LMFDB - Embedded newform 275.2.b.d.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,2,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.2.b.d.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +O(q^{10})$	\(q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +1.00000 q^{11} +5.17157i q^{12} -6.82843i q^{13} -0.828427 q^{14} +3.00000 q^{16} -1.17157i q^{17} -2.07107i q^{18} -5.65685 q^{21} +0.414214i q^{22} +2.82843i q^{23} -4.48528 q^{24} +2.82843 q^{26} -5.65685i q^{27} +3.65685i q^{28} -7.65685 q^{29} +4.41421i q^{32} +2.82843i q^{33} +0.485281 q^{34} -9.14214 q^{36} -3.65685i q^{37} +19.3137 q^{39} +6.00000 q^{41} -2.34315i q^{42} -6.00000i q^{43} +1.82843 q^{44} -1.17157 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} +3.31371 q^{51} -12.4853i q^{52} +0.343146i q^{53} +2.34315 q^{54} -3.17157 q^{56} -3.17157i q^{58} +9.65685 q^{59} +13.3137 q^{61} -10.0000i q^{63} +4.17157 q^{64} -1.17157 q^{66} +4.48528i q^{67} -2.14214i q^{68} -8.00000 q^{69} -11.3137 q^{71} -7.92893i q^{72} -6.82843i q^{73} +1.51472 q^{74} +2.00000i q^{77} +8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.48528i q^{82} -6.00000i q^{83} -10.3431 q^{84} +2.48528 q^{86} -21.6569i q^{87} +1.58579i q^{88} -9.31371 q^{89} +13.6569 q^{91} +5.17157i q^{92} -1.17157 q^{94} -12.4853 q^{96} +7.65685i q^{97} +1.24264i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9	$ 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9 + 4 * q^11 + 8 * q^14 + 12 * q^16 + 16 * q^24 - 8 * q^29 - 32 * q^34 + 20 * q^36 + 32 * q^39 + 24 * q^41 - 4 * q^44 - 16 * q^46 + 12 * q^49 - 32 * q^51 + 32 * q^54 - 24 * q^56 + 16 * q^59 + 8 * q^61 + 28 * q^64 - 16 * q^66 - 32 * q^69 + 40 * q^74 - 16 * q^79 + 4 * q^81 - 64 * q^84 - 24 * q^86 + 8 * q^89 + 32 * q^91 - 16 * q^94 - 16 * q^96 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$	2.82843i	1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	−1.17157	−0.478293
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.58579i	0.560660i
$9$	−5.00000	−1.66667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	5.17157i	1.49290i
$13$	− 6.82843i	− 1.89386i	−0.321433	−	0.946932i	$-0.604164\pi$
$13$	− 6.82843i	− 1.89386i	0.321433	−	0.946932i	$-0.395836\pi$
$14$	−0.828427	−0.221406
$15$	0	0
$16$	3.00000	0.750000
$17$	− 1.17157i	− 0.284148i	−0.989856	−	0.142074i	$-0.954623\pi$
$17$	− 1.17157i	− 0.284148i	0.989856	−	0.142074i	$-0.0453771\pi$
$18$	− 2.07107i	− 0.488155i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−5.65685	−1.23443
$22$	0.414214i	0.0883106i
$23$	2.82843i	0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$	2.82843i	0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$	−4.48528	−0.915554
$25$	0	0
$26$	2.82843	0.554700
$27$	− 5.65685i	− 1.08866i
$28$	3.65685i	0.691080i
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	4.41421i	0.780330i
$33$	2.82843i	0.492366i
$34$	0.485281	0.0832251
$35$	0	0
$36$	−9.14214	−1.52369
$37$	− 3.65685i	− 0.601183i	−0.953753	−	0.300592i	$-0.902816\pi$
$37$	− 3.65685i	− 0.601183i	0.953753	−	0.300592i	$-0.0971841\pi$
$38$	0	0
$39$	19.3137	3.09267
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 2.34315i	− 0.361555i
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	1.82843	0.275646
$45$	0	0
$46$	−1.17157	−0.172739
$47$	2.82843i	0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$	2.82843i	0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	8.48528i	1.22474i
$49$	3.00000	0.428571
$50$	0	0
$51$	3.31371	0.464012
$52$	− 12.4853i	− 1.73140i
$53$	0.343146i	0.0471347i	0.999722	+	0.0235673i	$0.00750241\pi$
$53$	0.343146i	0.0471347i	−0.999722	+	0.0235673i	$0.992498\pi$
$54$	2.34315	0.318862
$55$	0	0
$56$	−3.17157	−0.423819
$57$	0	0
$58$	− 3.17157i	− 0.416448i
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0	0
$63$	− 10.0000i	− 1.25988i
$64$	4.17157	0.521447
$65$	0	0
$66$	−1.17157	−0.144211
$67$	4.48528i	0.547964i	0.961735	+	0.273982i	$0.0883409\pi$
$67$	4.48528i	0.547964i	−0.961735	+	0.273982i	$0.911659\pi$
$68$	− 2.14214i	− 0.259772i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−11.3137	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$72$	− 7.92893i	− 0.934434i
$73$	− 6.82843i	− 0.799207i	−0.916688	−	0.399603i	$-0.869148\pi$
$73$	− 6.82843i	− 0.799207i	0.916688	−	0.399603i	$-0.130852\pi$
$74$	1.51472	0.176082
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	8.00000i	0.905822i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.48528i	0.274453i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	−10.3431	−1.12853
$85$	0	0
$86$	2.48528	0.267995
$87$	− 21.6569i	− 2.32186i
$88$	1.58579i	0.169045i
$89$	−9.31371	−0.987251	−0.493626	−	0.869675i	$-0.664329\pi$
$89$	−9.31371	−0.987251	−0.493626	+	0.869675i	$0.664329\pi$
$90$	0	0
$91$	13.6569	1.43163
$92$	5.17157i	0.539174i
$93$	0	0
$94$	−1.17157	−0.120839
$95$	0	0
$96$	−12.4853	−1.27427
$97$	7.65685i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$97$	7.65685i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$98$	1.24264i	0.125526i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−13.3137	−1.32476	−0.662382	−	0.749166i	$-0.730454\pi$
