LMFDB - Embedded newform 169.10.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,10,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$87.0410563117$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	\(q-25.3541 q^{2} -120.880 q^{3} +130.831 q^{4} +1004.42 q^{5} +3064.81 q^{6} +3164.79 q^{7} +9664.20 q^{8} -5071.03 q^{9} -25466.3 q^{10} +29216.4 q^{11} -15814.9 q^{12} -80240.5 q^{14} -121415. q^{15} -312013. q^{16} -227844. q^{17} +128571. q^{18} +416712. q^{19} +131410. q^{20} -382560. q^{21} -740755. q^{22} -2.24212e6 q^{23} -1.16821e6 q^{24} -944259. q^{25} +2.99227e6 q^{27} +414054. q^{28} -141462. q^{29} +3.07836e6 q^{30} +1.84540e6 q^{31} +2.96274e6 q^{32} -3.53167e6 q^{33} +5.77678e6 q^{34} +3.17879e6 q^{35} -663449. q^{36} -145102. q^{37} -1.05654e7 q^{38} +9.70695e6 q^{40} -1.15955e7 q^{41} +9.69947e6 q^{42} +3.71797e7 q^{43} +3.82242e6 q^{44} -5.09346e6 q^{45} +5.68471e7 q^{46} +3.05545e7 q^{47} +3.77161e7 q^{48} -3.03377e7 q^{49} +2.39408e7 q^{50} +2.75418e7 q^{51} +8.17666e7 q^{53} -7.58663e7 q^{54} +2.93456e7 q^{55} +3.05852e7 q^{56} -5.03721e7 q^{57} +3.58664e6 q^{58} +3.46046e7 q^{59} -1.58848e7 q^{60} -1.76707e8 q^{61} -4.67886e7 q^{62} -1.60487e7 q^{63} +8.46329e7 q^{64} +8.95425e7 q^{66} -1.88835e8 q^{67} -2.98091e7 q^{68} +2.71028e8 q^{69} -8.05955e7 q^{70} +4.23785e8 q^{71} -4.90074e7 q^{72} +1.11014e8 q^{73} +3.67892e6 q^{74} +1.14142e8 q^{75} +5.45190e7 q^{76} +9.24637e7 q^{77} +2.46113e8 q^{79} -3.13393e8 q^{80} -2.61892e8 q^{81} +2.93995e8 q^{82} -7.57907e8 q^{83} -5.00508e7 q^{84} -2.28852e8 q^{85} -9.42659e8 q^{86} +1.70999e7 q^{87} +2.82353e8 q^{88} +1.61392e8 q^{89} +1.29140e8 q^{90} -2.93340e8 q^{92} -2.23072e8 q^{93} -7.74683e8 q^{94} +4.18555e8 q^{95} -3.58136e8 q^{96} -7.15528e8 q^{97} +7.69186e8 q^{98} -1.48157e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 65 q^{2} + q^{3} + 7169 q^{4} - 3238 q^{5} - 8490 q^{6} - 17378 q^{7} - 54204 q^{8} + 191118 q^{9} + 11697 q^{10} - 164171 q^{11} - 181941 q^{12} - 77651 q^{14} - 614110 q^{15} + 3012565 q^{16}+ \cdots - 5866875443 q^{99}+O(q^{100}) $ 27 * q - 65 * q^2 + q^3 + 7169 * q^4 - 3238 * q^5 - 8490 * q^6 - 17378 * q^7 - 54204 * q^8 + 191118 * q^9 + 11697 * q^10 - 164171 * q^11 - 181941 * q^12 - 77651 * q^14 - 614110 * q^15 + 3012565 * q^16 + 105667 * q^17 - 584849 * q^18 - 2140031 * q^19 - 3722382 * q^20 - 2953252 * q^21 + 3789362 * q^22 + 3586420 * q^23 - 6334338 * q^24 + 10278737 * q^25 - 4332980 * q^27 - 5293616 * q^28 - 4203616 * q^29 - 34788079 * q^30 - 18634294 * q^31 - 38778128 * q^32 - 15170356 * q^33 - 10546914 * q^34 + 16260796 * q^35 + 112071987 * q^36 - 7214770 * q^37 - 60128169 * q^38 - 46270051 * q^40 - 67544533 * q^41 - 31249012 * q^42 - 55585357 * q^43 - 246583161 * q^44 - 172236942 * q^45 - 168715272 * q^46 - 175569972 * q^47 - 48573948 * q^48 + 144680057 * q^49 - 3455750 * q^50 + 122191382 * q^51 + 105504478 * q^53 + 359394409 * q^54 - 2471426 * q^55 + 260765426 * q^56 + 307372146 * q^57 + 777873375 * q^58 - 267562255 * q^59 + 745460915 * q^60 - 371334848 * q^61 + 52175323 * q^62 - 431551952 * q^63 + 642079732 * q^64 + 973549503 * q^66 - 213848603 * q^67 - 278629116 * q^68 - 201174310 * q^69 + 122562066 * q^70 - 970709202 * q^71 - 773619646 * q^72 - 271733895 * q^73 - 1068711482 * q^74 + 1236853937 * q^75 - 2585634844 * q^76 - 244285066 * q^77 + 1052001020 * q^79 - 3293699123 * q^80 - 456846001 * q^81 - 3832425 * q^82 - 1491498691 * q^83 - 5153327787 * q^84 - 2713007958 * q^85 - 3421126005 * q^86 - 1622618562 * q^87 + 6574070508 * q^88 - 4616501291 * q^89 - 2822436128 * q^90 + 3684890562 * q^92 - 3324407374 * q^93 - 5582614303 * q^94 - 2892286998 * q^95 - 12902380548 * q^96 - 3944038157 * q^97 - 7822213576 * q^98 - 5866875443 * 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 $

$2$	−25.3541	−1.12050	−0.560252	−	0.828322i	$-0.689296\pi$
$2$	−25.3541	−1.12050	−0.560252	+	0.828322i	$0.689296\pi$
$3$	−120.880	−0.861606	−0.430803	−	0.902446i	$-0.641770\pi$
$3$	−120.880	−0.861606	−0.430803	+	0.902446i	$0.641770\pi$
$4$	130.831	0.255530
$5$	1004.42	0.718707	0.359353	−	0.933202i	$-0.382997\pi$
$5$	1004.42	0.718707	0.359353	+	0.933202i	$0.382997\pi$
$6$	3064.81	0.965433
$7$	3164.79	0.498200	0.249100	−	0.968478i	$-0.419865\pi$
$7$	3164.79	0.498200	0.249100	+	0.968478i	$0.419865\pi$
$8$	9664.20	0.834182
$9$	−5071.03	−0.257635
$10$	−25466.3	−0.805314
$11$	29216.4	0.601671	0.300836	−	0.953676i	$-0.402734\pi$
$11$	29216.4	0.601671	0.300836	+	0.953676i	$0.402734\pi$
$12$	−15814.9	−0.220166
$13$	0	0
$13$	0	0
$14$	−80240.5	−0.558236
$15$	−121415.	−0.619242
$16$	−312013.	−1.19023
$17$	−227844.	−0.661633	−0.330817	−	0.943695i	$-0.607324\pi$
$17$	−227844.	−0.661633	−0.330817	+	0.943695i	$0.607324\pi$
$18$	128571.	0.288681
$19$	416712.	0.733576	0.366788	−	0.930305i	$-0.380458\pi$
$19$	416712.	0.733576	0.366788	+	0.930305i	$0.380458\pi$
$20$	131410.	0.183651
$21$	−382560.	−0.429252
$22$	−740755.	−0.674175
$23$	−2.24212e6	−1.67065	−0.835323	−	0.549760i	$-0.814719\pi$
$23$	−2.24212e6	−1.67065	−0.835323	+	0.549760i	$0.814719\pi$
$24$	−1.16821e6	−0.718736
$25$	−944259.	−0.483460
$26$	0	0
$27$	2.99227e6	1.08359
$28$	414054.	0.127305
$29$	−141462.	−0.0371405	−0.0185703	−	0.999828i	$-0.505911\pi$
$29$	−141462.	−0.0371405	−0.0185703	+	0.999828i	$0.505911\pi$
$30$	3.07836e6	0.693864
$31$	1.84540e6	0.358892	0.179446	−	0.983768i	$-0.442570\pi$
$31$	1.84540e6	0.358892	0.179446	+	0.983768i	$0.442570\pi$
$32$	2.96274e6	0.499481
$33$	−3.53167e6	−0.518404
$34$	5.77678e6	0.741363
$35$	3.17879e6	0.358060
$36$	−663449.	−0.0658334
$37$	−145102.	−0.0127281	−0.00636406	−	0.999980i	$-0.502026\pi$
$37$	−145102.	−0.0127281	−0.00636406	+	0.999980i	$0.502026\pi$
$38$	−1.05654e7	−0.821975
$39$	0	0
$40$	9.70695e6	0.599532
$41$	−1.15955e7	−0.640861	−0.320430	−	0.947272i	$-0.603827\pi$
$41$	−1.15955e7	−0.640861	−0.320430	+	0.947272i	$0.603827\pi$
$42$	9.69947e6	0.480979
$43$	3.71797e7	1.65843	0.829217	−	0.558927i	$-0.188787\pi$
$43$	3.71797e7	1.65843	0.829217	+	0.558927i	$0.188787\pi$
$44$	3.82242e6	0.153745
$45$	−5.09346e6	−0.185164
$46$	5.68471e7	1.87197
$47$	3.05545e7	0.913345	0.456673	−	0.889635i	$-0.349041\pi$
$47$	3.05545e7	0.913345	0.456673	+	0.889635i	$0.349041\pi$
