LMFDB - Embedded newform 275.2.b.d.199.4

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			199.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.2.b.d.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{2} +2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} -2.00000i q^{7} -4.41421i q^{8} -5.00000 q^{9} +O(q^{10})$	\(q+2.41421i q^{2} +2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} -2.00000i q^{7} -4.41421i q^{8} -5.00000 q^{9} +1.00000 q^{11} -10.8284i q^{12} +1.17157i q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843i q^{17} -12.0711i q^{18} +5.65685 q^{21} +2.41421i q^{22} +2.82843i q^{23} +12.4853 q^{24} -2.82843 q^{26} -5.65685i q^{27} +7.65685i q^{28} +3.65685 q^{29} -1.58579i q^{32} +2.82843i q^{33} -16.4853 q^{34} +19.1421 q^{36} -7.65685i q^{37} -3.31371 q^{39} +6.00000 q^{41} +13.6569i q^{42} +6.00000i q^{43} -3.82843 q^{44} -6.82843 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} -19.3137 q^{51} -4.48528i q^{52} -11.6569i q^{53} +13.6569 q^{54} -8.82843 q^{56} +8.82843i q^{58} -1.65685 q^{59} -9.31371 q^{61} +10.0000i q^{63} +9.82843 q^{64} -6.82843 q^{66} +12.4853i q^{67} -26.1421i q^{68} -8.00000 q^{69} +11.3137 q^{71} +22.0711i q^{72} +1.17157i q^{73} +18.4853 q^{74} -2.00000i q^{77} -8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +14.4853i q^{82} +6.00000i q^{83} -21.6569 q^{84} -14.4853 q^{86} +10.3431i q^{87} -4.41421i q^{88} +13.3137 q^{89} +2.34315 q^{91} -10.8284i q^{92} -6.82843 q^{94} +4.48528 q^{96} +3.65685i q^{97} +7.24264i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$	2.82843i	1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	−6.82843	−2.78769
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 4.41421i	− 1.56066i
$9$	−5.00000	−1.66667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 10.8284i	− 3.12590i
$13$	1.17157i	0.324936i	0.986714	+	0.162468i	$0.0519454\pi$
$13$	1.17157i	0.324936i	−0.986714	+	0.162468i	$0.948055\pi$
$14$	4.82843	1.29045
$15$	0	0
$16$	3.00000	0.750000
$17$	6.82843i	1.65614i	0.560627	+	0.828068i	$0.310560\pi$
$17$	6.82843i	1.65614i	−0.560627	+	0.828068i	$0.689440\pi$
$18$	− 12.0711i	− 2.84518i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	5.65685	1.23443
$22$	2.41421i	0.514712i
$23$	2.82843i	0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$	2.82843i	0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$	12.4853	2.54855
$25$	0	0
$26$	−2.82843	−0.554700
$27$	− 5.65685i	− 1.08866i
$28$	7.65685i	1.44701i
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.58579i	− 0.280330i
$33$	2.82843i	0.492366i
$34$	−16.4853	−2.82720
$35$	0	0
$36$	19.1421	3.19036
$37$	− 7.65685i	− 1.25878i	−0.777090	−	0.629390i	$-0.783305\pi$
$37$	− 7.65685i	− 1.25878i	0.777090	−	0.629390i	$-0.216695\pi$
$38$	0	0
$39$	−3.31371	−0.530618
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	13.6569i	2.10730i
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−3.82843	−0.577157
$45$	0	0
$46$	−6.82843	−1.00680
$47$	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
$51$	−19.3137	−2.70446
$52$	− 4.48528i	− 0.621997i
$53$	− 11.6569i	− 1.60119i	−0.599204	−	0.800596i	$-0.704516\pi$
$53$	− 11.6569i	− 1.60119i	0.599204	−	0.800596i	$-0.295484\pi$
$54$	13.6569	1.85846
$55$	0	0
$56$	−8.82843	−1.17975
$57$	0	0
$58$	8.82843i	1.15923i
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	10.0000i	1.25988i
$64$	9.82843	1.22855
$65$	0	0
$66$	−6.82843	−0.840521
$67$	12.4853i	1.52532i	0.646800	+	0.762660i	$0.276107\pi$
$67$	12.4853i	1.52532i	−0.646800	+	0.762660i	$0.723893\pi$
$68$	− 26.1421i	− 3.17020i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	22.0711i	2.60110i
$73$	1.17157i	0.137122i	0.997647	+	0.0685611i	$0.0218408\pi$
$73$	1.17157i	0.137122i	−0.997647	+	0.0685611i	$0.978159\pi$
$74$	18.4853	2.14887
$75$	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
$82$	14.4853i	1.59963i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	−21.6569	−2.36296
$85$	0	0
$86$	−14.4853	−1.56199
$87$	10.3431i	1.10890i
$88$	− 4.41421i	− 0.470557i
$89$	13.3137	1.41125	0.705625	−	0.708585i	$-0.250666\pi$
$89$	13.3137	1.41125	0.705625	+	0.708585i	$0.250666\pi$
$90$	0	0
