LMFDB - Embedded newform 169.8.a.i.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)

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);

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")

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			1.3
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)

$f(q)$	$=$	\(q-16.7260 q^{2} -13.7721 q^{3} +151.758 q^{4} +0.583029 q^{5} +230.352 q^{6} -274.333 q^{7} -397.369 q^{8} -1997.33 q^{9} -9.75172 q^{10} -1092.42 q^{11} -2090.03 q^{12} +4588.49 q^{14} -8.02956 q^{15} -12778.6 q^{16} +24367.8 q^{17} +33407.2 q^{18} +29441.5 q^{19} +88.4791 q^{20} +3778.16 q^{21} +18271.7 q^{22} -87568.7 q^{23} +5472.63 q^{24} -78124.7 q^{25} +57627.1 q^{27} -41632.2 q^{28} -98544.5 q^{29} +134.302 q^{30} +46345.4 q^{31} +264598. q^{32} +15044.9 q^{33} -407575. q^{34} -159.944 q^{35} -303110. q^{36} +217133. q^{37} -492438. q^{38} -231.678 q^{40} -623135. q^{41} -63193.3 q^{42} -751529. q^{43} -165783. q^{44} -1164.50 q^{45} +1.46467e6 q^{46} -1.15731e6 q^{47} +175988. q^{48} -748284. q^{49} +1.30671e6 q^{50} -335597. q^{51} +498300. q^{53} -963869. q^{54} -636.912 q^{55} +109012. q^{56} -405473. q^{57} +1.64825e6 q^{58} +1.11248e6 q^{59} -1218.55 q^{60} -624881. q^{61} -775171. q^{62} +547934. q^{63} -2.78999e6 q^{64} -251641. q^{66} -28848.2 q^{67} +3.69801e6 q^{68} +1.20601e6 q^{69} +2675.22 q^{70} +3.65711e6 q^{71} +793677. q^{72} +5.20600e6 q^{73} -3.63176e6 q^{74} +1.07594e6 q^{75} +4.46798e6 q^{76} +299687. q^{77} -2.37817e6 q^{79} -7450.29 q^{80} +3.57451e6 q^{81} +1.04225e7 q^{82} +1.77288e6 q^{83} +573364. q^{84} +14207.2 q^{85} +1.25700e7 q^{86} +1.35717e6 q^{87} +434094. q^{88} -2.72509e6 q^{89} +19477.4 q^{90} -1.32892e7 q^{92} -638275. q^{93} +1.93570e7 q^{94} +17165.3 q^{95} -3.64407e6 q^{96} -838595. q^{97} +1.25158e7 q^{98} +2.18192e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−16.7260	−1.47838	−0.739190	−	0.673497i	$-0.764791\pi$
$2$	−16.7260	−1.47838	−0.739190	+	0.673497i	$0.764791\pi$
$3$	−13.7721	−0.294494	−0.147247	−	0.989100i	$-0.547041\pi$
$3$	−13.7721	−0.294494	−0.147247	+	0.989100i	$0.547041\pi$
$4$	151.758	1.18561
$5$	0.583029	0.00208591	0.00104295	−	0.999999i	$-0.499668\pi$
$5$	0.583029	0.00208591	0.00104295	+	0.999999i	$0.499668\pi$
$6$	230.352	0.435374
$7$	−274.333	−0.302298	−0.151149	−	0.988511i	$-0.548297\pi$
$7$	−274.333	−0.302298	−0.151149	+	0.988511i	$0.548297\pi$
$8$	−397.369	−0.274397
$9$	−1997.33	−0.913273
$10$	−9.75172	−0.00308377
$11$	−1092.42	−0.247465	−0.123733	−	0.992316i	$-0.539487\pi$
$11$	−1092.42	−0.247465	−0.123733	+	0.992316i	$0.539487\pi$
$12$	−2090.03	−0.349154
$13$	0	0
$13$	0	0
$14$	4588.49	0.446911
$15$	−8.02956	−0.000614288 0
$16$	−12778.6	−0.779944
$17$	24367.8	1.20295	0.601473	−	0.798893i	$-0.294581\pi$
$17$	24367.8	1.20295	0.601473	+	0.798893i	$0.294581\pi$
$18$	33407.2	1.35016
$19$	29441.5	0.984743	0.492372	−	0.870385i	$-0.336130\pi$
$19$	29441.5	0.984743	0.492372	+	0.870385i	$0.336130\pi$
$20$	88.4791	0.00247307
$21$	3778.16	0.0890251
$22$	18271.7	0.365848
$23$	−87568.7	−1.50073	−0.750363	−	0.661026i	$-0.770121\pi$
$23$	−87568.7	−1.50073	−0.750363	+	0.661026i	$0.770121\pi$
$24$	5472.63	0.0808084
$25$	−78124.7	−0.999996
$26$	0	0
$27$	57627.1	0.563448
$28$	−41632.2	−0.358407
$29$	−98544.5	−0.750308	−0.375154	−	0.926963i	$-0.622410\pi$
$29$	−98544.5	−0.750308	−0.375154	+	0.926963i	$0.622410\pi$
$30$	134.302	0.000908151 0
$31$	46345.4	0.279409	0.139705	−	0.990193i	$-0.455385\pi$
$31$	46345.4	0.279409	0.139705	+	0.990193i	$0.455385\pi$
$32$	264598.	1.42745
$33$	15044.9	0.0728771
$34$	−407575.	−1.77841
$35$	−159.944	−0.000630566 0
$36$	−303110.	−1.08278
$37$	217133.	0.704727	0.352363	−	0.935863i	$-0.385378\pi$
$37$	217133.	0.704727	0.352363	+	0.935863i	$0.385378\pi$
$38$	−492438.	−1.45582
$39$	0	0
$40$	−231.678	−0.000572367 0
$41$	−623135.	−1.41201	−0.706006	−	0.708206i	$-0.749505\pi$
$41$	−623135.	−1.41201	−0.706006	+	0.708206i	$0.749505\pi$
$42$	−63193.3	−0.131613
$43$	−751529.	−1.44147	−0.720736	−	0.693210i	$-0.756196\pi$
$43$	−751529.	−1.44147	−0.720736	+	0.693210i	$0.756196\pi$
$44$	−165783.	−0.293397
$45$	−1164.50	−0.00190500
$46$	1.46467e6	2.21864
$47$	−1.15731e6	−1.62594	−0.812972	−	0.582303i	$-0.802152\pi$
$47$	−1.15731e6	−1.62594	−0.812972	+	0.582303i	$0.802152\pi$
$48$	175988.	0.229689
$49$	−748284.	−0.908616
$50$	1.30671e6	1.47837
$51$	−335597.	−0.354260
$52$	0	0
$53$	498300.	0.459753	0.229877	−	0.973220i	$-0.426168\pi$
$53$	498300.	0.459753	0.229877	+	0.973220i	$0.426168\pi$
$54$	−963869.	−0.832990
$55$	−636.912	−0.000516190 0
$56$	109012.	0.0829497
$57$	−405473.	−0.290001
$58$	1.64825e6	1.10924
$59$	1.11248e6	0.705194	0.352597	−	0.935775i	$-0.385299\pi$
$59$	1.11248e6	0.705194	0.352597	+	0.935775i	$0.385299\pi$
$60$	−1218.55	−0.000728304 0
$61$	−624881.	−0.352487	−0.176243	−	0.984347i	$-0.556395\pi$
$61$	−624881.	−0.352487	−0.176243	+	0.984347i	$0.556395\pi$
$62$	−775171.	−0.413073
$63$	547934.	0.276081
$64$	−2.78999e6	−1.33037
$65$	0	0
$66$	−251641.	−0.107740
$67$	−28848.2	−0.0117181	−0.00585904	−	0.999983i	$-0.501865\pi$
$67$	−28848.2	−0.0117181	−0.00585904	+	0.999983i	$0.501865\pi$
$68$	3.69801e6	1.42622
$69$	1.20601e6	0.441955
$70$	2675.22	0.000932217 0
$71$	3.65711e6	1.21265	0.606323	−	0.795218i	$-0.292644\pi$
$71$	3.65711e6	1.21265	0.606323	+	0.795218i	$0.292644\pi$
$72$	793677.	0.250600
$73$	5.20600e6	1.56630	0.783148	−	0.621835i	$-0.213613\pi$
$73$	5.20600e6	1.56630	0.783148	+	0.621835i	$0.213613\pi$
$74$	−3.63176e6	−1.04185
$75$	1.07594e6	0.294493
$76$	4.46798e6	1.16752
$77$	299687.	0.0748083
$78$	0	0
$79$	−2.37817e6	−0.542685	−0.271343	−	0.962483i	$-0.587468\pi$
$79$	−2.37817e6	−0.542685	−0.271343	+	0.962483i	$0.587468\pi$