$48$	3.77161e7	1.02551
$49$	−3.03377e7	−0.751796
$50$	2.39408e7	0.541719
$51$	2.75418e7	0.570067
$52$	0	0
$53$	8.17666e7	1.42342	0.711712	−	0.702471i	$-0.247920\pi$
$53$	8.17666e7	1.42342	0.711712	+	0.702471i	$0.247920\pi$
$54$	−7.58663e7	−1.21416
$55$	2.93456e7	0.432425
$56$	3.05852e7	0.415590
$57$	−5.03721e7	−0.632053
$58$	3.58664e6	0.0416161
$59$	3.46046e7	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$59$	3.46046e7	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$60$	−1.58848e7	−0.158235
$61$	−1.76707e8	−1.63406	−0.817031	−	0.576594i	$-0.804381\pi$
$61$	−1.76707e8	−1.63406	−0.817031	+	0.576594i	$0.804381\pi$
$62$	−4.67886e7	−0.402140
$63$	−1.60487e7	−0.128354
$64$	8.46329e7	0.630564
$65$	0	0
$66$	8.95425e7	0.580873
$67$	−1.88835e8	−1.14484	−0.572420	−	0.819961i	$-0.693995\pi$
$67$	−1.88835e8	−1.14484	−0.572420	+	0.819961i	$0.693995\pi$
$68$	−2.98091e7	−0.169067
$69$	2.71028e8	1.43944
$70$	−8.05955e7	−0.401208
$71$	4.23785e8	1.97917	0.989586	−	0.143946i	$-0.0459792\pi$
$71$	4.23785e8	1.97917	0.989586	+	0.143946i	$0.0459792\pi$
$72$	−4.90074e7	−0.214914
$73$	1.11014e8	0.457535	0.228768	−	0.973481i	$-0.426530\pi$
$73$	1.11014e8	0.457535	0.228768	+	0.973481i	$0.426530\pi$
$74$	3.67892e6	0.0142619
$75$	1.14142e8	0.416552
$76$	5.45190e7	0.187450
$77$	9.24637e7	0.299753
$78$	0	0
$79$	2.46113e8	0.710906	0.355453	−	0.934694i	$-0.384327\pi$
$79$	2.46113e8	0.710906	0.355453	+	0.934694i	$0.384327\pi$
$80$	−3.13393e8	−0.855430
$81$	−2.61892e8	−0.675989
$82$	2.93995e8	0.718087
$83$	−7.57907e8	−1.75293	−0.876465	−	0.481466i	$-0.840104\pi$
$83$	−7.57907e8	−1.75293	−0.876465	+	0.481466i	$0.840104\pi$
$84$	−5.00508e7	−0.109687
$85$	−2.28852e8	−0.475520
$86$	−9.42659e8	−1.85828
$87$	1.70999e7	0.0320005
$88$	2.82353e8	0.501903
$89$	1.61392e8	0.272663	0.136332	−	0.990663i	$-0.456469\pi$
$89$	1.61392e8	0.272663	0.136332	+	0.990663i	$0.456469\pi$
$90$	1.29140e8	0.207477
$91$	0	0
$92$	−2.93340e8	−0.426900
$93$	−2.23072e8	−0.309223
$94$	−7.74683e8	−1.02341
$95$	4.18555e8	0.527226
$96$	−3.58136e8	−0.430356
$97$	−7.15528e8	−0.820642	−0.410321	−	0.911941i	$-0.634583\pi$
$97$	−7.15528e8	−0.820642	−0.410321	+	0.911941i	$0.634583\pi$
$98$	7.69186e8	0.842391
$99$	−1.48157e8	−0.155011
$100$	−1.23539e8	−0.123539
$101$	1.51099e9	1.44483	0.722413	−	0.691461i	$-0.243033\pi$
$101$	1.51099e9	1.44483	0.722413	+	0.691461i	$0.243033\pi$
$102$	−6.98297e8	−0.638763
$103$	−1.35000e9	−1.18186	−0.590930	−	0.806723i	$-0.701239\pi$
$103$	−1.35000e9	−1.18186	−0.590930	+	0.806723i	$0.701239\pi$
$104$	0	0
$105$	−3.84252e8	−0.308507
$106$	−2.07312e9	−1.59495
$107$	6.31042e8	0.465405	0.232703	−	0.972548i	$-0.425243\pi$
$107$	6.31042e8	0.465405	0.232703	+	0.972548i	$0.425243\pi$
$108$	3.91482e8	0.276889
$109$	1.13360e9	0.769205	0.384602	−	0.923082i	$-0.374339\pi$
$109$	1.13360e9	0.769205	0.384602	+	0.923082i	$0.374339\pi$
$110$	−7.44032e8	−0.484534
$111$	1.75399e7	0.0109666
$112$	−9.87456e8	−0.592975
$113$	−2.04397e9	−1.17929	−0.589646	−	0.807662i	$-0.700733\pi$
$113$	−2.04397e9	−1.17929	−0.589646	+	0.807662i	$0.700733\pi$
$114$	1.27714e9	0.708218
$115$	−2.25204e9	−1.20070
$116$	−1.85076e7	−0.00949051
$117$	0	0
$118$	−8.77370e8	−0.416595
$119$	−7.21079e8	−0.329626
$120$	−1.17338e9	−0.516561
$121$	−1.50435e9	−0.637992
$122$	4.48024e9	1.83097
$123$	1.40167e9	0.552169
$124$	2.41436e8	0.0917076
$125$	−2.91020e9	−1.06617
$126$	4.06902e8	0.143821
$127$	2.51544e9	0.858019	0.429010	−	0.903300i	$-0.358863\pi$
$127$	2.51544e9	0.858019	0.429010	+	0.903300i	$0.358863\pi$
$128$	−3.66272e9	−1.20603
$129$	−4.49428e9	−1.42892
$130$	0	0
$131$	−5.08242e9	−1.50782	−0.753910	−	0.656977i	$-0.771835\pi$
$131$	−5.08242e9	−1.50782	−0.753910	+	0.656977i	$0.771835\pi$
$132$	−4.62054e8	−0.132468
$133$	1.31881e9	0.365468
$134$	4.78773e9	1.28280
$135$	3.00550e9	0.778781
$136$	−2.20193e9	−0.551923
$137$	7.21423e9	1.74963	0.874817	−	0.484453i	$-0.160981\pi$
$137$	7.21423e9	1.74963	0.874817	+	0.484453i	$0.160981\pi$
$138$	−6.87167e9	−1.61290
$139$	−5.09371e9	−1.15736	−0.578679	−	0.815556i	$-0.696432\pi$
$139$	−5.09371e9	−1.15736	−0.578679	+	0.815556i	$0.696432\pi$
$140$	4.15885e8	0.0914950
$141$	−3.69343e9	−0.786944
$142$	−1.07447e10	−2.21767
$143$	0	0
$144$	1.58223e9	0.306646
$145$	−1.42087e8	−0.0266932
$146$	−2.81466e9	−0.512670
$147$	3.66722e9	0.647752
$148$	−1.89838e7	−0.00325242
$149$	−5.20324e9	−0.864839	−0.432420	−	0.901672i	$-0.642340\pi$
$149$	−5.20324e9	−0.864839	−0.432420	+	0.901672i	$0.642340\pi$
$150$	−2.89397e9	−0.466749
$151$	5.55379e8	0.0869347	0.0434674	−	0.999055i	$-0.486160\pi$
$151$	5.55379e8	0.0869347	0.0434674	+	0.999055i	$0.486160\pi$
$152$	4.02719e9	0.611935
$153$	1.15540e9	0.170460
$154$	−2.34434e9	−0.335874
$155$	1.85357e9	0.257938
$156$	0	0
$157$	−9.35342e9	−1.22863	−0.614316	−	0.789060i	$-0.710568\pi$
$157$	−9.35342e9	−1.22863	−0.614316	+	0.789060i	$0.710568\pi$
$158$	−6.23997e9	−0.796573
$159$	−9.88394e9	−1.22643
$160$	2.97585e9	0.358980
$161$	−7.09585e9	−0.832316
$162$	6.64004e9	0.757449
$163$	5.95867e9	0.661157	0.330579	−	0.943778i	$-0.392756\pi$
$163$	5.95867e9	0.661157	0.330579	+	0.943778i	$0.392756\pi$
$164$	−1.51706e9	−0.163759
$165$	−3.54730e9	−0.372580
$166$	1.92161e10	1.96416
$167$	−4.28074e9	−0.425888	−0.212944	−	0.977064i	$-0.568305\pi$
$167$	−4.28074e9	−0.425888	−0.212944	+	0.977064i	$0.568305\pi$
$168$	−3.69714e9	−0.358075
$169$	0	0
$170$	5.80234e9	0.532823
$171$	−2.11316e9	−0.188995
$172$	4.86427e9	0.423779
$173$	7.28827e8	0.0618610	0.0309305	−	0.999522i	$-0.490153\pi$
$173$	7.28827e8	0.0618610	0.0309305	+	0.999522i	$0.490153\pi$
$174$	−4.33553e8	−0.0358567
$175$	−2.98838e9	−0.240860
$176$	−9.11588e9	−0.716130
$177$	−4.18301e9	−0.320338
$178$	−4.09195e9	−0.305520
$179$	4.87158e8	0.0354675	0.0177338	−	0.999843i	$-0.494355\pi$
$179$	4.87158e8	0.0354675	0.0177338	+	0.999843i	$0.494355\pi$
$180$	−6.66384e8	−0.0473149
$181$	1.69472e10	1.17366	0.586832	−	0.809709i	$-0.300375\pi$
$181$	1.69472e10	1.17366	0.586832	+	0.809709i	$0.300375\pi$
$182$	0	0
$183$	2.13603e10	1.40792
$184$	−2.16683e10	−1.39362
$185$	−1.45743e8	−0.00914779
$186$	5.65580e9	0.346486
$187$	−6.65677e9	−0.398086
$188$	3.99749e9	0.233387
$189$	9.46990e9	0.539843
$190$	−1.06121e10	−0.590759
$191$	−9.05255e9	−0.492176	−0.246088	−	0.969247i	$-0.579145\pi$
