	0.691398i	$0.243005\pi$
$450$	−5.59608e9	−2.89495
$451$	−2.60332e9	−1.33632
$452$	1.07853e9	0.549350
$453$	2.56594e9	1.29689
$454$	6.80797e9	3.41446
$455$	0	0
$456$	−7.60081e9	−3.75390
$457$	−1.88973e9	−0.926177	−0.463088	−	0.886312i	$-0.653259\pi$
$457$	−1.88973e9	−0.926177	−0.463088	+	0.886312i	$0.653259\pi$
$458$	1.62372e9	0.789738
$459$	−3.91284e8	−0.188863
$460$	−6.71431e9	−3.21624
$461$	−7.18617e8	−0.341621	−0.170810	−	0.985304i	$-0.554639\pi$
$461$	−7.18617e8	−0.341621	−0.170810	+	0.985304i	$0.554639\pi$
$462$	4.98144e9	2.35022
$463$	3.35895e8	0.157279	0.0786393	−	0.996903i	$-0.474942\pi$
$463$	3.35895e8	0.157279	0.0786393	+	0.996903i	$0.474942\pi$
$464$	1.26355e9	0.587190
$465$	4.01369e9	1.85122
$466$	2.98018e8	0.136424
$467$	1.62006e8	0.0736076	0.0368038	−	0.999323i	$-0.488282\pi$
$467$	1.62006e8	0.0736076	0.0368038	+	0.999323i	$0.488282\pi$
$468$	0	0
$469$	−4.09485e9	−1.83287
$470$	1.84028e9	0.817601
$471$	1.19056e9	0.525021
$472$	6.35536e9	2.78191
$473$	−6.52044e8	−0.283311
$474$	7.37704e9	3.18169
$475$	−3.85374e9	−1.64989
$476$	1.59230e9	0.676705
$477$	−4.12508e9	−1.74028
$478$	3.16953e9	1.32738
$479$	−1.87256e9	−0.778505	−0.389253	−	0.921131i	$-0.627267\pi$
$479$	−1.87256e9	−0.778505	−0.389253	+	0.921131i	$0.627267\pi$
$480$	1.65004e8	0.0681007
$481$	0	0
$482$	2.54273e9	1.03427
$483$	4.60079e9	1.85788
$484$	−1.89179e9	−0.758428
$485$	1.55904e9	0.620528
$486$	−5.70779e9	−2.25549
$487$	−2.79396e9	−1.09615	−0.548073	−	0.836431i	$-0.684638\pi$
$487$	−2.79396e9	−1.09615	−0.548073	+	0.836431i	$0.684638\pi$
$488$	1.29133e9	0.502998
$489$	−5.25246e9	−2.03133
$490$	−1.52851e9	−0.586923
$491$	−1.87416e8	−0.0714532	−0.0357266	−	0.999362i	$-0.511375\pi$
$491$	−1.87416e8	−0.0714532	−0.0357266	+	0.999362i	$0.511375\pi$
$492$	1.39260e10	5.27168
$493$	4.62986e8	0.174022
$494$	0	0
$495$	4.41508e9	1.63614
$496$	−2.24952e9	−0.827761
$497$	−2.57521e8	−0.0940946
$498$	7.57101e9	2.74695
$499$	3.05663e9	1.10126	0.550631	−	0.834749i	$-0.314387\pi$
$499$	3.05663e9	1.10126	0.550631	+	0.834749i	$0.314387\pi$
$500$	1.58997e9	0.568845
$501$	7.09867e9	2.52200
$502$	2.82784e9	0.997682
$503$	−1.10435e9	−0.386916	−0.193458	−	0.981109i	$-0.561970\pi$
$503$	−1.10435e9	−0.386916	−0.193458	+	0.981109i	$0.561970\pi$
$504$	−7.80740e9	−2.71644
$505$	4.43268e9	1.53160
$506$	4.31376e9	1.48023
$507$	0	0
$508$	8.62098e9	2.91765
$509$	2.84435e8	0.0956029	0.0478014	−	0.998857i	$-0.484779\pi$
$509$	2.84435e8	0.0956029	0.0478014	+	0.998857i	$0.484779\pi$
$510$	3.62276e9	1.20933
$511$	5.41257e9	1.79444
$512$	5.35633e9	1.76369
$513$	2.63126e9	0.860505
$514$	−4.88388e9	−1.58633
$515$	−3.34318e9	−1.07854
$516$	3.48799e9	1.11764
$517$	−7.89335e8	−0.251214
$518$	−3.39392e9	−1.07287
$519$	−4.03827e8	−0.126797
$520$	0	0
$521$	−6.21830e8	−0.192637	−0.0963184	−	0.995351i	$-0.530707\pi$
$521$	−6.21830e8	−0.192637	−0.0963184	+	0.995351i	$0.530707\pi$
$522$	−4.52108e9	−1.39122
$523$	−1.32430e9	−0.404790	−0.202395	−	0.979304i	$-0.564873\pi$
$523$	−1.32430e9	−0.404790	−0.202395	+	0.979304i	$0.564873\pi$
$524$	−7.70641e9	−2.33988
$525$	−6.78398e9	−2.04610
$526$	−3.75821e9	−1.12598
$527$	−8.24266e8	−0.245318
$528$	−4.24074e9	−1.25378
$529$	5.79302e8	0.170141
$530$	1.09315e10	3.18943