$101$	−13.3137	−1.32476	−0.662382	+	0.749166i	$0.730454\pi$
$102$	1.37258i	0.135906i
$103$	1.17157i	0.115439i	0.998333	+	0.0577193i	$0.0183828\pi$
$103$	1.17157i	0.115439i	−0.998333	+	0.0577193i	$0.981617\pi$
$104$	10.8284	1.06181
$105$	0	0
$106$	−0.142136	−0.0138054
$107$	3.65685i	0.353521i	0.984254	+	0.176761i	$0.0565619\pi$
$107$	3.65685i	0.353521i	−0.984254	+	0.176761i	$0.943438\pi$
$108$	− 10.3431i	− 0.995270i
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	0	0
$111$	10.3431	0.981728
$112$	6.00000i	0.566947i
$113$	8.34315i	0.784857i	0.919782	+	0.392429i	$0.128365\pi$
$113$	8.34315i	0.784857i	−0.919782	+	0.392429i	$0.871635\pi$
$114$	0	0
$115$	0	0
$116$	−14.0000	−1.29987
$117$	34.1421i	3.15644i
$118$	4.00000i	0.368230i
$119$	2.34315	0.214796
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.51472i	0.499279i
$123$	16.9706i	1.53018i
$124$	0	0
$125$	0	0
$126$	4.14214	0.369011
$127$	− 15.6569i	− 1.38932i	−0.719338	−	0.694661i	$-0.755555\pi$
$127$	− 15.6569i	− 1.38932i	0.719338	−	0.694661i	$-0.244445\pi$
$128$	10.5563i	0.933058i
$129$	16.9706	1.49417
$130$	0	0
$131$	11.3137	0.988483	0.494242	−	0.869325i	$-0.335446\pi$
$131$	11.3137	0.988483	0.494242	+	0.869325i	$0.335446\pi$
$132$	5.17157i	0.450128i
$133$	0	0
$134$	−1.85786	−0.160495
$135$	0	0
$136$	1.85786	0.159311
$137$	− 22.9706i	− 1.96251i	−0.192720	−	0.981254i	$-0.561731\pi$
$137$	− 22.9706i	− 1.96251i	0.192720	−	0.981254i	$-0.438269\pi$
$138$	− 3.31371i	− 0.282082i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	− 4.68629i	− 0.393265i
$143$	− 6.82843i	− 0.571022i
$144$	−15.0000	−1.25000
$145$	0	0
$146$	2.82843	0.234082
$147$	8.48528i	0.699854i
$148$	− 6.68629i	− 0.549610i
$149$	−11.6569	−0.954967	−0.477483	−	0.878641i	$-0.658451\pi$
$149$	−11.6569	−0.954967	−0.477483	+	0.878641i	$0.658451\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	5.85786i	0.473580i
$154$	−0.828427	−0.0667566
$155$	0	0
$156$	35.3137	2.82736
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 1.65685i	− 0.131812i
$159$	−0.970563	−0.0769706
$160$	0	0
$161$	−5.65685	−0.445823
$162$	0.414214i	0.0325437i
$163$	− 0.485281i	− 0.0380102i	−0.999819	−	0.0190051i	$-0.993950\pi$
$163$	− 0.485281i	− 0.0380102i	0.999819	−	0.0190051i	$-0.00604987\pi$
$164$	10.9706	0.856657
$165$	0	0
$166$	2.48528	0.192895
$167$	− 10.9706i	− 0.848928i	−0.905445	−	0.424464i	$-0.860463\pi$
$167$	− 10.9706i	− 0.848928i	0.905445	−	0.424464i	$-0.139537\pi$
$168$	− 8.97056i	− 0.692094i
$169$	−33.6274	−2.58672
$170$	0	0
$171$	0	0
$172$	− 10.9706i	− 0.836498i
$173$	6.14214i	0.466978i	0.972359	+	0.233489i	$0.0750143\pi$
$173$	6.14214i	0.466978i	−0.972359	+	0.233489i	$0.924986\pi$
$174$	8.97056	0.680057
$175$	0	0
$176$	3.00000	0.226134
$177$	27.3137i	2.05302i
$178$	− 3.85786i	− 0.289159i
$179$	1.65685	0.123839	0.0619196	−	0.998081i	$-0.480278\pi$
$179$	1.65685	0.123839	0.0619196	+	0.998081i	$0.480278\pi$
$180$	0	0
$181$	−1.31371	−0.0976472	−0.0488236	−	0.998807i	$-0.515547\pi$
$181$	−1.31371	−0.0976472	−0.0488236	+	0.998807i	$0.515547\pi$
$182$	5.65685i	0.419314i
$183$	37.6569i	2.78367i
$184$	−4.48528	−0.330659
$185$	0	0
$186$	0	0
$187$	− 1.17157i	− 0.0856739i
$188$	5.17157i	0.377176i
$189$	11.3137	0.822951
$190$	0	0
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	11.7990i	0.851519i
$193$	− 6.82843i	− 0.491521i	−0.969331	−	0.245760i	$-0.920962\pi$
$193$	− 6.82843i	− 0.491521i	0.969331	−	0.245760i	$-0.0790377\pi$
$194$	−3.17157	−0.227706
$195$	0	0
$196$	5.48528	0.391806
$197$	5.17157i	0.368459i	0.982883	+	0.184230i	$0.0589790\pi$
$197$	5.17157i	0.368459i	−0.982883	+	0.184230i	$0.941021\pi$
$198$	− 2.07107i	− 0.147184i
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	−12.6863	−0.894822
$202$	− 5.51472i	− 0.388014i
$203$	− 15.3137i	− 1.07481i
$204$	6.05887	0.424206
$205$	0	0
$206$	−0.485281	−0.0338112
$207$	− 14.1421i	− 0.982946i
$208$	− 20.4853i	− 1.42040i
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0.627417i	0.0430912i
$213$	− 32.0000i	− 2.19260i
$214$	−1.51472	−0.103544
$215$	0	0
$216$	8.97056	0.610369
$217$	0	0
$218$	− 1.51472i	− 0.102590i
$219$	19.3137	1.30510
$220$	0	0
$221$	−8.00000	−0.538138
$222$	4.28427i	0.287541i
$223$	− 5.17157i	− 0.346314i	−0.984894	−	0.173157i	$-0.944603\pi$
$223$	− 5.17157i	− 0.346314i	0.984894	−	0.173157i	$-0.0553968\pi$
$224$	−8.82843	−0.589874
$225$	0	0
$226$	−3.45584	−0.229879
$227$	− 2.68629i	− 0.178295i	−0.996018	−	0.0891477i	$-0.971586\pi$
$227$	− 2.68629i	− 0.178295i	0.996018	−	0.0891477i	$-0.0284143\pi$
$228$	0	0
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	− 12.1421i	− 0.797170i
$233$	22.1421i	1.45058i	0.688444	+	0.725290i	$0.258294\pi$
$233$	22.1421i	1.45058i	−0.688444	+	0.725290i	$0.741706\pi$
$234$	−14.1421	−0.924500
$235$	0	0
$236$	17.6569	1.14936
$237$	− 11.3137i	− 0.734904i
$238$	0.970563i	0.0629122i
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0.414214i	0.0266267i
$243$	− 14.1421i	− 0.907218i