$191$	−9.05255e9	−0.492176	−0.246088	+	0.969247i	$0.579145\pi$
$192$	−1.02304e10	−0.543298
$193$	1.81178e10	0.939936	0.469968	−	0.882683i	$-0.344265\pi$
$193$	1.81178e10	0.939936	0.469968	+	0.882683i	$0.344265\pi$
$194$	1.81416e10	0.919533
$195$	0	0
$196$	−3.96912e9	−0.192106
$197$	−2.43883e10	−1.15367	−0.576837	−	0.816859i	$-0.695713\pi$
$197$	−2.43883e10	−1.15367	−0.576837	+	0.816859i	$0.695713\pi$
$198$	3.75639e9	0.173691
$199$	−1.50980e9	−0.0682467	−0.0341233	−	0.999418i	$-0.510864\pi$
$199$	−1.50980e9	−0.0682467	−0.0341233	+	0.999418i	$0.510864\pi$
$200$	−9.12550e9	−0.403294
$201$	2.28263e10	0.986401
$202$	−3.83099e10	−1.61893
$203$	−4.47697e8	−0.0185034
$204$	3.60333e9	0.145669
$205$	−1.16468e10	−0.460591
$206$	3.42281e10	1.32428
$207$	1.13699e10	0.430416
$208$	0	0
$209$	1.21748e10	0.441371
$210$	9.74238e9	0.345683
$211$	−1.09075e9	−0.0378839	−0.0189419	−	0.999821i	$-0.506030\pi$
$211$	−1.09075e9	−0.0378839	−0.0189419	+	0.999821i	$0.506030\pi$
$212$	1.06976e10	0.363727
$213$	−5.12272e10	−1.70527
$214$	−1.59995e10	−0.521488
$215$	3.73442e10	1.19193
$216$	2.89179e10	0.903908
$217$	5.84032e9	0.178800
$218$	−2.87415e10	−0.861897
$219$	−1.34194e10	−0.394215
$220$	3.83932e9	0.110498
$221$	0	0
$222$	−4.44708e8	−0.0122882
$223$	5.47429e10	1.48237	0.741183	−	0.671303i	$-0.234265\pi$
$223$	5.47429e10	1.48237	0.741183	+	0.671303i	$0.234265\pi$
$224$	9.37646e9	0.248841
$225$	4.78836e9	0.124556
$226$	5.18230e10	1.32140
$227$	1.28165e10	0.320372	0.160186	−	0.987087i	$-0.448791\pi$
$227$	1.28165e10	0.320372	0.160186	+	0.987087i	$0.448791\pi$
$228$	−6.59025e9	−0.161508
$229$	5.66455e10	1.36115	0.680574	−	0.732679i	$-0.261730\pi$
$229$	5.66455e10	1.36115	0.680574	+	0.732679i	$0.261730\pi$
$230$	5.70985e10	1.34539
$231$	−1.11770e10	−0.258269
$232$	−1.36711e9	−0.0309820
$233$	−2.87274e9	−0.0638549	−0.0319275	−	0.999490i	$-0.510165\pi$
$233$	−2.87274e9	−0.0638549	−0.0319275	+	0.999490i	$0.510165\pi$
$234$	0	0
$235$	3.06897e10	0.656428
$236$	4.52737e9	0.0950040
$237$	−2.97501e10	−0.612521
$238$	1.82823e10	0.369347
$239$	−8.93396e9	−0.177114	−0.0885571	−	0.996071i	$-0.528226\pi$
$239$	−8.93396e9	−0.177114	−0.0885571	+	0.996071i	$0.528226\pi$
$240$	3.78829e10	0.737043
$241$	3.23205e10	0.617164	0.308582	−	0.951198i	$-0.400146\pi$
$241$	3.23205e10	0.617164	0.308582	+	0.951198i	$0.400146\pi$
$242$	3.81415e10	0.714873
$243$	−2.72393e10	−0.501149
$244$	−2.31187e10	−0.417551
$245$	−3.04719e10	−0.540321
$246$	−3.55381e10	−0.618708
$247$	0	0
$248$	1.78343e10	0.299381
$249$	9.16158e10	1.51033
$250$	7.37856e10	1.19465
$251$	−1.10159e11	−1.75181	−0.875907	−	0.482480i	$-0.839736\pi$
$251$	−1.10159e11	−1.75181	−0.875907	+	0.482480i	$0.839736\pi$
$252$	−2.09968e9	−0.0327982
$253$	−6.55067e10	−1.00518
$254$	−6.37767e10	−0.961414
$255$	2.76636e10	0.409711
$256$	4.95329e10	0.720798
$257$	−4.02685e10	−0.575793	−0.287897	−	0.957661i	$-0.592956\pi$
$257$	−4.02685e10	−0.575793	−0.287897	+	0.957661i	$0.592956\pi$
$258$	1.13949e11	1.60111
$259$	−4.59217e8	−0.00634116
$260$	0	0
$261$	7.17356e8	0.00956869
$262$	1.28860e11	1.68952
$263$	6.45831e10	0.832373	0.416187	−	0.909279i	$-0.363366\pi$
$263$	6.45831e10	0.832373	0.416187	+	0.909279i	$0.363366\pi$
$264$	−3.41308e10	−0.432443
$265$	8.21283e10	1.02303
$266$	−3.34372e10	−0.409508
$267$	−1.95091e10	−0.234928
$268$	−2.47055e10	−0.292541
$269$	−4.36313e10	−0.508057	−0.254029	−	0.967197i	$-0.581756\pi$
$269$	−4.36313e10	−0.508057	−0.254029	+	0.967197i	$0.581756\pi$
$270$	−7.62019e10	−0.872627
$271$	−7.70570e10	−0.867861	−0.433931	−	0.900946i	$-0.642874\pi$
$271$	−7.70570e10	−0.867861	−0.433931	+	0.900946i	$0.642874\pi$
$272$	7.10902e10	0.787499
$273$	0	0
$274$	−1.82910e11	−1.96047
$275$	−2.75878e10	−0.290884
$276$	3.54589e10	0.367819
$277$	−1.80766e11	−1.84484	−0.922418	−	0.386192i	$-0.873790\pi$
$277$	−1.80766e11	−1.84484	−0.922418	+	0.386192i	$0.873790\pi$
$278$	1.29147e11	1.29682
$279$	−9.35809e9	−0.0924631
$280$	3.07205e10	0.298687
$281$	8.13599e10	0.778453	0.389226	−	0.921142i	$-0.372742\pi$
$281$	8.13599e10	0.778453	0.389226	+	0.921142i	$0.372742\pi$
$282$	9.36437e10	0.881774
$283$	−2.40131e10	−0.222540	−0.111270	−	0.993790i	$-0.535492\pi$
$283$	−2.40131e10	−0.222540	−0.111270	+	0.993790i	$0.535492\pi$
$284$	5.54444e10	0.505737
$285$	−5.05950e10	−0.454261
$286$	0	0
$287$	−3.66975e10	−0.319277
$288$	−1.50241e10	−0.128684
$289$	−6.66750e10	−0.562241
$290$	3.60250e9	0.0299098
$291$	8.64930e10	0.707070
$292$	1.45241e10	0.116914
$293$	−1.69890e11	−1.34668	−0.673338	−	0.739335i	$-0.735140\pi$
$293$	−1.69890e11	−1.34668	−0.673338	+	0.739335i	$0.735140\pi$
$294$	−9.29791e10	−0.725809
$295$	3.47577e10	0.267210
$296$	−1.40229e9	−0.0106176
$297$	8.74232e10	0.651962
$298$	1.31923e11	0.969056
$299$	0	0
$300$	1.49333e10	0.106442
$301$	1.17666e11	0.826232
$302$	−1.40811e10	−0.0974107
$303$	−1.82649e11	−1.24487
$304$	−1.30019e11	−0.873127
$305$	−1.77488e11	−1.17441
$306$	−2.92942e10	−0.191001
$307$	−2.81612e9	−0.0180938	−0.00904688	−	0.999959i	$-0.502880\pi$
$307$	−2.81612e9	−0.0180938	−0.00904688	+	0.999959i	$0.502880\pi$
$308$	1.20971e10	0.0765958
$309$	1.63188e11	1.01830
$310$	−4.69955e10	−0.289021
$311$	3.49938e10	0.212114	0.106057	−	0.994360i	$-0.466177\pi$
$311$	3.49938e10	0.212114	0.106057	+	0.994360i	$0.466177\pi$
$312$	0	0
$313$	8.64720e10	0.509244	0.254622	−	0.967041i	$-0.418049\pi$
$313$	8.64720e10	0.509244	0.254622	+	0.967041i	$0.418049\pi$
$314$	2.37148e11	1.37669
$315$	−1.61197e10	−0.0922488
$316$	3.21992e10	0.181658
$317$	−3.25849e11	−1.81238	−0.906192	−	0.422866i	$-0.861024\pi$
$317$	−3.25849e11	−1.81238	−0.906192	+	0.422866i	$0.861024\pi$
$318$	2.50599e11	1.37422
$319$	−4.13300e9	−0.0223464
$320$	8.50073e10	0.453191
$321$	−7.62803e10	−0.400996
$322$	1.79909e11	0.932614
$323$	−9.49453e10	−0.485358
$324$	−3.42637e10	−0.172735
$325$	0	0
$326$	−1.51077e11	−0.740830
$327$	−1.37030e11	−0.662752
$328$	−1.12062e11	−0.534594
$329$	9.66987e10	0.455029
$330$	8.99386e10	0.417478
$331$	−2.79545e11	−1.28004	−0.640022	−	0.768356i	$-0.721075\pi$
$331$	−2.79545e11	−1.28004	−0.640022	+	0.768356i	$0.721075\pi$
$332$	−9.91579e10	−0.447926
$333$	7.35814e8	0.00327921
$334$	1.08534e11	0.477209
$335$	−1.89670e11	−0.822805
