LMFDB - Embedded newform 169.8.a.i.1.7

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:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	\(q-8.58294 q^{2} +18.1259 q^{3} -54.3331 q^{4} -37.6136 q^{5} -155.574 q^{6} -711.860 q^{7} +1564.95 q^{8} -1858.45 q^{9} +322.835 q^{10} -4443.35 q^{11} -984.838 q^{12} +6109.86 q^{14} -681.782 q^{15} -6477.28 q^{16} -12189.3 q^{17} +15951.0 q^{18} -3742.40 q^{19} +2043.66 q^{20} -12903.1 q^{21} +38137.0 q^{22} -36942.6 q^{23} +28366.3 q^{24} -76710.2 q^{25} -73327.6 q^{27} +38677.5 q^{28} +52831.5 q^{29} +5851.69 q^{30} +159829. q^{31} -144720. q^{32} -80539.9 q^{33} +104620. q^{34} +26775.6 q^{35} +100975. q^{36} -306798. q^{37} +32120.8 q^{38} -58863.5 q^{40} +720391. q^{41} +110747. q^{42} -822785. q^{43} +241421. q^{44} +69903.0 q^{45} +317076. q^{46} +157589. q^{47} -117407. q^{48} -316798. q^{49} +658400. q^{50} -220942. q^{51} -1.36471e6 q^{53} +629367. q^{54} +167130. q^{55} -1.11403e6 q^{56} -67834.6 q^{57} -453450. q^{58} -1.62822e6 q^{59} +37043.3 q^{60} +1.21911e6 q^{61} -1.37180e6 q^{62} +1.32296e6 q^{63} +2.07122e6 q^{64} +691269. q^{66} +3.98372e6 q^{67} +662280. q^{68} -669619. q^{69} -229814. q^{70} +267877. q^{71} -2.90839e6 q^{72} +4.29822e6 q^{73} +2.63323e6 q^{74} -1.39045e6 q^{75} +203336. q^{76} +3.16304e6 q^{77} +5.94629e6 q^{79} +243634. q^{80} +2.73530e6 q^{81} -6.18307e6 q^{82} -2.95302e6 q^{83} +701067. q^{84} +458481. q^{85} +7.06192e6 q^{86} +957621. q^{87} -6.95363e6 q^{88} +1.20011e7 q^{89} -599973. q^{90} +2.00720e6 q^{92} +2.89705e6 q^{93} -1.35258e6 q^{94} +140765. q^{95} -2.62319e6 q^{96} +4.66114e6 q^{97} +2.71906e6 q^{98} +8.25774e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.58294	−0.758632	−0.379316	−	0.925267i	$-0.623841\pi$
$2$	−8.58294	−0.758632	−0.379316	+	0.925267i	$0.623841\pi$
$3$	18.1259	0.387593	0.193797	−	0.981042i	$-0.437920\pi$
$3$	18.1259	0.387593	0.193797	+	0.981042i	$0.437920\pi$
$4$	−54.3331	−0.424477
$5$	−37.6136	−0.134570	−0.0672852	−	0.997734i	$-0.521434\pi$
$5$	−37.6136	−0.134570	−0.0672852	+	0.997734i	$0.521434\pi$
$6$	−155.574	−0.294041
$7$	−711.860	−0.784425	−0.392213	−	0.919875i	$-0.628290\pi$
$7$	−711.860	−0.784425	−0.392213	+	0.919875i	$0.628290\pi$
$8$	1564.95	1.08065
$9$	−1858.45	−0.849771
$10$	322.835	0.102089
$11$	−4443.35	−1.00655	−0.503275	−	0.864126i	$-0.667872\pi$
$11$	−4443.35	−1.00655	−0.503275	+	0.864126i	$0.667872\pi$
$12$	−984.838	−0.164525
$13$	0	0
$13$	0	0
$14$	6109.86	0.595090
$15$	−681.782	−0.0521586
$16$	−6477.28	−0.395342
$17$	−12189.3	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$17$	−12189.3	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$18$	15951.0	0.644664
$19$	−3742.40	−0.125174	−0.0625868	−	0.998040i	$-0.519935\pi$
$19$	−3742.40	−0.125174	−0.0625868	+	0.998040i	$0.519935\pi$
$20$	2043.66	0.0571221
$21$	−12903.1	−0.304038
$22$	38137.0	0.763602
$23$	−36942.6	−0.633111	−0.316555	−	0.948574i	$-0.602526\pi$
$23$	−36942.6	−0.633111	−0.316555	+	0.948574i	$0.602526\pi$
$24$	28366.3	0.418854
$25$	−76710.2	−0.981891
$26$	0	0
$27$	−73327.6	−0.716959
$28$	38677.5	0.332971
$29$	52831.5	0.402254	0.201127	−	0.979565i	$-0.435540\pi$
$29$	52831.5	0.402254	0.201127	+	0.979565i	$0.435540\pi$
$30$	5851.69	0.0395692
$31$	159829.	0.963584	0.481792	−	0.876286i	$-0.339986\pi$
$31$	159829.	0.963584	0.481792	+	0.876286i	$0.339986\pi$
$32$	−144720.	−0.780735
$33$	−80539.9	−0.390132
$34$	104620.	0.456496
$35$	26775.6	0.105560
$36$	100975.	0.360709
$37$	−306798.	−0.995741	−0.497871	−	0.867251i	$-0.665885\pi$
$37$	−306798.	−0.995741	−0.497871	+	0.867251i	$0.665885\pi$
$38$	32120.8	0.0949608
$39$	0	0
$40$	−58863.5	−0.145424
$41$	720391.	1.63239	0.816196	−	0.577774i	$-0.196079\pi$
$41$	720391.	1.63239	0.816196	+	0.577774i	$0.196079\pi$
$42$	110747.	0.230653
$43$	−822785.	−1.57814	−0.789072	−	0.614301i	$-0.789438\pi$
$43$	−822785.	−1.57814	−0.789072	+	0.614301i	$0.789438\pi$
$44$	241421.	0.427258
$45$	69903.0	0.114354
$46$	317076.	0.480298
$47$	157589.	0.221404	0.110702	−	0.993854i	$-0.464690\pi$
$47$	157589.	0.221404	0.110702	+	0.993854i	$0.464690\pi$
$48$	−117407.	−0.153232
$49$	−316798.	−0.384677
$50$	658400.	0.744894
$51$	−220942.	−0.233229
$52$	0	0
$53$	−1.36471e6	−1.25914	−0.629572	−	0.776942i	$-0.716770\pi$
$53$	−1.36471e6	−1.25914	−0.629572	+	0.776942i	$0.716770\pi$
$54$	629367.	0.543908
$55$	167130.	0.135452
$56$	−1.11403e6	−0.847692
$57$	−67834.6	−0.0485165
$58$	−453450.	−0.305163
$59$	−1.62822e6	−1.03212	−0.516060	−	0.856552i	$-0.672602\pi$
$59$	−1.62822e6	−1.03212	−0.516060	+	0.856552i	$0.672602\pi$
$60$	37043.3	0.0221401
$61$	1.21911e6	0.687683	0.343842	−	0.939028i	$-0.388272\pi$
$61$	1.21911e6	0.687683	0.343842	+	0.939028i	$0.388272\pi$
$62$	−1.37180e6	−0.731006
$63$	1.32296e6	0.666582
$64$	2.07122e6	0.987633
$65$	0	0
$66$	691269.	0.295967
$67$	3.98372e6	1.61818	0.809090	−	0.587685i	$-0.199961\pi$
$67$	3.98372e6	1.61818	0.809090	+	0.587685i	$0.199961\pi$
$68$	662280.	0.255423
$69$	−669619.	−0.245390
$70$	−229814.	−0.0800816
$71$	267877.	0.0888241	0.0444121	−	0.999013i	$-0.485859\pi$
$71$	267877.	0.0888241	0.0444121	+	0.999013i	$0.485859\pi$
$72$	−2.90839e6	−0.918309
$73$	4.29822e6	1.29318	0.646590	−	0.762838i	$-0.276195\pi$
$73$	4.29822e6	1.29318	0.646590	+	0.762838i	$0.276195\pi$
$74$	2.63323e6	0.755401
$75$	−1.39045e6	−0.380574
$76$	203336.	0.0531334
$77$	3.16304e6	0.789564
$78$	0	0
$79$	5.94629e6	1.35691	0.678455	−	0.734642i	$-0.262650\pi$
$79$	5.94629e6	1.35691	0.678455	+	0.734642i	$0.262650\pi$
$80$	243634.	0.0532014
$81$	2.73530e6	0.571883
$82$	−6.18307e6	−1.23839
$83$	−2.95302e6	−0.566883	−0.283441	−	0.958990i	$-0.591476\pi$