$531$	−7.68681e9	−2.22800
$532$	−1.07077e10	−3.08323
$533$	0	0
$534$	−1.23687e10	−3.51503
$535$	−9.74103e8	−0.275022
$536$	1.03126e10	2.89262
$537$	−5.52089e9	−1.53850
$538$	5.70684e9	1.58000
$539$	6.55609e8	0.180337
$540$	−6.75982e9	−1.84738
$541$	2.63957e9	0.716708	0.358354	−	0.933586i	$-0.383338\pi$
$541$	2.63957e9	0.716708	0.358354	+	0.933586i	$0.383338\pi$
$542$	−1.24504e9	−0.335881
$543$	−3.71883e9	−0.996797
$544$	−3.38859e7	−0.00902449
$545$	1.19588e8	0.0316447
$546$	0	0
$547$	−3.91983e9	−1.02403	−0.512013	−	0.858977i	$-0.671100\pi$
$547$	−3.91983e9	−1.02403	−0.512013	+	0.858977i	$0.671100\pi$
$548$	−4.56134e9	−1.18402
$549$	−1.56186e9	−0.402846
$550$	−6.36075e9	−1.63019
$551$	−3.11344e9	−0.792885
$552$	−1.15868e10	−2.93208
$553$	5.21828e9	1.31217
$554$	−4.42938e9	−1.10677
$555$	−5.15515e9	−1.28002
$556$	−5.77269e9	−1.42435
$557$	4.28372e9	1.05034	0.525168	−	0.850999i	$-0.324003\pi$
$557$	4.28372e9	1.05034	0.525168	+	0.850999i	$0.324003\pi$
$558$	8.04899e9	1.96120
$559$	0	0
$560$	6.99373e9	1.68287
$561$	−1.55388e9	−0.371576
$562$	1.30945e8	0.0311180
$563$	−7.15933e9	−1.69080	−0.845402	−	0.534131i	$-0.820639\pi$
$563$	−7.15933e9	−1.69080	−0.845402	+	0.534131i	$0.820639\pi$
$564$	4.22241e9	0.991022
$565$	1.73578e9	0.404879
$566$	−3.17999e9	−0.737171
$567$	−2.10831e9	−0.485729
$568$	6.48549e8	0.148499
$569$	−5.31555e9	−1.20964	−0.604818	−	0.796364i	$-0.706754\pi$
$569$	−5.31555e9	−1.20964	−0.604818	+	0.796364i	$0.706754\pi$
$570$	−2.43620e10	−5.50999
$571$	−1.41625e9	−0.318356	−0.159178	−	0.987250i	$-0.550884\pi$
$571$	−1.41625e9	−0.318356	−0.159178	+	0.987250i	$0.550884\pi$
$572$	0	0
$573$	−8.64938e8	−0.192063
$574$	1.47553e10	3.25654
$575$	−5.87470e9	−1.28869
$576$	−6.25945e9	−1.36476
$577$	3.09413e9	0.670538	0.335269	−	0.942123i	$-0.391173\pi$
$577$	3.09413e9	0.670538	0.335269	+	0.942123i	$0.391173\pi$
$578$	7.30839e9	1.57425
$579$	−9.94703e9	−2.12970
$580$	7.99856e9	1.70221
$581$	5.35549e9	1.13288
$582$	5.35809e9	1.12663
$583$	−4.68874e9	−0.979978
$584$	−1.36312e10	−2.83197
$585$	0	0
$586$	6.92853e9	1.42233
$587$	5.10437e8	0.104162	0.0520809	−	0.998643i	$-0.483415\pi$
$587$	5.10437e8	0.104162	0.0520809	+	0.998643i	$0.483415\pi$
$588$	−3.50706e9	−0.711415
$589$	5.54293e9	1.11773
$590$	2.03701e10	4.08329
$591$	6.76945e9	1.34895
$592$	2.88927e9	0.572351
$593$	6.99937e9	1.37838	0.689188	−	0.724583i	$-0.257968\pi$
$593$	6.99937e9	1.37838	0.689188	+	0.724583i	$0.257968\pi$
$594$	4.34300e9	0.850231
$595$	2.56263e9	0.498742
$596$	−4.19680e9	−0.812002
$597$	−9.32162e9	−1.79300
$598$	0	0
$599$	1.10847e9	0.210733	0.105366	−	0.994433i	$-0.466398\pi$
$599$	1.10847e9	0.210733	0.105366	+	0.994433i	$0.466398\pi$
$600$	1.70850e10	3.22914
$601$	−3.46969e9	−0.651973	−0.325987	−	0.945374i	$-0.605696\pi$
$601$	−3.46969e9	−0.651973	−0.325987	+	0.945374i	$0.605696\pi$
$602$	3.69570e9	0.690414
$603$	−1.24731e10	−2.31667
$604$	−9.10361e9	−1.68106
$605$	−3.04463e9	−0.558973
$606$	1.52342e10	2.78077
$607$	−8.78548e9	−1.59443	−0.797215	−	0.603696i	$-0.793694\pi$
$607$	−8.78548e9	−1.59443	−0.797215	+	0.603696i	$0.793694\pi$
$608$	2.27872e8	0.0411177
$609$	−5.48079e9	−0.983291
$610$	4.13893e9	0.738302
$611$	0	0
$612$	4.85022e9	0.855325