$336$	1.19364e11	0.510911
$337$	−2.35825e11	−0.995992	−0.497996	−	0.867179i	$-0.665931\pi$
$337$	−2.35825e11	−0.995992	−0.497996	+	0.867179i	$0.665931\pi$
$338$	0	0
$339$	2.47075e11	1.01609
$340$	−2.99410e10	−0.121510
$341$	5.39160e10	0.215935
$342$	5.35773e10	0.211769
$343$	−2.23723e11	−0.872746
$344$	3.59312e11	1.38344
$345$	2.72227e11	1.03453
$346$	−1.84788e10	−0.0693155
$347$	−2.53375e11	−0.938168	−0.469084	−	0.883153i	$-0.655416\pi$
$347$	−2.53375e11	−0.938168	−0.469084	+	0.883153i	$0.655416\pi$
$348$	2.23720e9	0.00817708
$349$	1.08825e11	0.392658	0.196329	−	0.980538i	$-0.437098\pi$
$349$	1.08825e11	0.392658	0.196329	+	0.980538i	$0.437098\pi$
$350$	7.57678e10	0.269885
$351$	0	0
$352$	8.65605e10	0.300523
$353$	−2.80993e11	−0.963186	−0.481593	−	0.876395i	$-0.659942\pi$
$353$	−2.80993e11	−0.963186	−0.481593	+	0.876395i	$0.659942\pi$
$354$	1.06057e11	0.358941
$355$	4.25660e11	1.42244
$356$	2.11151e10	0.0696736
$357$	8.71640e10	0.284008
$358$	−1.23515e10	−0.0397415
$359$	−2.86996e11	−0.911906	−0.455953	−	0.890004i	$-0.650702\pi$
$359$	−2.86996e11	−0.911906	−0.455953	+	0.890004i	$0.650702\pi$
$360$	−4.92242e10	−0.154460
$361$	−1.49039e11	−0.461867
$362$	−4.29681e11	−1.31510
$363$	1.81846e11	0.549698
$364$	0	0
$365$	1.11505e11	0.328834
$366$	−5.41571e11	−1.57758
$367$	−4.11627e11	−1.18442	−0.592211	−	0.805783i	$-0.701745\pi$
$367$	−4.11627e11	−1.18442	−0.592211	+	0.805783i	$0.701745\pi$
$368$	6.99571e11	1.98846
$369$	5.88013e10	0.165108
$370$	3.69520e9	0.0102501
$371$	2.58774e11	0.709151
$372$	−2.91848e10	−0.0790158
$373$	1.69982e11	0.454687	0.227344	−	0.973815i	$-0.426996\pi$
$373$	1.69982e11	0.454687	0.227344	+	0.973815i	$0.426996\pi$
$374$	1.68777e11	0.446057
$375$	3.51785e11	0.918621
$376$	2.95285e11	0.761896
$377$	0	0
$378$	−2.40101e11	−0.604896
$379$	−4.96621e11	−1.23637	−0.618185	−	0.786033i	$-0.712132\pi$
$379$	−4.96621e11	−1.23637	−0.618185	+	0.786033i	$0.712132\pi$
$380$	5.47601e10	0.134722
$381$	−3.04066e11	−0.739275
$382$	2.29519e11	0.551486
$383$	3.28319e11	0.779652	0.389826	−	0.920888i	$-0.372535\pi$
$383$	3.28319e11	0.779652	0.389826	+	0.920888i	$0.372535\pi$
$384$	4.42749e11	1.03912
$385$	9.28728e10	0.215434
$386$	−4.59362e11	−1.05320
$387$	−1.88539e11	−0.427270
$388$	−9.36134e10	−0.209698
$389$	−2.26591e11	−0.501729	−0.250865	−	0.968022i	$-0.580715\pi$
$389$	−2.26591e11	−0.501729	−0.250865	+	0.968022i	$0.580715\pi$
$390$	0	0
$391$	5.10854e11	1.10535
$392$	−2.93189e11	−0.627135
$393$	6.14363e11	1.29915
$394$	6.18343e11	1.29270
$395$	2.47201e11	0.510933
$396$	−1.93836e10	−0.0396101
$397$	−2.03383e10	−0.0410920	−0.0205460	−	0.999789i	$-0.506540\pi$
$397$	−2.03383e10	−0.0410920	−0.0205460	+	0.999789i	$0.506540\pi$
$398$	3.82797e10	0.0764707
$399$	−1.59417e11	−0.314889
$400$	2.94621e11	0.575431
$401$	−4.71475e11	−0.910561	−0.455280	−	0.890348i	$-0.650461\pi$
$401$	−4.71475e11	−0.910561	−0.455280	+	0.890348i	$0.650461\pi$
$402$	−5.78741e11	−1.10527
$403$	0	0
$404$	1.97685e11	0.369196
$405$	−2.63051e11	−0.485838
$406$	1.13510e10	0.0207332
$407$	−4.23934e9	−0.00765815
$408$	2.66169e11	0.475540
$409$	−7.28439e11	−1.28718	−0.643589	−	0.765372i	$-0.722555\pi$
$409$	−7.28439e11	−1.28718	−0.643589	+	0.765372i	$0.722555\pi$
$410$	2.95295e11	0.516094
$411$	−8.72056e11	−1.50750
$412$	−1.76622e11	−0.302001
$413$	1.09517e11	0.185227
$414$	−2.88273e11	−0.482284
$415$	−7.61259e11	−1.25984
$416$	0	0
$417$	6.15728e11	0.997187
$418$	−3.08682e11	−0.494558
$419$	−3.44730e11	−0.546407	−0.273204	−	0.961956i	$-0.588083\pi$
$419$	−3.44730e11	−0.546407	−0.273204	+	0.961956i	$0.588083\pi$
$420$	−5.02722e10	−0.0788327
$421$	−9.53236e11	−1.47887	−0.739436	−	0.673226i	$-0.764908\pi$
$421$	−9.53236e11	−1.47887	−0.739436	+	0.673226i	$0.764908\pi$
$422$	2.76550e10	0.0424490
$423$	−1.54943e11	−0.235310
$424$	7.90208e11	1.18740
$425$	2.15144e11	0.319874
$426$	1.29882e12	1.91076
$427$	−5.59240e11	−0.814090
$428$	8.25600e10	0.118925
$429$	0	0
$430$	−9.46829e11	−1.33556
$431$	2.77781e10	0.0387753	0.0193876	−	0.999812i	$-0.493828\pi$
$431$	2.77781e10	0.0387753	0.0193876	+	0.999812i	$0.493828\pi$
$432$	−9.33625e11	−1.28972
$433$	1.26496e12	1.72934	0.864669	−	0.502342i	$-0.167528\pi$
$433$	1.26496e12	1.72934	0.864669	+	0.502342i	$0.167528\pi$
$434$	−1.48076e11	−0.200346
$435$	1.71755e10	0.0229990
$436$	1.48311e11	0.196555
$437$	−9.34320e11	−1.22554
$438$	3.40236e11	0.441720
$439$	3.92097e11	0.503852	0.251926	−	0.967747i	$-0.418936\pi$
$439$	3.92097e11	0.503852	0.251926	+	0.967747i	$0.418936\pi$
$440$	2.83602e11	0.360721
$441$	1.53843e11	0.193689
$442$	0	0
$443$	−1.04786e12	−1.29267	−0.646336	−	0.763053i	$-0.723700\pi$
$443$	−1.04786e12	−1.29267	−0.646336	+	0.763053i	$0.723700\pi$
$444$	2.29477e9	0.00280230
$445$	1.62106e11	0.195965
$446$	−1.38796e12	−1.66100
$447$	6.28967e11	0.745151
$448$	2.67845e11	0.314147
$449$	−7.07476e11	−0.821491	−0.410746	−	0.911750i	$-0.634732\pi$
$449$	−7.07476e11	−0.821491	−0.410746	+	0.911750i	$0.634732\pi$
$450$	−1.21405e11	−0.139566
$451$	−3.38780e11	−0.385587
$452$	−2.67415e11	−0.301344
$453$	−6.71342e10	−0.0749035
$454$	−3.24952e11	−0.358978
$455$	0	0
$456$	−4.86806e11	−0.527247
$457$	1.77127e12	1.89960	0.949800	−	0.312858i	$-0.101286\pi$
$457$	1.77127e12	1.89960	0.949800	+	0.312858i	$0.101286\pi$
$458$	−1.43620e12	−1.52517
$459$	−6.81770e11	−0.716937
$460$	−2.94637e11	−0.306816
$461$	−9.29500e11	−0.958507	−0.479254	−	0.877677i	$-0.659093\pi$
$461$	−9.29500e11	−0.958507	−0.479254	+	0.877677i	$0.659093\pi$
$462$	2.83383e11	0.289391
$463$	−1.39217e12	−1.40792	−0.703959	−	0.710240i	$-0.748586\pi$
$463$	−1.39217e12	−1.40792	−0.703959	+	0.710240i	$0.748586\pi$
$464$	4.41379e10	0.0442059
$465$	−2.24059e11	−0.222241
$466$	7.28357e10	0.0715497
$467$	1.04856e11	0.102016	0.0510079	−	0.998698i	$-0.483757\pi$
$467$	1.04856e11	0.102016	0.0510079	+	0.998698i	$0.483757\pi$
$468$	0	0
$469$	−5.97622e11	−0.570360
$470$	−7.78110e11	−0.735530
$471$	1.13064e12	1.05860
$472$	3.34426e11	0.310142
$473$	1.08626e12	0.997832
$474$	7.54288e11	0.686332
$475$	−3.93484e11	−0.354655
$476$	−9.43397e10	−0.0842293
$477$	−4.14641e11	−0.366724
$478$	2.26513e11	0.198457
$479$	−8.01177e11	−0.695374	−0.347687	−	0.937611i	$-0.613033\pi$
$479$	−8.01177e11	−0.695374	−0.347687	+	0.937611i	$0.613033\pi$