$83$	−2.95302e6	−0.566883	−0.283441	+	0.958990i	$0.591476\pi$
$84$	701067.	0.129057
$85$	458481.	0.0809759
$86$	7.06192e6	1.19723
$87$	957621.	0.155911
$88$	−6.95363e6	−1.08773
$89$	1.20011e7	1.80449	0.902245	−	0.431223i	$-0.141918\pi$
$89$	1.20011e7	1.80449	0.902245	+	0.431223i	$0.141918\pi$
$90$	−599973.	−0.0867527
$91$	0	0
$92$	2.00720e6	0.268741
$93$	2.89705e6	0.373479
$94$	−1.35258e6	−0.167964
$95$	140765.	0.0168447
$96$	−2.62319e6	−0.302608
$97$	4.66114e6	0.518551	0.259275	−	0.965803i	$-0.416516\pi$
$97$	4.66114e6	0.518551	0.259275	+	0.965803i	$0.416516\pi$
$98$	2.71906e6	0.291828
$99$	8.25774e6	0.855338
$100$	4.16790e6	0.416790
$101$	−1.93897e7	−1.87261	−0.936304	−	0.351191i	$-0.885777\pi$
$101$	−1.93897e7	−1.87261	−0.936304	+	0.351191i	$0.885777\pi$
$102$	1.89633e6	0.176935
$103$	1.97787e7	1.78348	0.891740	−	0.452547i	$-0.149485\pi$
$103$	1.97787e7	1.78348	0.891740	+	0.452547i	$0.149485\pi$
$104$	0	0
$105$	485333.	0.0409145
$106$	1.17133e7	0.955228
$107$	1.45526e7	1.14841	0.574206	−	0.818711i	$-0.305311\pi$
$107$	1.45526e7	1.14841	0.574206	+	0.818711i	$0.305311\pi$
$108$	3.98411e6	0.304333
$109$	6.20545e6	0.458966	0.229483	−	0.973313i	$-0.426296\pi$
$109$	6.20545e6	0.458966	0.229483	+	0.973313i	$0.426296\pi$
$110$	−1.43447e6	−0.102758
$111$	−5.56100e6	−0.385943
$112$	4.61092e6	0.310116
$113$	7.16247e6	0.466969	0.233484	−	0.972361i	$-0.424987\pi$
$113$	7.16247e6	0.466969	0.233484	+	0.972361i	$0.424987\pi$
$114$	582221.	0.0368062
$115$	1.38954e6	0.0851980
$116$	−2.87050e6	−0.170747
$117$	0	0
$118$	1.39749e7	0.783000
$119$	8.67704e6	0.472017
$120$	−1.06696e6	−0.0563654
$121$	256155.	0.0131448
$122$	−1.04636e7	−0.521699
$123$	1.30578e7	0.632705
$124$	−8.68400e6	−0.409019
$125$	5.82391e6	0.266704
$126$	−1.13549e7	−0.505691
$127$	1.56555e7	0.678196	0.339098	−	0.940751i	$-0.389878\pi$
$127$	1.56555e7	0.678196	0.339098	+	0.940751i	$0.389878\pi$
$128$	747029.	0.0314849
$129$	−1.49138e7	−0.611678
$130$	0	0
$131$	−2.61761e7	−1.01731	−0.508657	−	0.860969i	$-0.669858\pi$
$131$	−2.61761e7	−1.01731	−0.508657	+	0.860969i	$0.669858\pi$
$132$	4.37598e6	0.165602
$133$	2.66407e6	0.0981894
$134$	−3.41920e7	−1.22760
$135$	2.75811e6	0.0964815
$136$	−1.90756e7	−0.650268
$137$	6.57353e6	0.218412	0.109206	−	0.994019i	$-0.465169\pi$
$137$	6.57353e6	0.218412	0.109206	+	0.994019i	$0.465169\pi$
$138$	5.74730e6	0.186160
$139$	1.10161e7	0.347919	0.173959	−	0.984753i	$-0.444344\pi$
$139$	1.10161e7	0.347919	0.173959	+	0.984753i	$0.444344\pi$
$140$	−1.45480e6	−0.0448080
$141$	2.85646e6	0.0858146
$142$	−2.29917e6	−0.0673849
$143$	0	0
$144$	1.20377e7	0.335950
$145$	−1.98718e6	−0.0541315
$146$	−3.68914e7	−0.981048
$147$	−5.74227e6	−0.149098
$148$	1.66693e7	0.422669
$149$	−2.61742e7	−0.648218	−0.324109	−	0.946020i	$-0.605064\pi$
$149$	−2.61742e7	−0.648218	−0.324109	+	0.946020i	$0.605064\pi$
$150$	1.19341e7	0.288716
$151$	−6.37855e7	−1.50766	−0.753829	−	0.657071i	$-0.771795\pi$
$151$	−6.37855e7	−1.50766	−0.753829	+	0.657071i	$0.771795\pi$
$152$	−5.85669e6	−0.135269
$153$	2.26531e7	0.511338
$154$	−2.71482e7	−0.598989
$155$	−6.01174e6	−0.129670
$156$	0	0
$157$	−6.74650e7	−1.39133	−0.695664	−	0.718367i	$-0.744890\pi$
$157$	−6.74650e7	−1.39133	−0.695664	+	0.718367i	$0.744890\pi$
$158$	−5.10367e7	−1.02940
$159$	−2.47367e7	−0.488036
$160$	5.44344e6	0.105064
$161$	2.62979e7	0.496628
$162$	−2.34769e7	−0.433849
$163$	−4.30912e7	−0.779348	−0.389674	−	0.920953i	$-0.627412\pi$
$163$	−4.30912e7	−0.779348	−0.389674	+	0.920953i	$0.627412\pi$
$164$	−3.91410e7	−0.692913
$165$	3.02939e6	0.0525003
$166$	2.53456e7	0.430056
$167$	−6.27530e7	−1.04262	−0.521311	−	0.853367i	$-0.674557\pi$
$167$	−6.27530e7	−1.04262	−0.521311	+	0.853367i	$0.674557\pi$
$168$	−2.01928e7	−0.328560
$169$	0	0
$170$	−3.93512e6	−0.0614309
$171$	6.95507e6	0.106369
$172$	4.47044e7	0.669886
$173$	9.73052e7	1.42881	0.714405	−	0.699732i	$-0.246697\pi$
$173$	9.73052e7	1.42881	0.714405	+	0.699732i	$0.246697\pi$
$174$	−8.21921e6	−0.118279
$175$	5.46069e7	0.770220
$176$	2.87808e7	0.397932
$177$	−2.95130e7	−0.400043
$178$	−1.03004e8	−1.36894
$179$	9.62895e7	1.25485	0.627427	−	0.778675i	$-0.284108\pi$
$179$	9.62895e7	1.25485	0.627427	+	0.778675i	$0.284108\pi$
$180$	−3.79804e6	−0.0485407
$181$	−9.20560e7	−1.15392	−0.576962	−	0.816771i	$-0.695762\pi$
$181$	−9.20560e7	−1.15392	−0.576962	+	0.816771i	$0.695762\pi$
$182$	0	0
$183$	2.20975e7	0.266541
$184$	−5.78134e7	−0.684174
$185$	1.15398e7	0.133997
$186$	−2.48652e7	−0.283333
$187$	5.41611e7	0.605678
$188$	−8.56232e6	−0.0939808
$189$	5.21990e7	0.562401
$190$	−1.20818e6	−0.0127789
$191$	1.37682e7	0.142976	0.0714878	−	0.997441i	$-0.477225\pi$
$191$	1.37682e7	0.142976	0.0714878	+	0.997441i	$0.477225\pi$
$192$	3.75428e7	0.382800
$193$	1.24460e7	0.124618	0.0623089	−	0.998057i	$-0.480154\pi$
$193$	1.24460e7	0.124618	0.0623089	+	0.998057i	$0.480154\pi$
$194$	−4.00063e7	−0.393389
$195$	0	0
$196$	1.72126e7	0.163287
$197$	−1.12835e8	−1.05151	−0.525754	−	0.850636i	$-0.676217\pi$
$197$	−1.12835e8	−1.05151	−0.525754	+	0.850636i	$0.676217\pi$
$198$	−7.08757e7	−0.648887
$199$	1.48457e8	1.33541	0.667704	−	0.744427i	$-0.267277\pi$
$199$	1.48457e8	1.33541	0.667704	+	0.744427i	$0.267277\pi$
$200$	−1.20048e8	−1.06108
$201$	7.22087e7	0.627196
$202$	1.66421e8	1.42062
$203$	−3.76086e7	−0.315538
$204$	1.20044e7	0.0990003
$205$	−2.70965e7	−0.219672
$206$	−1.69760e8	−1.35301
$207$	6.86559e7	0.537999
$208$	0	0
$209$	1.66288e7	0.125994
$210$	−4.16559e6	−0.0310391
$211$	−1.45474e8	−1.06610	−0.533048	−	0.846085i	$-0.678954\pi$
$211$	−1.45474e8	−1.06610	−0.533048	+	0.846085i	$0.678954\pi$
$212$	7.41490e7	0.534478
$213$	4.85552e6	0.0344276
$214$	−1.24904e8	−0.871223
$215$	3.09479e7	0.212372