$80$	−7450.29	−0.00162689
$81$	3.57451e6	0.747341
$82$	1.04225e7	2.08749
$83$	1.77288e6	0.340335	0.170168	−	0.985415i	$-0.445569\pi$
$83$	1.77288e6	0.340335	0.170168	+	0.985415i	$0.445569\pi$
$84$	573364.	0.105549
$85$	14207.2	0.00250923
$86$	1.25700e7	2.13104
$87$	1.35717e6	0.220961
$88$	434094.	0.0679038
$89$	−2.72509e6	−0.409746	−0.204873	−	0.978789i	$-0.565678\pi$
$89$	−2.72509e6	−0.409746	−0.204873	+	0.978789i	$0.565678\pi$
$90$	19477.4	0.00281632
$91$	0	0
$92$	−1.32892e7	−1.77927
$93$	−638275.	−0.0822844
$94$	1.93570e7	2.40376
$95$	17165.3	0.00205408
$96$	−3.64407e6	−0.420376
$97$	−838595.	−0.0932934	−0.0466467	−	0.998911i	$-0.514853\pi$
$97$	−838595.	−0.0932934	−0.0466467	+	0.998911i	$0.514853\pi$
$98$	1.25158e7	1.34328
$99$	2.18192e6	0.226003
$100$	−1.18560e7	−1.18560
$101$	1.35004e7	1.30383	0.651916	−	0.758291i	$-0.273965\pi$
$101$	1.35004e7	1.30383	0.651916	+	0.758291i	$0.273965\pi$
$102$	5.61318e6	0.523731
$103$	1.30297e7	1.17491	0.587456	−	0.809256i	$-0.300130\pi$
$103$	1.30297e7	1.17491	0.587456	+	0.809256i	$0.300130\pi$
$104$	0	0
$105$	2202.78	0.000185698 0
$106$	−8.33454e6	−0.679690
$107$	2.04841e7	1.61649	0.808246	−	0.588844i	$-0.200417\pi$
$107$	2.04841e7	1.61649	0.808246	+	0.588844i	$0.200417\pi$
$108$	8.74536e6	0.668028
$109$	−1.74867e7	−1.29335	−0.646673	−	0.762768i	$-0.723840\pi$
$109$	−1.74867e7	−1.29335	−0.646673	+	0.762768i	$0.723840\pi$
$110$	10653.0	0.000763125 0
$111$	−2.99039e6	−0.207538
$112$	3.50559e6	0.235775
$113$	6.72636e6	0.438536	0.219268	−	0.975665i	$-0.429633\pi$
$113$	6.72636e6	0.438536	0.219268	+	0.975665i	$0.429633\pi$
$114$	6.78192e6	0.428732
$115$	−51055.1	−0.00313038
$116$	−1.49549e7	−0.889570
$117$	0	0
$118$	−1.86072e7	−1.04254
$119$	−6.68491e6	−0.363648
$120$	3190.70	0.000168559 0
$121$	−1.82938e7	−0.938761
$122$	1.04517e7	0.521109
$123$	8.58190e6	0.415830
$124$	7.03327e6	0.331269
$125$	−91098.1	−0.00417181
$126$	−9.16472e6	−0.408152
$127$	−3.10681e7	−1.34587	−0.672934	−	0.739703i	$-0.734966\pi$
$127$	−3.10681e7	−1.34587	−0.672934	+	0.739703i	$0.734966\pi$
$128$	1.27967e7	0.539341
$129$	1.03502e7	0.424505
$130$	0	0
$131$	1.55397e7	0.603940	0.301970	−	0.953317i	$-0.402356\pi$
$131$	1.55397e7	0.603940	0.301970	+	0.953317i	$0.402356\pi$
$132$	2.28318e6	0.0864036
$133$	−8.07680e6	−0.297686
$134$	482514.	0.0173238
$135$	33598.3	0.00117530
$136$	−9.68304e6	−0.330085
$137$	1.75618e7	0.583508	0.291754	−	0.956493i	$-0.405761\pi$
$137$	1.75618e7	0.583508	0.291754	+	0.956493i	$0.405761\pi$
$138$	−2.01716e7	−0.653378
$139$	2.32529e7	0.734386	0.367193	−	0.930145i	$-0.380319\pi$
$139$	2.32529e7	0.734386	0.367193	+	0.930145i	$0.380319\pi$
$140$	−24272.8	−0.000747604 0
$141$	1.59386e7	0.478831
$142$	−6.11687e7	−1.79275
$143$	0	0
$144$	2.55230e7	0.712301
$145$	−57454.3	−0.00156507
$146$	−8.70753e7	−2.31558
$147$	1.03055e7	0.267582
$148$	3.29517e7	0.835529
$149$	7.31136e7	1.81070	0.905350	−	0.424666i	$-0.139609\pi$
$149$	7.31136e7	1.81070	0.905350	+	0.424666i	$0.139609\pi$
$150$	−1.79962e7	−0.435372
$151$	−1.74324e7	−0.412039	−0.206019	−	0.978548i	$-0.566051\pi$
$151$	−1.74324e7	−0.412039	−0.206019	+	0.978548i	$0.566051\pi$
$152$	−1.16992e7	−0.270211
$153$	−4.86706e7	−1.09862
$154$	−5.01255e6	−0.110595
$155$	27020.7	0.000582822 0
$156$	0	0
$157$	8.56670e7	1.76671	0.883354	−	0.468706i	$-0.155280\pi$
$157$	8.56670e7	1.76671	0.883354	+	0.468706i	$0.155280\pi$
$158$	3.97772e7	0.802295
$159$	−6.86265e6	−0.135395
$160$	154268.	0.00297753
$161$	2.40230e7	0.453667
$162$	−5.97871e7	−1.10485
$163$	−4.30758e7	−0.779070	−0.389535	−	0.921012i	$-0.627364\pi$
$163$	−4.30758e7	−0.779070	−0.389535	+	0.921012i	$0.627364\pi$
$164$	−9.45655e7	−1.67409
$165$	8771.63	0.000152015 0
$166$	−2.96532e7	−0.503145
$167$	−6.57708e7	−1.09276	−0.546381	−	0.837537i	$-0.683995\pi$
$167$	−6.57708e7	−1.09276	−0.546381	+	0.837537i	$0.683995\pi$
$168$	−1.50132e6	−0.0244282
$169$	0	0
$170$	−237628.	−0.00370960
$171$	−5.88044e7	−0.899340
$172$	−1.14050e8	−1.70902
$173$	−1.02460e8	−1.50451	−0.752255	−	0.658872i	$-0.771034\pi$
$173$	−1.02460e8	−1.50451	−0.752255	+	0.658872i	$0.771034\pi$
$174$	−2.26999e7	−0.326665
$175$	2.14322e7	0.302297
$176$	1.39596e7	0.193009
$177$	−1.53212e7	−0.207675
$178$	4.55797e7	0.605761
$179$	6.44115e7	0.839417	0.419708	−	0.907659i	$-0.362132\pi$
$179$	6.44115e7	0.839417	0.419708	+	0.907659i	$0.362132\pi$
$180$	−176722.	−0.00225859
$181$	9.36975e7	1.17450	0.587250	−	0.809406i	$-0.300211\pi$
$181$	9.36975e7	1.17450	0.587250	+	0.809406i	$0.300211\pi$
$182$	0	0
$183$	8.60594e6	0.103805
$184$	3.47971e7	0.411795
$185$	126595.	0.00147000
$186$	1.06758e7	0.121648
$187$	−2.66199e7	−0.297687
$188$	−1.75630e8	−1.92773
$189$	−1.58090e7	−0.170329
$190$	−287106.	−0.00303672
$191$	1.03934e8	1.07929	0.539647	−	0.841891i	$-0.318558\pi$
$191$	1.03934e8	1.07929	0.539647	+	0.841891i	$0.318558\pi$
$192$	3.84241e7	0.391786
$193$	−4.66141e6	−0.0466731	−0.0233365	−	0.999728i	$-0.507429\pi$
$193$	−4.66141e6	−0.0466731	−0.0233365	+	0.999728i	$0.507429\pi$
$194$	1.40263e7	0.137923
$195$	0	0
$196$	−1.13558e8	−1.07726
$197$	1.01127e8	0.942402	0.471201	−	0.882026i	$-0.343821\pi$
$197$	1.01127e8	0.942402	0.471201	+	0.882026i	$0.343821\pi$
$198$	−3.64947e7	−0.334119
$199$	−1.29148e8	−1.16172	−0.580860	−	0.814003i	$-0.697284\pi$
$199$	−1.29148e8	−1.16172	−0.580860	+	0.814003i	$0.697284\pi$
$200$	3.10444e7	0.274396
$201$	397301.	0.00345091
$202$	−2.25807e8	−1.92756
$203$	2.70340e7	0.226817
$204$	−5.09294e7	−0.420013
$205$	−363306.	−0.00294533
$206$	−2.17935e8	−1.73696
$207$	1.74903e8	1.37057
$208$	0	0
$209$	−3.21625e7	−0.243690
$210$	−36843.5	−0.000274532 0
$211$	1.43953e8	1.05495	0.527475	−	0.849571i	$-0.323139\pi$
$211$	1.43953e8	1.05495	0.527475	+	0.849571i	$0.323139\pi$