$91$	2.34315	0.245628
$92$	− 10.8284i	− 1.12894i
$93$	0	0
$94$	−6.82843	−0.704298
$95$	0	0
$96$	4.48528	0.457777
$97$	3.65685i	0.371297i	0.982616	+	0.185649i	$0.0594386\pi$
$97$	3.65685i	0.371297i	−0.982616	+	0.185649i	$0.940561\pi$
$98$	7.24264i	0.731617i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	9.31371	0.926749	0.463374	−	0.886163i	$-0.346639\pi$
$101$	9.31371	0.926749	0.463374	+	0.886163i	$0.346639\pi$
$102$	− 46.6274i	− 4.61680i
$103$	− 6.82843i	− 0.672825i	−0.941715	−	0.336412i	$-0.890786\pi$
$103$	− 6.82843i	− 0.672825i	0.941715	−	0.336412i	$-0.109214\pi$
$104$	5.17157	0.507114
$105$	0	0
$106$	28.1421	2.73341
$107$	7.65685i	0.740216i	0.928989	+	0.370108i	$0.120679\pi$
$107$	7.65685i	0.740216i	−0.928989	+	0.370108i	$0.879321\pi$
$108$	21.6569i	2.08393i
$109$	7.65685	0.733394	0.366697	−	0.930341i	$-0.380489\pi$
$109$	7.65685	0.733394	0.366697	+	0.930341i	$0.380489\pi$
$110$	0	0
$111$	21.6569	2.05558
$112$	− 6.00000i	− 0.566947i
$113$	− 19.6569i	− 1.84916i	−0.380986	−	0.924581i	$-0.624416\pi$
$113$	− 19.6569i	− 1.84916i	0.380986	−	0.924581i	$-0.375584\pi$
$114$	0	0
$115$	0	0
$116$	−14.0000	−1.29987
$117$	− 5.85786i	− 0.541560i
$118$	− 4.00000i	− 0.368230i
$119$	13.6569	1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 22.4853i	− 2.03572i
$123$	16.9706i	1.53018i
$124$	0	0
$125$	0	0
$126$	−24.1421	−2.15075
$127$	4.34315i	0.385392i	0.981259	+	0.192696i	$0.0617231\pi$
$127$	4.34315i	0.385392i	−0.981259	+	0.192696i	$0.938277\pi$
$128$	20.5563i	1.81694i
$129$	−16.9706	−1.49417
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	− 10.8284i	− 0.942494i
$133$	0	0
$134$	−30.1421	−2.60388
$135$	0	0
$136$	30.1421	2.58467
$137$	− 10.9706i	− 0.937278i	−0.883390	−	0.468639i	$-0.844744\pi$
$137$	− 10.9706i	− 0.937278i	0.883390	−	0.468639i	$-0.155256\pi$
$138$	− 19.3137i	− 1.64409i
$139$	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
$142$	27.3137i	2.29212i
$143$	1.17157i	0.0979718i
$144$	−15.0000	−1.25000
$145$	0	0
$146$	−2.82843	−0.234082
$147$	8.48528i	0.699854i
$148$	29.3137i	2.40957i
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	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
$153$	− 34.1421i	− 2.76023i
$154$	4.82843	0.389086
$155$	0	0
$156$	12.6863	1.01572
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	− 9.65685i	− 0.768258i
$159$	32.9706	2.61474
$160$	0	0
$161$	5.65685	0.445823
$162$	2.41421i	0.189679i
$163$	− 16.4853i	− 1.29123i	−0.763664	−	0.645613i	$-0.776602\pi$
$163$	− 16.4853i	− 1.29123i	0.763664	−	0.645613i	$-0.223398\pi$
$164$	−22.9706	−1.79370
$165$	0	0
$166$	−14.4853	−1.12428
$167$	− 22.9706i	− 1.77752i	−0.458377	−	0.888758i	$-0.651569\pi$
$167$	− 22.9706i	− 1.77752i	0.458377	−	0.888758i	$-0.348431\pi$
$168$	− 24.9706i	− 1.92652i
$169$	11.6274	0.894417
$170$	0	0
$171$	0	0
$172$	− 22.9706i	− 1.75149i
$173$	22.1421i	1.68344i	0.539918	+	0.841718i	$0.318455\pi$
$173$	22.1421i	1.68344i	−0.539918	+	0.841718i	$0.681545\pi$
$174$	−24.9706	−1.89301
$175$	0	0
$176$	3.00000	0.226134
$177$	− 4.68629i	− 0.352243i
$178$	32.1421i	2.40915i
$179$	−9.65685	−0.721787	−0.360894	−	0.932607i	$-0.617528\pi$
$179$	−9.65685	−0.721787	−0.360894	+	0.932607i	$0.617528\pi$
$180$	0	0
$181$	21.3137	1.58424	0.792118	−	0.610368i	$-0.208979\pi$
$181$	21.3137	1.58424	0.792118	+	0.610368i	$0.208979\pi$
$182$	5.65685i	0.419314i
$183$	− 26.3431i	− 1.94734i
$184$	12.4853	0.920427
$185$	0	0
$186$	0	0
$187$	6.82843i	0.499344i
$188$	− 10.8284i	− 0.789744i
$189$	−11.3137	−0.822951
$190$	0	0
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	27.7990i	2.00622i
$193$	1.17157i	0.0843317i	0.999111	+	0.0421658i	$0.0134258\pi$
$193$	1.17157i	0.0843317i	−0.999111	+	0.0421658i	$0.986574\pi$
$194$	−8.82843	−0.633844
$195$	0	0
$196$	−11.4853	−0.820377
$197$	− 10.8284i	− 0.771493i	−0.922605	−	0.385747i	$-0.873944\pi$
$197$	− 10.8284i	− 0.771493i	0.922605	−	0.385747i	$-0.126056\pi$
$198$	− 12.0711i	− 0.857853i
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	−35.3137	−2.49084
$202$	22.4853i	1.58206i
$203$	− 7.31371i	− 0.513322i
$204$	73.9411	5.17691
$205$	0	0
$206$	16.4853	1.14858
$207$	− 14.1421i	− 0.982946i
$208$	3.51472i	0.243702i