$216$	−1.14754e8	−0.774785
$217$	−1.13776e8	−0.755860
$218$	−5.32610e7	−0.348187
$219$	7.79094e7	0.501228
$220$	−9.08069e6	−0.0574963
$221$	0	0
$222$	4.77298e7	0.292789
$223$	−2.87816e8	−1.73799	−0.868996	−	0.494819i	$-0.835234\pi$
$223$	−2.87816e8	−1.73799	−0.868996	+	0.494819i	$0.835234\pi$
$224$	1.03020e8	0.612428
$225$	1.42562e8	0.834383
$226$	−6.14750e7	−0.354258
$227$	9.37576e7	0.532005	0.266003	−	0.963972i	$-0.414297\pi$
$227$	9.37576e7	0.532005	0.266003	+	0.963972i	$0.414297\pi$
$228$	3.68566e6	0.0205941
$229$	1.30682e8	0.719103	0.359551	−	0.933125i	$-0.382930\pi$
$229$	1.30682e8	0.719103	0.359551	+	0.933125i	$0.382930\pi$
$230$	−1.19264e7	−0.0646339
$231$	5.73331e7	0.306030
$232$	8.26789e7	0.434697
$233$	5.84615e7	0.302778	0.151389	−	0.988474i	$-0.451625\pi$
$233$	5.84615e7	0.302778	0.151389	+	0.988474i	$0.451625\pi$
$234$	0	0
$235$	−5.92750e6	−0.0297944
$236$	8.84660e7	0.438111
$237$	1.07782e8	0.525929
$238$	−7.44746e7	−0.358087
$239$	6.20670e7	0.294082	0.147041	−	0.989130i	$-0.453025\pi$
$239$	6.20670e7	0.294082	0.147041	+	0.989130i	$0.453025\pi$
$240$	4.41609e6	0.0206205
$241$	−3.17938e8	−1.46313	−0.731565	−	0.681771i	$-0.761210\pi$
$241$	−3.17938e8	−1.46313	−0.731565	+	0.681771i	$0.761210\pi$
$242$	−2.19856e6	−0.00997207
$243$	2.09947e8	0.938617
$244$	−6.62380e7	−0.291906
$245$	1.19159e7	0.0517662
$246$	−1.12074e8	−0.479990
$247$	0	0
$248$	2.50125e8	1.04130
$249$	−5.35263e7	−0.219720
$250$	−4.99863e7	−0.202330
$251$	−4.54805e7	−0.181538	−0.0907689	−	0.995872i	$-0.528932\pi$
$251$	−4.54805e7	−0.181538	−0.0907689	+	0.995872i	$0.528932\pi$
$252$	−7.18803e7	−0.282949
$253$	1.64149e8	0.637258
$254$	−1.34371e8	−0.514501
$255$	8.31041e6	0.0313857
$256$	−2.71527e8	−1.01152
$257$	1.85747e8	0.682583	0.341292	−	0.939957i	$-0.389136\pi$
$257$	1.85747e8	0.682583	0.341292	+	0.939957i	$0.389136\pi$
$258$	1.28004e8	0.464039
$259$	2.18397e8	0.781084
$260$	0	0
$261$	−9.81847e7	−0.341824
$262$	2.24668e8	0.771767
$263$	−1.32964e6	−0.00450702	−0.00225351	−	0.999997i	$-0.500717\pi$
$263$	−1.32964e6	−0.00450702	−0.00225351	+	0.999997i	$0.500717\pi$
$264$	−1.26041e8	−0.421598
$265$	5.13317e7	0.169444
$266$	−2.28655e7	−0.0744896
$267$	2.17531e8	0.699409
$268$	−2.16448e8	−0.686880
$269$	4.40803e8	1.38074	0.690369	−	0.723457i	$-0.257448\pi$
$269$	4.40803e8	1.38074	0.690369	+	0.723457i	$0.257448\pi$
$270$	−2.36727e7	−0.0731940
$271$	−4.43852e7	−0.135471	−0.0677353	−	0.997703i	$-0.521577\pi$
$271$	−4.43852e7	−0.135471	−0.0677353	+	0.997703i	$0.521577\pi$
$272$	7.89533e7	0.237891
$273$	0	0
$274$	−5.64203e7	−0.165695
$275$	3.40850e8	0.988323
$276$	3.63825e7	0.104162
$277$	−5.63635e8	−1.59338	−0.796689	−	0.604389i	$-0.793417\pi$
$277$	−5.63635e8	−1.59338	−0.796689	+	0.604389i	$0.793417\pi$
$278$	−9.45509e7	−0.263942
$279$	−2.97034e8	−0.818826
$280$	4.19026e7	0.114074
$281$	6.29310e8	1.69197	0.845984	−	0.533208i	$-0.179014\pi$
$281$	6.29310e8	1.69197	0.845984	+	0.533208i	$0.179014\pi$
$282$	−2.45168e7	−0.0651017
$283$	−4.86094e8	−1.27487	−0.637437	−	0.770502i	$-0.720006\pi$
$283$	−4.86094e8	−1.27487	−0.637437	+	0.770502i	$0.720006\pi$
$284$	−1.45546e7	−0.0377038
$285$	2.55150e6	0.00652888
$286$	0	0
$287$	−5.12818e8	−1.28049
$288$	2.68955e8	0.663446
$289$	−2.61761e8	−0.637914
$290$	1.70559e7	0.0410659
$291$	8.44876e7	0.200987
$292$	−2.33536e8	−0.548925
$293$	−1.58617e8	−0.368396	−0.184198	−	0.982889i	$-0.558969\pi$
$293$	−1.58617e8	−0.368396	−0.184198	+	0.982889i	$0.558969\pi$
$294$	4.92856e7	0.113111
$295$	6.12431e7	0.138893
$296$	−4.80125e8	−1.07605
$297$	3.25820e8	0.721656
$298$	2.24651e8	0.491759
$299$	0	0
$300$	7.55472e7	0.161545
$301$	5.85708e8	1.23794
$302$	5.47468e8	1.14376
$303$	−3.51457e8	−0.725810
$304$	2.42406e7	0.0494864
$305$	−4.58551e7	−0.0925418
$306$	−1.94430e8	−0.387917
$307$	3.43683e8	0.677912	0.338956	−	0.940802i	$-0.389926\pi$
$307$	3.43683e8	0.677912	0.338956	+	0.940802i	$0.389926\pi$
$308$	−1.71858e8	−0.335152
$309$	3.58508e8	0.691265
$310$	5.15984e7	0.0983718
$311$	−3.43924e8	−0.648338	−0.324169	−	0.945999i	$-0.605085\pi$
$311$	−3.43924e8	−0.648338	−0.324169	+	0.945999i	$0.605085\pi$
$312$	0	0
$313$	−2.30082e8	−0.424109	−0.212054	−	0.977258i	$-0.568015\pi$
$313$	−2.30082e8	−0.424109	−0.212054	+	0.977258i	$0.568015\pi$
$314$	5.79048e8	1.05551
$315$	−4.97611e7	−0.0897022
$316$	−3.23080e8	−0.575977
$317$	5.16781e8	0.911169	0.455584	−	0.890193i	$-0.349430\pi$
$317$	5.16781e8	0.911169	0.455584	+	0.890193i	$0.349430\pi$
$318$	2.12314e8	0.370240
$319$	−2.34749e8	−0.404889
$320$	−7.79059e7	−0.132906
$321$	2.63780e8	0.445117
$322$	−2.25714e8	−0.376758
$323$	4.56171e7	0.0753215
$324$	−1.48617e8	−0.242751
$325$	0	0
$326$	3.69849e8	0.591239
$327$	1.12480e8	0.177892
$328$	1.12738e9	1.76405
$329$	−1.12182e8	−0.173675
$330$	−2.60011e7	−0.0398284
$331$	1.09918e9	1.66598	0.832990	−	0.553287i	$-0.186627\pi$
$331$	1.09918e9	1.66598	0.832990	+	0.553287i	$0.186627\pi$
$332$	1.60447e8	0.240629
$333$	5.70169e8	0.846152
$334$	5.38606e8	0.790967
$335$	−1.49842e8	−0.217759
$336$	8.35773e7	0.120199
$337$	9.93632e8	1.41423	0.707116	−	0.707097i	$-0.249996\pi$
$337$	9.93632e8	1.41423	0.707116	+	0.707097i	$0.249996\pi$
$338$	0	0
$339$	1.29826e8	0.180994
$340$	−2.49107e7	−0.0343724
$341$	−7.10176e8	−0.969896
$342$	−5.96950e7	−0.0806950
$343$	8.11763e8	1.08618
$344$	−1.28762e9	−1.70543
$345$	2.51868e7	0.0330222
$346$	−8.35165e8	−1.08394
$347$	9.94060e7	0.127720	0.0638601	−	0.997959i	$-0.479659\pi$
$347$	9.94060e7	0.127720	0.0638601	+	0.997959i	$0.479659\pi$
$348$	−5.20305e7	−0.0661806
$349$	−2.62555e8	−0.330621	−0.165311	−	0.986242i	$-0.552863\pi$
$349$	−2.62555e8	−0.330621	−0.165311	+	0.986242i	$0.552863\pi$