$244$	24.3431	1.55841
$245$	0	0
$246$	−7.02944	−0.448181
$247$	0	0
$248$	0	0
$249$	16.9706	1.07547
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	− 18.2843i	− 1.15180i
$253$	2.82843i	0.177822i
$254$	6.48528	0.406923
$255$	0	0
$256$	3.97056	0.248160
$257$	− 13.3137i	− 0.830486i	−0.909710	−	0.415243i	$-0.863696\pi$
$257$	− 13.3137i	− 0.830486i	0.909710	−	0.415243i	$-0.136304\pi$
$258$	7.02944i	0.437634i
$259$	7.31371	0.454452
$260$	0	0
$261$	38.2843	2.36974
$262$	4.68629i	0.289520i
$263$	22.9706i	1.41643i	0.705999	+	0.708213i	$0.250498\pi$
$263$	22.9706i	1.41643i	−0.705999	+	0.708213i	$0.749502\pi$
$264$	−4.48528	−0.276050
$265$	0	0
$266$	0	0
$267$	− 26.3431i	− 1.61217i
$268$	8.20101i	0.500956i
$269$	5.31371	0.323983	0.161991	−	0.986792i	$-0.448208\pi$
$269$	5.31371	0.323983	0.161991	+	0.986792i	$0.448208\pi$
$270$	0	0
$271$	−15.3137	−0.930242	−0.465121	−	0.885247i	$-0.653989\pi$
$271$	−15.3137	−0.930242	−0.465121	+	0.885247i	$0.653989\pi$
$272$	− 3.51472i	− 0.213111i
$273$	38.6274i	2.33784i
$274$	9.51472	0.574805
$275$	0	0
$276$	−14.6274	−0.880467
$277$	− 1.17157i	− 0.0703930i	−0.999380	−	0.0351965i	$-0.988794\pi$
$277$	− 1.17157i	− 0.0703930i	0.999380	−	0.0351965i	$-0.0112057\pi$
$278$	1.65685i	0.0993715i
$279$	0	0
$280$	0	0
$281$	−5.31371	−0.316989	−0.158495	−	0.987360i	$-0.550664\pi$
$281$	−5.31371	−0.316989	−0.158495	+	0.987360i	$0.550664\pi$
$282$	− 3.31371i	− 0.197328i
$283$	− 12.6274i	− 0.750622i	−0.926899	−	0.375311i	$-0.877536\pi$
$283$	− 12.6274i	− 0.750622i	0.926899	−	0.375311i	$-0.122464\pi$
$284$	−20.6863	−1.22751
$285$	0	0
$286$	2.82843	0.167248
$287$	12.0000i	0.708338i
$288$	− 22.0711i	− 1.30055i
$289$	15.6274	0.919260
$290$	0	0
$291$	−21.6569	−1.26955
$292$	− 12.4853i	− 0.730646i
$293$	− 14.8284i	− 0.866286i	−0.901325	−	0.433143i	$-0.857405\pi$
$293$	− 14.8284i	− 0.866286i	0.901325	−	0.433143i	$-0.142595\pi$
$294$	−3.51472	−0.204983
$295$	0	0
$296$	5.79899	0.337059
$297$	− 5.65685i	− 0.328244i
$298$	− 4.82843i	− 0.279703i
$299$	19.3137	1.11694
$300$	0	0
$301$	12.0000	0.691669
$302$	− 4.97056i	− 0.286024i
$303$	− 37.6569i	− 2.16333i
$304$	0	0
$305$	0	0
$306$	−2.42641	−0.138708
$307$	27.6569i	1.57846i	0.614098	+	0.789230i	$0.289520\pi$
$307$	27.6569i	1.57846i	−0.614098	+	0.789230i	$0.710480\pi$
$308$	3.65685i	0.208369i
$309$	−3.31371	−0.188510
$310$	0	0
$311$	27.3137	1.54882	0.774409	−	0.632685i	$-0.218047\pi$
$311$	27.3137	1.54882	0.774409	+	0.632685i	$0.218047\pi$
$312$	30.6274i	1.73394i
$313$	21.3137i	1.20472i	0.798224	+	0.602361i	$0.205773\pi$
$313$	21.3137i	1.20472i	−0.798224	+	0.602361i	$0.794227\pi$
$314$	−5.79899	−0.327256
$315$	0	0
$316$	−7.31371	−0.411428
$317$	− 21.3137i	− 1.19710i	−0.801087	−	0.598549i	$-0.795744\pi$
$317$	− 21.3137i	− 1.19710i	0.801087	−	0.598549i	$-0.204256\pi$
$318$	− 0.402020i	− 0.0225442i
$319$	−7.65685	−0.428702
$320$	0	0
$321$	−10.3431	−0.577298
$322$	− 2.34315i	− 0.130578i
$323$	0	0
$324$	1.82843	0.101579
$325$	0	0
$326$	0.201010	0.0111329
$327$	− 10.3431i	− 0.571977i
$328$	9.51472i	0.525362i
$329$	−5.65685	−0.311872
$330$	0	0
$331$	15.3137	0.841718	0.420859	−	0.907126i	$-0.361729\pi$
$331$	15.3137	0.841718	0.420859	+	0.907126i	$0.361729\pi$
$332$	− 10.9706i	− 0.602088i
$333$	18.2843i	1.00197i
$334$	4.54416	0.248645
$335$	0	0
$336$	−16.9706	−0.925820
$337$	3.51472i	0.191459i	0.995407	+	0.0957295i	$0.0305184\pi$
$337$	3.51472i	0.191459i	−0.995407	+	0.0957295i	$0.969482\pi$
$338$	− 13.9289i	− 0.757634i
$339$	−23.5980	−1.28167
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	9.51472	0.512999
$345$	0	0
$346$	−2.54416	−0.136775
$347$	22.9706i	1.23312i	0.787306	+	0.616562i	$0.211475\pi$
$347$	22.9706i	1.23312i	−0.787306	+	0.616562i	$0.788525\pi$
$348$	− 39.5980i	− 2.12267i
$349$	6.97056	0.373126	0.186563	−	0.982443i	$-0.440265\pi$
$349$	6.97056	0.373126	0.186563	+	0.982443i	$0.440265\pi$
$350$	0	0
$351$	−38.6274	−2.06178
$352$	4.41421i	0.235278i
$353$	− 1.31371i	− 0.0699216i	−0.999389	−	0.0349608i	$-0.988869\pi$
$353$	− 1.31371i	− 0.0699216i	0.999389	−	0.0349608i	$-0.0111306\pi$
$354$	−11.3137	−0.601317
$355$	0	0
$356$	−17.0294	−0.902558
$357$	6.62742i	0.350760i
$358$	0.686292i	0.0362716i
$359$	−23.3137	−1.23045	−0.615225	−	0.788351i	$-0.710935\pi$
$359$	−23.3137	−1.23045	−0.615225	+	0.788351i	$0.710935\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 0.544156i	− 0.0286002i
$363$	2.82843i	0.148454i
$364$	24.9706	1.30881
$365$	0	0
$366$	−15.5980	−0.815319
$367$	− 8.48528i	− 0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$	− 8.48528i	− 0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$	8.48528i	0.442326i
$369$	−30.0000	−1.56174
$370$	0	0
$371$	−0.686292	−0.0356305
$372$	0	0
$373$	− 3.79899i	− 0.196704i	−0.995152	−	0.0983521i	$-0.968643\pi$
$373$	− 3.79899i	− 0.196704i	0.995152	−	0.0983521i	$-0.0313571\pi$
$374$	0.485281	0.0250933
$375$	0	0
$376$	−4.48528	−0.231311
$377$	52.2843i	2.69278i
$378$	4.68629i	0.241037i
$379$	−22.3431	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$379$	−22.3431	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$380$	0	0