$209$	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	$0.685655\pi$
$212$	44.6274i	3.06502i
$213$	32.0000i	2.19260i
$214$	−18.4853	−1.26363
$215$	0	0
$216$	−24.9706	−1.69903
$217$	0	0
$218$	18.4853i	1.25198i
$219$	−3.31371	−0.223920
$220$	0	0
$221$	−8.00000	−0.538138
$222$	52.2843i	3.50909i
$223$	10.8284i	0.725125i	0.931959	+	0.362563i	$0.118098\pi$
$223$	10.8284i	0.725125i	−0.931959	+	0.362563i	$0.881902\pi$
$224$	−3.17157	−0.211910
$225$	0	0
$226$	47.4558	3.15672
$227$	25.3137i	1.68013i	0.542486	+	0.840065i	$0.317483\pi$
$227$	25.3137i	1.68013i	−0.542486	+	0.840065i	$0.682517\pi$
$228$	0	0
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	− 16.1421i	− 1.05978i
$233$	6.14214i	0.402385i	0.979552	+	0.201192i	$0.0644816\pi$
$233$	6.14214i	0.402385i	−0.979552	+	0.201192i	$0.935518\pi$
$234$	14.1421	0.924500
$235$	0	0
$236$	6.34315	0.412904
$237$	− 11.3137i	− 0.734904i
$238$	32.9706i	2.13716i
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	2.41421i	0.155192i
$243$	− 14.1421i	− 0.907218i
$244$	35.6569	2.28270
$245$	0	0
$246$	−40.9706	−2.61219
$247$	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	$0.376365\pi$
$252$	− 38.2843i	− 2.41168i
$253$	2.82843i	0.177822i
$254$	−10.4853	−0.657905
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 9.31371i	− 0.580973i	−0.956879	−	0.290487i	$-0.906183\pi$
$257$	− 9.31371i	− 0.580973i	0.956879	−	0.290487i	$-0.0938172\pi$
$258$	− 40.9706i	− 2.55072i
$259$	−15.3137	−0.951548
$260$	0	0
$261$	−18.2843	−1.13177
$262$	− 27.3137i	− 1.68745i
$263$	10.9706i	0.676474i	0.941061	+	0.338237i	$0.109831\pi$
$263$	10.9706i	0.676474i	−0.941061	+	0.338237i	$0.890169\pi$
$264$	12.4853	0.768416
$265$	0	0
$266$	0	0
$267$	37.6569i	2.30456i
$268$	− 47.7990i	− 2.91979i
$269$	−17.3137	−1.05564	−0.527818	−	0.849358i	$-0.676990\pi$
$269$	−17.3137	−1.05564	−0.527818	+	0.849358i	$0.676990\pi$
$270$	0	0
$271$	7.31371	0.444276	0.222138	−	0.975015i	$-0.428696\pi$
$271$	7.31371	0.444276	0.222138	+	0.975015i	$0.428696\pi$
$272$	20.4853i	1.24210i
$273$	6.62742i	0.401110i
$274$	26.4853	1.60003
$275$	0	0
$276$	30.6274	1.84355
$277$	6.82843i	0.410280i	0.978733	+	0.205140i	$0.0657650\pi$
$277$	6.82843i	0.410280i	−0.978733	+	0.205140i	$0.934235\pi$
$278$	9.65685i	0.579180i
$279$	0	0
$280$	0	0
$281$	17.3137	1.03285	0.516425	−	0.856333i	$-0.327263\pi$
$281$	17.3137	1.03285	0.516425	+	0.856333i	$0.327263\pi$
$282$	− 19.3137i	− 1.15011i
$283$	− 32.6274i	− 1.93950i	−0.244103	−	0.969749i	$-0.578493\pi$
$283$	− 32.6274i	− 1.93950i	0.244103	−	0.969749i	$-0.421507\pi$
$284$	−43.3137	−2.57020
$285$	0	0
$286$	−2.82843	−0.167248
$287$	− 12.0000i	− 0.708338i
$288$	7.92893i	0.467217i
$289$	−29.6274	−1.74279
$290$	0	0
$291$	−10.3431	−0.606326
$292$	− 4.48528i	− 0.262481i
$293$	9.17157i	0.535809i	0.963445	+	0.267905i	$0.0863312\pi$
$293$	9.17157i	0.535809i	−0.963445	+	0.267905i	$0.913669\pi$
$294$	−20.4853	−1.19473
$295$	0	0
$296$	−33.7990	−1.96453
$297$	− 5.65685i	− 0.328244i
$298$	− 0.828427i	− 0.0479895i
$299$	−3.31371	−0.191637
$300$	0	0
$301$	12.0000	0.691669
$302$	− 28.9706i	− 1.66707i
$303$	26.3431i	1.51337i
$304$	0	0
$305$	0	0
$306$	82.4264	4.71200
$307$	− 16.3431i	− 0.932753i	−0.884586	−	0.466376i	$-0.845559\pi$
$307$	− 16.3431i	− 0.932753i	0.884586	−	0.466376i	$-0.154441\pi$
$308$	7.65685i	0.436290i
$309$	19.3137	1.09872
$310$	0	0
$311$	4.68629	0.265735	0.132868	−	0.991134i	$-0.457581\pi$
$311$	4.68629	0.265735	0.132868	+	0.991134i	$0.457581\pi$
$312$	14.6274i	0.828114i
$313$	1.31371i	0.0742552i	0.999311	+	0.0371276i	$0.0118208\pi$
$313$	1.31371i	0.0742552i	−0.999311	+	0.0371276i	$0.988179\pi$
$314$	33.7990	1.90739
$315$	0	0
$316$	15.3137	0.861463
$317$	− 1.31371i	− 0.0737852i	−0.999319	−	0.0368926i	$-0.988254\pi$
$317$	− 1.31371i	− 0.0737852i	0.999319	−	0.0368926i	$-0.0117459\pi$
$318$	79.5980i	4.46363i
$319$	3.65685	0.204745
$320$	0	0
$321$	−21.6569	−1.20877
$322$	13.6569i	0.761067i
$323$	0	0
$324$	−3.82843	−0.212690
$325$	0	0
$326$	39.7990	2.20426
$327$	21.6569i	1.19763i
$328$	− 26.4853i	− 1.46241i
$329$	5.65685	0.311872
$330$	0	0
$331$	−7.31371	−0.401998	−0.200999	−	0.979591i	$-0.564419\pi$
$331$	−7.31371	−0.401998	−0.200999	+	0.979591i	$0.564419\pi$