$350$	−4.68688e8	−0.584314
$351$	0	0
$352$	6.43041e8	0.785850
$353$	−7.65807e7	−0.0926633	−0.0463316	−	0.998926i	$-0.514753\pi$
$353$	−7.65807e7	−0.0926633	−0.0463316	+	0.998926i	$0.514753\pi$
$354$	2.53308e8	0.303485
$355$	−1.00758e7	−0.0119531
$356$	−6.52054e8	−0.765965
$357$	1.57280e8	0.182951
$358$	−8.26448e8	−0.951973
$359$	1.28834e9	1.46960	0.734801	−	0.678282i	$-0.237275\pi$
$359$	1.28834e9	1.46960	0.734801	+	0.678282i	$0.237275\pi$
$360$	1.09395e8	0.123577
$361$	−8.79866e8	−0.984332
$362$	7.90112e8	0.875404
$363$	4.64305e6	0.00509484
$364$	0	0
$365$	−1.61672e8	−0.174024
$366$	−1.89662e8	−0.202207
$367$	−9.75428e8	−1.03006	−0.515032	−	0.857171i	$-0.672220\pi$
$367$	−9.75428e8	−1.03006	−0.515032	+	0.857171i	$0.672220\pi$
$368$	2.39288e8	0.250295
$369$	−1.33881e9	−1.38716
$370$	−9.90452e7	−0.101655
$371$	9.71484e8	0.987705
$372$	−1.57406e8	−0.158533
$373$	−7.26009e8	−0.724371	−0.362186	−	0.932106i	$-0.617969\pi$
$373$	−7.26009e8	−0.724371	−0.362186	+	0.932106i	$0.617969\pi$
$374$	−4.64861e8	−0.459487
$375$	1.05564e8	0.103373
$376$	2.46620e8	0.239261
$377$	0	0
$378$	−4.48021e8	−0.426655
$379$	−1.53421e9	−1.44760	−0.723800	−	0.690009i	$-0.757606\pi$
$379$	−1.53421e9	−1.44760	−0.723800	+	0.690009i	$0.757606\pi$
$380$	−7.64820e6	−0.00715018
$381$	2.83772e8	0.262864
$382$	−1.18172e8	−0.108466
$383$	3.52677e7	0.0320760	0.0160380	−	0.999871i	$-0.494895\pi$
$383$	3.52677e7	0.0320760	0.0160380	+	0.999871i	$0.494895\pi$
$384$	1.35406e7	0.0122033
$385$	−1.18973e8	−0.106252
$386$	−1.06823e8	−0.0945390
$387$	1.52910e9	1.34106
$388$	−2.53254e8	−0.220113
$389$	5.04613e8	0.434645	0.217323	−	0.976100i	$-0.430268\pi$
$389$	5.04613e8	0.434645	0.217323	+	0.976100i	$0.430268\pi$
$390$	0	0
$391$	4.50302e8	0.380965
$392$	−4.95775e8	−0.415703
$393$	−4.74466e8	−0.394304
$394$	9.68458e8	0.797709
$395$	−2.23661e8	−0.182600
$396$	−4.48668e8	−0.363071
$397$	2.26708e8	0.181845	0.0909224	−	0.995858i	$-0.471018\pi$
$397$	2.26708e8	0.181845	0.0909224	+	0.995858i	$0.471018\pi$
$398$	−1.27420e9	−1.01308
$399$	4.82888e7	0.0380576
$400$	4.96874e8	0.388183
$401$	−2.33490e8	−0.180827	−0.0904134	−	0.995904i	$-0.528819\pi$
$401$	−2.33490e8	−0.180827	−0.0904134	+	0.995904i	$0.528819\pi$
$402$	−6.19763e8	−0.475811
$403$	0	0
$404$	1.05350e9	0.794879
$405$	−1.02884e8	−0.0769585
$406$	3.22793e8	0.239377
$407$	1.36321e9	1.00226
$408$	−3.45764e8	−0.252040
$409$	1.88840e9	1.36478	0.682388	−	0.730990i	$-0.260941\pi$
$409$	1.88840e9	1.36478	0.682388	+	0.730990i	$0.260941\pi$
$410$	2.32568e8	0.166650
$411$	1.19152e8	0.0846551
$412$	−1.07464e9	−0.757047
$413$	1.15906e9	0.809621
$414$	−5.89270e8	−0.408144
$415$	1.11074e8	0.0762857
$416$	0	0
$417$	1.99678e8	0.134851
$418$	−1.42724e8	−0.0955829
$419$	5.93294e7	0.0394023	0.0197011	−	0.999806i	$-0.493729\pi$
$419$	5.93294e7	0.0394023	0.0197011	+	0.999806i	$0.493729\pi$
$420$	−2.63696e7	−0.0173673
$421$	4.12500e7	0.0269424	0.0134712	−	0.999909i	$-0.495712\pi$
$421$	4.12500e7	0.0269424	0.0134712	+	0.999909i	$0.495712\pi$
$422$	1.24859e9	0.808775
$423$	−2.92872e8	−0.188142
$424$	−2.13571e9	−1.36070
$425$	9.35040e8	0.590839
$426$	−4.16747e7	−0.0261179
$427$	−8.67836e8	−0.539436
$428$	−7.90688e8	−0.487475
$429$	0	0
$430$	−2.65624e8	−0.161112
$431$	−1.56893e9	−0.943917	−0.471958	−	0.881621i	$-0.656453\pi$
$431$	−1.56893e9	−0.943917	−0.471958	+	0.881621i	$0.656453\pi$
$432$	4.74964e8	0.283444
$433$	−1.76006e8	−0.104188	−0.0520941	−	0.998642i	$-0.516590\pi$
$433$	−1.76006e8	−0.104188	−0.0520941	+	0.998642i	$0.516590\pi$
$434$	9.76532e8	0.573419
$435$	−3.60196e7	−0.0209810
$436$	−3.37161e8	−0.194821
$437$	1.38254e8	0.0792488
$438$	−6.68692e8	−0.380248
$439$	1.88359e9	1.06257	0.531287	−	0.847192i	$-0.321708\pi$
$439$	1.88359e9	1.06257	0.531287	+	0.847192i	$0.321708\pi$
$440$	2.61551e8	0.146377
$441$	5.88754e8	0.326888
$442$	0	0
$443$	−2.29191e9	−1.25252	−0.626260	−	0.779614i	$-0.715415\pi$
$443$	−2.29191e9	−1.25252	−0.626260	+	0.779614i	$0.715415\pi$
$444$	3.02146e8	0.163824
$445$	−4.51403e8	−0.242831
$446$	2.47031e9	1.31850
$447$	−4.74432e8	−0.251245
$448$	−1.47442e9	−0.774724
$449$	2.77965e9	1.44920	0.724599	−	0.689170i	$-0.242025\pi$
$449$	2.77965e9	1.44920	0.724599	+	0.689170i	$0.242025\pi$
$450$	−1.22360e9	−0.632990
$451$	−3.20095e9	−1.64309
$452$	−3.89159e8	−0.198218
$453$	−1.15617e9	−0.584358
$454$	−8.04716e8	−0.403596
$455$	0	0
$456$	−1.06158e8	−0.0524295
$457$	−2.68742e9	−1.31713	−0.658566	−	0.752523i	$-0.728837\pi$
$457$	−2.68742e9	−1.31713	−0.658566	+	0.752523i	$0.728837\pi$
$458$	−1.12163e9	−0.545535
$459$	8.93809e8	0.431420
$460$	−7.54981e7	−0.0361646
$461$	−2.79625e9	−1.32930	−0.664650	−	0.747155i	$-0.731419\pi$
$461$	−2.79625e9	−1.32930	−0.664650	+	0.747155i	$0.731419\pi$
$462$	−4.92087e8	−0.232164
$463$	−8.32472e7	−0.0389795	−0.0194898	−	0.999810i	$-0.506204\pi$
$463$	−8.32472e7	−0.0389795	−0.0194898	+	0.999810i	$0.506204\pi$
$464$	−3.42205e8	−0.159028
$465$	−1.08968e8	−0.0502592
$466$	−5.01771e8	−0.229697
$467$	−3.64730e9	−1.65715	−0.828576	−	0.559876i	$-0.810849\pi$
$467$	−3.64730e9	−1.65715	−0.828576	+	0.559876i	$0.810849\pi$
$468$	0	0
$469$	−2.83585e9	−1.26934
$470$	5.08754e7	0.0226030
$471$	−1.22287e9	−0.539269
$472$	−2.54809e9	−1.11537
$473$	3.65592e9	1.58848
$474$	−9.25088e8	−0.398987
$475$	2.87081e8	0.122907
$476$	−4.71450e8	−0.200360
$477$	2.53625e9	1.06999
$478$	−5.32718e8	−0.223100
$479$	−2.04827e9	−0.851556	−0.425778	−	0.904828i	$-0.640000\pi$
$479$	−2.04827e9	−0.851556	−0.425778	+	0.904828i	$0.640000\pi$
$480$	9.86675e7	0.0407221
$481$	0	0
$482$	2.72885e9	1.10998
$483$	4.76675e8	0.192490
$484$	−1.39177e7	−0.00557967
$485$	−1.75322e8	−0.0697816
$486$	−1.80197e9	−0.712065