$212$	7.56208e7	0.545087
$213$	−5.03663e7	−0.357118
$214$	−3.42616e8	−2.38979
$215$	−438163.	−0.00300678
$216$	−2.28993e7	−0.154608
$217$	−1.27141e7	−0.0844649
$218$	2.92481e8	1.91206
$219$	−7.16977e7	−0.461265
$220$	−96656.2	−0.000611998 0
$221$	0	0
$222$	5.00171e7	0.306820
$223$	9.50963e7	0.574244	0.287122	−	0.957894i	$-0.407301\pi$
$223$	9.50963e7	0.574244	0.287122	+	0.957894i	$0.407301\pi$
$224$	−7.25879e7	−0.431515
$225$	1.56041e8	0.913269
$226$	−1.12505e8	−0.648323
$227$	2.36462e8	1.34175	0.670874	−	0.741571i	$-0.265919\pi$
$227$	2.36462e8	1.34175	0.670874	+	0.741571i	$0.265919\pi$
$228$	−6.15336e7	−0.343827
$229$	2.40501e8	1.32340	0.661702	−	0.749767i	$-0.269834\pi$
$229$	2.40501e8	1.32340	0.661702	+	0.749767i	$0.269834\pi$
$230$	853946.	0.00462789
$231$	−4.12732e6	−0.0220306
$232$	3.91586e7	0.205882
$233$	2.23639e8	1.15825	0.579123	−	0.815240i	$-0.303395\pi$
$233$	2.23639e8	1.15825	0.579123	+	0.815240i	$0.303395\pi$
$234$	0	0
$235$	−674743.	−0.00339157
$236$	1.68827e8	0.836082
$237$	3.27525e7	0.159818
$238$	1.11812e8	0.537610
$239$	2.60257e8	1.23313	0.616567	−	0.787302i	$-0.288523\pi$
$239$	2.60257e8	1.23313	0.616567	+	0.787302i	$0.288523\pi$
$240$	102606.	0.000479110 0
$241$	−2.92081e8	−1.34414	−0.672069	−	0.740489i	$-0.734594\pi$
$241$	−2.92081e8	−1.34414	−0.672069	+	0.740489i	$0.734594\pi$
$242$	3.05981e8	1.38785
$243$	−1.75259e8	−0.783535
$244$	−9.48305e7	−0.417911
$245$	−436272.	−0.00189529
$246$	−1.43540e8	−0.614754
$247$	0	0
$248$	−1.84162e7	−0.0766691
$249$	−2.44164e7	−0.100227
$250$	1.52370e6	0.00616752
$251$	1.00598e8	0.401542	0.200771	−	0.979638i	$-0.435655\pi$
$251$	1.00598e8	0.401542	0.200771	+	0.979638i	$0.435655\pi$
$252$	8.31532e7	0.327323
$253$	9.56616e7	0.371378
$254$	5.19644e8	1.98970
$255$	−195663.	−0.000738955 0
$256$	1.43081e8	0.533018
$257$	−1.67536e7	−0.0615662	−0.0307831	−	0.999526i	$-0.509800\pi$
$257$	−1.67536e7	−0.0615662	−0.0307831	+	0.999526i	$0.509800\pi$
$258$	−1.73116e8	−0.627580
$259$	−5.95669e7	−0.213038
$260$	0	0
$261$	1.96826e8	0.685236
$262$	−2.59917e8	−0.892853
$263$	−2.50229e7	−0.0848187	−0.0424094	−	0.999100i	$-0.513503\pi$
$263$	−2.50229e7	−0.0848187	−0.0424094	+	0.999100i	$0.513503\pi$
$264$	−5.97839e6	−0.0199973
$265$	290523.	0.000959004 0
$266$	1.35092e8	0.440093
$267$	3.75302e7	0.120668
$268$	−4.37793e6	−0.0138930
$269$	2.47248e7	0.0774462	0.0387231	−	0.999250i	$-0.487671\pi$
$269$	2.47248e7	0.0774462	0.0387231	+	0.999250i	$0.487671\pi$
$270$	−561964.	−0.00173754
$271$	5.81639e8	1.77526	0.887628	−	0.460561i	$-0.152352\pi$
$271$	5.81639e8	1.77526	0.887628	+	0.460561i	$0.152352\pi$
$272$	−3.11387e8	−0.938229
$273$	0	0
$274$	−2.93738e8	−0.862647
$275$	8.53448e7	0.247464
$276$	1.83021e8	0.523985
$277$	−2.85853e8	−0.808097	−0.404048	−	0.914738i	$-0.632397\pi$
$277$	−2.85853e8	−0.808097	−0.404048	+	0.914738i	$0.632397\pi$
$278$	−3.88926e8	−1.08570
$279$	−9.25670e7	−0.255177
$280$	63557.0	0.000173026 0
$281$	−5.88850e8	−1.58319	−0.791593	−	0.611048i	$-0.790748\pi$
$281$	−5.88850e8	−1.58319	−0.791593	+	0.611048i	$0.790748\pi$
$282$	−2.66588e8	−0.707894
$283$	−4.95887e8	−1.30056	−0.650279	−	0.759695i	$-0.725348\pi$
$283$	−4.95887e8	−1.30056	−0.650279	+	0.759695i	$0.725348\pi$
$284$	5.54995e8	1.43772
$285$	−236403.	−0.000604916 0
$286$	0	0
$287$	1.70947e8	0.426849
$288$	−5.28488e8	−1.30365
$289$	1.83453e8	0.447077
$290$	960979.	0.00231377
$291$	1.15492e7	0.0274744
$292$	7.90050e8	1.85701
$293$	5.41027e8	1.25656	0.628279	−	0.777988i	$-0.283760\pi$
$293$	5.41027e8	1.25656	0.628279	+	0.777988i	$0.283760\pi$
$294$	−1.72369e8	−0.395588
$295$	648606.	0.00147097
$296$	−8.62822e7	−0.193375
$297$	−6.29529e7	−0.139434
$298$	−1.22290e9	−2.67690
$299$	0	0
$300$	1.63283e8	0.349153
$301$	2.06169e8	0.435754
$302$	2.91574e8	0.609149
$303$	−1.85929e8	−0.383971
$304$	−3.76222e8	−0.768044
$305$	−364324.	−0.000735255 0
$306$	8.14062e8	1.62417
$307$	−4.24834e8	−0.837982	−0.418991	−	0.907990i	$-0.637616\pi$
$307$	−4.24834e8	−0.837982	−0.418991	+	0.907990i	$0.637616\pi$
$308$	4.54797e7	0.0886932
$309$	−1.79447e8	−0.346005
$310$	−451947.	−0.000861632 0
$311$	2.89109e8	0.545005	0.272503	−	0.962155i	$-0.412149\pi$
$311$	2.89109e8	0.545005	0.272503	+	0.962155i	$0.412149\pi$
$312$	0	0
$313$	−9.46652e8	−1.74496	−0.872479	−	0.488651i	$-0.837489\pi$
$313$	−9.46652e8	−1.74496	−0.872479	+	0.488651i	$0.837489\pi$
$314$	−1.43286e9	−2.61187
$315$	319461.	0.000575879 0
$316$	−3.60906e8	−0.643412
$317$	4.18587e8	0.738038	0.369019	−	0.929422i	$-0.379694\pi$
$317$	4.18587e8	0.738038	0.369019	+	0.929422i	$0.379694\pi$
$318$	1.14784e8	0.200165
$319$	1.07652e8	0.185675
$320$	−1.62664e6	−0.00277503
$321$	−2.82110e8	−0.476048
$322$	−4.01808e8	−0.670692
$323$	7.17427e8	1.18459
$324$	5.42459e8	0.886052
$325$	0	0
$326$	7.20483e8	1.15176
$327$	2.40829e8	0.380883
$328$	2.47615e8	0.387452
$329$	3.17488e8	0.491520
$330$	−146714.	−0.000224736 0
$331$	4.12347e8	0.624978	0.312489	−	0.949921i	$-0.398837\pi$
$331$	4.12347e8	0.624978	0.312489	+	0.949921i	$0.398837\pi$
$332$	2.69049e8	0.403504
$333$	−4.33687e8	−0.643608
$334$	1.10008e9	1.61552
$335$	−16819.3	−2.44429e−5 0
$336$	−4.82795e7	−0.0694345
$337$	−5.74808e8	−0.818123	−0.409061	−	0.912507i	$-0.634144\pi$
$337$	−5.74808e8	−0.818123	−0.409061	+	0.912507i	$0.634144\pi$
$338$	0	0
$339$	−9.26363e7	−0.129146
$340$	2.15605e6	0.00297496
$341$	−5.06285e7	−0.0691441
$342$	9.83561e8	1.32957
$343$	4.31205e8	0.576971
$344$	2.98635e8	0.395536
$345$	703138.	0.000921878 0
$346$	1.71375e9	2.22424
$347$	−1.46161e9	−1.87792	−0.938962	−	0.344020i	$-0.888211\pi$
$347$	−1.46161e9	−1.87792	−0.938962	+	0.344020i	$0.888211\pi$
$348$	2.05961e8	0.261973
$349$	4.00590e8	0.504441	0.252221	−	0.967670i	$-0.418839\pi$