$332$	− 22.9706i	− 1.26067i
$333$	38.2843i	2.09797i
$334$	55.4558	3.03441
$335$	0	0
$336$	16.9706	0.925820
$337$	− 20.4853i	− 1.11590i	−0.829873	−	0.557952i	$-0.811587\pi$
$337$	− 20.4853i	− 1.11590i	0.829873	−	0.557952i	$-0.188413\pi$
$338$	28.0711i	1.52686i
$339$	55.5980	3.01967
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	26.4853	1.42799
$345$	0	0
$346$	−53.4558	−2.87380
$347$	10.9706i	0.588931i	0.955662	+	0.294465i	$0.0951416\pi$
$347$	10.9706i	0.588931i	−0.955662	+	0.294465i	$0.904858\pi$
$348$	− 39.5980i	− 2.12267i
$349$	−26.9706	−1.44370	−0.721851	−	0.692049i	$-0.756708\pi$
$349$	−26.9706	−1.44370	−0.721851	+	0.692049i	$0.756708\pi$
$350$	0	0
$351$	6.62742	0.353745
$352$	− 1.58579i	− 0.0845227i
$353$	− 21.3137i	− 1.13441i	−0.823575	−	0.567207i	$-0.808024\pi$
$353$	− 21.3137i	− 1.13441i	0.823575	−	0.567207i	$-0.191976\pi$
$354$	11.3137	0.601317
$355$	0	0
$356$	−50.9706	−2.70143
$357$	38.6274i	2.04438i
$358$	− 23.3137i	− 1.23217i
$359$	−0.686292	−0.0362211	−0.0181105	−	0.999836i	$-0.505765\pi$
$359$	−0.686292	−0.0362211	−0.0181105	+	0.999836i	$0.505765\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	51.4558i	2.70446i
$363$	2.82843i	0.148454i
$364$	−8.97056	−0.470185
$365$	0	0
$366$	63.5980	3.32432
$367$	− 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
$371$	−23.3137	−1.21039
$372$	0	0
$373$	− 35.7990i	− 1.85360i	−0.375554	−	0.926801i	$-0.622547\pi$
$373$	− 35.7990i	− 1.85360i	0.375554	−	0.926801i	$-0.377453\pi$
$374$	−16.4853	−0.852434
$375$	0	0
$376$	12.4853	0.643879
$377$	4.28427i	0.220651i
$378$	− 27.3137i	− 1.40487i
$379$	−33.6569	−1.72884	−0.864418	−	0.502773i	$-0.832313\pi$
$379$	−33.6569	−1.72884	−0.864418	+	0.502773i	$0.832313\pi$
$380$	0	0
$381$	−12.2843	−0.629342
$382$	8.00000i	0.409316i
$383$	5.85786i	0.299323i	0.988737	+	0.149661i	$0.0478184\pi$
$383$	5.85786i	0.299323i	−0.988737	+	0.149661i	$0.952182\pi$
$384$	−58.1421	−2.96705
$385$	0	0
$386$	−2.82843	−0.143963
$387$	− 30.0000i	− 1.52499i
$388$	− 14.0000i	− 0.710742i
$389$	−20.6274	−1.04585	−0.522926	−	0.852378i	$-0.675160\pi$
$389$	−20.6274	−1.04585	−0.522926	+	0.852378i	$0.675160\pi$
$390$	0	0
$391$	−19.3137	−0.976736
$392$	− 13.2426i	− 0.668854i
$393$	− 32.0000i	− 1.61419i
$394$	26.1421	1.31702
$395$	0	0
$396$	19.1421	0.961929
$397$	− 9.31371i	− 0.467442i	−0.972304	−	0.233721i	$-0.924910\pi$
$397$	− 9.31371i	− 0.467442i	0.972304	−	0.233721i	$-0.0750902\pi$
$398$	− 24.9706i	− 1.25166i
$399$	0	0
$400$	0	0
$401$	−5.31371	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$401$	−5.31371	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$402$	− 85.2548i	− 4.25212i
$403$	0	0
$404$	−35.6569	−1.77399
$405$	0	0
$406$	17.6569	0.876295
$407$	− 7.65685i	− 0.379536i
$408$	85.2548i	4.22074i
$409$	−1.02944	−0.0509024	−0.0254512	−	0.999676i	$-0.508102\pi$
$409$	−1.02944	−0.0509024	−0.0254512	+	0.999676i	$0.508102\pi$
$410$	0	0
$411$	31.0294	1.53057
$412$	26.1421i	1.28793i
$413$	3.31371i	0.163057i
$414$	34.1421	1.67799
$415$	0	0
$416$	1.85786	0.0910893
$417$	11.3137i	0.554035i
$418$	0	0
$419$	25.6569	1.25342	0.626710	−	0.779253i	$-0.284401\pi$
$419$	25.6569	1.25342	0.626710	+	0.779253i	$0.284401\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	− 38.6274i	− 1.88035i
$423$	− 14.1421i	− 0.687614i
$424$	−51.4558	−2.49892
$425$	0	0
$426$	−77.2548	−3.74301
$427$	18.6274i	0.901444i
$428$	− 29.3137i	− 1.41693i
$429$	−3.31371	−0.159987
$430$	0	0
$431$	−11.3137	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$431$	−11.3137	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$432$	− 16.9706i	− 0.816497i
$433$	7.65685i	0.367965i	0.982930	+	0.183982i	$0.0588990\pi$
$433$	7.65685i	0.367965i	−0.982930	+	0.183982i	$0.941101\pi$
$434$	0	0
$435$	0	0
$436$	−29.3137	−1.40387
$437$	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
$442$	− 19.3137i	− 0.918659i
$443$	26.8284i	1.27466i	0.770592	+	0.637329i	$0.219961\pi$
$443$	26.8284i	1.27466i	−0.770592	+	0.637329i	$0.780039\pi$
$444$	−82.9117	−3.93481
$445$	0	0
$446$	−26.1421	−1.23787
$447$	− 0.970563i	− 0.0459060i
$448$	− 19.6569i	− 0.928699i
$449$	−28.6274	−1.35101	−0.675506	−	0.737355i	$-0.736075\pi$