$487$	1.60868e9	0.631129	0.315564	−	0.948904i	$-0.397806\pi$
$487$	1.60868e9	0.631129	0.315564	+	0.948904i	$0.397806\pi$
$488$	1.90785e9	0.743148
$489$	−7.81068e8	−0.302070
$490$	−1.02274e8	−0.0392715
$491$	2.08343e9	0.794316	0.397158	−	0.917750i	$-0.369997\pi$
$491$	2.08343e9	0.794316	0.397158	+	0.917750i	$0.369997\pi$
$492$	−7.09469e8	−0.268569
$493$	−6.43977e8	−0.242050
$494$	0	0
$495$	−3.10603e8	−0.115103
$496$	−1.03526e9	−0.380945
$497$	−1.90691e8	−0.0696759
$498$	4.59413e8	0.166687
$499$	1.01332e9	0.365087	0.182543	−	0.983198i	$-0.441567\pi$
$499$	1.01332e9	0.365087	0.182543	+	0.983198i	$0.441567\pi$
$500$	−3.16431e8	−0.113210
$501$	−1.13746e9	−0.404113
$502$	3.90356e8	0.137720
$503$	1.20560e9	0.422391	0.211195	−	0.977444i	$-0.432264\pi$
$503$	1.20560e9	0.422391	0.211195	+	0.977444i	$0.432264\pi$
$504$	2.07037e9	0.720345
$505$	7.29317e8	0.251998
$506$	−1.40888e9	−0.483445
$507$	0	0
$508$	−8.50614e8	−0.287879
$509$	3.99769e9	1.34368	0.671842	−	0.740694i	$-0.265503\pi$
$509$	3.99769e9	1.34368	0.671842	+	0.740694i	$0.265503\pi$
$510$	−7.13278e7	−0.0238102
$511$	−3.05973e9	−1.01440
$512$	2.23488e9	0.735885
$513$	2.74422e8	0.0897444
$514$	−1.59425e9	−0.517830
$515$	−7.43949e8	−0.240004
$516$	8.10310e8	0.259643
$517$	−7.00225e8	−0.222854
$518$	−1.87449e9	−0.592556
$519$	1.76375e9	0.553798
$520$	0	0
$521$	2.01184e9	0.623249	0.311624	−	0.950205i	$-0.399127\pi$
$521$	2.01184e9	0.623249	0.311624	+	0.950205i	$0.399127\pi$
$522$	8.42714e8	0.259318
$523$	5.43348e8	0.166082	0.0830409	−	0.996546i	$-0.473537\pi$
$523$	5.43348e8	0.166082	0.0830409	+	0.996546i	$0.473537\pi$
$524$	1.42223e9	0.431827
$525$	9.89803e8	0.298532
$526$	1.14122e7	0.00341917
$527$	−1.94820e9	−0.579823
$528$	5.21680e8	0.154236
$529$	−2.04007e9	−0.599171
$530$	−4.40577e8	−0.128545
$531$	3.02596e9	0.877066
$532$	−1.44747e8	−0.0416791
$533$	0	0
$534$	−1.86705e9	−0.530594
$535$	−5.47376e8	−0.154542
$536$	6.23434e9	1.74869
$537$	1.74534e9	0.486373
$538$	−3.78339e9	−1.04747
$539$	1.40764e9	0.387197
$540$	−1.49857e8	−0.0409542
$541$	4.04447e8	0.109818	0.0549088	−	0.998491i	$-0.482513\pi$
$541$	4.04447e8	0.109818	0.0549088	+	0.998491i	$0.482513\pi$
$542$	3.80955e8	0.102772
$543$	−1.66860e9	−0.447253
$544$	1.76403e9	0.469796
$545$	−2.33409e8	−0.0617633
$546$	0	0
$547$	−7.98136e8	−0.208507	−0.104254	−	0.994551i	$-0.533245\pi$
$547$	−7.98136e8	−0.208507	−0.104254	+	0.994551i	$0.533245\pi$
$548$	−3.57160e8	−0.0927110
$549$	−2.26566e9	−0.584374
$550$	−2.92550e9	−0.749774
$551$	−1.97717e8	−0.0503516
$552$	−1.04792e9	−0.265181
$553$	−4.23293e9	−1.06439
$554$	4.83765e9	1.20879
$555$	2.09169e8	0.0519365
$556$	−5.98541e8	−0.147684
$557$	1.48425e9	0.363927	0.181963	−	0.983305i	$-0.441755\pi$
$557$	1.48425e9	0.363927	0.181963	+	0.983305i	$0.441755\pi$
$558$	2.54943e9	0.621188
$559$	0	0
$560$	−1.73433e8	−0.0417325
$561$	9.81721e8	0.234757
$562$	−5.40133e9	−1.28358
$563$	1.39349e9	0.329097	0.164548	−	0.986369i	$-0.447383\pi$
$563$	1.39349e9	0.329097	0.164548	+	0.986369i	$0.447383\pi$
$564$	−1.55200e8	−0.0364263
$565$	−2.69406e8	−0.0628402
$566$	4.17212e9	0.967161
$567$	−1.94715e9	−0.448599
$568$	4.19215e8	0.0959882
$569$	−8.69007e9	−1.97756	−0.988781	−	0.149370i	$-0.952275\pi$
$569$	−8.69007e9	−1.97756	−0.988781	+	0.149370i	$0.952275\pi$
$570$	−2.18994e7	−0.00495302
$571$	5.03913e9	1.13274	0.566369	−	0.824152i	$-0.308348\pi$
$571$	5.03913e9	1.13274	0.566369	+	0.824152i	$0.308348\pi$
$572$	0	0
$573$	2.49563e8	0.0554164
$574$	4.40148e9	0.971421
$575$	2.83387e9	0.621646
$576$	−3.84925e9	−0.839262
$577$	−4.02573e9	−0.872427	−0.436214	−	0.899843i	$-0.643681\pi$
$577$	−4.02573e9	−0.872427	−0.436214	+	0.899843i	$0.643681\pi$
$578$	2.24668e9	0.483942
$579$	2.25596e8	0.0483010
$580$	1.07970e8	0.0229776
$581$	2.10214e9	0.444677
$582$	−7.25153e8	−0.152475
$583$	6.06389e9	1.26739
$584$	6.72652e9	1.39748
$585$	0	0
$586$	1.36141e9	0.279477
$587$	2.51557e9	0.513337	0.256669	−	0.966499i	$-0.417375\pi$
$587$	2.51557e9	0.513337	0.256669	+	0.966499i	$0.417375\pi$
$588$	3.11995e8	0.0632888
$589$	−5.98145e8	−0.120615
$590$	−5.25646e8	−0.105369
$591$	−2.04525e9	−0.407558
$592$	1.98722e9	0.393658
$593$	8.91464e9	1.75555	0.877774	−	0.479076i	$-0.159028\pi$
$593$	8.91464e9	1.75555	0.877774	+	0.479076i	$0.159028\pi$
$594$	−2.79649e9	−0.547471
$595$	−3.26375e8	−0.0635195
$596$	1.42212e9	0.275154
$597$	2.69092e9	0.517596
$598$	0	0
$599$	7.56365e9	1.43793	0.718964	−	0.695047i	$-0.244616\pi$
$599$	7.56365e9	1.43793	0.718964	+	0.695047i	$0.244616\pi$
$600$	−2.17598e9	−0.411269
$601$	6.55777e9	1.23224	0.616120	−	0.787652i	$-0.288704\pi$
$601$	6.55777e9	1.23224	0.616120	+	0.787652i	$0.288704\pi$
$602$	−5.02710e9	−0.939138
$603$	−7.40354e9	−1.37508
$604$	3.46566e9	0.639966
$605$	−9.63491e6	−0.00176890
$606$	3.01654e9	0.550623
$607$	8.10496e9	1.47092	0.735462	−	0.677566i	$-0.236965\pi$
$607$	8.10496e9	1.47092	0.735462	+	0.677566i	$0.236965\pi$
$608$	5.41601e8	0.0977275
$609$	−6.81692e8	−0.122300
$610$	3.93572e8	0.0702052
$611$	0	0
$612$	−1.23081e9	−0.217051
$613$	2.95977e9	0.518975	0.259487	−	0.965746i	$-0.416446\pi$
$613$	2.95977e9	0.518975	0.259487	+	0.965746i	$0.416446\pi$
$614$	−2.94981e9	−0.514286
$615$	−4.91149e8	−0.0851433
$616$	4.95002e9	0.853246
$617$	4.15027e9	0.711341	0.355671	−	0.934611i	$-0.384253\pi$
$617$	4.15027e9	0.711341	0.355671	+	0.934611i	$0.384253\pi$
$618$	−3.07706e9	−0.524416
$619$	−4.05061e9	−0.686441	−0.343220	−	0.939255i	$-0.611518\pi$
$619$	−4.05061e9	−0.686441	−0.343220	+	0.939255i	$0.611518\pi$
$620$	3.26636e8	0.0550419
$621$	2.70891e9	0.453914
$622$	2.95188e9	0.491850
$623$	−8.54308e9	−1.41549
$624$	0	0
$625$	5.77393e9	0.946000
$626$	1.97478e9	0.321743