$381$	44.2843	2.26875
$382$	− 8.00000i	− 0.409316i
$383$	− 34.1421i	− 1.74458i	−0.488987	−	0.872291i	$-0.662634\pi$
$383$	− 34.1421i	− 1.74458i	0.488987	−	0.872291i	$-0.337366\pi$
$384$	−29.8579	−1.52368
$385$	0	0
$386$	2.82843	0.143963
$387$	30.0000i	1.52499i
$388$	14.0000i	0.710742i
$389$	24.6274	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$389$	24.6274	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$390$	0	0
$391$	3.31371	0.167581
$392$	4.75736i	0.240283i
$393$	32.0000i	1.61419i
$394$	−2.14214	−0.107919
$395$	0	0
$396$	−9.14214	−0.459410
$397$	− 13.3137i	− 0.668196i	−0.942538	−	0.334098i	$-0.891568\pi$
$397$	− 13.3137i	− 0.668196i	0.942538	−	0.334098i	$-0.108432\pi$
$398$	− 8.97056i	− 0.449654i
$399$	0	0
$400$	0	0
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	− 5.25483i	− 0.262087i
$403$	0	0
$404$	−24.3431	−1.21112
$405$	0	0
$406$	6.34315	0.314805
$407$	− 3.65685i	− 0.181264i
$408$	5.25483i	0.260153i
$409$	−34.9706	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$409$	−34.9706	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$410$	0	0
$411$	64.9706	3.20476
$412$	2.14214i	0.105535i
$413$	19.3137i	0.950365i
$414$	5.85786	0.287898
$415$	0	0
$416$	30.1421	1.47784
$417$	11.3137i	0.554035i
$418$	0	0
$419$	14.3431	0.700709	0.350354	−	0.936617i	$-0.386061\pi$
$419$	14.3431	0.700709	0.350354	+	0.936617i	$0.386061\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	− 6.62742i	− 0.322618i
$423$	− 14.1421i	− 0.687614i
$424$	−0.544156	−0.0264265
$425$	0	0
$426$	13.2548	0.642199
$427$	26.6274i	1.28859i
$428$	6.68629i	0.323194i
$429$	19.3137	0.932475
$430$	0	0
$431$	11.3137	0.544962	0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137	0.544962	0.272481	+	0.962161i	$0.412156\pi$
$432$	− 16.9706i	− 0.816497i
$433$	3.65685i	0.175737i	0.996132	+	0.0878686i	$0.0280056\pi$
$433$	3.65685i	0.175737i	−0.996132	+	0.0878686i	$0.971994\pi$
$434$	0	0
$435$	0	0
$436$	−6.68629	−0.320215
$437$	0	0
$438$	8.00000i	0.382255i
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	− 3.31371i	− 0.157617i
$443$	− 21.1716i	− 1.00589i	−0.864318	−	0.502946i	$-0.832249\pi$
$443$	− 21.1716i	− 1.00589i	0.864318	−	0.502946i	$-0.167751\pi$
$444$	18.9117	0.897509
$445$	0	0
$446$	2.14214	0.101433
$447$	− 32.9706i	− 1.55945i
$448$	8.34315i	0.394177i
$449$	16.6274	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$449$	16.6274	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	15.2548i	0.717527i
$453$	− 33.9411i	− 1.59469i
$454$	1.11270	0.0522215
$455$	0	0
$456$	0	0
$457$	16.4853i	0.771149i	0.922677	+	0.385574i	$0.125997\pi$
$457$	16.4853i	0.771149i	−0.922677	+	0.385574i	$0.874003\pi$
$458$	8.82843i	0.412525i
$459$	−6.62742	−0.309341
$460$	0	0
$461$	−32.6274	−1.51961	−0.759805	−	0.650151i	$-0.774706\pi$
$461$	−32.6274	−1.51961	−0.759805	+	0.650151i	$0.774706\pi$
$462$	− 2.34315i	− 0.109013i
$463$	22.1421i	1.02903i	0.857481	+	0.514516i	$0.172028\pi$
$463$	22.1421i	1.02903i	−0.857481	+	0.514516i	$0.827972\pi$
$464$	−22.9706	−1.06638
$465$	0	0
$466$	−9.17157	−0.424865
$467$	9.17157i	0.424410i	0.977225	+	0.212205i	$0.0680644\pi$
$467$	9.17157i	0.424410i	−0.977225	+	0.212205i	$0.931936\pi$
$468$	62.4264i	2.88566i
$469$	−8.97056	−0.414222
$470$	0	0
$471$	−39.5980	−1.82458
$472$	15.3137i	0.704871i
$473$	− 6.00000i	− 0.275880i
$474$	4.68629	0.215248
$475$	0	0
$476$	4.28427	0.196369
$477$	− 1.71573i	− 0.0785578i
$478$	0.284271i	0.0130023i
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−24.9706	−1.13856
$482$	2.48528i	0.113201i
$483$	− 16.0000i	− 0.728025i
$484$	1.82843	0.0831103
$485$	0	0
$486$	5.85786	0.265718
$487$	7.51472i	0.340524i	0.985399	+	0.170262i	$0.0544615\pi$
$487$	7.51472i	0.340524i	−0.985399	+	0.170262i	$0.945539\pi$
$488$	21.1127i	0.955727i
$489$	1.37258	0.0620703
$490$	0	0
$491$	−23.3137	−1.05213	−0.526066	−	0.850443i	$-0.676334\pi$
$491$	−23.3137	−1.05213	−0.526066	+	0.850443i	$0.676334\pi$
$492$	31.0294i	1.39892i
$493$	8.97056i	0.404014i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 22.6274i	− 1.01498i
$498$	7.02944i	0.314997i
$499$	1.65685	0.0741710	0.0370855	−	0.999312i	$-0.488193\pi$
$499$	1.65685	0.0741710	0.0370855	+	0.999312i	$0.488193\pi$
$500$	0	0
$501$	31.0294	1.38629
$502$	4.97056i	0.221847i
$503$	− 28.6274i	− 1.27643i	−0.769857	−	0.638217i	$-0.779672\pi$
$503$	− 28.6274i	− 1.27643i	0.769857	−	0.638217i	$-0.220328\pi$
$504$	15.8579	0.706365
$505$	0	0
$506$	−1.17157	−0.0520828
$507$	− 95.1127i	− 4.22410i
$508$	− 28.6274i	− 1.27014i
$509$	−9.31371	−0.412823	−0.206411	−	0.978465i	$-0.566179\pi$
$509$	−9.31371	−0.412823	−0.206411	+	0.978465i	$0.566179\pi$
$510$	0	0
$511$	13.6569	0.604144
$512$	22.7574i	1.00574i
$513$	0	0
$514$	5.51472	0.243244
$515$	0	0
$516$	31.0294	1.36599
$517$	2.82843i	0.124394i
$518$	3.02944i	0.133106i
$519$	−17.3726	−0.762572
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	15.8579i	0.694080i
$523$	37.5980i	1.64404i	0.569455	+	0.822022i	$0.307154\pi$
$523$	37.5980i	1.64404i	−0.569455	+	0.822022i	$0.692846\pi$