$449$	−28.6274	−1.35101	−0.675506	+	0.737355i	$0.736075\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	75.2548i	3.53969i
$453$	− 33.9411i	− 1.59469i
$454$	−61.1127	−2.86816
$455$	0	0
$456$	0	0
$457$	0.485281i	0.0227005i	0.999936	+	0.0113503i	$0.00361298\pi$
$457$	0.485281i	0.0227005i	−0.999936	+	0.0113503i	$0.996387\pi$
$458$	− 3.17157i	− 0.148198i
$459$	38.6274	1.80297
$460$	0	0
$461$	12.6274	0.588117	0.294059	−	0.955787i	$-0.404994\pi$
$461$	12.6274	0.588117	0.294059	+	0.955787i	$0.404994\pi$
$462$	13.6569i	0.635374i
$463$	6.14214i	0.285449i	0.989762	+	0.142725i	$0.0455863\pi$
$463$	6.14214i	0.285449i	−0.989762	+	0.142725i	$0.954414\pi$
$464$	10.9706	0.509296
$465$	0	0
$466$	−14.8284	−0.686914
$467$	− 14.8284i	− 0.686178i	−0.939303	−	0.343089i	$-0.888527\pi$
$467$	− 14.8284i	− 0.686178i	0.939303	−	0.343089i	$-0.111473\pi$
$468$	22.4264i	1.03666i
$469$	24.9706	1.15303
$470$	0	0
$471$	39.5980	1.82458
$472$	7.31371i	0.336641i
$473$	6.00000i	0.275880i
$474$	27.3137	1.25456
$475$	0	0
$476$	−52.2843	−2.39645
$477$	58.2843i	2.66865i
$478$	56.2843i	2.57438i
$479$	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
$481$	8.97056	0.409022
$482$	14.4853i	0.659786i
$483$	16.0000i	0.728025i
$484$	−3.82843	−0.174019
$485$	0	0
$486$	34.1421	1.54872
$487$	− 24.4853i	− 1.10953i	−0.832006	−	0.554767i	$-0.812807\pi$
$487$	− 24.4853i	− 1.10953i	0.832006	−	0.554767i	$-0.187193\pi$
$488$	41.1127i	1.86108i
$489$	46.6274	2.10856
$490$	0	0
$491$	−0.686292	−0.0309719	−0.0154860	−	0.999880i	$-0.504930\pi$
$491$	−0.686292	−0.0309719	−0.0154860	+	0.999880i	$0.504930\pi$
$492$	− 64.9706i	− 2.92910i
$493$	24.9706i	1.12462i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 22.6274i	− 1.01498i
$498$	− 40.9706i	− 1.83593i
$499$	−9.65685	−0.432300	−0.216150	−	0.976360i	$-0.569350\pi$
$499$	−9.65685	−0.432300	−0.216150	+	0.976360i	$0.569350\pi$
$500$	0	0
$501$	64.9706	2.90267
$502$	28.9706i	1.29302i
$503$	− 16.6274i	− 0.741380i	−0.928757	−	0.370690i	$-0.879121\pi$
$503$	− 16.6274i	− 0.741380i	0.928757	−	0.370690i	$-0.120879\pi$
$504$	44.1421	1.96625
$505$	0	0
$506$	−6.82843	−0.303561
$507$	32.8873i	1.46058i
$508$	− 16.6274i	− 0.737722i
$509$	13.3137	0.590120	0.295060	−	0.955479i	$-0.404660\pi$
$509$	13.3137	0.590120	0.295060	+	0.955479i	$0.404660\pi$
$510$	0	0
$511$	2.34315	0.103655
$512$	− 31.2426i	− 1.38074i
$513$	0	0
$514$	22.4853	0.991783
$515$	0	0
$516$	64.9706	2.86017
$517$	2.82843i	0.124394i
$518$	− 36.9706i	− 1.62439i
$519$	−62.6274	−2.74904
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	− 44.1421i	− 1.93205i
$523$	41.5980i	1.81895i	0.415756	+	0.909476i	$0.363517\pi$
$523$	41.5980i	1.81895i	−0.415756	+	0.909476i	$0.636483\pi$
$524$	43.3137	1.89217
$525$	0	0
$526$	−26.4853	−1.15481
$527$	0	0
$528$	8.48528i	0.369274i
$529$	15.0000	0.652174
$530$	0	0
$531$	8.28427	0.359507
$532$	0	0
$533$	7.02944i	0.304479i
$534$	−90.9117	−3.93413
$535$	0	0
$536$	55.1127	2.38051
$537$	− 27.3137i	− 1.17867i
$538$	− 41.7990i	− 1.80208i
$539$	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	$0.458830\pi$
$542$	17.6569i	0.758427i
$543$	60.2843i	2.58705i
$544$	10.8284	0.464265
$545$	0	0
$546$	−16.0000	−0.684737
$547$	− 34.0000i	− 1.45374i	−0.686778	−	0.726868i	$-0.740975\pi$
$547$	− 34.0000i	− 1.45374i	0.686778	−	0.726868i	$-0.259025\pi$
$548$	42.0000i	1.79415i
$549$	46.5685	1.98750
$550$	0	0
$551$	0	0
$552$	35.3137i	1.50305i
$553$	8.00000i	0.340195i
$554$	−16.4853	−0.700392
$555$	0	0
$556$	−15.3137	−0.649446
$557$	9.85786i	0.417691i	0.977949	+	0.208846i	$0.0669706\pi$
$557$	9.85786i	0.417691i	−0.977949	+	0.208846i	$0.933029\pi$
$558$	0	0
$559$	−7.02944	−0.297314
$560$	0	0
$561$	−19.3137	−0.815425
$562$	41.7990i	1.76318i
$563$	− 0.343146i	− 0.0144619i	−0.999974	−	0.00723093i	$-0.997698\pi$
$563$	− 0.343146i	− 0.0144619i	0.999974	−	0.00723093i	$-0.00230170\pi$
$564$	30.6274	1.28965
$565$	0	0
$566$	78.7696	3.31093
$567$	− 2.00000i	− 0.0839921i
$568$	− 49.9411i	− 2.09548i
$569$	−31.6569	−1.32712	−0.663562	−	0.748121i	$-0.730956\pi$
$569$	−31.6569	−1.32712	−0.663562	+	0.748121i	$0.730956\pi$
$570$	0	0
$571$	−21.9411	−0.918208	−0.459104	−	0.888383i	$-0.651829\pi$
$571$	−21.9411	−0.918208	−0.459104	+	0.888383i	$0.651829\pi$