$627$	3.01413e8	0.0488343
$628$	3.66558e9	0.590587
$629$	3.73964e9	0.599173
$630$	4.27097e8	0.0680510
$631$	−3.55581e9	−0.563425	−0.281712	−	0.959499i	$-0.590902\pi$
$631$	−3.55581e9	−0.563425	−0.281712	+	0.959499i	$0.590902\pi$
$632$	9.30567e9	1.46635
$633$	−2.63685e9	−0.413212
$634$	−4.43550e9	−0.691242
$635$	−5.88861e8	−0.0912651
$636$	1.34402e9	0.207160
$637$	0	0
$638$	2.01483e9	0.307162
$639$	−4.97836e8	−0.0754802
$640$	−2.80984e7	−0.00423694
$641$	2.95831e9	0.443650	0.221825	−	0.975086i	$-0.428799\pi$
$641$	2.95831e9	0.443650	0.221825	+	0.975086i	$0.428799\pi$
$642$	−2.26401e9	−0.337680
$643$	−2.39725e9	−0.355611	−0.177806	−	0.984066i	$-0.556900\pi$
$643$	−2.39725e9	−0.355611	−0.177806	+	0.984066i	$0.556900\pi$
$644$	−1.42885e9	−0.210807
$645$	5.60960e8	0.0823138
$646$	−3.91529e8	−0.0571413
$647$	−6.84211e9	−0.993174	−0.496587	−	0.867987i	$-0.665414\pi$
$647$	−6.84211e9	−0.993174	−0.496587	+	0.867987i	$0.665414\pi$
$648$	4.28062e9	0.618008
$649$	7.23473e9	1.03888
$650$	0	0
$651$	−2.06230e9	−0.292966
$652$	2.34128e9	0.330816
$653$	1.09310e10	1.53626	0.768129	−	0.640295i	$-0.221188\pi$
$653$	1.09310e10	1.53626	0.768129	+	0.640295i	$0.221188\pi$
$654$	−9.65407e8	−0.134955
$655$	9.84576e8	0.136900
$656$	−4.66618e9	−0.645354
$657$	−7.98803e9	−1.09891
$658$	9.62849e8	0.131755
$659$	4.64584e9	0.632362	0.316181	−	0.948699i	$-0.397599\pi$
$659$	4.64584e9	0.632362	0.316181	+	0.948699i	$0.397599\pi$
$660$	−1.64596e8	−0.0222852
$661$	5.27114e8	0.0709904	0.0354952	−	0.999370i	$-0.488699\pi$
$661$	5.27114e8	0.0709904	0.0354952	+	0.999370i	$0.488699\pi$
$662$	−9.43418e9	−1.26387
$663$	0	0
$664$	−4.62134e9	−0.612604
$665$	−1.00205e8	−0.0132134
$666$	−4.89373e9	−0.641918
$667$	−1.95173e9	−0.254671
$668$	3.40956e9	0.442569
$669$	−5.21694e9	−0.673634
$670$	1.28608e9	0.165199
$671$	−5.41693e9	−0.692188
$672$	1.86734e9	0.237373
$673$	1.17630e9	0.148752	0.0743762	−	0.997230i	$-0.476303\pi$
$673$	1.17630e9	0.148752	0.0743762	+	0.997230i	$0.476303\pi$
$674$	−8.52828e9	−1.07288
$675$	5.62498e9	0.703976
$676$	0	0
$677$	−4.93872e9	−0.611722	−0.305861	−	0.952076i	$-0.598944\pi$
$677$	−4.93872e9	−0.611722	−0.305861	+	0.952076i	$0.598944\pi$
$678$	−1.11429e9	−0.137308
$679$	−3.31808e9	−0.406764
$680$	7.17503e8	0.0875069
$681$	1.69945e9	0.206202
$682$	6.09540e9	0.735795
$683$	1.25808e10	1.51090	0.755450	−	0.655207i	$-0.227418\pi$
$683$	1.25808e10	1.51090	0.755450	+	0.655207i	$0.227418\pi$
$684$	−3.77890e8	−0.0451512
$685$	−2.47254e8	−0.0293918
$686$	−6.96732e9	−0.824008
$687$	2.36873e9	0.278720
$688$	5.32941e9	0.623907
$689$	0	0
$690$	−2.16177e8	−0.0250517
$691$	5.61694e9	0.647630	0.323815	−	0.946121i	$-0.395035\pi$
$691$	5.61694e9	0.647630	0.323815	+	0.946121i	$0.395035\pi$
$692$	−5.28689e9	−0.606498
$693$	−5.87835e9	−0.670949
$694$	−8.53196e8	−0.0968926
$695$	−4.14357e8	−0.0468196
$696$	1.49863e9	0.168486
$697$	−8.78103e9	−0.982269
$698$	2.25349e9	0.250820
$699$	1.05967e9	0.117355
$700$	−2.96696e9	−0.326941
$701$	−1.47184e10	−1.61380	−0.806898	−	0.590691i	$-0.798855\pi$
$701$	−1.47184e10	−1.61380	−0.806898	+	0.590691i	$0.798855\pi$
$702$	0	0
$703$	1.14816e9	0.124641
$704$	−9.20313e9	−0.994103
$705$	−1.07442e8	−0.0115481
$706$	6.57287e8	0.0702973
$707$	1.38028e10	1.46892
$708$	1.60353e9	0.169809
$709$	−2.79809e9	−0.294849	−0.147424	−	0.989073i	$-0.547098\pi$
$709$	−2.79809e9	−0.294849	−0.147424	+	0.989073i	$0.547098\pi$
$710$	8.64801e7	0.00906801
$711$	−1.10509e10	−1.15306
$712$	1.87811e10	1.95003
$713$	−5.90449e9	−0.610055
$714$	−1.34992e9	−0.138792
$715$	0	0
$716$	−5.23171e9	−0.532657
$717$	1.12502e9	0.113984
$718$	−1.10577e10	−1.11489
$719$	−2.42687e9	−0.243498	−0.121749	−	0.992561i	$-0.538850\pi$
$719$	−2.42687e9	−0.243498	−0.121749	+	0.992561i	$0.538850\pi$
$720$	−4.52781e8	−0.0452090
$721$	−1.40797e10	−1.39901
$722$	7.55184e9	0.746746
$723$	−5.76293e9	−0.567100
$724$	5.00169e9	0.489814
$725$	−4.05272e9	−0.394969
$726$	−3.98511e7	−0.00386511
$727$	−8.43860e9	−0.814517	−0.407258	−	0.913313i	$-0.633515\pi$
$727$	−8.43860e9	−0.814517	−0.407258	+	0.913313i	$0.633515\pi$
$728$	0	0
$729$	−2.17660e9	−0.208081
$730$	1.38762e9	0.132020
$731$	1.00291e10	0.949626
$732$	−1.20063e9	−0.113141
$733$	−1.80224e10	−1.69024	−0.845122	−	0.534573i	$-0.820473\pi$
$733$	−1.80224e10	−1.69024	−0.845122	+	0.534573i	$0.820473\pi$
$734$	8.37204e9	0.781439
$735$	2.15987e8	0.0200642
$736$	5.34633e9	0.494292
$737$	−1.77010e10	−1.62878
$738$	1.14909e10	1.05234
$739$	−8.22224e9	−0.749436	−0.374718	−	0.927139i	$-0.622260\pi$
$739$	−8.22224e9	−0.749436	−0.374718	+	0.927139i	$0.622260\pi$
$740$	−6.26991e8	−0.0568788
$741$	0	0
$742$	−8.33820e9	−0.749305
$743$	−5.43950e9	−0.486517	−0.243258	−	0.969962i	$-0.578216\pi$
$743$	−5.43950e9	−0.486517	−0.243258	+	0.969962i	$0.578216\pi$
$744$	4.53375e9	0.403601
$745$	9.84504e8	0.0872310
$746$	6.23130e9	0.549532
$747$	5.48804e9	0.481721
$748$	−2.94274e9	−0.257096
$749$	−1.03594e10	−0.900844
$750$	−9.06048e8	−0.0784218
$751$	2.16087e10	1.86161	0.930807	−	0.365512i	$-0.119106\pi$
$751$	2.16087e10	1.86161	0.930807	+	0.365512i	$0.119106\pi$
$752$	−1.02075e9	−0.0875302
$753$	−8.24377e8	−0.0703628
$754$	0	0
$755$	2.39920e9	0.202886
$756$	−2.83613e9	−0.238726
$757$	−2.97846e9	−0.249549	−0.124775	−	0.992185i	$-0.539821\pi$
$757$	−2.97846e9	−0.249549	−0.124775	+	0.992185i	$0.539821\pi$
$758$	1.31681e10	1.09820
$759$	2.97535e9	0.246997
$760$	2.20291e8	0.0182033
$761$	2.61767e9	0.215312	0.107656	−	0.994188i	$-0.465665\pi$
$761$	2.61767e9	0.215312	0.107656	+	0.994188i	$0.465665\pi$
$762$	−2.43560e9	−0.199417
$763$	−4.41741e9	−0.360025
$764$	−7.48071e8	−0.0606899
$765$	−8.52065e8	−0.0688110
$766$	−3.02700e8	−0.0243339