$524$	20.6863	0.903685
$525$	0	0
$526$	−9.51472	−0.414861
$527$	0	0
$528$	8.48528i	0.369274i
$529$	15.0000	0.652174
$530$	0	0
$531$	−48.2843	−2.09536
$532$	0	0
$533$	− 40.9706i	− 1.77463i
$534$	10.9117	0.472195
$535$	0	0
$536$	−7.11270	−0.307222
$537$	4.68629i	0.202228i
$538$	2.20101i	0.0948923i
$539$	3.00000	0.129219
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	− 6.34315i	− 0.272461i
$543$	− 3.71573i	− 0.159457i
$544$	5.17157	0.221729
$545$	0	0
$546$	−16.0000	−0.684737
$547$	34.0000i	1.45374i	0.686778	+	0.726868i	$0.259025\pi$
$547$	34.0000i	1.45374i	−0.686778	+	0.726868i	$0.740975\pi$
$548$	− 42.0000i	− 1.79415i
$549$	−66.5685	−2.84108
$550$	0	0
$551$	0	0
$552$	− 12.6863i	− 0.539964i
$553$	− 8.00000i	− 0.340195i
$554$	0.485281	0.0206176
$555$	0	0
$556$	7.31371	0.310170
$557$	− 38.1421i	− 1.61613i	−0.589090	−	0.808067i	$-0.700514\pi$
$557$	− 38.1421i	− 1.61613i	0.589090	−	0.808067i	$-0.299486\pi$
$558$	0	0
$559$	−40.9706	−1.73287
$560$	0	0
$561$	3.31371	0.139905
$562$	− 2.20101i	− 0.0928440i
$563$	11.6569i	0.491278i	0.969361	+	0.245639i	$0.0789977\pi$
$563$	11.6569i	0.491278i	−0.969361	+	0.245639i	$0.921002\pi$
$564$	−14.6274	−0.615925
$565$	0	0
$566$	5.23045	0.219852
$567$	2.00000i	0.0839921i
$568$	− 17.9411i	− 0.752793i
$569$	−20.3431	−0.852829	−0.426415	−	0.904528i	$-0.640224\pi$
$569$	−20.3431	−0.852829	−0.426415	+	0.904528i	$0.640224\pi$
$570$	0	0
$571$	45.9411	1.92258	0.961288	−	0.275545i	$-0.0888584\pi$
$571$	45.9411	1.92258	0.961288	+	0.275545i	$0.0888584\pi$
$572$	− 12.4853i	− 0.522036i
$573$	− 54.6274i	− 2.28209i
$574$	−4.97056	−0.207467
$575$	0	0
$576$	−20.8579	−0.869078
$577$	− 6.97056i	− 0.290188i	−0.989418	−	0.145094i	$-0.953651\pi$
$577$	− 6.97056i	− 0.290188i	0.989418	−	0.145094i	$-0.0463485\pi$
$578$	6.47309i	0.269245i
$579$	19.3137	0.802650
$580$	0	0
$581$	12.0000	0.497844
$582$	− 8.97056i	− 0.371842i
$583$	0.343146i	0.0142116i
$584$	10.8284	0.448084
$585$	0	0
$586$	6.14214	0.253729
$587$	− 26.1421i	− 1.07900i	−0.841985	−	0.539501i	$-0.818613\pi$
$587$	− 26.1421i	− 1.07900i	0.841985	−	0.539501i	$-0.181387\pi$
$588$	15.5147i	0.639816i
$589$	0	0
$590$	0	0
$591$	−14.6274	−0.601692
$592$	− 10.9706i	− 0.450887i
$593$	20.4853i	0.841230i	0.907239	+	0.420615i	$0.138186\pi$
$593$	20.4853i	0.841230i	−0.907239	+	0.420615i	$0.861814\pi$
$594$	2.34315	0.0961404
$595$	0	0
$596$	−21.3137	−0.873044
$597$	− 61.2548i	− 2.50699i
$598$	8.00000i	0.327144i
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	−43.9411	−1.79240	−0.896198	−	0.443654i	$-0.853682\pi$
$601$	−43.9411	−1.79240	−0.896198	+	0.443654i	$0.853682\pi$
$602$	4.97056i	0.202585i
$603$	− 22.4264i	− 0.913274i
$604$	−21.9411	−0.892772
$605$	0	0
$606$	15.5980	0.633625
$607$	18.2843i	0.742136i	0.928606	+	0.371068i	$0.121008\pi$
$607$	18.2843i	0.742136i	−0.928606	+	0.371068i	$0.878992\pi$
$608$	0	0
$609$	43.3137	1.75516
$610$	0	0
$611$	19.3137	0.781349
$612$	10.7107i	0.432954i
$613$	25.4558i	1.02815i	0.857745	+	0.514076i	$0.171865\pi$
$613$	25.4558i	1.02815i	−0.857745	+	0.514076i	$0.828135\pi$
$614$	−11.4558	−0.462320
$615$	0	0
$616$	−3.17157	−0.127786
$617$	− 11.6569i	− 0.469287i	−0.972081	−	0.234644i	$-0.924608\pi$
$617$	− 11.6569i	− 0.469287i	0.972081	−	0.234644i	$-0.0753923\pi$
$618$	− 1.37258i	− 0.0552134i
$619$	25.6569	1.03124	0.515618	−	0.856819i	$-0.327562\pi$
$619$	25.6569	1.03124	0.515618	+	0.856819i	$0.327562\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	11.3137i	0.453638i
$623$	− 18.6274i	− 0.746292i
$624$	57.9411	2.31950
$625$	0	0
$626$	−8.82843	−0.352855
$627$	0	0
$628$	25.5980i	1.02147i
$629$	−4.28427	−0.170825
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	− 6.34315i	− 0.252317i
$633$	− 45.2548i	− 1.79872i
$634$	8.82843	0.350622
$635$	0	0
$636$	−1.77460	−0.0703676
$637$	− 20.4853i	− 0.811656i
$638$	− 3.17157i	− 0.125564i
$639$	56.5685	2.23782
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 4.28427i	− 0.169087i
$643$	49.4558i	1.95035i	0.221440	+	0.975174i	$0.428924\pi$
$643$	49.4558i	1.95035i	−0.221440	+	0.975174i	$0.571076\pi$
$644$	−10.3431	−0.407577
$645$	0	0
$646$	0	0
$647$	35.1127i	1.38042i	0.723608	+	0.690211i	$0.242482\pi$
$647$	35.1127i	1.38042i	−0.723608	+	0.690211i	$0.757518\pi$
$648$	1.58579i	0.0622956i
$649$	9.65685	0.379065
$650$	0	0
$651$	0	0
$652$	− 0.887302i	− 0.0347494i
$653$	0.343146i	0.0134283i	0.999977	+	0.00671417i	$0.00213720\pi$
$653$	0.343146i	0.0134283i	−0.999977	+	0.00671417i	$0.997863\pi$
$654$	4.28427	0.167528
$655$	0	0
$656$	18.0000	0.702782
$657$	34.1421i	1.33201i
$658$	− 2.34315i	− 0.0913453i
$659$	21.9411	0.854705	0.427352	−	0.904085i	$-0.359446\pi$
$659$	21.9411	0.854705	0.427352	+	0.904085i	$0.359446\pi$
$660$	0	0
$661$	−0.627417	−0.0244037	−0.0122018	−	0.999926i	$-0.503884\pi$
$661$	−0.627417	−0.0244037	−0.0122018	+	0.999926i	$0.503884\pi$
$662$	6.34315i	0.246533i
$663$	− 22.6274i	− 0.878776i
$664$	9.51472	0.369243
$665$	0	0
$666$	−7.57359	−0.293471
$667$	− 21.6569i	− 0.838557i
$668$	− 20.0589i	− 0.776101i
$669$	14.6274	0.565529
$670$	0	0