$572$	− 4.48528i	− 0.187539i
$573$	9.37258i	0.391545i
$574$	28.9706	1.20921
$575$	0	0
$576$	−49.1421	−2.04759
$577$	− 26.9706i	− 1.12280i	−0.827545	−	0.561400i	$-0.810263\pi$
$577$	− 26.9706i	− 1.12280i	0.827545	−	0.561400i	$-0.189737\pi$
$578$	− 71.5269i	− 2.97513i
$579$	−3.31371	−0.137713
$580$	0	0
$581$	12.0000	0.497844
$582$	− 24.9706i	− 1.03506i
$583$	− 11.6569i	− 0.482778i
$584$	5.17157	0.214001
$585$	0	0
$586$	−22.1421	−0.914683
$587$	− 2.14214i	− 0.0884154i	−0.999022	−	0.0442077i	$-0.985924\pi$
$587$	− 2.14214i	− 0.0884154i	0.999022	−	0.0442077i	$-0.0140763\pi$
$588$	− 32.4853i	− 1.33967i
$589$	0	0
$590$	0	0
$591$	30.6274	1.25984
$592$	− 22.9706i	− 0.944084i
$593$	− 3.51472i	− 0.144332i	−0.997393	−	0.0721661i	$-0.977009\pi$
$593$	− 3.51472i	− 0.144332i	0.997393	−	0.0721661i	$-0.0229912\pi$
$594$	13.6569	0.560348
$595$	0	0
$596$	1.31371	0.0538116
$597$	− 29.2548i	− 1.19732i
$598$	− 8.00000i	− 0.327144i
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	23.9411	0.976579	0.488289	−	0.872682i	$-0.337621\pi$
$601$	23.9411	0.976579	0.488289	+	0.872682i	$0.337621\pi$
$602$	28.9706i	1.18075i
$603$	− 62.4264i	− 2.54220i
$604$	45.9411	1.86932
$605$	0	0
$606$	−63.5980	−2.58349
$607$	38.2843i	1.55391i	0.629556	+	0.776955i	$0.283237\pi$
$607$	38.2843i	1.55391i	−0.629556	+	0.776955i	$0.716763\pi$
$608$	0	0
$609$	20.6863	0.838251
$610$	0	0
$611$	−3.31371	−0.134058
$612$	130.711i	5.28367i
$613$	25.4558i	1.02815i	0.857745	+	0.514076i	$0.171865\pi$
$613$	25.4558i	1.02815i	−0.857745	+	0.514076i	$0.828135\pi$
$614$	39.4558	1.59231
$615$	0	0
$616$	−8.82843	−0.355707
$617$	0.343146i	0.0138145i	0.999976	+	0.00690726i	$0.00219867\pi$
$617$	0.343146i	0.0138145i	−0.999976	+	0.00690726i	$0.997801\pi$
$618$	46.6274i	1.87563i
$619$	14.3431	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$619$	14.3431	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	11.3137i	0.453638i
$623$	− 26.6274i	− 1.06680i
$624$	−9.94113	−0.397964
$625$	0	0
$626$	−3.17157	−0.126762
$627$	0	0
$628$	53.5980i	2.13879i
$629$	52.2843	2.08471
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	17.6569i	0.702352i
$633$	− 45.2548i	− 1.79872i
$634$	3.17157	0.125959
$635$	0	0
$636$	−126.225	−5.00516
$637$	3.51472i	0.139258i
$638$	8.82843i	0.349521i
$639$	−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	$0.298155\pi$
$642$	− 52.2843i	− 2.06350i
$643$	1.45584i	0.0574129i	0.999588	+	0.0287064i	$0.00913880\pi$
$643$	1.45584i	0.0574129i	−0.999588	+	0.0287064i	$0.990861\pi$
$644$	−21.6569	−0.853400
$645$	0	0
$646$	0	0
$647$	27.1127i	1.06591i	0.846144	+	0.532955i	$0.178919\pi$
$647$	27.1127i	1.06591i	−0.846144	+	0.532955i	$0.821081\pi$
$648$	− 4.41421i	− 0.173407i
$649$	−1.65685	−0.0650372
$650$	0	0
$651$	0	0
$652$	63.1127i	2.47168i
$653$	− 11.6569i	− 0.456168i	−0.973641	−	0.228084i	$-0.926754\pi$
$653$	− 11.6569i	− 0.456168i	0.973641	−	0.228084i	$-0.0732461\pi$
$654$	−52.2843	−2.04448
$655$	0	0
$656$	18.0000	0.702782
$657$	− 5.85786i	− 0.228537i
$658$	13.6569i	0.532400i
$659$	−45.9411	−1.78961	−0.894806	−	0.446455i	$-0.852686\pi$
$659$	−45.9411	−1.78961	−0.894806	+	0.446455i	$0.852686\pi$
$660$	0	0
$661$	44.6274	1.73581	0.867903	−	0.496734i	$-0.165468\pi$
$661$	44.6274	1.73581	0.867903	+	0.496734i	$0.165468\pi$
$662$	− 17.6569i	− 0.686253i
$663$	− 22.6274i	− 0.878776i
$664$	26.4853	1.02783
$665$	0	0
$666$	−92.4264	−3.58145
$667$	10.3431i	0.400488i
$668$	87.9411i	3.40254i
$669$	−30.6274	−1.18412
$670$	0	0
$671$	−9.31371	−0.359552
$672$	− 8.97056i	− 0.346047i
$673$	12.4853i	0.481272i	0.970615	+	0.240636i	$0.0773560\pi$
$673$	12.4853i	0.481272i	−0.970615	+	0.240636i	$0.922644\pi$
$674$	49.4558	1.90497
$675$	0	0
$676$	−44.5147	−1.71210
$677$	22.8284i	0.877368i	0.898641	+	0.438684i	$0.144555\pi$
$677$	22.8284i	0.877368i	−0.898641	+	0.438684i	$0.855445\pi$
$678$	134.225i	5.15490i
$679$	7.31371	0.280674
$680$	0	0
$681$	−71.5980	−2.74364
$682$	0	0
$683$	7.79899i	0.298420i	0.988806	+	0.149210i	$0.0476731\pi$
$683$	7.79899i	0.298420i	−0.988806	+	0.149210i	$0.952327\pi$
$684$	0	0
$685$	0	0
$686$	48.2843	1.84350
$687$	− 3.71573i	− 0.141764i
$688$	18.0000i	0.686244i