$613$	−1.72321e9	−0.302152	−0.151076	−	0.988522i	$-0.548274\pi$
$613$	−1.72321e9	−0.302152	−0.151076	+	0.988522i	$0.548274\pi$
$614$	1.33425e9	0.232620
$615$	2.24124e10	3.88531
$616$	−8.87423e9	−1.52967
$617$	−3.95049e9	−0.677101	−0.338550	−	0.940948i	$-0.609937\pi$
$617$	−3.95049e9	−0.677101	−0.338550	+	0.940948i	$0.609937\pi$
$618$	−1.14898e10	−1.95818
$619$	−3.95281e9	−0.669866	−0.334933	−	0.942242i	$-0.608714\pi$
$619$	−3.95281e9	−0.669866	−0.334933	+	0.942242i	$0.608714\pi$
$620$	−1.42400e10	−2.39961
$621$	4.01114e9	0.672120
$622$	1.37077e10	2.28400
$623$	−8.74919e9	−1.44964
$624$	0	0
$625$	−4.71237e9	−0.772074
$626$	−1.65186e10	−2.69130
$627$	1.04494e10	1.69299
$628$	−4.22393e9	−0.680547
$629$	1.05868e9	0.169624
$630$	−2.50241e10	−3.98719
$631$	−6.84671e9	−1.08487	−0.542437	−	0.840097i	$-0.682498\pi$
$631$	−6.84671e9	−1.08487	−0.542437	+	0.840097i	$0.682498\pi$
$632$	−1.31419e10	−2.07085
$633$	1.11547e10	1.74802
$634$	−1.94550e10	−3.03193
$635$	1.38745e10	2.15035
$636$	2.50816e10	3.86594
$637$	0	0
$638$	−5.13885e9	−0.783419
$639$	−7.84421e8	−0.118931
$640$	1.68790e10	2.54517
$641$	3.12250e9	0.468273	0.234136	−	0.972204i	$-0.424774\pi$
$641$	3.12250e9	0.468273	0.234136	+	0.972204i	$0.424774\pi$
$642$	−3.34779e9	−0.499327
$643$	8.57775e9	1.27243	0.636216	−	0.771511i	$-0.280499\pi$
$643$	8.57775e9	1.27243	0.636216	+	0.771511i	$0.280499\pi$
$644$	−1.63230e10	−2.40823
$645$	5.61354e9	0.823717
$646$	5.00307e9	0.730167
$647$	8.70898e9	1.26416	0.632080	−	0.774903i	$-0.282201\pi$
$647$	8.70898e9	1.26416	0.632080	+	0.774903i	$0.282201\pi$
$648$	5.30965e9	0.766572
$649$	−8.73716e9	−1.25462
$650$	0	0
$651$	9.75757e9	1.38614
$652$	1.86350e10	2.63307
$653$	−9.71192e9	−1.36493	−0.682463	−	0.730920i	$-0.739091\pi$
$653$	−9.71192e9	−1.36493	−0.682463	+	0.730920i	$0.739091\pi$
$654$	4.10999e8	0.0574538
$655$	−1.24026e10	−1.72452
$656$	−1.25613e10	−1.73729
$657$	1.64870e10	2.26810
$658$	4.47385e9	0.612197
$659$	3.34780e9	0.455680	0.227840	−	0.973699i	$-0.426834\pi$
$659$	3.34780e9	0.455680	0.227840	+	0.973699i	$0.426834\pi$
$660$	−2.68449e10	−3.63461
$661$	−8.98798e9	−1.21048	−0.605239	−	0.796044i	$-0.706922\pi$
$661$	−8.98798e9	−1.21048	−0.605239	+	0.796044i	$0.706922\pi$
$662$	−2.87142e9	−0.384675
$663$	0	0
$664$	−1.34874e10	−1.78789
$665$	−1.72329e10	−2.27239
$666$	−1.03381e10	−1.35606
$667$	−4.74618e9	−0.619303
$668$	−2.51851e10	−3.26909
$669$	9.42414e9	1.21689
$670$	3.30538e10	4.24580
$671$	−1.77528e9	−0.226849
$672$	4.01138e8	0.0509919
$673$	−1.90573e9	−0.240995	−0.120498	−	0.992714i	$-0.538449\pi$
$673$	−1.90573e9	−0.240995	−0.120498	+	0.992714i	$0.538449\pi$
$674$	−1.02519e10	−1.28971
$675$	−5.91452e9	−0.740213
$676$	0	0
$677$	1.10831e10	1.37278	0.686388	−	0.727236i	$-0.259195\pi$
$677$	1.10831e10	1.37278	0.686388	+	0.727236i	$0.259195\pi$
$678$	5.96552e9	0.735096
$679$	3.79014e9	0.464634
$680$	−6.45381e9	−0.787109
$681$	2.51394e10	3.05029
$682$	9.14882e9	1.10438
$683$	5.53250e9	0.664429	0.332215	−	0.943204i	$-0.392204\pi$
$683$	5.53250e9	0.664429	0.332215	+	0.943204i	$0.392204\pi$
$684$	−3.26162e10	−3.89706
$685$	−7.34098e9	−0.872644
$686$	1.25401e10	1.48309
$687$	5.99584e9	0.705507
$688$	−3.14618e9	−0.368319
$689$	0	0