$671$	13.3137	0.513970
$672$	− 24.9706i	− 0.963260i
$673$	4.48528i	0.172895i	0.996256	+	0.0864474i	$0.0275515\pi$
$673$	4.48528i	0.172895i	−0.996256	+	0.0864474i	$0.972449\pi$
$674$	−1.45584	−0.0560770
$675$	0	0
$676$	−61.4853	−2.36482
$677$	− 17.1716i	− 0.659957i	−0.943988	−	0.329979i	$-0.892958\pi$
$677$	− 17.1716i	− 0.659957i	0.943988	−	0.329979i	$-0.107042\pi$
$678$	− 9.77460i	− 0.375391i
$679$	−15.3137	−0.587686
$680$	0	0
$681$	7.59798	0.291155
$682$	0	0
$683$	31.7990i	1.21675i	0.793648	+	0.608377i	$0.208179\pi$
$683$	31.7990i	1.21675i	−0.793648	+	0.608377i	$0.791821\pi$
$684$	0	0
$685$	0	0
$686$	−8.28427	−0.316295
$687$	60.2843i	2.29999i
$688$	− 18.0000i	− 0.686244i
$689$	2.34315	0.0892667
$690$	0	0
$691$	−16.6863	−0.634776	−0.317388	−	0.948296i	$-0.602806\pi$
$691$	−16.6863	−0.634776	−0.317388	+	0.948296i	$0.602806\pi$
$692$	11.2304i	0.426918i
$693$	− 10.0000i	− 0.379869i
$694$	−9.51472	−0.361174
$695$	0	0
$696$	34.3431	1.30177
$697$	− 7.02944i	− 0.266259i
$698$	2.88730i	0.109286i
$699$	−62.6274	−2.36879
$700$	0	0
$701$	32.6274	1.23232	0.616160	−	0.787621i	$-0.288687\pi$
$701$	32.6274	1.23232	0.616160	+	0.787621i	$0.288687\pi$
$702$	− 16.0000i	− 0.603881i
$703$	0	0
$704$	4.17157	0.157222
$705$	0	0
$706$	0.544156	0.0204796
$707$	− 26.6274i	− 1.00143i
$708$	49.9411i	1.87690i
$709$	20.6274	0.774679	0.387339	−	0.921937i	$-0.373394\pi$
$709$	20.6274	0.774679	0.387339	+	0.921937i	$0.373394\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	− 14.7696i	− 0.553512i
$713$	0	0
$714$	−2.74517	−0.102735
$715$	0	0
$716$	3.02944	0.113215
$717$	1.94113i	0.0724927i
$718$	− 9.65685i	− 0.360391i
$719$	−29.6569	−1.10601	−0.553007	−	0.833177i	$-0.686520\pi$
$719$	−29.6569	−1.10601	−0.553007	+	0.833177i	$0.686520\pi$
$720$	0	0
$721$	−2.34315	−0.0872633
$722$	− 7.87006i	− 0.292893i
$723$	16.9706i	0.631142i
$724$	−2.40202	−0.0892704
$725$	0	0
$726$	−1.17157	−0.0434811
$727$	36.4853i	1.35316i	0.736367	+	0.676582i	$0.236540\pi$
$727$	36.4853i	1.35316i	−0.736367	+	0.676582i	$0.763460\pi$
$728$	21.6569i	0.802656i
$729$	43.0000	1.59259
$730$	0	0
$731$	−7.02944	−0.259993
$732$	68.8528i	2.54487i
$733$	33.4558i	1.23572i	0.786288	+	0.617860i	$0.212000\pi$
$733$	33.4558i	1.23572i	−0.786288	+	0.617860i	$0.788000\pi$
$734$	3.51472	0.129731
$735$	0	0
$736$	−12.4853	−0.460214
$737$	4.48528i	0.165217i
$738$	− 12.4264i	− 0.457422i
$739$	37.9411	1.39569	0.697843	−	0.716250i	$-0.254143\pi$
$739$	37.9411	1.39569	0.697843	+	0.716250i	$0.254143\pi$
$740$	0	0
$741$	0	0
$742$	− 0.284271i	− 0.0104359i
$743$	29.5980i	1.08584i	0.839783	+	0.542922i	$0.182682\pi$
$743$	29.5980i	1.08584i	−0.839783	+	0.542922i	$0.817318\pi$
$744$	0	0
$745$	0	0
$746$	1.57359	0.0576133
$747$	30.0000i	1.09764i
$748$	− 2.14214i	− 0.0783242i
$749$	−7.31371	−0.267237
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	8.48528i	0.309426i
$753$	33.9411i	1.23688i
$754$	−21.6569	−0.788696
$755$	0	0
$756$	20.6863	0.752353
$757$	9.31371i	0.338512i	0.985572	+	0.169256i	$0.0541365\pi$
$757$	9.31371i	0.338512i	−0.985572	+	0.169256i	$0.945863\pi$
$758$	− 9.25483i	− 0.336151i
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	18.3431i	0.664502i
$763$	− 7.31371i	− 0.264774i
$764$	−35.3137	−1.27761
$765$	0	0
$766$	14.1421	0.510976
$767$	− 65.9411i	− 2.38100i
$768$	11.2304i	0.405244i
$769$	−14.9706	−0.539852	−0.269926	−	0.962881i	$-0.586999\pi$
$769$	−14.9706	−0.539852	−0.269926	+	0.962881i	$0.586999\pi$
$770$	0	0
$771$	37.6569	1.35618
$772$	− 12.4853i	− 0.449355i
$773$	− 30.2843i	− 1.08925i	−0.838680	−	0.544625i	$-0.816672\pi$
$773$	− 30.2843i	− 1.08925i	0.838680	−	0.544625i	$-0.183328\pi$
$774$	−12.4264	−0.446658
$775$	0	0
$776$	−12.1421	−0.435877
$777$	20.6863i	0.742117i
$778$	10.2010i	0.365724i
$779$	0	0
$780$	0	0
$781$	−11.3137	−0.404836
$782$	1.37258i	0.0490835i
$783$	43.3137i	1.54791i
$784$	9.00000	0.321429
$785$	0	0
$786$	−13.2548	−0.472784
$787$	− 18.9706i	− 0.676228i	−0.941105	−	0.338114i	$-0.890211\pi$
$787$	− 18.9706i	− 0.676228i	0.941105	−	0.338114i	$-0.109789\pi$
$788$	9.45584i	0.336850i
$789$	−64.9706	−2.31301
$790$	0	0
$791$	−16.6863	−0.593296
$792$	− 7.92893i	− 0.281742i
$793$	− 90.9117i	− 3.22837i
$794$	5.51472	0.195710
$795$	0	0
$796$	−39.5980	−1.40351
$797$	12.6274i	0.447286i	0.974671	+	0.223643i	$0.0717950\pi$
$797$	12.6274i	0.447286i	−0.974671	+	0.223643i	$0.928205\pi$
$798$	0	0
$799$	3.31371	0.117231
$800$	0	0
$801$	46.5685	1.64542
$802$	7.17157i	0.253237i
$803$	− 6.82843i	− 0.240970i
$804$	−23.1960	−0.818058
$805$	0	0
$806$	0	0
$807$	15.0294i	0.529061i
$808$	− 21.1127i	− 0.742742i
$809$	−22.9706	−0.807602	−0.403801	−	0.914847i	$-0.632311\pi$
$809$	−22.9706	−0.807602	−0.403801	+	0.914847i	$0.632311\pi$
$810$	0	0
$811$	−13.9411	−0.489539	−0.244770	−	0.969581i	$-0.578712\pi$
$811$	−13.9411	−0.489539	−0.244770	+	0.969581i	$0.578712\pi$
$812$	− 28.0000i	− 0.982607i
$813$	− 43.3137i	− 1.51908i
$814$	1.51472	0.0530909
$815$	0	0
$816$	9.94113	0.348009
$817$	0	0
$818$	− 14.4853i	− 0.506466i
$819$	−68.2843	−2.38605
$820$	0	0