$480$	−3.59720e11	−0.309300
$481$	0	0
$482$	−8.19457e11	−0.691535
$483$	8.57747e11	0.717129
$484$	−1.96816e11	−0.163026
$485$	−7.18693e11	−0.589801
$486$	6.90628e11	0.561540
$487$	−3.80130e11	−0.306233	−0.153116	−	0.988208i	$-0.548931\pi$
$487$	−3.80130e11	−0.306233	−0.153116	+	0.988208i	$0.548931\pi$
$488$	−1.70773e12	−1.36310
$489$	−7.20283e11	−0.569657
$490$	7.72588e11	0.605432
$491$	5.61326e10	0.0435861	0.0217931	−	0.999763i	$-0.493063\pi$
$491$	5.61326e10	0.0435861	0.0217931	+	0.999763i	$0.493063\pi$
$492$	1.83382e11	0.141096
$493$	3.22312e10	0.0245734
$494$	0	0
$495$	−1.48812e11	−0.111408
$496$	−5.75789e11	−0.427165
$497$	1.34119e12	0.986024
$498$	−2.32284e12	−1.69234
$499$	−1.63734e12	−1.18219	−0.591094	−	0.806603i	$-0.701304\pi$
$499$	−1.63734e12	−1.18219	−0.591094	+	0.806603i	$0.701304\pi$
$500$	−3.80745e11	−0.272439
$501$	5.17456e11	0.366947
$502$	2.79298e12	1.96292
$503$	−1.97834e12	−1.37799	−0.688994	−	0.724767i	$-0.741947\pi$
$503$	−1.97834e12	−1.37799	−0.688994	+	0.724767i	$0.741947\pi$
$504$	−1.55098e11	−0.107070
$505$	1.51768e12	1.03841
$506$	1.66086e12	1.12631
$507$	0	0
$508$	3.29098e11	0.219250
$509$	1.78443e12	1.17834	0.589169	−	0.808010i	$-0.299455\pi$
$509$	1.78443e12	1.17834	0.589169	+	0.808010i	$0.299455\pi$
$510$	−7.01386e11	−0.459083
$511$	3.51336e11	0.227944
$512$	6.19448e11	0.398373
$513$	1.24691e12	0.794892
$514$	1.02097e12	0.645179
$515$	−1.35597e12	−0.849411
$516$	−5.87993e11	−0.365131
$517$	8.92692e11	0.549534
$518$	1.16430e10	0.00710529
$519$	−8.81006e10	−0.0532998
$520$	0	0
$521$	1.61000e12	0.957319	0.478659	−	0.878001i	$-0.341123\pi$
$521$	1.61000e12	0.957319	0.478659	+	0.878001i	$0.341123\pi$
$522$	−1.81879e10	−0.0107218
$523$	3.33912e12	1.95152	0.975762	−	0.218833i	$-0.0702251\pi$
$523$	3.33912e12	1.95152	0.975762	+	0.218833i	$0.0702251\pi$
$524$	−6.64939e11	−0.385293
$525$	3.61236e11	0.207527
$526$	−1.63745e12	−0.932678
$527$	−4.20464e11	−0.237455
$528$	1.10193e12	0.617022
$529$	3.22596e12	1.79106
$530$	−2.08229e12	−1.14630
$531$	−1.75481e11	−0.0957867
$532$	1.72541e11	0.0933879
$533$	0	0
$534$	4.94635e11	0.263238
$535$	6.33833e11	0.334490
$536$	−1.82493e12	−0.955005
$537$	−5.88876e10	−0.0305590
$538$	1.10623e12	0.569280
$539$	−8.86357e11	−0.452334
$540$	3.93214e11	0.199002
$541$	3.66759e12	1.84074	0.920372	−	0.391044i	$-0.127886\pi$
$541$	3.66759e12	1.84074	0.920372	+	0.391044i	$0.127886\pi$
$542$	1.95371e12	0.972442
$543$	−2.04857e12	−1.01124
$544$	−6.75043e11	−0.330473
$545$	1.13862e12	0.552833
$546$	0	0
$547$	2.06591e12	0.986664	0.493332	−	0.869841i	$-0.335779\pi$
$547$	2.06591e12	0.986664	0.493332	+	0.869841i	$0.335779\pi$
$548$	9.43847e11	0.447084
$549$	8.96084e11	0.420991
$550$	6.99465e11	0.325937
$551$	−5.89488e10	−0.0272454
$552$	2.61927e12	1.20075
$553$	7.78896e11	0.354174
$554$	4.58316e12	2.06715
$555$	1.76175e10	0.00788180
$556$	−6.66417e11	−0.295739
$557$	3.46024e12	1.52320	0.761601	−	0.648046i	$-0.224414\pi$
$557$	3.46024e12	1.52320	0.761601	+	0.648046i	$0.224414\pi$
$558$	2.37266e11	0.103605
$559$	0	0
$560$	−9.91824e11	−0.426175
$561$	8.04671e11	0.342993
$562$	−2.06281e12	−0.872259
$563$	2.77992e12	1.16612	0.583061	−	0.812428i	$-0.301855\pi$
$563$	2.77992e12	1.16612	0.583061	+	0.812428i	$0.301855\pi$
$564$	−4.83216e11	−0.201088
$565$	−2.05301e12	−0.847565
$566$	6.08831e11	0.249358
$567$	−8.28834e11	−0.336778
$568$	4.09554e12	1.65099
$569$	−3.98689e12	−1.59452	−0.797259	−	0.603638i	$-0.793717\pi$
$569$	−3.98689e12	−1.59452	−0.797259	+	0.603638i	$0.793717\pi$
$570$	1.28279e12	0.509001
$571$	−7.46531e11	−0.293890	−0.146945	−	0.989145i	$-0.546944\pi$
$571$	−7.46531e11	−0.293890	−0.146945	+	0.989145i	$0.546944\pi$
$572$	0	0
$573$	1.09427e12	0.424062
$574$	9.30432e11	0.357751
$575$	2.11714e12	0.807691
$576$	−4.29176e11	−0.162455
$577$	−4.50853e12	−1.69334	−0.846669	−	0.532119i	$-0.821396\pi$
$577$	−4.50853e12	−1.69334	−0.846669	+	0.532119i	$0.821396\pi$
$578$	1.69049e12	0.629994
$579$	−2.19008e12	−0.809855
$580$	−1.85895e10	−0.00682090
$581$	−2.39862e12	−0.873310
$582$	−2.19295e12	−0.792275
$583$	2.38892e12	0.856434
$584$	1.07286e12	0.381668
$585$	0	0
$586$	4.30741e12	1.50896
$587$	−3.22309e12	−1.12047	−0.560236	−	0.828333i	$-0.689290\pi$
$587$	−3.22309e12	−1.12047	−0.560236	+	0.828333i	$0.689290\pi$
$588$	4.79787e11	0.165520
$589$	7.69002e11	0.263274
$590$	−8.81251e11	−0.299410
$591$	2.94806e12	0.994013
$592$	4.52736e10	0.0151495
$593$	−3.88467e12	−1.29006	−0.645028	−	0.764159i	$-0.723154\pi$
$593$	−3.88467e12	−1.29006	−0.645028	+	0.764159i	$0.723154\pi$
$594$	−2.21654e12	−0.730527
$595$	−7.24269e11	−0.236904
$596$	−6.80746e11	−0.220992
$597$	1.82505e11	0.0588017
$598$	0	0
$599$	6.29932e11	0.199928	0.0999639	−	0.994991i	$-0.468127\pi$
$599$	6.29932e11	0.199928	0.0999639	+	0.994991i	$0.468127\pi$
$600$	1.10309e12	0.347481
$601$	2.00132e12	0.625722	0.312861	−	0.949799i	$-0.398713\pi$
$601$	2.00132e12	0.625722	0.312861	+	0.949799i	$0.398713\pi$
$602$	−2.98332e12	−0.925797
$603$	9.57585e11	0.294951
$604$	7.26610e10	0.0222144
$605$	−1.51101e12	−0.458529
$606$	4.63089e12	1.39488
$607$	−5.18880e11	−0.155138	−0.0775690	−	0.996987i	$-0.524716\pi$
$607$	−5.18880e11	−0.155138	−0.0775690	+	0.996987i	$0.524716\pi$
$608$	1.23461e12	0.366407
$609$	5.41176e10	0.0159427
$610$	4.50006e12	1.31593
$611$	0	0
$612$	1.51163e11	0.0435576
$613$	4.65724e12	1.33216	0.666080	−	0.745881i	$-0.267971\pi$
$613$	4.65724e12	1.33216	0.666080	+	0.745881i	$0.267971\pi$
$614$	7.14003e10	0.0202741
$615$	1.40787e12	0.396848
$616$	8.93588e11	0.250048
$617$	3.25346e12	0.903779	0.451890	−	0.892074i	$-0.350750\pi$
$617$	3.25346e12	0.903779	0.451890	+	0.892074i	$0.350750\pi$
$618$	−4.13749e12	−1.14101
$619$	8.84891e11	0.242260	0.121130	−	0.992637i	$-0.461348\pi$
$619$	8.84891e11	0.242260	0.121130	+	0.992637i	$0.461348\pi$
$620$	2.42504e11	0.0659109
$621$	−6.70903e12	−1.81029
$622$	−8.87238e11	−0.237675
$623$	5.10772e11	0.135841
$624$	0	0
$625$	−1.07882e12	−0.282806
$626$	−2.19242e12	−0.570610
$627$	−1.47169e12	−0.380288
$628$	−1.22372e12	−0.313952
$629$	3.30605e10	0.00842135
$630$	4.08702e11	0.103365
$631$	3.91487e12	0.983073	0.491536	−	0.870857i	$-0.336436\pi$
$631$	3.91487e12	0.983073	0.491536	+	0.870857i	$0.336436\pi$
$632$	2.37848e12	0.593025