$349$	4.00590e8	0.504441	0.252221	+	0.967670i	$0.418839\pi$
$350$	−3.58474e8	−0.446910
$351$	0	0
$352$	−2.89051e8	−0.353244
$353$	5.97796e8	0.723338	0.361669	−	0.932307i	$-0.382207\pi$
$353$	5.97796e8	0.723338	0.361669	+	0.932307i	$0.382207\pi$
$354$	2.56261e8	0.307023
$355$	2.13220e6	0.00252947
$356$	−4.13553e8	−0.485798
$357$	9.20655e7	0.107092
$358$	−1.07734e9	−1.24098
$359$	6.48576e8	0.739827	0.369913	−	0.929066i	$-0.379387\pi$
$359$	6.48576e8	0.739827	0.369913	+	0.929066i	$0.379387\pi$
$360$	462737.	0.000522728 0
$361$	−2.70670e7	−0.0302806
$362$	−1.56718e9	−1.73636
$363$	2.51945e8	0.276460
$364$	0	0
$365$	3.03525e6	0.00326715
$366$	−1.43943e8	−0.153464
$367$	6.24121e8	0.659080	0.329540	−	0.944142i	$-0.393106\pi$
$367$	6.24121e8	0.659080	0.329540	+	0.944142i	$0.393106\pi$
$368$	1.11901e9	1.17048
$369$	1.24461e9	1.28955
$370$	−2.11742e6	−0.00217321
$371$	−1.36700e8	−0.138983
$372$	−9.68631e7	−0.0975569
$373$	−1.14420e8	−0.114162	−0.0570810	−	0.998370i	$-0.518179\pi$
$373$	−1.14420e8	−0.114162	−0.0570810	+	0.998370i	$0.518179\pi$
$374$	4.45243e8	0.440095
$375$	1.25462e6	0.00122857
$376$	4.59878e8	0.446154
$377$	0	0
$378$	2.64421e8	0.251811
$379$	1.79026e9	1.68919	0.844594	−	0.535407i	$-0.179842\pi$
$379$	1.79026e9	1.68919	0.844594	+	0.535407i	$0.179842\pi$
$380$	2.60496e6	0.00243534
$381$	4.27874e8	0.396350
$382$	−1.73839e9	−1.59561
$383$	−1.71291e9	−1.55789	−0.778947	−	0.627090i	$-0.784246\pi$
$383$	−1.71291e9	−1.55789	−0.778947	+	0.627090i	$0.784246\pi$
$384$	−1.76238e8	−0.158833
$385$	174726.	0.000156043 0
$386$	7.79665e7	0.0690005
$387$	1.50105e9	1.31646
$388$	−1.27263e8	−0.110609
$389$	1.68291e9	1.44957	0.724783	−	0.688978i	$-0.241940\pi$
$389$	1.68291e9	1.44957	0.724783	+	0.688978i	$0.241940\pi$
$390$	0	0
$391$	−2.13386e9	−1.80529
$392$	2.97345e8	0.249322
$393$	−2.14015e8	−0.177857
$394$	−1.69145e9	−1.39323
$395$	−1.38654e6	−0.00113199
$396$	3.31123e8	0.267951
$397$	−2.12886e9	−1.70758	−0.853789	−	0.520619i	$-0.825701\pi$
$397$	−2.12886e9	−1.70758	−0.853789	+	0.520619i	$0.825701\pi$
$398$	2.16012e9	1.71746
$399$	1.11235e8	0.0876668
$400$	9.98323e8	0.779940
$401$	9.05120e8	0.700972	0.350486	−	0.936568i	$-0.386016\pi$
$401$	9.05120e8	0.700972	0.350486	+	0.936568i	$0.386016\pi$
$402$	−6.64524e6	−0.00510175
$403$	0	0
$404$	2.04879e9	1.54583
$405$	2.08404e6	0.00155889
$406$	−4.52170e8	−0.335321
$407$	−2.37200e8	−0.174395
$408$	1.33356e8	0.0972080
$409$	−1.60956e9	−1.16326	−0.581629	−	0.813454i	$-0.697585\pi$
$409$	−1.60956e9	−1.16326	−0.581629	+	0.813454i	$0.697585\pi$
$410$	6.07664e6	0.00435432
$411$	−2.41863e8	−0.171840
$412$	1.97736e9	1.39298
$413$	−3.05189e8	−0.213179
$414$	−2.92543e9	−2.02623
$415$	1.03364e6	0.000709908 0
$416$	0	0
$417$	−3.20241e8	−0.216273
$418$	5.37948e8	0.360266
$419$	−2.87655e8	−0.191039	−0.0955196	−	0.995428i	$-0.530451\pi$
$419$	−2.87655e8	−0.191039	−0.0955196	+	0.995428i	$0.530451\pi$
$420$	334288.	0.000220165 0
$421$	1.52458e7	0.00995781	0.00497891	−	0.999988i	$-0.498415\pi$
$421$	1.52458e7	0.00995781	0.00497891	+	0.999988i	$0.498415\pi$
$422$	−2.40775e9	−1.55962
$423$	2.31152e9	1.48493
$424$	−1.98009e8	−0.126155
$425$	−1.90373e9	−1.20294
$426$	8.42424e8	0.527955
$427$	1.71426e8	0.106556
$428$	3.10862e9	1.91652
$429$	0	0
$430$	7.32870e6	0.00444516
$431$	−9.22436e8	−0.554965	−0.277483	−	0.960731i	$-0.589500\pi$
$431$	−9.22436e8	−0.554965	−0.277483	+	0.960731i	$0.589500\pi$
$432$	−7.36394e8	−0.439458
$433$	−1.88411e9	−1.11532	−0.557660	−	0.830069i	$-0.688301\pi$
$433$	−1.88411e9	−1.11532	−0.557660	+	0.830069i	$0.688301\pi$
$434$	2.12655e8	0.124871
$435$	791269.	0.000460905 0
$436$	−2.65374e9	−1.53340
$437$	−2.57816e9	−1.47783
$438$	1.19921e9	0.681925
$439$	−8.65096e8	−0.488021	−0.244010	−	0.969773i	$-0.578463\pi$
$439$	−8.65096e8	−0.488021	−0.244010	+	0.969773i	$0.578463\pi$
$440$	253089.	0.000141641 0
$441$	1.49457e9	0.829814
$442$	0	0
$443$	9.40807e8	0.514147	0.257074	−	0.966392i	$-0.417242\pi$
$443$	9.40807e8	0.514147	0.257074	+	0.966392i	$0.417242\pi$
$444$	−4.53815e8	−0.246058
$445$	−1.58880e6	−0.000854694 0
$446$	−1.59058e9	−0.848951
$447$	−1.00693e9	−0.533241
$448$	7.65386e8	0.402168
$449$	−8.79790e7	−0.0458687	−0.0229344	−	0.999737i	$-0.507301\pi$
$449$	−8.79790e7	−0.0458687	−0.0229344	+	0.999737i	$0.507301\pi$
$450$	−2.60993e9	−1.35016
$451$	6.80724e8	0.349424
$452$	1.02078e9	0.519931
$453$	2.40081e8	0.121343
$454$	−3.95506e9	−1.98361
$455$	0	0
$456$	1.61123e8	0.0795755
$457$	1.95139e9	0.956396	0.478198	−	0.878252i	$-0.341290\pi$
$457$	1.95139e9	0.956396	0.478198	+	0.878252i	$0.341290\pi$
$458$	−4.02261e9	−1.95649
$459$	1.40425e9	0.677797
$460$	−7.74801e6	−0.00371140
$461$	−1.15818e8	−0.0550585	−0.0275293	−	0.999621i	$-0.508764\pi$
$461$	−1.15818e8	−0.0550585	−0.0275293	+	0.999621i	$0.508764\pi$
$462$	6.90335e7	0.0325696
$463$	−1.46488e9	−0.685914	−0.342957	−	0.939351i	$-0.611429\pi$
$463$	−1.46488e9	−0.685914	−0.342957	+	0.939351i	$0.611429\pi$
$464$	1.25926e9	0.585198
$465$	−372133.	−0.000171638 0
$466$	−3.74057e9	−1.71233
$467$	−6.23541e8	−0.283306	−0.141653	−	0.989916i	$-0.545242\pi$
$467$	−6.23541e8	−0.283306	−0.141653	+	0.989916i	$0.545242\pi$
$468$	0	0
$469$	7.91402e6	0.00354236
$470$	1.12857e7	0.00501403
$471$	−1.17982e9	−0.520286
$472$	−4.42064e8	−0.193503
$473$	8.20984e8	0.356714
$474$	−5.47817e8	−0.236271
$475$	−2.30011e9	−0.984739
$476$	−1.01449e9	−0.431144
$477$	−9.95268e8	−0.419880
$478$	−4.35305e9	−1.82304
$479$	2.10997e9	0.877207	0.438604	−	0.898681i	$-0.355473\pi$
$479$	2.10997e9	0.877207	0.438604	+	0.898681i	$0.355473\pi$
$480$	−2.12460e6	−0.000876865 0
$481$	0	0
$482$	4.88534e9	1.98715
$483$	−3.30848e8	−0.133602
$484$	−2.77622e9	−1.11300
$485$	−488925.	−0.000194602 0