$689$	13.6569	0.520285
$690$	0	0
$691$	−39.3137	−1.49556	−0.747782	−	0.663944i	$-0.768881\pi$
$691$	−39.3137	−1.49556	−0.747782	+	0.663944i	$0.768881\pi$
$692$	− 84.7696i	− 3.22245i
$693$	10.0000i	0.379869i
$694$	−26.4853	−1.00537
$695$	0	0
$696$	45.6569	1.73062
$697$	40.9706i	1.55187i
$698$	− 65.1127i	− 2.46455i
$699$	−17.3726	−0.657091
$700$	0	0
$701$	−12.6274	−0.476931	−0.238465	−	0.971151i	$-0.576644\pi$
$701$	−12.6274	−0.476931	−0.238465	+	0.971151i	$0.576644\pi$
$702$	16.0000i	0.603881i
$703$	0	0
$704$	9.82843	0.370423
$705$	0	0
$706$	51.4558	1.93657
$707$	− 18.6274i	− 0.700556i
$708$	17.9411i	0.674269i
$709$	−24.6274	−0.924902	−0.462451	−	0.886645i	$-0.653030\pi$
$709$	−24.6274	−0.924902	−0.462451	+	0.886645i	$0.653030\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	− 58.7696i	− 2.20248i
$713$	0	0
$714$	−93.2548	−3.48997
$715$	0	0
$716$	36.9706	1.38165
$717$	65.9411i	2.46262i
$718$	− 1.65685i	− 0.0618333i
$719$	−18.3431	−0.684084	−0.342042	−	0.939685i	$-0.611118\pi$
$719$	−18.3431	−0.684084	−0.342042	+	0.939685i	$0.611118\pi$
$720$	0	0
$721$	−13.6569	−0.508608
$722$	− 45.8701i	− 1.70711i
$723$	16.9706i	0.631142i
$724$	−81.5980	−3.03257
$725$	0	0
$726$	−6.82843	−0.253427
$727$	− 19.5147i	− 0.723761i	−0.932225	−	0.361880i	$-0.882135\pi$
$727$	− 19.5147i	− 0.723761i	0.932225	−	0.361880i	$-0.117865\pi$
$728$	− 10.3431i	− 0.383342i
$729$	43.0000	1.59259
$730$	0	0
$731$	−40.9706	−1.51535
$732$	100.853i	3.72763i
$733$	17.4558i	0.644746i	0.946613	+	0.322373i	$0.104481\pi$
$733$	17.4558i	0.644746i	−0.946613	+	0.322373i	$0.895519\pi$
$734$	20.4853	0.756126
$735$	0	0
$736$	4.48528	0.165330
$737$	12.4853i	0.459901i
$738$	− 72.4264i	− 2.66605i
$739$	−29.9411	−1.10140	−0.550701	−	0.834703i	$-0.685640\pi$
$739$	−29.9411	−1.10140	−0.550701	+	0.834703i	$0.685640\pi$
$740$	0	0
$741$	0	0
$742$	− 56.2843i	− 2.06626i
$743$	49.5980i	1.81957i	0.415076	+	0.909787i	$0.363755\pi$
$743$	49.5980i	1.81957i	−0.415076	+	0.909787i	$0.636245\pi$
$744$	0	0
$745$	0	0
$746$	86.4264	3.16430
$747$	− 30.0000i	− 1.09764i
$748$	− 26.1421i	− 0.955851i
$749$	15.3137	0.559551
$750$	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
$754$	−10.3431	−0.376675
$755$	0	0
$756$	43.3137	1.57530
$757$	13.3137i	0.483895i	0.970289	+	0.241947i	$0.0777862\pi$
$757$	13.3137i	0.483895i	−0.970289	+	0.241947i	$0.922214\pi$
$758$	− 81.2548i	− 2.95131i
$759$	−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	$0.682996\pi$
$762$	− 29.6569i	− 1.07435i
$763$	− 15.3137i	− 0.554393i
$764$	−12.6863	−0.458974
$765$	0	0
$766$	−14.1421	−0.510976
$767$	− 1.94113i	− 0.0700900i
$768$	− 84.7696i	− 3.05886i
$769$	18.9706	0.684096	0.342048	−	0.939682i	$-0.388879\pi$
$769$	18.9706	0.684096	0.342048	+	0.939682i	$0.388879\pi$
$770$	0	0
$771$	26.3431	0.948725
$772$	− 4.48528i	− 0.161429i
$773$	− 26.2843i	− 0.945380i	−0.881229	−	0.472690i	$-0.843283\pi$
$773$	− 26.2843i	− 0.945380i	0.881229	−	0.472690i	$-0.156717\pi$
$774$	72.4264	2.60331
$775$	0	0
$776$	16.1421	0.579469
$777$	− 43.3137i	− 1.55387i
$778$	− 49.7990i	− 1.78538i
$779$	0	0
$780$	0	0
$781$	11.3137	0.404836
$782$	− 46.6274i	− 1.66739i
$783$	− 20.6863i	− 0.739268i
$784$	9.00000	0.321429
$785$	0	0
$786$	77.2548	2.75559
$787$	− 14.9706i	− 0.533643i	−0.963746	−	0.266821i	$-0.914027\pi$
$787$	− 14.9706i	− 0.533643i	0.963746	−	0.266821i	$-0.0859734\pi$
$788$	41.4558i	1.47680i
$789$	−31.0294	−1.10468
$790$	0	0
$791$	−39.3137	−1.39783
$792$	22.0711i	0.784261i
$793$	− 10.9117i	− 0.387485i
$794$	22.4853	0.797973
$795$	0	0
$796$	39.5980	1.40351
$797$	32.6274i	1.15572i	0.816135	+	0.577861i	$0.196113\pi$
$797$	32.6274i	1.15572i	−0.816135	+	0.577861i	$0.803887\pi$
$798$	0	0
$799$	−19.3137	−0.683270
$800$	0	0
$801$	−66.5685	−2.35208
$802$	− 12.8284i	− 0.452988i
$803$	1.17157i	0.0413439i
$804$	135.196	4.76799
$805$	0	0
$806$	0	0
$807$	− 48.9706i	− 1.72385i
$808$	− 41.1127i	− 1.44634i
$809$	10.9706	0.385704	0.192852	−	0.981228i	$-0.438226\pi$
$809$	10.9706	0.385704	0.192852	+	0.981228i	$0.438226\pi$
$810$	0	0
$811$	53.9411	1.89413	0.947065	−	0.321043i	$-0.104033\pi$
$811$	53.9411	1.89413	0.947065	+	0.321043i	$0.104033\pi$
$812$	28.0000i	0.982607i
$813$	20.6863i	0.725500i