$633$	1.31850e11	0.0326410
$634$	8.26162e12	2.03078
$635$	2.52657e12	0.616664
$636$	−1.29313e12	−0.313390
$637$	0	0
$638$	1.04789e11	0.0250392
$639$	−2.14903e12	−0.509903
$640$	−3.67892e12	−0.866782
$641$	−6.32532e12	−1.47986	−0.739932	−	0.672682i	$-0.765142\pi$
$641$	−6.32532e12	−1.47986	−0.739932	+	0.672682i	$0.765142\pi$
$642$	1.93402e12	0.449318
$643$	−6.11217e12	−1.41009	−0.705044	−	0.709164i	$-0.749073\pi$
$643$	−6.11217e12	−1.41009	−0.705044	+	0.709164i	$0.749073\pi$
$644$	−9.28360e11	−0.212682
$645$	−4.51416e12	−1.02697
$646$	2.40725e12	0.543846
$647$	6.01346e12	1.34913	0.674567	−	0.738214i	$-0.264330\pi$
$647$	6.01346e12	1.34913	0.674567	+	0.738214i	$0.264330\pi$
$648$	−2.53098e12	−0.563898
$649$	1.01102e12	0.223697
$650$	0	0
$651$	−7.05977e11	−0.154055
$652$	7.79580e11	0.168945
$653$	2.64895e12	0.570117	0.285058	−	0.958510i	$-0.407987\pi$
$653$	2.64895e12	0.570117	0.285058	+	0.958510i	$0.407987\pi$
$654$	3.47428e12	0.742616
$655$	−5.10490e12	−1.08368
$656$	3.61796e12	0.762774
$657$	−5.62955e11	−0.117877
$658$	−2.45171e12	−0.509862
$659$	−6.06796e12	−1.25331	−0.626655	−	0.779297i	$-0.715576\pi$
$659$	−6.06796e12	−1.25331	−0.626655	+	0.779297i	$0.715576\pi$
$660$	−4.64097e11	−0.0952054
$661$	−2.97125e12	−0.605386	−0.302693	−	0.953088i	$-0.597886\pi$
$661$	−2.97125e12	−0.605386	−0.302693	+	0.953088i	$0.597886\pi$
$662$	7.08761e12	1.43430
$663$	0	0
$664$	−7.32456e12	−1.46226
$665$	1.32464e12	0.262664
$666$	−1.86559e10	−0.00367437
$667$	3.17175e11	0.0620486
$668$	−5.60055e11	−0.108827
$669$	−6.61732e12	−1.27722
$670$	4.80891e12	0.921956
$671$	−5.16272e12	−0.983168
$672$	−1.13343e12	−0.214403
$673$	2.96918e12	0.557916	0.278958	−	0.960303i	$-0.410011\pi$
$673$	2.96918e12	0.557916	0.278958	+	0.960303i	$0.410011\pi$
$674$	5.97914e12	1.11601
$675$	−2.82547e12	−0.523871
$676$	0	0
$677$	−2.66655e12	−0.487866	−0.243933	−	0.969792i	$-0.578438\pi$
$677$	−2.66655e12	−0.487866	−0.243933	+	0.969792i	$0.578438\pi$
$678$	−6.26437e12	−1.13853
$679$	−2.26450e12	−0.408844
$680$	−2.21167e12	−0.396671
$681$	−1.54926e12	−0.276034
$682$	−1.36699e12	−0.241956
$683$	−2.08876e12	−0.367278	−0.183639	−	0.982994i	$-0.558788\pi$
$683$	−2.08876e12	−0.367278	−0.183639	+	0.982994i	$0.558788\pi$
$684$	−2.76467e11	−0.0482938
$685$	7.24614e12	1.25747
$686$	5.67231e12	0.977915
$687$	−6.84730e12	−1.17277
$688$	−1.16005e13	−1.97392
$689$	0	0
$690$	−6.90207e12	−1.15920
$691$	2.67314e12	0.446036	0.223018	−	0.974814i	$-0.428409\pi$
$691$	2.67314e12	0.446036	0.223018	+	0.974814i	$0.428409\pi$
$692$	9.53534e10	0.0158073
$693$	−4.68886e11	−0.0772268
$694$	6.42410e12	1.05122
$695$	−5.11624e12	−0.831801
$696$	1.65257e11	0.0266942
$697$	2.64197e12	0.424015
$698$	−2.75916e12	−0.439975
$699$	3.47257e11	0.0550178
$700$	−3.90974e11	−0.0615470
$701$	−6.13467e12	−0.959533	−0.479766	−	0.877396i	$-0.659279\pi$
$701$	−6.13467e12	−0.959533	−0.479766	+	0.877396i	$0.659279\pi$
$702$	0	0
$703$	−6.04656e10	−0.00933704
$704$	2.47267e12	0.379392
$705$	−3.70977e12	−0.565582
$706$	7.12434e12	1.07925
$707$	4.78197e12	0.719813
$708$	−5.47268e11	−0.0818560
$709$	−9.96058e11	−0.148039	−0.0740196	−	0.997257i	$-0.523583\pi$
$709$	−9.96058e11	−0.148039	−0.0740196	+	0.997257i	$0.523583\pi$
$710$	−1.07922e13	−1.59385
$711$	−1.24804e12	−0.183154
$712$	1.55972e12	0.227451
$713$	−4.13762e12	−0.599581
$714$	−2.20997e12	−0.318232
$715$	0	0
$716$	6.37355e10	0.00906301
$717$	1.07994e12	0.152603
$718$	7.27652e12	1.02180
$719$	−5.70305e11	−0.0795842	−0.0397921	−	0.999208i	$-0.512670\pi$
$719$	−5.70305e11	−0.0795842	−0.0397921	+	0.999208i	$0.512670\pi$
$720$	1.58922e12	0.220389
$721$	−4.27247e12	−0.588803
$722$	3.77875e12	0.517524
$723$	−3.90690e12	−0.531752
$724$	2.21722e12	0.299906
$725$	1.33576e11	0.0179560
$726$	−4.61054e12	−0.615939
$727$	−1.45252e13	−1.92849	−0.964245	−	0.265014i	$-0.914623\pi$
$727$	−1.45252e13	−1.92849	−0.964245	+	0.265014i	$0.914623\pi$
$728$	0	0
$729$	8.44751e12	1.10778
$730$	−2.82711e12	−0.368460
$731$	−8.47118e12	−1.09728
$732$	2.79459e12	0.359765
$733$	1.00657e13	1.28788	0.643939	−	0.765077i	$-0.277299\pi$
$733$	1.00657e13	1.28788	0.643939	+	0.765077i	$0.277299\pi$
$734$	1.04364e13	1.32715
$735$	3.68344e12	0.465544
$736$	−6.64283e12	−0.834455
$737$	−5.51706e12	−0.688817
$738$	−1.49085e12	−0.185004
$739$	−8.88151e12	−1.09544	−0.547718	−	0.836663i	$-0.684503\pi$
$739$	−8.88151e12	−1.09544	−0.547718	+	0.836663i	$0.684503\pi$
$740$	−1.90678e10	−0.00233753
$741$	0	0
$742$	−6.56099e12	−0.794606
$743$	−7.65719e12	−0.921765	−0.460882	−	0.887461i	$-0.652467\pi$
$743$	−7.65719e12	−0.921765	−0.460882	+	0.887461i	$0.652467\pi$
$744$	−2.15581e12	−0.257949
$745$	−5.22625e12	−0.621566
$746$	−4.30974e12	−0.509479
$747$	3.84337e12	0.451616
$748$	−8.70914e11	−0.101723
$749$	1.99712e12	0.231865
$750$	−8.91920e12	−1.02932
$751$	7.39052e12	0.847804	0.423902	−	0.905708i	$-0.360660\pi$
$751$	7.39052e12	0.847804	0.423902	+	0.905708i	$0.360660\pi$
$752$	−9.53340e12	−1.08710
$753$	1.33160e13	1.50937
$754$	0	0
$755$	5.57836e11	0.0624806
$756$	1.23896e12	0.137946
$757$	−1.27866e12	−0.141522	−0.0707609	−	0.997493i	$-0.522543\pi$
$757$	−1.27866e12	−0.141522	−0.0707609	+	0.997493i	$0.522543\pi$
$758$	1.25914e13	1.38536
$759$	7.91845e12	0.866068
$760$	4.04500e12	0.439802
$761$	−6.99851e12	−0.756440	−0.378220	−	0.925716i	$-0.623464\pi$
$761$	−6.99851e12	−0.756440	−0.378220	+	0.925716i	$0.623464\pi$
$762$	7.70933e12	0.828360
$763$	3.58762e12	0.383218
$764$	−1.18436e12	−0.125766
$765$	1.16051e12	0.122511
$766$	−8.32423e12	−0.873604
$767$	0	0
$768$	−5.98753e12	−0.621044
$769$	−1.08626e13	−1.12012	−0.560059	−	0.828453i	$-0.689222\pi$
$769$	−1.08626e13	−1.12012	−0.560059	+	0.828453i	$0.689222\pi$
$770$	−2.35471e12	−0.241395
$771$	4.86766e12	0.496107
$772$	2.37038e12	0.240182
$773$	8.10088e12	0.816064	0.408032	−	0.912968i	$-0.366215\pi$
$773$	8.10088e12	0.816064	0.408032	+	0.912968i	$0.366215\pi$
$774$	4.78025e12	0.478758
$775$	−1.74254e12	−0.173510
$776$	−6.91500e12	−0.684565
$777$	5.55101e10	0.00546358
$778$	5.74502e12	0.562190
$779$	−4.83200e12	−0.470120
$780$	0	0
$781$	1.23815e13	1.19081
$782$	−1.29523e13	−1.23855
$783$	−4.23291e11	−0.0402449
$784$	9.46575e12	0.894814
$785$	−9.39480e12	−0.883027
$786$	−1.55766e13	−1.45570