$690$	−3.71378e10	−4.30372
$691$	9.96854e9	1.14937	0.574683	−	0.818376i	$-0.305125\pi$
$691$	9.96854e9	1.14937	0.574683	+	0.818376i	$0.305125\pi$
$692$	1.43272e9	0.164358
$693$	1.07334e10	1.22510
$694$	−1.32298e9	−0.150243
$695$	−9.29051e9	−1.04977
$696$	1.38030e10	1.55182
$697$	−4.60269e9	−0.514869
$698$	−1.42187e10	−1.58258
$699$	1.10048e9	0.121874
$700$	2.40687e10	2.65221
$701$	1.56464e9	0.171554	0.0857769	−	0.996314i	$-0.472663\pi$
$701$	1.56464e9	0.171554	0.0857769	+	0.996314i	$0.472663\pi$
$702$	0	0
$703$	−7.11930e9	−0.772847
$704$	−7.11475e9	−0.768520
$705$	6.79550e9	0.730398
$706$	−8.47749e9	−0.906673
$707$	1.07762e10	1.14682
$708$	4.67379e10	4.94940
$709$	−4.49879e9	−0.474061	−0.237031	−	0.971502i	$-0.576174\pi$
$709$	−4.49879e9	−0.474061	−0.237031	+	0.971502i	$0.576174\pi$
$710$	2.07872e9	0.217967
$711$	1.58951e10	1.65852
$712$	2.20343e10	2.28780
$713$	8.44973e9	0.873030
$714$	8.80721e9	0.905512
$715$	0	0
$716$	1.95874e10	1.99425
$717$	1.17039e10	1.18581
$718$	2.16336e10	2.18118
$719$	−1.28814e9	−0.129245	−0.0646223	−	0.997910i	$-0.520584\pi$
$719$	−1.28814e9	−0.129245	−0.0646223	+	0.997910i	$0.520584\pi$
$720$	2.13033e10	2.12707
$721$	−8.12753e9	−0.807579
$722$	−1.61030e10	−1.59231
$723$	9.38940e9	0.923962
$724$	1.31939e10	1.29208
$725$	6.99836e9	0.682046
$726$	−1.04638e10	−1.01487
$727$	1.64110e9	0.158403	0.0792017	−	0.996859i	$-0.474763\pi$
$727$	1.64110e9	0.158403	0.0792017	+	0.996859i	$0.474763\pi$
$728$	0	0
$729$	−1.64929e10	−1.57671
$730$	−4.36905e10	−4.15678
$731$	−1.15282e9	−0.109157
$732$	9.49653e9	0.894903
$733$	6.09340e9	0.571473	0.285737	−	0.958308i	$-0.407762\pi$
$733$	6.09340e9	0.571473	0.285737	+	0.958308i	$0.407762\pi$
$734$	−2.58384e10	−2.41174
$735$	−5.64424e9	−0.524324
$736$	3.47372e8	0.0321161
$737$	−1.41775e10	−1.30456
$738$	4.49455e10	4.11613
$739$	−1.10220e10	−1.00462	−0.502311	−	0.864687i	$-0.667517\pi$
$739$	−1.10220e10	−1.00462	−0.502311	+	0.864687i	$0.667517\pi$
$740$	1.82898e10	1.65919
$741$	0	0
$742$	2.65752e10	2.38816
$743$	1.44250e9	0.129020	0.0645098	−	0.997917i	$-0.479452\pi$
$743$	1.44250e9	0.129020	0.0645098	+	0.997917i	$0.479452\pi$
$744$	−2.45738e10	−2.18760
$745$	−6.75430e9	−0.598457
$746$	−2.82748e10	−2.49353
$747$	1.63131e10	1.43190
$748$	5.51296e9	0.481648
$749$	−2.36812e9	−0.205929
$750$	8.79435e9	0.761183
$751$	−4.23430e9	−0.364789	−0.182395	−	0.983225i	$-0.558385\pi$
$751$	−4.23430e9	−0.364789	−0.182395	+	0.983225i	$0.558385\pi$
$752$	−3.80863e9	−0.326593
$753$	1.04422e10	0.891273
$754$	0	0
$755$	−1.46513e10	−1.23897
$756$	−1.64336e10	−1.38327
$757$	−4.85119e9	−0.406456	−0.203228	−	0.979131i	$-0.565143\pi$
$757$	−4.85119e9	−0.406456	−0.203228	+	0.979131i	$0.565143\pi$
$758$	2.00099e9	0.166879
$759$	1.59292e10	1.32235
$760$	4.33999e10	3.58625
$761$	−1.68175e10	−1.38329	−0.691647	−	0.722236i	$-0.743115\pi$
$761$	−1.68175e10	−1.38329	−0.691647	+	0.722236i	$0.743115\pi$
$762$	4.76838e10	3.90417
$763$	2.90727e8	0.0236946
$764$	3.06868e9	0.248958
$765$	7.80589e9	0.630387
$766$	3.81108e10	3.06371
$767$	0	0
$768$	3.90608e10	3.11155
$769$	−1.62454e9	−0.128821	−0.0644107	−	0.997923i	$-0.520517\pi$
$769$	−1.62454e9	−0.128821	−0.0644107	+	0.997923i	$0.520517\pi$
$770$	−2.84435e10	−2.24525