$486$	2.93138e9	1.15836
$487$	3.08900e9	1.21190	0.605949	−	0.795503i	$-0.292793\pi$
$487$	3.08900e9	1.21190	0.605949	+	0.795503i	$0.292793\pi$
$488$	2.48309e8	0.0967214
$489$	5.93245e8	0.229431
$490$	7.29706e6	0.00280196
$491$	−3.63222e9	−1.38480	−0.692399	−	0.721515i	$-0.743446\pi$
$491$	−3.63222e9	−1.38480	−0.692399	+	0.721515i	$0.743446\pi$
$492$	1.30237e9	0.493010
$493$	−2.40132e9	−0.902579
$494$	0	0
$495$	1.27212e6	0.000471423 0
$496$	−5.92229e8	−0.217923
$497$	−1.00327e9	−0.366581
$498$	4.08388e8	0.148173
$499$	4.71935e9	1.70032	0.850160	−	0.526525i	$-0.176505\pi$
$499$	4.71935e9	1.70032	0.850160	+	0.526525i	$0.176505\pi$
$500$	−1.38248e7	−0.00494612
$501$	9.05805e8	0.321812
$502$	−1.68260e9	−0.593632
$503$	7.89607e8	0.276645	0.138323	−	0.990387i	$-0.455829\pi$
$503$	7.89607e8	0.276645	0.138323	+	0.990387i	$0.455829\pi$
$504$	−2.17732e8	−0.0757558
$505$	7.87113e6	0.00271968
$506$	−1.60003e9	−0.549037
$507$	0	0
$508$	−4.71483e9	−1.59567
$509$	9.69489e7	0.0325860	0.0162930	−	0.999867i	$-0.494814\pi$
$509$	9.69489e7	0.0325860	0.0162930	+	0.999867i	$0.494814\pi$
$510$	3.27265e6	0.00109246
$511$	−1.42818e9	−0.473489
$512$	−4.03115e9	−1.32734
$513$	1.69663e9	0.554852
$514$	2.80220e8	0.0910182
$515$	7.59671e6	0.00245076
$516$	1.57072e9	0.503296
$517$	1.26426e9	0.402365
$518$	9.96314e8	0.314950
$519$	1.41110e9	0.443069
$520$	0	0
$521$	4.10985e9	1.27319	0.636596	−	0.771197i	$-0.280342\pi$
$521$	4.10985e9	1.27319	0.636596	+	0.771197i	$0.280342\pi$
$522$	−3.29210e9	−1.01304
$523$	2.38880e8	0.0730169	0.0365084	−	0.999333i	$-0.488376\pi$
$523$	2.38880e8	0.0730169	0.0365084	+	0.999333i	$0.488376\pi$
$524$	2.35827e9	0.716036
$525$	−2.95167e8	−0.0890247
$526$	4.18531e8	0.125394
$527$	1.12934e9	0.336114
$528$	−1.92253e8	−0.0568400
$529$	4.26345e9	1.25218
$530$	−4.85928e6	−0.00141777
$531$	−2.22198e9	−0.644034
$532$	−1.22572e9	−0.352939
$533$	0	0
$534$	−6.27729e8	−0.178393
$535$	1.19428e7	0.00337186
$536$	1.14634e7	0.00321541
$537$	−8.87083e8	−0.247203
$538$	−4.13546e8	−0.114495
$539$	8.17439e8	0.224851
$540$	5.09880e6	0.00139344
$541$	5.59019e8	0.151788	0.0758938	−	0.997116i	$-0.475819\pi$
$541$	5.59019e8	0.151788	0.0758938	+	0.997116i	$0.475819\pi$
$542$	−9.72847e9	−2.62450
$543$	−1.29041e9	−0.345883
$544$	6.44767e9	1.71714
$545$	−1.01952e7	−0.00269780
$546$	0	0
$547$	−7.01923e8	−0.183372	−0.0916861	−	0.995788i	$-0.529226\pi$
$547$	−7.01923e8	−0.183372	−0.0916861	+	0.995788i	$0.529226\pi$
$548$	2.66514e9	0.691812
$549$	1.24809e9	0.321917
$550$	−1.42747e9	−0.365846
$551$	−2.90130e9	−0.738860
$552$	−4.79231e8	−0.121271
$553$	6.52412e8	0.164053
$554$	4.78116e9	1.19467
$555$	−1.74348e6	−0.000432905 0
$556$	3.52880e9	0.870693
$557$	−3.51904e9	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$557$	−3.51904e9	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$558$	1.54827e9	0.377248
$559$	0	0
$560$	2.04386e6	0.000491806 0
$561$	3.66612e8	0.0876672
$562$	9.84907e9	2.34055
$563$	−3.27161e8	−0.0772648	−0.0386324	−	0.999253i	$-0.512300\pi$
$563$	−3.27161e8	−0.0772648	−0.0386324	+	0.999253i	$0.512300\pi$
$564$	2.41880e9	0.567705
$565$	3.92166e6	0.000914746 0
$566$	8.29418e9	1.92272
$567$	−9.80607e8	−0.225920
$568$	−1.45323e9	−0.332747
$569$	5.92681e9	1.34874	0.674369	−	0.738394i	$-0.264416\pi$
$569$	5.92681e9	1.34874	0.674369	+	0.738394i	$0.264416\pi$
$570$	3.95406e6	0.000894296 0
$571$	3.50991e9	0.788988	0.394494	−	0.918899i	$-0.370920\pi$
$571$	3.50991e9	0.788988	0.394494	+	0.918899i	$0.370920\pi$
$572$	0	0
$573$	−1.43139e9	−0.317846
$574$	−2.85925e9	−0.631045
$575$	6.84128e9	1.50072
$576$	5.57252e9	1.21499
$577$	−9.14479e9	−1.98179	−0.990897	−	0.134621i	$-0.957018\pi$
$577$	−9.14479e9	−1.98179	−0.990897	+	0.134621i	$0.957018\pi$
$578$	−3.06843e9	−0.660950
$579$	6.41975e7	0.0137450
$580$	−8.71913e6	−0.00185556
$581$	−4.86361e8	−0.102883
$582$	−1.93172e8	−0.0406176
$583$	−5.44351e8	−0.113773
$584$	−2.06871e9	−0.429787
$585$	0	0
$586$	−9.04920e9	−1.85767
$587$	−1.65538e9	−0.337804	−0.168902	−	0.985633i	$-0.554022\pi$
$587$	−1.65538e9	−0.337804	−0.168902	+	0.985633i	$0.554022\pi$
$588$	1.56393e9	0.317247
$589$	1.36448e9	0.275146
$590$	−1.08486e7	−0.00217465
$591$	−1.39274e9	−0.277532
$592$	−2.77466e9	−0.549647
$593$	7.99836e9	1.57510	0.787552	−	0.616248i	$-0.211348\pi$
$593$	7.99836e9	1.57510	0.787552	+	0.616248i	$0.211348\pi$
$594$	1.05295e9	0.206136
$595$	−3.89750e6	−0.000758537 0
$596$	1.10956e10	2.14678
$597$	1.77864e9	0.342120
$598$	0	0
$599$	−8.80866e8	−0.167462	−0.0837309	−	0.996488i	$-0.526684\pi$
$599$	−8.80866e8	−0.167462	−0.0837309	+	0.996488i	$0.526684\pi$
$600$	−4.27547e8	−0.0808080
$601$	2.97804e9	0.559589	0.279795	−	0.960060i	$-0.409734\pi$
$601$	2.97804e9	0.559589	0.279795	+	0.960060i	$0.409734\pi$
$602$	−3.44838e9	−0.644210
$603$	5.76193e7	0.0107018
$604$	−2.64550e9	−0.488516
$605$	−1.06658e7	−0.00195817
$606$	3.10985e9	0.567655
$607$	−3.31888e8	−0.0602325	−0.0301163	−	0.999546i	$-0.509588\pi$
$607$	−3.31888e8	−0.0602325	−0.0301163	+	0.999546i	$0.509588\pi$
$608$	7.79016e9	1.40567
$609$	−3.72316e8	−0.0667962
$610$	6.09367e6	0.00108699
$611$	0	0
$612$	−7.38613e9	−1.30253
$613$	−3.93427e9	−0.689846	−0.344923	−	0.938631i	$-0.612095\pi$
$613$	−3.93427e9	−0.689846	−0.344923	+	0.938631i	$0.612095\pi$
$614$	7.10575e9	1.23886
$615$	5.00350e6	0.000867383 0
$616$	−1.19086e8	−0.0205272
$617$	7.24044e9	1.24099	0.620493	−	0.784212i	$-0.286933\pi$
$617$	7.24044e9	1.24099	0.620493	+	0.784212i	$0.286933\pi$
$618$	3.00142e9	0.511526
$619$	1.16885e10	1.98080	0.990401	−	0.138220i	$-0.0441383\pi$
$619$	1.16885e10	1.98080	0.990401	+	0.138220i	$0.0441383\pi$
$620$	4.10060e6	0.000690998 0
$621$	−5.04633e9	−0.845581
$622$	−4.83563e9	−0.805725
$623$	7.47582e8	0.123866
$624$	0	0