$767$	0	0
$768$	−4.92169e9	−0.392058
$769$	−3.25859e9	−0.258397	−0.129199	−	0.991619i	$-0.541240\pi$
$769$	−3.25859e9	−0.258397	−0.129199	+	0.991619i	$0.541240\pi$
$770$	1.02114e9	0.0806062
$771$	3.36684e9	0.264565
$772$	−6.76230e8	−0.0528974
$773$	−1.25784e10	−0.979485	−0.489743	−	0.871867i	$-0.662909\pi$
$773$	−1.25784e10	−0.979485	−0.489743	+	0.871867i	$0.662909\pi$
$774$	−1.31242e10	−1.01737
$775$	−1.22605e10	−0.946134
$776$	7.29448e9	0.560374
$777$	3.95866e9	0.302743
$778$	−4.33106e9	−0.329736
$779$	−2.69599e9	−0.204333
$780$	0	0
$781$	−1.19027e9	−0.0894060
$782$	−3.86492e9	−0.289013
$783$	−3.87401e9	−0.288399
$784$	2.05199e9	0.152079
$785$	2.53760e9	0.187232
$786$	4.07232e9	0.299132
$787$	7.59812e9	0.555641	0.277821	−	0.960633i	$-0.410388\pi$
$787$	7.59812e9	0.555641	0.277821	+	0.960633i	$0.410388\pi$
$788$	6.13068e9	0.446341
$789$	−2.41010e7	−0.00174689
$790$	1.91967e9	0.138526
$791$	−5.09867e9	−0.366302
$792$	1.29230e10	0.924325
$793$	0	0
$794$	−1.94583e9	−0.137953
$795$	9.30436e8	0.0656752
$796$	−8.06611e9	−0.566850
$797$	2.54466e10	1.78043	0.890216	−	0.455539i	$-0.150554\pi$
$797$	2.54466e10	1.78043	0.890216	+	0.455539i	$0.150554\pi$
$798$	−4.14460e8	−0.0288717
$799$	−1.92090e9	−0.133227
$800$	1.11015e10	0.766597
$801$	−2.23034e10	−1.53340
$802$	2.00403e9	0.137181
$803$	−1.90985e10	−1.30165
$804$	−3.92332e9	−0.266230
$805$	−9.89160e8	−0.0668314
$806$	0	0
$807$	7.98997e9	0.535165
$808$	−3.03440e10	−2.02364
$809$	−1.30758e10	−0.868255	−0.434128	−	0.900851i	$-0.642943\pi$
$809$	−1.30758e10	−0.868255	−0.434128	+	0.900851i	$0.642943\pi$
$810$	8.83051e8	0.0583832
$811$	−2.55569e10	−1.68242	−0.841211	−	0.540707i	$-0.818157\pi$
$811$	−2.55569e10	−1.68242	−0.841211	+	0.540707i	$0.818157\pi$
$812$	2.04339e9	0.133939
$813$	−8.04523e8	−0.0525075
$814$	−1.17004e10	−0.760350
$815$	1.62081e9	0.104877
$816$	1.43110e9	0.0922052
$817$	3.07919e9	0.197542
$818$	−1.62080e10	−1.03536
$819$	0	0
$820$	1.47223e9	0.0932457
$821$	−1.19206e10	−0.751788	−0.375894	−	0.926663i	$-0.622664\pi$
$821$	−1.19206e10	−0.751788	−0.375894	+	0.926663i	$0.622664\pi$
$822$	−1.02267e9	−0.0642221
$823$	−2.15124e9	−0.134521	−0.0672604	−	0.997735i	$-0.521426\pi$
$823$	−2.15124e9	−0.134521	−0.0672604	+	0.997735i	$0.521426\pi$
$824$	3.09528e10	1.92733
$825$	6.17823e9	0.383067
$826$	−9.94817e9	−0.614205
$827$	8.67859e9	0.533556	0.266778	−	0.963758i	$-0.414041\pi$
$827$	8.67859e9	0.533556	0.266778	+	0.963758i	$0.414041\pi$
$828$	−3.73029e9	−0.228368
$829$	1.09237e10	0.665931	0.332966	−	0.942939i	$-0.391951\pi$
$829$	1.09237e10	0.665931	0.332966	+	0.942939i	$0.391951\pi$
$830$	−9.53339e8	−0.0578728
$831$	−1.02164e10	−0.617583
$832$	0	0
$833$	3.86153e9	0.231474
$834$	−1.71383e9	−0.102302
$835$	2.36037e9	0.140306
$836$	−9.03493e8	−0.0534814
$837$	−1.17199e10	−0.690850
$838$	−5.09221e8	−0.0298918
$839$	2.72391e10	1.59231	0.796153	−	0.605096i	$-0.206865\pi$
$839$	2.72391e10	1.59231	0.796153	+	0.605096i	$0.206865\pi$
$840$	7.59525e8	0.0442145
$841$	−1.44587e10	−0.838192
$842$	−3.54047e8	−0.0204394
$843$	1.14068e10	0.655796
$844$	7.90404e9	0.452534
$845$	0	0
$846$	2.51371e9	0.142731
$847$	−1.82347e8	−0.0103111
$848$	8.83963e9	0.497793
$849$	−8.81091e9	−0.494133
$850$	−8.02540e9	−0.448229
$851$	1.13339e10	0.630414
$852$	−2.63815e8	−0.0146137
$853$	3.05226e10	1.68384	0.841918	−	0.539606i	$-0.181427\pi$
$853$	3.05226e10	1.68384	0.841918	+	0.539606i	$0.181427\pi$
$854$	7.44859e9	0.409234
$855$	−2.61605e8	−0.0143141
$856$	2.27742e10	1.24104
$857$	1.11731e10	0.606373	0.303187	−	0.952931i	$-0.401949\pi$
$857$	1.11731e10	0.606373	0.303187	+	0.952931i	$0.401949\pi$
$858$	0	0
$859$	2.74544e10	1.47787	0.738934	−	0.673778i	$-0.235330\pi$
$859$	2.74544e10	1.47787	0.738934	+	0.673778i	$0.235330\pi$
$860$	−1.68149e9	−0.0901469
$861$	−9.29530e9	−0.496309
$862$	1.34661e10	0.716086
$863$	−1.32881e9	−0.0703761	−0.0351880	−	0.999381i	$-0.511203\pi$
$863$	−1.32881e9	−0.0703761	−0.0351880	+	0.999381i	$0.511203\pi$
$864$	1.06120e10	0.559755
$865$	−3.66000e9	−0.192276
$866$	1.51065e9	0.0790406
$867$	−4.74466e9	−0.247251
$868$	6.18179e9	0.320845
$869$	−2.64214e10	−1.36580
$870$	3.09154e8	0.0159169
$871$	0	0
$872$	9.71125e9	0.495984
$873$	−8.66250e9	−0.440650
$874$	−1.18663e9	−0.0601207
$875$	−4.14581e9	−0.209209
$876$	−4.23306e9	−0.212760
$877$	−2.02332e10	−1.01290	−0.506450	−	0.862269i	$-0.669043\pi$
$877$	−2.02332e10	−1.01290	−0.506450	+	0.862269i	$0.669043\pi$
$878$	−1.61667e10	−0.806103
$879$	−2.87509e9	−0.142788
$880$	−1.08255e9	−0.0535499
$881$	−3.68108e10	−1.81367	−0.906837	−	0.421482i	$-0.861510\pi$
$881$	−3.68108e10	−1.81367	−0.906837	+	0.421482i	$0.861510\pi$
$882$	−5.05324e9	−0.247987
$883$	−1.87326e10	−0.915662	−0.457831	−	0.889039i	$-0.651374\pi$
$883$	−1.87326e10	−0.915662	−0.457831	+	0.889039i	$0.651374\pi$
$884$	0	0
$885$	1.11009e9	0.0538340
$886$	1.96714e10	0.950203
$887$	2.70884e10	1.30332	0.651661	−	0.758511i	$-0.274073\pi$
$887$	2.70884e10	1.30332	0.651661	+	0.758511i	$0.274073\pi$
$888$	−8.70272e9	−0.417071
$889$	−1.11446e10	−0.531994
$890$	3.87437e9	0.184219
$891$	−1.21539e10	−0.575629
$892$	1.56379e10	0.737738
$893$	−5.89763e8	−0.0277139
$894$	4.07202e9	0.190603
$895$	−3.62179e9	−0.168866
$896$	−5.31780e8	−0.0246976
$897$	0	0
$898$	−2.38576e10	−1.09941
$899$	8.44401e9	0.387605
$900$	−7.74584e9	−0.354176
$901$	1.66348e10	0.757672
$902$	2.74735e10	1.24650
$903$	1.06165e10	0.479816
$904$	1.12089e10	0.504632
$905$	3.46256e9	0.155284
$906$	9.92337e9	0.443313
$907$	9.38357e9	0.417583	0.208791	−	0.977960i	$-0.433047\pi$
$907$	9.38357e9	0.417583	0.208791	+	0.977960i	$0.433047\pi$
$908$	−5.09414e9	−0.225824
$909$	3.60348e10	1.59129
$910$	0	0