$771$	−1.80345e10	−1.41714
$772$	3.52907e10	2.76058
$773$	2.06113e10	1.60501	0.802503	−	0.596648i	$-0.203501\pi$
$773$	2.06113e10	1.60501	0.802503	+	0.596648i	$0.203501\pi$
$774$	1.12573e10	0.872652
$775$	−1.24593e10	−0.961478
$776$	−9.54523e9	−0.733281
$777$	−1.25325e10	−0.958442
$778$	−3.44705e10	−2.62433
$779$	3.09517e10	2.34587
$780$	0	0
$781$	−8.91607e8	−0.0669722
$782$	7.62674e9	0.570316
$783$	−4.77835e9	−0.355723
$784$	3.16339e9	0.234448
$785$	−6.79795e9	−0.501573
$786$	−4.26252e10	−3.13103
$787$	7.97469e9	0.583179	0.291590	−	0.956543i	$-0.405816\pi$
$787$	7.97469e9	0.583179	0.291590	+	0.956543i	$0.405816\pi$
$788$	−2.40171e10	−1.74855
$789$	−1.38777e10	−1.00589
$790$	−4.21222e10	−3.03960
$791$	4.21981e9	0.303162
$792$	−2.70314e10	−1.93344
$793$	0	0
$794$	4.36220e10	3.09267
$795$	4.03661e10	2.84926
$796$	3.30719e10	2.32414
$797$	−2.05018e10	−1.43446	−0.717228	−	0.696839i	$-0.754589\pi$
$797$	−2.05018e10	−1.43446	−0.717228	+	0.696839i	$0.754589\pi$
$798$	−5.92258e10	−4.12573
$799$	−1.39555e9	−0.0967902
$800$	−5.12209e8	−0.0353698
$801$	−2.66505e10	−1.83228
$802$	3.17819e10	2.17556
$803$	1.87398e10	1.27720
$804$	7.58399e10	5.14638
$805$	−2.62700e10	−1.77491
$806$	0	0
$807$	2.10734e10	1.41149
$808$	−2.71391e10	−1.80990
$809$	−1.78384e10	−1.18450	−0.592252	−	0.805753i	$-0.701761\pi$
$809$	−1.78384e10	−1.18450	−0.592252	+	0.805753i	$0.701761\pi$
$810$	1.70184e10	1.12518
$811$	−1.20810e10	−0.795297	−0.397649	−	0.917538i	$-0.630174\pi$
$811$	−1.20810e10	−0.795297	−0.397649	+	0.917538i	$0.630174\pi$
$812$	1.94451e10	1.27457
$813$	−4.59750e9	−0.300057
$814$	−1.17507e10	−0.763620
$815$	2.99910e10	1.94062
$816$	−7.49765e9	−0.483069
$817$	7.75235e9	0.497343
$818$	−5.05947e10	−3.23198
$819$	0	0
$820$	−7.95162e10	−5.03625
$821$	2.10707e10	1.32886	0.664428	−	0.747352i	$-0.268675\pi$
$821$	2.10707e10	1.32886	0.664428	+	0.747352i	$0.268675\pi$
$822$	−2.52294e10	−1.58437
$823$	4.97330e9	0.310989	0.155494	−	0.987837i	$-0.450303\pi$
$823$	4.97330e9	0.310989	0.155494	+	0.987837i	$0.450303\pi$
$824$	2.04687e10	1.27451
$825$	−2.34880e10	−1.45632
$826$	4.95211e10	3.05746
$827$	−4.23151e9	−0.260151	−0.130076	−	0.991504i	$-0.541522\pi$
$827$	−4.23151e9	−0.260151	−0.130076	+	0.991504i	$0.541522\pi$
$828$	−4.97206e10	−3.04390
$829$	1.94458e10	1.18546	0.592729	−	0.805402i	$-0.298051\pi$
$829$	1.94458e10	1.18546	0.592729	+	0.805402i	$0.298051\pi$
$830$	−4.32297e10	−2.62427
$831$	−1.63561e10	−0.988729
$832$	0	0
$833$	1.15912e9	0.0694818
$834$	−3.19295e10	−1.90595
$835$	−4.05327e10	−2.40937
$836$	−3.70730e10	−2.19450
$837$	8.50701e9	0.501462
$838$	−3.04924e10	−1.78994
$839$	−1.48186e9	−0.0866244	−0.0433122	−	0.999062i	$-0.513791\pi$
$839$	−1.48186e9	−0.0866244	−0.0433122	+	0.999062i	$0.513791\pi$
$840$	7.63995e10	4.44747
$841$	−1.15959e10	−0.672231
$842$	−2.64003e10	−1.52411
$843$	4.83533e8	0.0277990
$844$	−3.95754e10	−2.26583
$845$	0	0
$846$	1.36276e10	0.773790
$847$	−7.40172e9	−0.418544
$848$	−2.26237e10	−1.27403
$849$	−1.17426e10	−0.658547
$850$	−1.12458e10	−0.628095
$851$	−1.08528e10	−0.603652
$852$	4.76949e9	0.264200
$853$	5.95360e9	0.328442	0.164221	−	0.986424i	$-0.447489\pi$
$853$	5.95360e9	0.328442	0.164221	+	0.986424i	$0.447489\pi$