$625$	6.10344e9	0.999987
$626$	1.58337e10	2.57971
$627$	4.42946e8	0.0717652
$628$	1.30006e10	2.09462
$629$	5.29107e9	0.847747
$630$	−5.34330e6	−0.000851368 0
$631$	−6.38796e8	−0.101218	−0.0506092	−	0.998719i	$-0.516116\pi$
$631$	−6.38796e8	−0.101218	−0.0506092	+	0.998719i	$0.516116\pi$
$632$	9.45013e8	0.148911
$633$	−1.98254e9	−0.310677
$634$	−7.00127e9	−1.09110
$635$	−1.81136e7	−0.00280736
$636$	−1.04146e9	−0.160525
$637$	0	0
$638$	−1.80058e9	−0.274498
$639$	−7.30446e9	−1.10748
$640$	7.46086e6	0.00112502
$641$	6.15856e8	0.0923584	0.0461792	−	0.998933i	$-0.485295\pi$
$641$	6.15856e8	0.0923584	0.0461792	+	0.998933i	$0.485295\pi$
$642$	4.71856e9	0.703779
$643$	4.86092e9	0.721075	0.360537	−	0.932745i	$-0.382593\pi$
$643$	4.86092e9	0.721075	0.360537	+	0.932745i	$0.382593\pi$
$644$	3.64568e9	0.537870
$645$	6.03444e6	0.000885479 0
$646$	−1.19997e10	−1.75128
$647$	7.67484e9	1.11405	0.557025	−	0.830496i	$-0.311943\pi$
$647$	7.67484e9	1.11405	0.557025	+	0.830496i	$0.311943\pi$
$648$	−1.42040e9	−0.205068
$649$	−1.21529e9	−0.174511
$650$	0	0
$651$	1.75100e8	0.0248744
$652$	−6.53707e9	−0.923670
$653$	−5.86264e9	−0.823942	−0.411971	−	0.911197i	$-0.635160\pi$
$653$	−5.86264e9	−0.823942	−0.411971	+	0.911197i	$0.635160\pi$
$654$	−4.02809e9	−0.563089
$655$	9.06012e6	0.00125976
$656$	7.96279e9	1.10129
$657$	−1.03981e10	−1.43046
$658$	−5.31028e9	−0.726653
$659$	1.17798e10	1.60338	0.801692	−	0.597737i	$-0.203933\pi$
$659$	1.17798e10	1.60338	0.801692	+	0.597737i	$0.203933\pi$
$660$	1.33116e6	0.000180230 0
$661$	−3.65943e9	−0.492843	−0.246421	−	0.969163i	$-0.579255\pi$
$661$	−3.65943e9	−0.492843	−0.246421	+	0.969163i	$0.579255\pi$
$662$	−6.89690e9	−0.923955
$663$	0	0
$664$	−7.04490e8	−0.0933870
$665$	−4.70901e6	−0.000620946 0
$666$	7.25383e9	0.951497
$667$	8.62942e9	1.12601
$668$	−9.98123e9	−1.29559
$669$	−1.30968e9	−0.169112
$670$	281319.	3.61358e−5 0
$671$	6.82631e8	0.0872283
$672$	9.99691e8	0.127079
$673$	−8.09730e9	−1.02397	−0.511985	−	0.858994i	$-0.671090\pi$
$673$	−8.09730e9	−1.02397	−0.511985	+	0.858994i	$0.671090\pi$
$674$	9.61422e9	1.20950
$675$	−4.50210e9	−0.563445
$676$	0	0
$677$	6.04454e9	0.748691	0.374346	−	0.927289i	$-0.377867\pi$
$677$	6.04454e9	0.748691	0.374346	+	0.927289i	$0.377867\pi$
$678$	1.54943e9	0.190927
$679$	2.30055e8	0.0282024
$680$	−5.64549e6	−0.000688527 0
$681$	−3.25659e9	−0.395137
$682$	8.46811e8	0.102221
$683$	6.73839e9	0.809252	0.404626	−	0.914482i	$-0.367402\pi$
$683$	6.73839e9	0.809252	0.404626	+	0.914482i	$0.367402\pi$
$684$	−8.92402e9	−1.06626
$685$	1.02390e7	0.00121715
$686$	−7.21231e9	−0.852982
$687$	−3.31221e9	−0.389735
$688$	9.60348e9	1.12427
$689$	0	0
$690$	−1.17607e7	−0.00136289
$691$	−3.86203e9	−0.445289	−0.222645	−	0.974900i	$-0.571469\pi$
$691$	−3.86203e9	−0.445289	−0.222645	+	0.974900i	$0.571469\pi$
$692$	−1.55492e10	−1.78376
$693$	−5.98573e8	−0.0683204
$694$	2.44468e10	2.77629
$695$	1.35571e7	0.00153186
$696$	−5.39297e8	−0.0606311
$697$	−1.51845e10	−1.69857
$698$	−6.70024e9	−0.745756
$699$	−3.07998e9	−0.341097
$700$	3.25250e9	0.358405
$701$	−6.81601e9	−0.747338	−0.373669	−	0.927562i	$-0.621900\pi$
$701$	−6.81601e9	−0.747338	−0.373669	+	0.927562i	$0.621900\pi$
$702$	0	0
$703$	6.39274e9	0.693975
$704$	3.04783e9	0.329220
$705$	9.29265e6	0.000998798 0
$706$	−9.99871e9	−1.06937
$707$	−3.70361e9	−0.394146
$708$	−2.32510e9	−0.246221
$709$	7.39468e9	0.779215	0.389608	−	0.920981i	$-0.372611\pi$
$709$	7.39468e9	0.779215	0.389608	+	0.920981i	$0.372611\pi$
$710$	−3.56632e7	−0.00373952
$711$	4.74999e9	0.495620
$712$	1.08287e9	0.112433
$713$	−4.05841e9	−0.419317
$714$	−1.53988e9	−0.158323
$715$	0	0
$716$	9.77493e9	0.995218
$717$	−3.58430e9	−0.363151
$718$	−1.08481e10	−1.09375
$719$	−1.85081e10	−1.85700	−0.928500	−	0.371333i	$-0.878901\pi$
$719$	−1.85081e10	−1.85700	−0.928500	+	0.371333i	$0.878901\pi$
$720$	1.48807e7	0.00148580
$721$	−3.57449e9	−0.355173
$722$	4.52721e8	0.0447663
$723$	4.02258e9	0.395841
$724$	1.42193e10	1.39249
$725$	7.69876e9	0.750304
$726$	−4.21401e9	−0.408712
$727$	−4.04178e9	−0.390124	−0.195062	−	0.980791i	$-0.562491\pi$
$727$	−4.04178e9	−0.390124	−0.195062	+	0.980791i	$0.562491\pi$
$728$	0	0
$729$	−5.40376e9	−0.516594
$730$	−5.07675e7	−0.00483009
$731$	−1.83131e10	−1.73401
$732$	1.30602e9	0.123072
$733$	−1.54392e10	−1.44797	−0.723985	−	0.689816i	$-0.757692\pi$
$733$	−1.54392e10	−1.44797	−0.723985	+	0.689816i	$0.757692\pi$
$734$	−1.04390e10	−0.974370
$735$	6.00839e6	0.000558152 0
$736$	−2.31705e10	−2.14221
$737$	3.15143e7	0.00289982
$738$	−2.08172e10	−1.90645
$739$	6.64104e8	0.0605313	0.0302657	−	0.999542i	$-0.490365\pi$
$739$	6.64104e8	0.0605313	0.0302657	+	0.999542i	$0.490365\pi$
$740$	1.92118e7	0.00174284
$741$	0	0
$742$	2.28644e9	0.205469
$743$	1.54094e9	0.137824	0.0689118	−	0.997623i	$-0.478047\pi$
$743$	1.54094e9	0.137824	0.0689118	+	0.997623i	$0.478047\pi$
$744$	2.53631e8	0.0225786
$745$	4.26274e7	0.00377695
$746$	1.91379e9	0.168775
$747$	−3.54103e9	−0.310819
$748$	−4.03977e9	−0.352940
$749$	−5.61947e9	−0.488663
$750$	−2.09846e7	−0.00181630
$751$	3.60098e9	0.310228	0.155114	−	0.987897i	$-0.450426\pi$
$751$	3.60098e9	0.310228	0.155114	+	0.987897i	$0.450426\pi$
$752$	1.47887e10	1.26814
$753$	−1.38545e9	−0.118252
$754$	0	0
$755$	−1.01636e7	−0.000859475 0
$756$	−2.39914e9	−0.201944
$757$	3.39784e8	0.0284686	0.0142343	−	0.999899i	$-0.495469\pi$
$757$	3.39784e8	0.0284686	0.0142343	+	0.999899i	$0.495469\pi$
$758$	−2.99438e10	−2.49726
$759$	−1.31746e9	−0.109369
$760$	−6.82096e6	−0.000563635 0
$761$	−1.96039e9	−0.161249	−0.0806243	−	0.996745i	$-0.525691\pi$
$761$	−1.96039e9	−0.161249	−0.0806243	+	0.996745i	$0.525691\pi$
$762$	−7.15661e9	−0.585956
$763$	4.79718e9	0.390976
$764$	1.57727e10	1.27962