$821$	−18.6863	−0.652156	−0.326078	−	0.945343i	$-0.605727\pi$
$821$	−18.6863	−0.652156	−0.326078	+	0.945343i	$0.605727\pi$
$822$	26.9117i	0.938653i
$823$	36.4853i	1.27180i	0.771773	+	0.635898i	$0.219370\pi$
$823$	36.4853i	1.27180i	−0.771773	+	0.635898i	$0.780630\pi$
$824$	−1.85786	−0.0647218
$825$	0	0
$826$	−8.00000	−0.278356
$827$	34.2843i	1.19218i	0.802917	+	0.596090i	$0.203280\pi$
$827$	34.2843i	1.19218i	−0.802917	+	0.596090i	$0.796720\pi$
$828$	− 25.8579i	− 0.898623i
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	3.31371	0.114951
$832$	− 28.4853i	− 0.987549i
$833$	− 3.51472i	− 0.121778i
$834$	−4.68629	−0.162273
$835$	0	0
$836$	0	0
$837$	0	0
$838$	5.94113i	0.205233i
$839$	−37.6569	−1.30006	−0.650029	−	0.759909i	$-0.725243\pi$
$839$	−37.6569	−1.30006	−0.650029	+	0.759909i	$0.725243\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	− 2.48528i	− 0.0856485i
$843$	− 15.0294i	− 0.517641i
$844$	−29.2548	−1.00699
$845$	0	0
$846$	5.85786	0.201398
$847$	2.00000i	0.0687208i
$848$	1.02944i	0.0353510i
$849$	35.7157	1.22576
$850$	0	0
$851$	10.3431	0.354558
$852$	− 58.5097i	− 2.00451i
$853$	− 32.4853i	− 1.11227i	−0.831090	−	0.556137i	$-0.812283\pi$
$853$	− 32.4853i	− 1.11227i	0.831090	−	0.556137i	$-0.187717\pi$
$854$	−11.0294	−0.377420
$855$	0	0
$856$	−5.79899	−0.198205
$857$	48.7696i	1.66594i	0.553321	+	0.832968i	$0.313360\pi$
$857$	48.7696i	1.66594i	−0.553321	+	0.832968i	$0.686640\pi$
$858$	8.00000i	0.273115i
$859$	32.2843	1.10153	0.550763	−	0.834662i	$-0.314337\pi$
$859$	32.2843	1.10153	0.550763	+	0.834662i	$0.314337\pi$
$860$	0	0
$861$	−33.9411	−1.15671
$862$	4.68629i	0.159616i
$863$	− 14.8284i	− 0.504766i	−0.967627	−	0.252383i	$-0.918786\pi$
$863$	− 14.8284i	− 0.504766i	0.967627	−	0.252383i	$-0.0812142\pi$
$864$	24.9706	0.849516
$865$	0	0
$866$	−1.51472	−0.0514722
$867$	44.2010i	1.50115i
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	30.6274	1.03777
$872$	− 5.79899i	− 0.196379i
$873$	− 38.2843i	− 1.29573i
$874$	0	0
$875$	0	0
$876$	35.3137	1.19314
$877$	− 1.45584i	− 0.0491604i	−0.999698	−	0.0245802i	$-0.992175\pi$
$877$	− 1.45584i	− 0.0491604i	0.999698	−	0.0245802i	$-0.00782490\pi$
$878$	6.62742i	0.223664i
$879$	41.9411	1.41464
$880$	0	0
$881$	−52.6274	−1.77306	−0.886531	−	0.462668i	$-0.846892\pi$
$881$	−52.6274	−1.77306	−0.886531	+	0.462668i	$0.846892\pi$
$882$	− 6.21320i	− 0.209209i
$883$	42.8284i	1.44129i	0.693304	+	0.720646i	$0.256155\pi$
$883$	42.8284i	1.44129i	−0.693304	+	0.720646i	$0.743845\pi$
$884$	−14.6274	−0.491973
$885$	0	0
$886$	8.76955	0.294619
$887$	18.2843i	0.613926i	0.951721	+	0.306963i	$0.0993127\pi$
$887$	18.2843i	0.613926i	−0.951721	+	0.306963i	$0.900687\pi$
$888$	16.4020i	0.550416i
$889$	31.3137	1.05023
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 9.45584i	− 0.316605i
$893$	0	0
$894$	13.6569	0.456754
$895$	0	0
$896$	−21.1127	−0.705326
$897$	54.6274i	1.82396i
$898$	6.88730i	0.229832i
$899$	0	0
$900$	0	0
$901$	0.402020	0.0133932
$902$	2.48528i	0.0827508i
$903$	33.9411i	1.12949i
$904$	−13.2304	−0.440038
$905$	0	0
$906$	14.0589	0.467075
$907$	44.4853i	1.47711i	0.674193	+	0.738555i	$0.264491\pi$
$907$	44.4853i	1.47711i	−0.674193	+	0.738555i	$0.735509\pi$
$908$	− 4.91169i	− 0.163000i
$909$	66.5685	2.20794
$910$	0	0
$911$	57.9411	1.91968	0.959838	−	0.280556i	$-0.0905189\pi$
$911$	57.9411	1.91968	0.959838	+	0.280556i	$0.0905189\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	−6.82843	−0.225864
$915$	0	0
$916$	38.9706	1.28762
$917$	22.6274i	0.747223i
$918$	− 2.74517i	− 0.0906040i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−78.2254	−2.57761
$922$	− 13.5147i	− 0.445084i
$923$	77.2548i	2.54287i
$924$	−10.3431	−0.340265
$925$	0	0
$926$	−9.17157	−0.301397
$927$	− 5.85786i	− 0.192398i
$928$	− 33.7990i	− 1.10951i
$929$	17.3137	0.568044	0.284022	−	0.958818i	$-0.408331\pi$
$929$	17.3137	0.568044	0.284022	+	0.958818i	$0.408331\pi$
$930$	0	0
$931$	0	0
$932$	40.4853i	1.32614i
$933$	77.2548i	2.52921i
$934$	−3.79899	−0.124307
$935$	0	0
$936$	−54.1421	−1.76969
$937$	− 49.4558i	− 1.61565i	−0.589421	−	0.807826i	$-0.700644\pi$
$937$	− 49.4558i	− 1.61565i	0.589421	−	0.807826i	$-0.299356\pi$
$938$	− 3.71573i	− 0.121323i
$939$	−60.2843	−1.96730
$940$	0	0
$941$	−29.3137	−0.955600	−0.477800	−	0.878469i	$-0.658566\pi$
$941$	−29.3137	−0.955600	−0.477800	+	0.878469i	$0.658566\pi$
$942$	− 16.4020i	− 0.534407i
$943$	16.9706i	0.552638i
$944$	28.9706	0.942912
$945$	0	0
$946$	2.48528	0.0808035
$947$	− 46.8284i	− 1.52172i	−0.648916	−	0.760860i	$-0.724778\pi$
$947$	− 46.8284i	− 1.52172i	0.648916	−	0.760860i	$-0.275222\pi$
$948$	− 20.6863i	− 0.671860i
$949$	−46.6274	−1.51359
$950$	0	0
$951$	60.2843	1.95485
$952$	3.71573i	0.120427i
$953$	58.8284i	1.90564i	0.303536	+	0.952820i	$0.401833\pi$
$953$	58.8284i	1.90564i	−0.303536	+	0.952820i	$0.598167\pi$
$954$	0.710678	0.0230091
$955$	0	0
$956$	1.25483	0.0405842
$957$	− 21.6569i	− 0.700067i
$958$	14.9117i	0.481775i
$959$	45.9411	1.48352
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 10.3431i	− 0.333476i
$963$	− 18.2843i	− 0.589202i
$964$	10.9706	0.353338