$854$	1.00620e10	0.552820
$855$	−5.24923e10	−2.87219
$856$	5.96395e9	0.324994
$857$	−8.37353e9	−0.454439	−0.227220	−	0.973844i	$-0.572963\pi$
$857$	−8.37353e9	−0.454439	−0.227220	+	0.973844i	$0.572963\pi$
$858$	0	0
$859$	−2.14894e10	−1.15677	−0.578386	−	0.815763i	$-0.696317\pi$
$859$	−2.14894e10	−1.15677	−0.578386	+	0.815763i	$0.696317\pi$
$860$	−1.99161e10	−1.06773
$861$	5.44862e10	2.90921
$862$	3.15574e10	1.67813
$863$	9.93195e9	0.526013	0.263007	−	0.964794i	$-0.415286\pi$
$863$	9.93195e9	0.526013	0.263007	+	0.964794i	$0.415286\pi$
$864$	3.49727e8	0.0184472
$865$	2.30581e9	0.121134
$866$	3.71657e10	1.94460
$867$	2.69873e10	1.40635
$868$	−3.46185e10	−1.79676
$869$	1.80671e10	0.933940
$870$	4.42412e10	2.27777
$871$	0	0
$872$	−7.32179e8	−0.0373946
$873$	1.15450e10	0.587277
$874$	−5.12875e10	−2.59849
$875$	6.22084e9	0.313921
$876$	−1.00245e11	−5.03847
$877$	−3.23745e10	−1.62071	−0.810355	−	0.585940i	$-0.800725\pi$
$877$	−3.23745e10	−1.62071	−0.810355	+	0.585940i	$0.800725\pi$
$878$	−2.13152e10	−1.06282
$879$	2.55846e10	1.27063
$880$	2.42142e10	1.19779
$881$	6.00548e9	0.295891	0.147946	−	0.988995i	$-0.452734\pi$
$881$	6.00548e9	0.295891	0.147946	+	0.988995i	$0.452734\pi$
$882$	−1.13189e10	−0.555473
$883$	−2.76852e9	−0.135327	−0.0676636	−	0.997708i	$-0.521554\pi$
$883$	−2.76852e9	−0.135327	−0.0676636	+	0.997708i	$0.521554\pi$
$884$	0	0
$885$	7.52195e10	3.64778
$886$	5.25788e10	2.53976
$887$	3.08999e10	1.48670	0.743351	−	0.668902i	$-0.233235\pi$
$887$	3.08999e10	1.48670	0.743351	+	0.668902i	$0.233235\pi$
$888$	3.15624e10	1.51260
$889$	3.37300e10	1.61013
$890$	7.06239e10	3.35804
$891$	−7.29955e9	−0.345720
$892$	−3.34356e10	−1.57736
$893$	9.38464e9	0.440999
$894$	−2.32131e10	−1.08656
$895$	3.15237e10	1.46980
$896$	4.10341e10	1.90575
$897$	0	0
$898$	−5.43871e10	−2.50627
$899$	−1.00659e10	−0.462056
$900$	7.33144e10	3.35228
$901$	−8.28972e9	−0.377575
$902$	5.10869e10	2.31786
$903$	1.36469e10	0.616777
$904$	−1.06273e10	−0.478448
$905$	2.12341e10	0.952280
$906$	−5.03533e10	−2.24946
$907$	2.63801e10	1.17395	0.586977	−	0.809604i	$-0.300318\pi$
$907$	2.63801e10	1.17395	0.586977	+	0.809604i	$0.300318\pi$
$908$	−8.91913e10	−3.95387
$909$	3.28247e10	1.44953
$910$	0	0
$911$	9.42968e9	0.413221	0.206611	−	0.978423i	$-0.433757\pi$
$911$	9.42968e9	0.413221	0.206611	+	0.978423i	$0.433757\pi$
$912$	5.04194e10	2.20098
$913$	1.85422e10	0.806329
$914$	3.70836e10	1.60646
$915$	1.52836e10	0.659557
$916$	−2.12724e10	−0.914498
$917$	−3.01517e10	−1.29128
$918$	7.67844e9	0.327585
$919$	3.27533e10	1.39204	0.696019	−	0.718023i	$-0.254953\pi$
$919$	3.27533e10	1.39204	0.696019	+	0.718023i	$0.254953\pi$
$920$	6.61594e10	2.80114
$921$	4.92691e9	0.207810
$922$	1.41019e10	0.592544
$923$	0	0
$924$	−6.52619e10	−2.72150
$925$	1.60027e10	0.664809
$926$	−6.59151e9	−0.272801
$927$	−2.47569e10	−1.02074
$928$	−4.13814e8	−0.0169976
$929$	−2.49414e10	−1.02062	−0.510312	−	0.859989i	$-0.670470\pi$
$929$	−2.49414e10	−1.02062	−0.510312	+	0.859989i	$0.670470\pi$
$930$	−7.87636e10	−3.21096
$931$	−7.79473e9	−0.316576
$932$	−3.90434e9	−0.157976
$933$	5.06175e10	2.04040
$934$	−3.17917e9	−0.127673
$935$	8.87251e9	0.354982
$936$	0	0
$937$	−1.59590e10	−0.633750	−0.316875	−	0.948467i	$-0.602634\pi$