$765$	−2.83764e7	−0.00229162
$766$	2.86500e10	2.30316
$767$	0	0
$768$	−1.97053e9	−0.156971
$769$	−1.11615e10	−0.885077	−0.442538	−	0.896750i	$-0.645922\pi$
$769$	−1.11615e10	−0.885077	−0.442538	+	0.896750i	$0.645922\pi$
$770$	−2.92246e6	−0.000230691 0
$771$	2.30733e8	0.0181309
$772$	−7.07404e8	−0.0553359
$773$	2.85420e9	0.222258	0.111129	−	0.993806i	$-0.464553\pi$
$773$	2.85420e9	0.222258	0.111129	+	0.993806i	$0.464553\pi$
$774$	−2.51065e10	−1.94622
$775$	−3.62072e9	−0.279408
$776$	3.33232e8	0.0255994
$777$	8.20364e8	0.0627383
$778$	−2.81483e10	−2.14301
$779$	−1.83461e10	−1.39047
$780$	0	0
$781$	−3.99510e9	−0.300088
$782$	3.56909e10	2.66891
$783$	−5.67884e9	−0.422759
$784$	9.56202e9	0.708669
$785$	4.99464e7	0.00368519
$786$	3.57961e9	0.262940
$787$	1.33429e10	0.975749	0.487874	−	0.872914i	$-0.337772\pi$
$787$	1.33429e10	0.975749	0.487874	+	0.872914i	$0.337772\pi$
$788$	1.53468e10	1.11732
$789$	3.44618e8	0.0249786
$790$	2.31913e7	0.00167351
$791$	−1.84526e9	−0.132569
$792$	−8.67027e8	−0.0620147
$793$	0	0
$794$	3.56073e10	2.52445
$795$	−4.00112e6	−0.000282421 0
$796$	−1.95992e10	−1.37734
$797$	8.42860e9	0.589727	0.294864	−	0.955539i	$-0.404726\pi$
$797$	8.42860e9	0.589727	0.294864	+	0.955539i	$0.404726\pi$
$798$	−1.86051e9	−0.129605
$799$	−2.82010e10	−1.95592
$800$	−2.06716e10	−1.42744
$801$	5.44289e9	0.374210
$802$	−1.51390e10	−1.03630
$803$	−5.68713e9	−0.387604
$804$	6.02935e7	0.00409142
$805$	1.40061e7	0.000946308 0
$806$	0	0
$807$	−3.40513e8	−0.0228075
$808$	−5.36465e9	−0.357768
$809$	9.44279e9	0.627019	0.313509	−	0.949585i	$-0.398495\pi$
$809$	9.44279e9	0.627019	0.313509	+	0.949585i	$0.398495\pi$
$810$	−3.48576e7	−0.00230462
$811$	1.75509e10	1.15539	0.577693	−	0.816254i	$-0.303953\pi$
$811$	1.75509e10	1.15539	0.577693	+	0.816254i	$0.303953\pi$
$812$	4.10262e9	0.268915
$813$	−8.01041e9	−0.522803
$814$	3.96740e9	0.257823
$815$	−2.51144e7	−0.00162507
$816$	4.28846e9	0.276303
$817$	−2.21262e10	−1.41948
$818$	2.69215e10	1.71974
$819$	0	0
$820$	−5.51344e7	−0.00349200
$821$	1.65515e10	1.04384	0.521922	−	0.852993i	$-0.325215\pi$
$821$	1.65515e10	1.04384	0.521922	+	0.852993i	$0.325215\pi$
$822$	4.04540e9	0.254045
$823$	2.59526e10	1.62286	0.811430	−	0.584449i	$-0.198689\pi$
$823$	2.59526e10	1.62286	0.811430	+	0.584449i	$0.198689\pi$
$824$	−5.17762e9	−0.322392
$825$	−1.17538e9	−0.0728768
$826$	5.10458e9	0.315159
$827$	2.53210e10	1.55672	0.778362	−	0.627816i	$-0.216051\pi$
$827$	2.53210e10	1.55672	0.778362	+	0.627816i	$0.216051\pi$
$828$	2.65429e10	1.62496
$829$	2.70822e10	1.65099	0.825494	−	0.564411i	$-0.190897\pi$
$829$	2.70822e10	1.65099	0.825494	+	0.564411i	$0.190897\pi$
$830$	−1.72887e7	−0.00104951
$831$	3.93680e9	0.237980
$832$	0	0
$833$	−1.82341e10	−1.09301
$834$	5.35634e9	0.319733
$835$	−3.83463e7	−0.00227940
$836$	−4.88090e9	−0.288920
$837$	2.67075e9	0.157432
$838$	4.81130e9	0.282428
$839$	−1.26152e10	−0.737442	−0.368721	−	0.929540i	$-0.620204\pi$
$839$	−1.26152e10	−0.737442	−0.368721	+	0.929540i	$0.620204\pi$
$840$	−875316.	−5.09550e−5 0
$841$	−7.53886e9	−0.437038
$842$	−2.55001e8	−0.0147214
$843$	8.10971e9	0.466239
$844$	2.18460e10	1.25076
$845$	0	0
$846$	−3.86624e10	−2.19529
$847$	5.01860e9	0.283786
$848$	−6.36757e9	−0.358582
$849$	6.82942e9	0.383007
$850$	3.18417e10	1.77840
$851$	−1.90141e10	−1.05760
$852$	−7.64347e9	−0.423401
$853$	−1.11666e9	−0.0616027	−0.0308013	−	0.999526i	$-0.509806\pi$
$853$	−1.11666e9	−0.0616027	−0.0308013	+	0.999526i	$0.509806\pi$
$854$	−2.86726e9	−0.157530
$855$	−3.42847e7	−0.00187594
$856$	−8.13976e9	−0.443561
$857$	−1.48481e10	−0.805821	−0.402911	−	0.915239i	$-0.632001\pi$
$857$	−1.48481e10	−0.805821	−0.402911	+	0.915239i	$0.632001\pi$
$858$	0	0
$859$	1.26319e10	0.679974	0.339987	−	0.940430i	$-0.389577\pi$
$859$	1.26319e10	0.679974	0.339987	+	0.940430i	$0.389577\pi$
$860$	−6.64946e7	−0.00356486
$861$	−2.35430e9	−0.125705
$862$	1.54286e10	0.820449
$863$	−2.29823e10	−1.21718	−0.608591	−	0.793484i	$-0.708265\pi$
$863$	−2.29823e10	−1.21718	−0.608591	+	0.793484i	$0.708265\pi$
$864$	1.52480e10	0.804294
$865$	−5.97374e7	−0.00313827
$866$	3.15136e10	1.64887
$867$	−2.52654e9	−0.131662
$868$	−1.92946e9	−0.100142
$869$	2.59796e9	0.134296
$870$	−1.32347e7	−0.000681393 0
$871$	0	0
$872$	6.94867e9	0.354890
$873$	1.67495e9	0.0852024
$874$	4.31222e10	2.18479
$875$	2.49913e7	0.00126113
$876$	−1.08807e10	−0.546879
$877$	−1.63120e10	−0.816599	−0.408299	−	0.912848i	$-0.633878\pi$
$877$	−1.63120e10	−0.816599	−0.408299	+	0.912848i	$0.633878\pi$
$878$	1.44696e10	0.721480
$879$	−7.45110e9	−0.370049
$880$	8.13883e6	0.000402599 0
$881$	2.07580e9	0.102275	0.0511374	−	0.998692i	$-0.483715\pi$
$881$	2.07580e9	0.102275	0.0511374	+	0.998692i	$0.483715\pi$
$882$	−2.49981e10	−1.22678
$883$	1.70041e10	0.831175	0.415587	−	0.909553i	$-0.363576\pi$
$883$	1.70041e10	0.831175	0.415587	+	0.909553i	$0.363576\pi$
$884$	0	0
$885$	−8.93268e6	−0.000433192 0
$886$	−1.57359e10	−0.760105
$887$	−1.27005e10	−0.611066	−0.305533	−	0.952182i	$-0.598835\pi$
$887$	−1.27005e10	−0.611066	−0.305533	+	0.952182i	$0.598835\pi$
$888$	1.18829e9	0.0569478
$889$	8.52303e9	0.406853
$890$	2.65743e7	0.00126356
$891$	−3.90486e9	−0.184941
$892$	1.44316e10	0.680828
$893$	−3.40729e10	−1.60114
$894$	1.68419e10	0.788332
$895$	3.75538e7	0.00175095
$896$	−3.51057e9	−0.163042
$897$	0	0
$898$	1.47153e9	0.0678114
$899$	−4.56708e9	−0.209643
$900$	2.36804e10	1.08278
$901$	1.21425e10	0.553058
$902$	−1.13858e10	−0.516582
$903$	−2.83939e9	−0.128327
$904$	−2.67285e9	−0.120333
$905$	5.46284e7	0.00244990
$906$	−4.01559e9	−0.179391
$907$	−5.79320e9	−0.257806	−0.128903	−	0.991657i	$-0.541146\pi$
$907$	−5.79320e9	−0.257806	−0.128903	+	0.991657i	$0.541146\pi$
$908$	3.58849e10	1.59079
$909$	−2.69647e10	−1.19076