$911$	2.34729e10	1.02862	0.514308	−	0.857606i	$-0.328049\pi$
$911$	2.34729e10	1.02862	0.514308	+	0.857606i	$0.328049\pi$
$912$	4.39384e8	0.0191806
$913$	1.31213e10	0.570596
$914$	2.30660e10	0.999219
$915$	−8.31167e8	−0.0358686
$916$	−7.10035e9	−0.305243
$917$	1.86337e10	0.798007
$918$	−7.67151e9	−0.327289
$919$	2.74313e10	1.16585	0.582924	−	0.812527i	$-0.301908\pi$
$919$	2.74313e10	1.16585	0.582924	+	0.812527i	$0.301908\pi$
$920$	2.17457e9	0.0920696
$921$	6.22957e9	0.262754
$922$	2.40001e10	1.00845
$923$	0	0
$924$	−3.11508e9	−0.129903
$925$	2.35345e10	0.977709
$926$	7.14506e8	0.0295711
$927$	−3.67578e10	−1.51555
$928$	−7.64578e9	−0.314054
$929$	−1.37674e10	−0.563375	−0.281688	−	0.959506i	$-0.590894\pi$
$929$	−1.37674e10	−0.563375	−0.281688	+	0.959506i	$0.590894\pi$
$930$	9.35270e8	0.0381283
$931$	1.18559e9	0.0481514
$932$	−3.17639e9	−0.128522
$933$	−6.23396e9	−0.251292
$934$	3.13046e10	1.25717
$935$	−2.03719e9	−0.0815063
$936$	0	0
$937$	−3.57657e10	−1.42029	−0.710147	−	0.704054i	$-0.751371\pi$
$937$	−3.57657e10	−1.42029	−0.710147	+	0.704054i	$0.751371\pi$
$938$	2.43399e10	0.962963
$939$	−4.17045e9	−0.164382
$940$	3.22060e8	0.0126470
$941$	1.46325e10	0.572471	0.286236	−	0.958159i	$-0.407596\pi$
$941$	1.46325e10	0.572471	0.286236	+	0.958159i	$0.407596\pi$
$942$	1.04958e10	0.409107
$943$	−2.66131e10	−1.03349
$944$	1.05464e10	0.408041
$945$	−1.96339e9	−0.0756825
$946$	−3.13785e10	−1.20507
$947$	3.26365e10	1.24876	0.624380	−	0.781121i	$-0.285352\pi$
$947$	3.26365e10	1.24876	0.624380	+	0.781121i	$0.285352\pi$
$948$	−5.85613e9	−0.223245
$949$	0	0
$950$	−2.46400e9	−0.0932411
$951$	9.36714e9	0.353163
$952$	1.35792e10	0.510087
$953$	−3.11753e10	−1.16677	−0.583385	−	0.812195i	$-0.698272\pi$
$953$	−3.11753e10	−1.16677	−0.583385	+	0.812195i	$0.698272\pi$
$954$	−2.17685e10	−0.811725
$955$	−5.17873e8	−0.0192403
$956$	−3.37229e9	−0.124831
$957$	−4.25504e9	−0.156932
$958$	1.75802e10	0.646018
$959$	−4.67944e9	−0.171328
$960$	−1.41212e9	−0.0515136
$961$	−1.96731e9	−0.0715058
$962$	0	0
$963$	−2.70453e10	−0.975888
$964$	1.72746e10	0.621066
$965$	−4.68139e8	−0.0167699
$966$	−4.09128e9	−0.146029
$967$	2.54520e9	0.0905169	0.0452585	−	0.998975i	$-0.485589\pi$
$967$	2.54520e9	0.0905169	0.0452585	+	0.998975i	$0.485589\pi$
$968$	4.00871e8	0.0142050
$969$	8.26853e8	0.0291941
$970$	1.50478e9	0.0529386
$971$	−3.03029e10	−1.06223	−0.531114	−	0.847300i	$-0.678226\pi$
$971$	−3.03029e10	−1.06223	−0.531114	+	0.847300i	$0.678226\pi$
$972$	−1.14071e10	−0.398421
$973$	−7.84195e9	−0.272916
$974$	−1.38072e10	−0.478795
$975$	0	0
$976$	−7.89652e9	−0.271870
$977$	1.42652e10	0.489381	0.244690	−	0.969601i	$-0.421314\pi$
$977$	1.42652e10	0.489381	0.244690	+	0.969601i	$0.421314\pi$
$978$	6.70387e9	0.229160
$979$	−5.33249e10	−1.81631
$980$	−6.47428e8	−0.0219736
$981$	−1.15325e10	−0.390016
$982$	−1.78819e10	−0.602593
$983$	3.44221e10	1.15585	0.577924	−	0.816091i	$-0.303863\pi$
$983$	3.44221e10	1.15585	0.577924	+	0.816091i	$0.303863\pi$
$984$	2.04348e10	0.683735
$985$	4.24414e9	0.141502
$986$	5.52722e9	0.183627
$987$	−2.03340e9	−0.0673151
$988$	0	0
$989$	3.03958e10	0.999140
$990$	2.66589e9	0.0873210
$991$	−1.26763e10	−0.413748	−0.206874	−	0.978368i	$-0.566329\pi$
$991$	−1.26763e10	−0.413748	−0.206874	+	0.978368i	$0.566329\pi$
$992$	−2.31305e10	−0.752304
$993$	1.99236e10	0.645723
$994$	1.63669e9	0.0528584
$995$	−5.58399e9	−0.179707
$996$	2.90825e9	0.0932661
$997$	2.08066e10	0.664917	0.332459	−	0.943118i	$-0.392122\pi$
$997$	2.08066e10	0.664917	0.332459	+	0.943118i	$0.392122\pi$
$998$	−8.69730e9	−0.276967
$999$	2.24968e10	0.713906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.i.1.7	yes	21
13.5	odd	4		169.8.b.f.168.28		42
13.8	odd	4		169.8.b.f.168.15		42
13.12	even	2		169.8.a.h.1.15	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.15	✓	21	13.12	even	2
169.8.a.i.1.7	yes	21	1.1	even	1	trivial
169.8.b.f.168.15		42	13.8	odd	4
169.8.b.f.168.28		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)

\(f(q)\)	\(=\)	\(q-8.58294 q^{2} +18.1259 q^{3} -54.3331 q^{4} -37.6136 q^{5} -155.574 q^{6} -711.860 q^{7} +1564.95 q^{8} -1858.45 q^{9} +322.835 q^{10} -4443.35 q^{11} -984.838 q^{12} +6109.86 q^{14} -681.782 q^{15} -6477.28 q^{16} -12189.3 q^{17} +15951.0 q^{18} -3742.40 q^{19} +2043.66 q^{20} -12903.1 q^{21} +38137.0 q^{22} -36942.6 q^{23} +28366.3 q^{24} -76710.2 q^{25} -73327.6 q^{27} +38677.5 q^{28} +52831.5 q^{29} +5851.69 q^{30} +159829. q^{31} -144720. q^{32} -80539.9 q^{33} +104620. q^{34} +26775.6 q^{35} +100975. q^{36} -306798. q^{37} +32120.8 q^{38} -58863.5 q^{40} +720391. q^{41} +110747. q^{42} -822785. q^{43} +241421. q^{44} +69903.0 q^{45} +317076. q^{46} +157589. q^{47} -117407. q^{48} -316798. q^{49} +658400. q^{50} -220942. q^{51} -1.36471e6 q^{53} +629367. q^{54} +167130. q^{55} -1.11403e6 q^{56} -67834.6 q^{57} -453450. q^{58} -1.62822e6 q^{59} +37043.3 q^{60} +1.21911e6 q^{61} -1.37180e6 q^{62} +1.32296e6 q^{63} +2.07122e6 q^{64} +691269. q^{66} +3.98372e6 q^{67} +662280. q^{68} -669619. q^{69} -229814. q^{70} +267877. q^{71} -2.90839e6 q^{72} +4.29822e6 q^{73} +2.63323e6 q^{74} -1.39045e6 q^{75} +203336. q^{76} +3.16304e6 q^{77} +5.94629e6 q^{79} +243634. q^{80} +2.73530e6 q^{81} -6.18307e6 q^{82} -2.95302e6 q^{83} +701067. q^{84} +458481. q^{85} +7.06192e6 q^{86} +957621. q^{87} -6.95363e6 q^{88} +1.20011e7 q^{89} -599973. q^{90} +2.00720e6 q^{92} +2.89705e6 q^{93} -1.35258e6 q^{94} +140765. q^{95} -2.62319e6 q^{96} +4.66114e6 q^{97} +2.71906e6 q^{98} +8.25774e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) \) 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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