$937$	−1.59590e10	−0.633750	−0.316875	+	0.948467i	$0.602634\pi$
$938$	8.03562e10	3.17914
$939$	−6.09972e10	−2.40425
$940$	−2.41095e10	−0.946763
$941$	−2.91329e8	−0.0113978	−0.00569888	−	0.999984i	$-0.501814\pi$
$941$	−2.91329e8	−0.0113978	−0.00569888	+	0.999984i	$0.501814\pi$
$942$	−2.33631e10	−0.910653
$943$	4.71832e10	1.83230
$944$	−4.21577e10	−1.63108
$945$	−2.64481e10	−1.01949
$946$	1.27955e10	0.491405
$947$	−1.58421e10	−0.606160	−0.303080	−	0.952965i	$-0.598015\pi$
$947$	−1.58421e10	−0.606160	−0.303080	+	0.952965i	$0.598015\pi$
$948$	−9.66467e10	−3.68432
$949$	0	0
$950$	7.56248e10	2.86175
$951$	−7.18405e10	−2.70856
$952$	−1.56897e10	−0.589366
$953$	4.30011e10	1.60937	0.804683	−	0.593705i	$-0.202335\pi$
$953$	4.30011e10	1.60937	0.804683	+	0.593705i	$0.202335\pi$
$954$	8.09495e10	3.01853
$955$	4.93871e9	0.183486
$956$	−4.15240e10	−1.53708
$957$	−1.89760e10	−0.699862
$958$	3.67466e10	1.35032
$959$	−1.78465e10	−0.653412
$960$	6.12519e10	2.23445
$961$	−9.59202e9	−0.348641
$962$	0	0
$963$	−7.21341e9	−0.260285
$964$	−3.33123e10	−1.19767
$965$	5.67966e10	2.03459
$966$	−9.02846e10	−3.22251
$967$	−4.06842e10	−1.44688	−0.723441	−	0.690386i	$-0.757441\pi$
$967$	−4.06842e10	−1.44688	−0.723441	+	0.690386i	$0.757441\pi$
$968$	1.86408e10	0.660541
$969$	1.84746e10	0.652290
$970$	−3.05942e10	−1.07631
$971$	5.95290e9	0.208671	0.104335	−	0.994542i	$-0.466729\pi$
$971$	5.95290e9	0.208671	0.104335	+	0.994542i	$0.466729\pi$
$972$	7.47779e10	2.61181
$973$	−2.25859e10	−0.786036
$974$	5.48279e10	1.90128
$975$	0	0
$976$	−8.56590e9	−0.294916
$977$	8.40433e9	0.288318	0.144159	−	0.989555i	$-0.453952\pi$
$977$	8.40433e9	0.288318	0.144159	+	0.989555i	$0.453952\pi$
$978$	1.03073e11	3.52337
$979$	−3.02921e10	−1.03179
$980$	2.00250e10	0.679643
$981$	8.85571e8	0.0299490
$982$	3.67780e9	0.123936
$983$	−3.76311e10	−1.26360	−0.631799	−	0.775132i	$-0.717683\pi$
$983$	−3.76311e10	−1.26360	−0.631799	+	0.775132i	$0.717683\pi$
$984$	−1.37220e11	−4.59129
$985$	−3.86529e10	−1.28871
$986$	−9.08552e9	−0.301843
$987$	1.65204e10	0.546902
$988$	0	0
$989$	1.18178e10	0.388462
$990$	−8.66404e10	−2.83790
$991$	4.38732e10	1.43199	0.715997	−	0.698103i	$-0.245972\pi$
$991$	4.38732e10	1.43199	0.715997	+	0.698103i	$0.245972\pi$
$992$	7.36723e8	0.0239615
$993$	−1.06031e10	−0.343647
$994$	5.05352e9	0.163208
$995$	5.32255e10	1.71293
$996$	−9.91879e10	−3.18091
$997$	2.15416e10	0.688408	0.344204	−	0.938895i	$-0.388149\pi$
$997$	2.15416e10	0.688408	0.344204	+	0.938895i	$0.388149\pi$
$998$	−5.99824e10	−1.91015
$999$	−1.09263e10	−0.346733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.h.1.3	✓	21
13.5	odd	4		169.8.b.f.168.38		42
13.8	odd	4		169.8.b.f.168.5		42
13.12	even	2		169.8.a.i.1.19	yes	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.3	✓	21	1.1	even	1	trivial
169.8.a.i.1.19	yes	21	13.12	even	2
169.8.b.f.168.5		42	13.8	odd	4