$910$	0	0
$911$	−2.92730e10	−1.28278	−0.641390	−	0.767215i	$-0.721642\pi$
$911$	−2.92730e10	−1.28278	−0.641390	+	0.767215i	$0.721642\pi$
$912$	5.18137e9	0.226185
$913$	−1.93673e9	−0.0842212
$914$	−3.26389e10	−1.41392
$915$	5.01752e6	0.000216528 0
$916$	3.64978e10	1.56904
$917$	−4.26307e9	−0.182570
$918$	−2.34874e10	−1.00204
$919$	1.74294e10	0.740759	0.370380	−	0.928880i	$-0.379228\pi$
$919$	1.74294e10	0.740759	0.370380	+	0.928880i	$0.379228\pi$
$920$	2.02877e7	0.000858967 0
$921$	5.85087e9	0.246781
$922$	1.93717e9	0.0813974
$923$	0	0
$924$	−6.26353e8	−0.0261196
$925$	−1.69635e10	−0.704723
$926$	2.45016e10	1.01404
$927$	−2.60246e10	−1.07301
$928$	−2.60746e10	−1.07103
$929$	−1.64663e10	−0.673814	−0.336907	−	0.941538i	$-0.609381\pi$
$929$	−1.64663e10	−0.673814	−0.336907	+	0.941538i	$0.609381\pi$
$930$	6.22428e6	0.000253746 0
$931$	−2.20306e10	−0.894753
$932$	3.39389e10	1.37322
$933$	−3.98165e9	−0.160501
$934$	1.04293e10	0.418834
$935$	−1.55202e7	−0.000620948 0
$936$	0	0
$937$	−1.22607e10	−0.486885	−0.243442	−	0.969915i	$-0.578277\pi$
$937$	−1.22607e10	−0.486885	−0.243442	+	0.969915i	$0.578277\pi$
$938$	−1.32370e8	−0.00523695
$939$	1.30374e10	0.513880
$940$	−1.02397e8	−0.00402107
$941$	−3.58066e10	−1.40088	−0.700438	−	0.713713i	$-0.747012\pi$
$941$	−3.58066e10	−1.40088	−0.700438	+	0.713713i	$0.747012\pi$
$942$	1.97336e10	0.769180
$943$	5.45671e10	2.11904
$944$	−1.42159e10	−0.550011
$945$	−9.21713e6	−0.000355291 0
$946$	−1.37317e10	−0.527359
$947$	3.11279e10	1.19104	0.595518	−	0.803342i	$-0.296947\pi$
$947$	3.11279e10	1.19104	0.595518	+	0.803342i	$0.296947\pi$
$948$	4.97044e9	0.189481
$949$	0	0
$950$	3.84716e10	1.45582
$951$	−5.76484e9	−0.217348
$952$	2.65638e9	0.0997840
$953$	6.95907e9	0.260451	0.130225	−	0.991484i	$-0.458430\pi$
$953$	6.95907e9	0.260451	0.130225	+	0.991484i	$0.458430\pi$
$954$	1.66468e10	0.620743
$955$	6.05964e7	0.00225131
$956$	3.94961e10	1.46201
$957$	−1.48259e9	−0.0546803
$958$	−3.52913e10	−1.29685
$959$	−4.81779e9	−0.176394
$960$	2.24024e7	0.000817230 0
$961$	−2.53647e10	−0.921931
$962$	0	0
$963$	−4.09135e10	−1.47630
$964$	−4.43255e10	−1.59362
$965$	−2.71774e6	−9.73558e−5 0
$966$	5.53375e9	0.197515
$967$	3.75275e10	1.33462	0.667310	−	0.744780i	$-0.267446\pi$
$967$	3.75275e10	1.33462	0.667310	+	0.744780i	$0.267446\pi$
$968$	7.26940e9	0.257593
$969$	−9.88050e9	−0.348856
$970$	8.17774e6	0.000287695 0
$971$	1.78987e9	0.0627414	0.0313707	−	0.999508i	$-0.490013\pi$
$971$	1.78987e9	0.0627414	0.0313707	+	0.999508i	$0.490013\pi$
$972$	−2.65969e10	−0.928965
$973$	−6.37904e9	−0.222004
$974$	−5.16665e10	−1.79165
$975$	0	0
$976$	7.98510e9	0.274920
$977$	−1.98276e10	−0.680203	−0.340101	−	0.940389i	$-0.610461\pi$
$977$	−1.98276e10	−0.680203	−0.340101	+	0.940389i	$0.610461\pi$
$978$	−9.92259e9	−0.339187
$979$	2.97693e9	0.101398
$980$	−6.62075e7	−0.00224707
$981$	3.49266e10	1.18118
$982$	6.07523e10	2.04726
$983$	3.24213e10	1.08866	0.544332	−	0.838870i	$-0.316783\pi$
$983$	3.24213e10	1.08866	0.544332	+	0.838870i	$0.316783\pi$
$984$	−3.41018e9	−0.114102
$985$	5.89601e7	0.00196576
$986$	4.01643e10	1.33435
$987$	−4.37248e9	−0.144750
$988$	0	0
$989$	6.58104e10	2.16326
$990$	−2.12775e7	−0.000696942 0
$991$	−2.45428e10	−0.801063	−0.400532	−	0.916283i	$-0.631175\pi$
$991$	−2.45428e10	−0.801063	−0.400532	+	0.916283i	$0.631175\pi$
$992$	1.22629e10	0.398843
$993$	−5.67890e9	−0.184052
$994$	1.67806e10	0.541946
$995$	−7.52970e7	−0.00242324
$996$	−3.70537e9	−0.118830
$997$	−1.08000e10	−0.345137	−0.172568	−	0.984998i	$-0.555207\pi$
$997$	−1.08000e10	−0.345137	−0.172568	+	0.984998i	$0.555207\pi$
$998$	−7.89357e10	−2.51372
$999$	1.25128e10	0.397077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.i.1.3	yes	21
13.5	odd	4		169.8.b.f.168.36		42
13.8	odd	4		169.8.b.f.168.7		42
13.12	even	2		169.8.a.h.1.19	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.19	✓	21	13.12	even	2
169.8.a.i.1.3	yes	21	1.1	even	1	trivial
169.8.b.f.168.7		42	13.8	odd	4
169.8.b.f.168.36		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)

\(f(q)\)	\(=\)	\(q-16.7260 q^{2} -13.7721 q^{3} +151.758 q^{4} +0.583029 q^{5} +230.352 q^{6} -274.333 q^{7} -397.369 q^{8} -1997.33 q^{9} -9.75172 q^{10} -1092.42 q^{11} -2090.03 q^{12} +4588.49 q^{14} -8.02956 q^{15} -12778.6 q^{16} +24367.8 q^{17} +33407.2 q^{18} +29441.5 q^{19} +88.4791 q^{20} +3778.16 q^{21} +18271.7 q^{22} -87568.7 q^{23} +5472.63 q^{24} -78124.7 q^{25} +57627.1 q^{27} -41632.2 q^{28} -98544.5 q^{29} +134.302 q^{30} +46345.4 q^{31} +264598. q^{32} +15044.9 q^{33} -407575. q^{34} -159.944 q^{35} -303110. q^{36} +217133. q^{37} -492438. q^{38} -231.678 q^{40} -623135. q^{41} -63193.3 q^{42} -751529. q^{43} -165783. q^{44} -1164.50 q^{45} +1.46467e6 q^{46} -1.15731e6 q^{47} +175988. q^{48} -748284. q^{49} +1.30671e6 q^{50} -335597. q^{51} +498300. q^{53} -963869. q^{54} -636.912 q^{55} +109012. q^{56} -405473. q^{57} +1.64825e6 q^{58} +1.11248e6 q^{59} -1218.55 q^{60} -624881. q^{61} -775171. q^{62} +547934. q^{63} -2.78999e6 q^{64} -251641. q^{66} -28848.2 q^{67} +3.69801e6 q^{68} +1.20601e6 q^{69} +2675.22 q^{70} +3.65711e6 q^{71} +793677. q^{72} +5.20600e6 q^{73} -3.63176e6 q^{74} +1.07594e6 q^{75} +4.46798e6 q^{76} +299687. q^{77} -2.37817e6 q^{79} -7450.29 q^{80} +3.57451e6 q^{81} +1.04225e7 q^{82} +1.77288e6 q^{83} +573364. q^{84} +14207.2 q^{85} +1.25700e7 q^{86} +1.35717e6 q^{87} +434094. q^{88} -2.72509e6 q^{89} +19477.4 q^{90} -1.32892e7 q^{92} -638275. q^{93} +1.93570e7 q^{94} +17165.3 q^{95} -3.64407e6 q^{96} -838595. q^{97} +1.25158e7 q^{98} +2.18192e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) \) 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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