$965$	0	0
$966$	6.62742	0.213234
$967$	− 18.9706i	− 0.610052i	−0.952344	−	0.305026i	$-0.901335\pi$
$967$	− 18.9706i	− 0.610052i	0.952344	−	0.305026i	$-0.0986652\pi$
$968$	1.58579i	0.0509691i
$969$	0	0
$970$	0	0
$971$	−31.3137	−1.00490	−0.502452	−	0.864605i	$-0.667569\pi$
$971$	−31.3137	−1.00490	−0.502452	+	0.864605i	$0.667569\pi$
$972$	− 25.8579i	− 0.829391i
$973$	8.00000i	0.256468i
$974$	−3.11270	−0.0997373
$975$	0	0
$976$	39.9411	1.27848
$977$	− 43.6569i	− 1.39671i	−0.715753	−	0.698353i	$-0.753916\pi$
$977$	− 43.6569i	− 1.39671i	0.715753	−	0.698353i	$-0.246084\pi$
$978$	0.568542i	0.0181800i
$979$	−9.31371	−0.297667
$980$	0	0
$981$	18.2843	0.583772
$982$	− 9.65685i	− 0.308163i
$983$	− 50.1421i	− 1.59929i	−0.600476	−	0.799643i	$-0.705022\pi$
$983$	− 50.1421i	− 1.59929i	0.600476	−	0.799643i	$-0.294978\pi$
$984$	−26.9117	−0.857913
$985$	0	0
$986$	−3.71573	−0.118333
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	9.94113	0.315790	0.157895	−	0.987456i	$-0.449529\pi$
$991$	9.94113	0.315790	0.157895	+	0.987456i	$0.449529\pi$
$992$	0	0
$993$	43.3137i	1.37452i
$994$	9.37258	0.297280
$995$	0	0
$996$	31.0294	0.983205
$997$	− 9.45584i	− 0.299470i	−0.988726	−	0.149735i	$-0.952158\pi$
$997$	− 9.45584i	− 0.299470i	0.988726	−	0.149735i	$-0.0478420\pi$
$998$	0.686292i	0.0217242i
$999$	−20.6863	−0.654485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.b.d.199.3		4
3.2	odd	2		2475.2.c.l.199.2		4
4.3	odd	2		4400.2.b.q.4049.1		4
5.2	odd	4		55.2.a.b.1.1	✓	2
5.3	odd	4		275.2.a.c.1.2		2
5.4	even	2	inner	275.2.b.d.199.2		4
15.2	even	4		495.2.a.b.1.2		2
15.8	even	4		2475.2.a.x.1.1		2
15.14	odd	2		2475.2.c.l.199.3		4
20.3	even	4		4400.2.a.bn.1.2		2
20.7	even	4		880.2.a.m.1.1		2
20.19	odd	2		4400.2.b.q.4049.4		4
35.27	even	4		2695.2.a.f.1.1		2
40.27	even	4		3520.2.a.bo.1.2		2
40.37	odd	4		3520.2.a.bn.1.1		2
55.2	even	20		605.2.g.l.81.1		8
55.7	even	20		605.2.g.l.511.2		8
55.17	even	20		605.2.g.l.366.1		8
55.27	odd	20		605.2.g.f.366.2		8
55.32	even	4		605.2.a.d.1.2		2
55.37	odd	20		605.2.g.f.511.1		8
55.42	odd	20		605.2.g.f.81.2		8
55.43	even	4		3025.2.a.o.1.1		2
55.47	odd	20		605.2.g.f.251.1		8
55.52	even	20		605.2.g.l.251.2		8
60.47	odd	4		7920.2.a.ch.1.1		2
65.12	odd	4		9295.2.a.g.1.2		2
165.32	odd	4		5445.2.a.y.1.1		2
220.87	odd	4		9680.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.1	✓	2	5.2	odd	4
275.2.a.c.1.2		2	5.3	odd	4
275.2.b.d.199.2		4	5.4	even	2	inner
275.2.b.d.199.3		4	1.1	even	1	trivial
495.2.a.b.1.2		2	15.2	even	4
605.2.a.d.1.2		2	55.32	even	4
605.2.g.f.81.2		8	55.42	odd	20
605.2.g.f.251.1		8	55.47	odd	20
605.2.g.f.366.2		8	55.27	odd	20
605.2.g.f.511.1		8	55.37	odd	20
605.2.g.l.81.1		8	55.2	even	20
605.2.g.l.251.2		8	55.52	even	20
605.2.g.l.366.1		8	55.17	even	20
605.2.g.l.511.2		8	55.7	even	20
880.2.a.m.1.1		2	20.7	even	4
2475.2.a.x.1.1		2	15.8	even	4
2475.2.c.l.199.2		4	3.2	odd	2
2475.2.c.l.199.3		4	15.14	odd	2
2695.2.a.f.1.1		2	35.27	even	4
3025.2.a.o.1.1		2	55.43	even	4
3520.2.a.bn.1.1		2	40.37	odd	4
3520.2.a.bo.1.2		2	40.27	even	4
4400.2.a.bn.1.2		2	20.3	even	4
4400.2.b.q.4049.1		4	4.3	odd	2
4400.2.b.q.4049.4		4	20.19	odd	2
5445.2.a.y.1.1		2	165.32	odd	4
7920.2.a.ch.1.1		2	60.47	odd	4
9295.2.a.g.1.2		2	65.12	odd	4
9680.2.a.bn.1.1		2	220.87	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +O(q^{10})\)	\(q+0.414214i q^{2} +2.82843i q^{3} +1.82843 q^{4} -1.17157 q^{6} +2.00000i q^{7} +1.58579i q^{8} -5.00000 q^{9} +1.00000 q^{11} +5.17157i q^{12} -6.82843i q^{13} -0.828427 q^{14} +3.00000 q^{16} -1.17157i q^{17} -2.07107i q^{18} -5.65685 q^{21} +0.414214i q^{22} +2.82843i q^{23} -4.48528 q^{24} +2.82843 q^{26} -5.65685i q^{27} +3.65685i q^{28} -7.65685 q^{29} +4.41421i q^{32} +2.82843i q^{33} +0.485281 q^{34} -9.14214 q^{36} -3.65685i q^{37} +19.3137 q^{39} +6.00000 q^{41} -2.34315i q^{42} -6.00000i q^{43} +1.82843 q^{44} -1.17157 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} +3.31371 q^{51} -12.4853i q^{52} +0.343146i q^{53} +2.34315 q^{54} -3.17157 q^{56} -3.17157i q^{58} +9.65685 q^{59} +13.3137 q^{61} -10.0000i q^{63} +4.17157 q^{64} -1.17157 q^{66} +4.48528i q^{67} -2.14214i q^{68} -8.00000 q^{69} -11.3137 q^{71} -7.92893i q^{72} -6.82843i q^{73} +1.51472 q^{74} +2.00000i q^{77} +8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +2.48528i q^{82} -6.00000i q^{83} -10.3431 q^{84} +2.48528 q^{86} -21.6569i q^{87} +1.58579i q^{88} -9.31371 q^{89} +13.6569 q^{91} +5.17157i q^{92} -1.17157 q^{94} -12.4853 q^{96} +7.65685i q^{97} +1.24264i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9	\( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9 + 4 * q^11 + 8 * q^14 + 12 * q^16 + 16 * q^24 - 8 * q^29 - 32 * q^34 + 20 * q^36 + 32 * q^39 + 24 * q^41 - 4 * q^44 - 16 * q^46 + 12 * q^49 - 32 * q^51 + 32 * q^54 - 24 * q^56 + 16 * q^59 + 8 * q^61 + 28 * q^64 - 16 * q^66 - 32 * q^69 + 40 * q^74 - 16 * q^79 + 4 * q^81 - 64 * q^84 - 24 * q^86 + 8 * q^89 + 32 * q^91 - 16 * q^94 - 16 * q^96 - 20 * q^99

Introduction

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