169.8.b.f.168.38		42	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-19.6237 q^{2} -72.4635 q^{3} +257.091 q^{4} +413.760 q^{5} +1422.00 q^{6} +1005.88 q^{7} -2533.24 q^{8} +3063.96 q^{9} -8119.51 q^{10} +3482.63 q^{11} -18629.7 q^{12} -19739.1 q^{14} -29982.5 q^{15} +16804.1 q^{16} +6157.31 q^{17} -60126.4 q^{18} -41406.0 q^{19} +106374. q^{20} -72889.6 q^{21} -68342.2 q^{22} -63119.9 q^{23} +183568. q^{24} +93072.0 q^{25} -63547.8 q^{27} +258602. q^{28} +75193.0 q^{29} +588368. q^{30} -133868. q^{31} -5503.36 q^{32} -252364. q^{33} -120829. q^{34} +416192. q^{35} +787717. q^{36} +171939. q^{37} +812541. q^{38} -1.04815e6 q^{40} -747517. q^{41} +1.43037e6 q^{42} -187227. q^{43} +895352. q^{44} +1.26774e6 q^{45} +1.23865e6 q^{46} -226649. q^{47} -1.21768e6 q^{48} +188251. q^{49} -1.82642e6 q^{50} -446180. q^{51} -1.34632e6 q^{53} +1.24705e6 q^{54} +1.44097e6 q^{55} -2.54814e6 q^{56} +3.00043e6 q^{57} -1.47557e6 q^{58} -2.50878e6 q^{59} -7.70822e6 q^{60} -509752. q^{61} +2.62699e6 q^{62} +3.08198e6 q^{63} -2.04292e6 q^{64} +4.95232e6 q^{66} -4.07091e6 q^{67} +1.58299e6 q^{68} +4.57389e6 q^{69} -8.16725e6 q^{70} -256015. q^{71} -7.76177e6 q^{72} +5.38093e6 q^{73} -3.37408e6 q^{74} -6.74433e6 q^{75} -1.06451e7 q^{76} +3.50311e6 q^{77} +5.18777e6 q^{79} +6.95285e6 q^{80} -2.09599e6 q^{81} +1.46691e7 q^{82} +5.32418e6 q^{83} -1.87392e7 q^{84} +2.54765e6 q^{85} +3.67410e6 q^{86} -5.44875e6 q^{87} -8.82235e6 q^{88} -8.69805e6 q^{89} -2.48779e7 q^{90} -1.62276e7 q^{92} +9.70053e6 q^{93} +4.44770e6 q^{94} -1.71321e7 q^{95} +398793. q^{96} +3.76799e6 q^{97} -3.69419e6 q^{98} +1.06706e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 31 q^{2} - 26 q^{3} + 1409 q^{4} - 680 q^{5} - 1470 q^{6} - 2929 q^{7} - 4716 q^{8} + 15465 q^{9} - 5167 q^{10} - 14824 q^{11} + 21795 q^{12} - 179 q^{14} - 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots - 37605493 q^{99}+O(q^{100}) \) 21 * q - 31 * q^2 - 26 * q^3 + 1409 * q^4 - 680 * q^5 - 1470 * q^6 - 2929 * q^7 - 4716 * q^8 + 15465 * q^9 - 5167 * q^10 - 14824 * q^11 + 21795 * q^12 - 179 * q^14 - 36398 * q^15 + 113205 * q^16 + 45016 * q^17 - 72967 * q^18 - 119316 * q^19 - 131130 * q^20 - 94178 * q^21 - 274174 * q^22 - 128105 * q^23 - 316902 * q^24 + 366537 * q^25 - 106643 * q^27 - 170352 * q^28 - 62329 * q^29 + 9761 * q^30 - 467367 * q^31 - 677464 * q^32 - 587690 * q^33 - 578614 * q^34 + 848860 * q^35 - 274245 * q^36 - 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 - 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 + 263025 * q^44 + 1938345 * q^45 + 3170392 * q^46 - 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 - 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 - 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 - 5590305 * q^57 - 7510959 * q^58 - 10746233 * q^59 - 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 - 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 - 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 - 28670666 * q^70 - 23267025 * q^71 - 49774322 * q^72 - 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 - 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 - 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 - 21455789 * q^83 - 60768429 * q^84 - 26216249 * q^85 - 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 - 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 - 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 - 66667956 * q^96 - 10916609 * q^97 - 41738840 * q^98 - 37605493 * q^99

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