$787$	1.93126e13	1.79454	0.897271	−	0.441480i	$-0.145546\pi$
$787$	1.93126e13	1.79454	0.897271	+	0.441480i	$0.145546\pi$
$788$	−3.19075e12	−0.294798
$789$	−7.80680e12	−0.717178
$790$	−6.26757e12	−0.572503
$791$	−6.46874e12	−0.587524
$792$	−1.43182e12	−0.129308
$793$	0	0
$794$	5.15660e11	0.0460438
$795$	−9.92767e12	−0.881445
$796$	−1.97529e11	−0.0174391
$797$	−1.16826e13	−1.02560	−0.512799	−	0.858509i	$-0.671391\pi$
$797$	−1.16826e13	−1.02560	−0.512799	+	0.858509i	$0.671391\pi$
$798$	4.04189e12	0.352835
$799$	−6.96166e12	−0.604300
$800$	−2.79759e12	−0.241479
$801$	−8.18423e11	−0.0702476
$802$	1.19538e13	1.02029
$803$	3.24343e12	0.275286
$804$	2.98640e12	0.252055
$805$	−7.12724e12	−0.598191
$806$	0	0
$807$	5.27415e12	0.437745
$808$	1.46025e13	1.20525
$809$	1.93154e12	0.158539	0.0792694	−	0.996853i	$-0.474741\pi$
$809$	1.93154e12	0.158539	0.0792694	+	0.996853i	$0.474741\pi$
$810$	6.66942e12	0.544384
$811$	−2.04410e13	−1.65923	−0.829616	−	0.558334i	$-0.811441\pi$
$811$	−2.04410e13	−1.65923	−0.829616	+	0.558334i	$0.811441\pi$
$812$	−5.85728e10	−0.00472818
$813$	9.31465e12	0.747755
$814$	1.07485e11	0.00858099
$815$	5.98502e12	0.475178
$816$	−8.59339e12	−0.678514
$817$	1.54932e13	1.21659
$818$	1.84689e13	1.44229
$819$	0	0
$820$	−1.52377e12	−0.117695
$821$	3.67240e12	0.282102	0.141051	−	0.990002i	$-0.454952\pi$
$821$	3.67240e12	0.282102	0.141051	+	0.990002i	$0.454952\pi$
$822$	2.21102e13	1.68916
$823$	1.15495e13	0.877530	0.438765	−	0.898602i	$-0.355416\pi$
$823$	1.15495e13	0.877530	0.438765	+	0.898602i	$0.355416\pi$
$824$	−1.30467e13	−0.985887
$825$	3.33481e12	0.250628
$826$	−2.77669e12	−0.207548
$827$	4.87130e11	0.0362135	0.0181067	−	0.999836i	$-0.494236\pi$
$827$	4.87130e11	0.0362135	0.0181067	+	0.999836i	$0.494236\pi$
$828$	1.48753e12	0.109984
$829$	3.88609e12	0.285770	0.142885	−	0.989739i	$-0.454362\pi$
$829$	3.88609e12	0.285770	0.142885	+	0.989739i	$0.454362\pi$
$830$	1.93011e13	1.41166
$831$	2.18510e13	1.58952
$832$	0	0
$833$	6.91226e12	0.497414
$834$	−1.56112e13	−1.11735
$835$	−4.29968e12	−0.306088
$836$	1.59285e12	0.112784
$837$	5.52194e12	0.388890
$838$	8.74034e12	0.612252
$839$	−1.15192e13	−0.802588	−0.401294	−	0.915949i	$-0.631439\pi$
$839$	−1.15192e13	−0.802588	−0.401294	+	0.915949i	$0.631439\pi$
$840$	−3.71349e12	−0.257351
$841$	−1.44871e13	−0.998621
$842$	2.41684e13	1.65708
$843$	−9.83479e12	−0.670720
$844$	−1.42704e11	−0.00968046
$845$	0	0
$846$	3.92844e12	0.263665
$847$	−4.76096e12	−0.317848
$848$	−2.55122e13	−1.69421
$849$	2.90270e12	0.191742
$850$	−5.45478e12	−0.358420
$851$	3.25336e11	0.0212642
$852$	−6.70212e12	−0.435746
$853$	2.71704e13	1.75721	0.878607	−	0.477545i	$-0.158473\pi$
$853$	2.71704e13	1.75721	0.878607	+	0.477545i	$0.158473\pi$
$854$	1.41790e13	0.912191
$855$	−2.12251e12	−0.135832
$856$	6.09851e12	0.388233
$857$	1.89111e13	1.19758	0.598789	−	0.800907i	$-0.295649\pi$
$857$	1.89111e13	1.19758	0.598789	+	0.800907i	$0.295649\pi$
$858$	0	0
$859$	−1.93022e12	−0.120959	−0.0604793	−	0.998169i	$-0.519263\pi$
$859$	−1.93022e12	−0.120959	−0.0604793	+	0.998169i	$0.519263\pi$
$860$	4.88579e12	0.304573
$861$	4.43599e12	0.275091
$862$	−7.04289e11	−0.0434478
$863$	−2.27253e13	−1.39463	−0.697317	−	0.716763i	$-0.745623\pi$
$863$	−2.27253e13	−1.39463	−0.697317	+	0.716763i	$0.745623\pi$
$864$	8.86531e12	0.541230
$865$	7.32051e11	0.0444599
$866$	−3.20719e13	−1.93773
$867$	8.05967e12	0.484431
$868$	7.64096e11	0.0456887
$869$	7.19052e12	0.427732
$870$	−4.35471e11	−0.0257705
$871$	0	0
$872$	1.09554e13	0.641657
$873$	3.62846e12	0.211426
$874$	2.36889e13	1.37323
$875$	−9.21018e12	−0.531168
$876$	−1.75567e12	−0.100734
$877$	1.75855e13	1.00382	0.501911	−	0.864920i	$-0.332631\pi$
$877$	1.75855e13	1.00382	0.501911	+	0.864920i	$0.332631\pi$
$878$	−9.94127e12	−0.564568
$879$	2.05363e13	1.16030
$880$	−9.15620e12	−0.514687
$881$	−1.91369e13	−1.07023	−0.535117	−	0.844778i	$-0.679733\pi$
$881$	−1.91369e13	−1.07023	−0.535117	+	0.844778i	$0.679733\pi$
$882$	−3.90056e12	−0.217029
$883$	1.58819e12	0.0879182	0.0439591	−	0.999033i	$-0.486003\pi$
$883$	1.58819e12	0.0879182	0.0439591	+	0.999033i	$0.486003\pi$
$884$	0	0
$885$	−4.20151e12	−0.230229
$886$	2.65677e13	1.44844
$887$	−1.40852e13	−0.764023	−0.382011	−	0.924158i	$-0.624768\pi$
$887$	−1.40852e13	−0.764023	−0.382011	+	0.924158i	$0.624768\pi$
$888$	1.69509e11	0.00914817
$889$	7.96084e12	0.427465
$890$	−4.11005e12	−0.219580
$891$	−7.65154e12	−0.406723
$892$	7.16208e12	0.378789
$893$	1.27324e13	0.670008
$894$	−1.59469e13	−0.834945
$895$	4.89313e11	0.0254908
$896$	−1.15917e13	−0.600845
$897$	0	0
$898$	1.79374e13	0.920485
$899$	−2.61054e11	−0.0133294
$900$	6.26467e11	0.0318278
$901$	−1.86300e13	−0.941785
$902$	8.58946e12	0.432052
$903$	−1.42235e13	−0.711887
$904$	−1.97533e13	−0.983744
$905$	1.70221e13	0.843520
$906$	1.70213e12	0.0839297
$907$	−9.90700e12	−0.486082	−0.243041	−	0.970016i	$-0.578145\pi$
$907$	−9.90700e12	−0.486082	−0.243041	+	0.970016i	$0.578145\pi$
$908$	1.67680e12	0.0818646
$909$	−7.66228e12	−0.372238
$910$	0	0
$911$	−2.10442e13	−1.01228	−0.506139	−	0.862452i	$-0.668928\pi$
$911$	−2.10442e13	−1.01228	−0.506139	+	0.862452i	$0.668928\pi$
$912$	1.57168e13	0.752291
$913$	−2.21433e13	−1.05469
$914$	−4.49090e13	−2.12851
$915$	2.14548e13	1.01188
$916$	7.41100e12	0.347814
$917$	−1.60848e13	−0.751197
$918$	1.72857e13	0.803331
$919$	−2.30007e13	−1.06371	−0.531853	−	0.846837i	$-0.678504\pi$
$919$	−2.30007e13	−1.06371	−0.531853	+	0.846837i	$0.678504\pi$
$920$	−2.17642e13	−1.00161
$921$	3.40413e11	0.0155897
$922$	2.35667e13	1.07401
$923$	0	0
$924$	−1.46230e12	−0.0659954
$925$	1.37013e11	0.00615355
$926$	3.52972e13	1.57758
$927$	6.84589e12	0.304488
$928$	−4.19114e11	−0.0185510
$929$	1.12282e13	0.494582	0.247291	−	0.968941i	$-0.420460\pi$
$929$	1.12282e13	0.494582	0.247291	+	0.968941i	$0.420460\pi$
$930$	5.68082e12	0.249022
$931$	−1.26421e13	−0.551499
$932$	−3.75844e11	−0.0163168
$933$	−4.23005e12	−0.182759
$934$	−2.65853e12	−0.114309
$935$	−6.68622e12	−0.286107
$936$	0	0
$937$	−2.59924e13	−1.10159	−0.550794	−	0.834641i	$-0.685675\pi$
$937$	−2.59924e13	−1.10159	−0.550794	+	0.834641i	$0.685675\pi$
$938$	1.51522e13	0.639091
$939$	−1.04527e13	−0.438768
$940$	4.01517e12	0.167737
$941$	−1.42934e13	−0.594268	−0.297134	−	0.954836i	$-0.596031\pi$
$941$	−1.42934e13	−0.594268	−0.297134	+	0.954836i	$0.596031\pi$
$942$	−2.86664e13	−1.18616
$943$	2.59986e13	1.07065
$944$	−1.07971e13	−0.442520
$945$	9.51179e12	0.387989
$946$	−2.75411e13	−1.11807
$947$	3.34514e13	1.35157	0.675787	−	0.737097i	$-0.263804\pi$
$947$	3.34514e13	1.35157	0.675787	+	0.737097i	$0.263804\pi$
$948$	−3.89224e12	−0.156517
$949$	0	0
$950$	9.97644e12	0.397392
$951$	3.93887e13	1.56156
$952$	−6.96865e12	−0.274968
$953$	−2.89560e13	−1.13716	−0.568579	−	0.822629i	$-0.692507\pi$
$953$	−2.89560e13	−1.13716	−0.568579	+	0.822629i	$0.692507\pi$
$954$	1.05128e13	0.410916
$955$	−9.09259e12	−0.353730
$956$	−1.16884e12	−0.0452580
$957$	4.99597e11	0.0192538
$958$	2.03131e13	0.779170
$959$	2.28315e13	0.871668
$960$	−1.02757e13	−0.390472
$961$	−2.30341e13	−0.871197
$962$	0	0
$963$	−3.20003e12	−0.119905
$964$	4.22853e12	0.157704
$965$	1.81980e13	0.675539
$966$	−2.17474e13	−0.803546
$967$	3.68608e13	1.35564	0.677822	−	0.735226i	$-0.262924\pi$
$967$	3.68608e13	1.35564	0.677822	+	0.735226i	$0.262924\pi$
$968$	−1.45383e13	−0.532201
$969$	1.14770e13	0.418187
$970$	1.82218e13	0.660874
$971$	3.60011e13	1.29966	0.649830	−	0.760080i	$-0.274840\pi$
$971$	3.60011e13	1.29966	0.649830	+	0.760080i	$0.274840\pi$
$972$	−3.56375e12	−0.128059
$973$	−1.61205e13	−0.576596
$974$	9.63786e12	0.343135
$975$	0	0
$976$	5.51347e13	1.94492
$977$	2.70378e13	0.949391	0.474695	−	0.880150i	$-0.342558\pi$
$977$	2.70378e13	0.949391	0.474695	+	0.880150i	$0.342558\pi$
$978$	1.82622e13	0.638303
$979$	4.71529e12	0.164054
$980$	−3.98668e12	−0.138068
$981$	−5.74854e12	−0.198174
$982$	−1.42319e12	−0.0488384
$983$	1.52920e13	0.522365	0.261182	−	0.965289i	$-0.415888\pi$
$983$	1.52920e13	0.522365	0.261182	+	0.965289i	$0.415888\pi$
$984$	1.35460e13	0.460610
$985$	−2.44962e13	−0.829154
$986$	−8.17194e11	−0.0275346
$987$	−1.16889e13	−0.392056
$988$	0	0
$989$	−8.33615e13	−2.77065
$990$	3.77301e12	0.124833
$991$	2.76896e13	0.911979	0.455990	−	0.889985i	$-0.349285\pi$
$991$	2.76896e13	0.911979	0.455990	+	0.889985i	$0.349285\pi$
$992$	5.46745e12	0.179260
$993$	3.37914e13	1.10289
$994$	−3.40048e13	−1.10484
$995$	−1.51648e12	−0.0490494
$996$	1.19862e13	0.385936
$997$	2.89579e13	0.928193	0.464096	−	0.885785i	$-0.346379\pi$
$997$	2.89579e13	0.928193	0.464096	+	0.885785i	$0.346379\pi$
$998$	4.15133e13	1.32465
$999$	−4.34183e11	−0.0137920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.g.1.8	✓	27
13.12	even	2		169.10.a.h.1.20	yes	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.10.a.g.1.8	✓	27	1.1	even	1	trivial
169.10.a.h.1.20	yes	27	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q-25.3541 q^{2} -120.880 q^{3} +130.831 q^{4} +1004.42 q^{5} +3064.81 q^{6} +3164.79 q^{7} +9664.20 q^{8} -5071.03 q^{9} -25466.3 q^{10} +29216.4 q^{11} -15814.9 q^{12} -80240.5 q^{14} -121415. q^{15} -312013. q^{16} -227844. q^{17} +128571. q^{18} +416712. q^{19} +131410. q^{20} -382560. q^{21} -740755. q^{22} -2.24212e6 q^{23} -1.16821e6 q^{24} -944259. q^{25} +2.99227e6 q^{27} +414054. q^{28} -141462. q^{29} +3.07836e6 q^{30} +1.84540e6 q^{31} +2.96274e6 q^{32} -3.53167e6 q^{33} +5.77678e6 q^{34} +3.17879e6 q^{35} -663449. q^{36} -145102. q^{37} -1.05654e7 q^{38} +9.70695e6 q^{40} -1.15955e7 q^{41} +9.69947e6 q^{42} +3.71797e7 q^{43} +3.82242e6 q^{44} -5.09346e6 q^{45} +5.68471e7 q^{46} +3.05545e7 q^{47} +3.77161e7 q^{48} -3.03377e7 q^{49} +2.39408e7 q^{50} +2.75418e7 q^{51} +8.17666e7 q^{53} -7.58663e7 q^{54} +2.93456e7 q^{55} +3.05852e7 q^{56} -5.03721e7 q^{57} +3.58664e6 q^{58} +3.46046e7 q^{59} -1.58848e7 q^{60} -1.76707e8 q^{61} -4.67886e7 q^{62} -1.60487e7 q^{63} +8.46329e7 q^{64} +8.95425e7 q^{66} -1.88835e8 q^{67} -2.98091e7 q^{68} +2.71028e8 q^{69} -8.05955e7 q^{70} +4.23785e8 q^{71} -4.90074e7 q^{72} +1.11014e8 q^{73} +3.67892e6 q^{74} +1.14142e8 q^{75} +5.45190e7 q^{76} +9.24637e7 q^{77} +2.46113e8 q^{79} -3.13393e8 q^{80} -2.61892e8 q^{81} +2.93995e8 q^{82} -7.57907e8 q^{83} -5.00508e7 q^{84} -2.28852e8 q^{85} -9.42659e8 q^{86} +1.70999e7 q^{87} +2.82353e8 q^{88} +1.61392e8 q^{89} +1.29140e8 q^{90} -2.93340e8 q^{92} -2.23072e8 q^{93} -7.74683e8 q^{94} +4.18555e8 q^{95} -3.58136e8 q^{96} -7.15528e8 q^{97} +7.69186e8 q^{98} -1.48157e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 65 q^{2} + q^{3} + 7169 q^{4} - 3238 q^{5} - 8490 q^{6} - 17378 q^{7} - 54204 q^{8} + 191118 q^{9} + 11697 q^{10} - 164171 q^{11} - 181941 q^{12} - 77651 q^{14} - 614110 q^{15} + 3012565 q^{16}+ \cdots - 5866875443 q^{99}+O(q^{100}) \) 27 * q - 65 * q^2 + q^3 + 7169 * q^4 - 3238 * q^5 - 8490 * q^6 - 17378 * q^7 - 54204 * q^8 + 191118 * q^9 + 11697 * q^10 - 164171 * q^11 - 181941 * q^12 - 77651 * q^14 - 614110 * q^15 + 3012565 * q^16 + 105667 * q^17 - 584849 * q^18 - 2140031 * q^19 - 3722382 * q^20 - 2953252 * q^21 + 3789362 * q^22 + 3586420 * q^23 - 6334338 * q^24 + 10278737 * q^25 - 4332980 * q^27 - 5293616 * q^28 - 4203616 * q^29 - 34788079 * q^30 - 18634294 * q^31 - 38778128 * q^32 - 15170356 * q^33 - 10546914 * q^34 + 16260796 * q^35 + 112071987 * q^36 - 7214770 * q^37 - 60128169 * q^38 - 46270051 * q^40 - 67544533 * q^41 - 31249012 * q^42 - 55585357 * q^43 - 246583161 * q^44 - 172236942 * q^45 - 168715272 * q^46 - 175569972 * q^47 - 48573948 * q^48 + 144680057 * q^49 - 3455750 * q^50 + 122191382 * q^51 + 105504478 * q^53 + 359394409 * q^54 - 2471426 * q^55 + 260765426 * q^56 + 307372146 * q^57 + 777873375 * q^58 - 267562255 * q^59 + 745460915 * q^60 - 371334848 * q^61 + 52175323 * q^62 - 431551952 * q^63 + 642079732 * q^64 + 973549503 * q^66 - 213848603 * q^67 - 278629116 * q^68 - 201174310 * q^69 + 122562066 * q^70 - 970709202 * q^71 - 773619646 * q^72 - 271733895 * q^73 - 1068711482 * q^74 + 1236853937 * q^75 - 2585634844 * q^76 - 244285066 * q^77 + 1052001020 * q^79 - 3293699123 * q^80 - 456846001 * q^81 - 3832425 * q^82 - 1491498691 * q^83 - 5153327787 * q^84 - 2713007958 * q^85 - 3421126005 * q^86 - 1622618562 * q^87 + 6574070508 * q^88 - 4616501291 * q^89 - 2822436128 * q^90 + 3684890562 * q^92 - 3324407374 * q^93 - 5582614303 * q^94 - 2892286998 * q^95 - 12902380548 * q^96 - 3944038157 * q^97 - 7822213576 * q^98 - 5866875443 * q^99

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