$814$	18.4853	0.647909
$815$	0	0
$816$	−57.9411	−2.02835
$817$	0	0
$818$	− 2.48528i	− 0.0868958i
$819$	−11.7157	−0.409381
$820$	0	0
$821$	−41.3137	−1.44186	−0.720929	−	0.693009i	$-0.756285\pi$
$821$	−41.3137	−1.44186	−0.720929	+	0.693009i	$0.756285\pi$
$822$	74.9117i	2.61285i
$823$	− 19.5147i	− 0.680240i	−0.940382	−	0.340120i	$-0.889532\pi$
$823$	− 19.5147i	− 0.680240i	0.940382	−	0.340120i	$-0.110468\pi$
$824$	−30.1421	−1.05005
$825$	0	0
$826$	−8.00000	−0.278356
$827$	22.2843i	0.774900i	0.921891	+	0.387450i	$0.126644\pi$
$827$	22.2843i	0.774900i	−0.921891	+	0.387450i	$0.873356\pi$
$828$	54.1421i	1.88157i
$829$	−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
$831$	−19.3137	−0.669985
$832$	11.5147i	0.399201i
$833$	20.4853i	0.709773i
$834$	−27.3137	−0.945796
$835$	0	0
$836$	0	0
$837$	0	0
$838$	61.9411i	2.13972i
$839$	−26.3431	−0.909466	−0.454733	−	0.890628i	$-0.650265\pi$
$839$	−26.3431	−0.909466	−0.454733	+	0.890628i	$0.650265\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	− 14.4853i	− 0.499196i
$843$	48.9706i	1.68664i
$844$	61.2548	2.10848
$845$	0	0
$846$	34.1421	1.17383
$847$	− 2.00000i	− 0.0687208i
$848$	− 34.9706i	− 1.20089i
$849$	92.2843	3.16719
$850$	0	0
$851$	21.6569	0.742387
$852$	− 122.510i	− 4.19711i
$853$	15.5147i	0.531214i	0.964081	+	0.265607i	$0.0855723\pi$
$853$	15.5147i	0.531214i	−0.964081	+	0.265607i	$0.914428\pi$
$854$	−44.9706	−1.53886
$855$	0	0
$856$	33.7990	1.15523
$857$	24.7696i	0.846112i	0.906104	+	0.423056i	$0.139043\pi$
$857$	24.7696i	0.846112i	−0.906104	+	0.423056i	$0.860957\pi$
$858$	− 8.00000i	− 0.273115i
$859$	−24.2843	−0.828569	−0.414284	−	0.910148i	$-0.635968\pi$
$859$	−24.2843	−0.828569	−0.414284	+	0.910148i	$0.635968\pi$
$860$	0	0
$861$	33.9411	1.15671
$862$	− 27.3137i	− 0.930309i
$863$	9.17157i	0.312204i	0.987741	+	0.156102i	$0.0498929\pi$
$863$	9.17157i	0.312204i	−0.987741	+	0.156102i	$0.950107\pi$
$864$	−8.97056	−0.305185
$865$	0	0
$866$	−18.4853	−0.628155
$867$	− 83.7990i	− 2.84596i
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−14.6274	−0.495631
$872$	− 33.7990i	− 1.14458i
$873$	− 18.2843i	− 0.618829i
$874$	0	0
$875$	0	0
$876$	12.6863	0.428630
$877$	− 49.4558i	− 1.67001i	−0.550246	−	0.835003i	$-0.685466\pi$
$877$	− 49.4558i	− 1.67001i	0.550246	−	0.835003i	$-0.314534\pi$
$878$	38.6274i	1.30361i
$879$	−25.9411	−0.874972
$880$	0	0
$881$	−7.37258	−0.248389	−0.124194	−	0.992258i	$-0.539635\pi$
$881$	−7.37258	−0.248389	−0.124194	+	0.992258i	$0.539635\pi$
$882$	− 36.2132i	− 1.21936i
$883$	− 37.1716i	− 1.25092i	−0.780255	−	0.625462i	$-0.784911\pi$
$883$	− 37.1716i	− 1.25092i	0.780255	−	0.625462i	$-0.215089\pi$
$884$	30.6274	1.03011
$885$	0	0
$886$	−64.7696	−2.17598
$887$	38.2843i	1.28546i	0.766093	+	0.642730i	$0.222198\pi$
$887$	38.2843i	1.28546i	−0.766093	+	0.642730i	$0.777802\pi$
$888$	− 95.5980i	− 3.20806i
$889$	8.68629	0.291329
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 41.4558i	− 1.38804i
$893$	0	0
$894$	2.34315	0.0783665
$895$	0	0
$896$	41.1127	1.37348
$897$	− 9.37258i	− 0.312941i
$898$	− 69.1127i	− 2.30632i
$899$	0	0
$900$	0	0
$901$	79.5980	2.65179
$902$	14.4853i	0.482307i
$903$	33.9411i	1.12949i
$904$	−86.7696	−2.88591
$905$	0	0
$906$	81.9411	2.72231
$907$	− 27.5147i	− 0.913611i	−0.889567	−	0.456806i	$-0.848993\pi$
$907$	− 27.5147i	− 0.913611i	0.889567	−	0.456806i	$-0.151007\pi$
$908$	− 96.9117i	− 3.21613i
$909$	−46.5685	−1.54458
$910$	0	0
$911$	−9.94113	−0.329364	−0.164682	−	0.986347i	$-0.552660\pi$
$911$	−9.94113	−0.329364	−0.164682	+	0.986347i	$0.552660\pi$
$912$	0	0
$913$	6.00000i	0.198571i
$914$	−1.17157	−0.0387522
$915$	0	0
$916$	5.02944	0.166177
$917$	22.6274i	0.747223i
$918$	93.2548i	3.07787i
$919$	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
$921$	46.2254	1.52318
$922$	30.4853i	1.00398i
$923$	13.2548i	0.436288i
$924$	−21.6569	−0.712458
$925$	0	0
$926$	−14.8284	−0.487292
$927$	34.1421i	1.12137i
$928$	− 5.79899i	− 0.190361i
$929$	−5.31371	−0.174337	−0.0871686	−	0.996194i	$-0.527782\pi$
$929$	−5.31371	−0.174337	−0.0871686	+	0.996194i	$0.527782\pi$
$930$	0	0
$931$	0	0
$932$	− 23.5147i	− 0.770250i
$933$	13.2548i	0.433944i
$934$	35.7990	1.17138
$935$	0	0
$936$	−25.8579	−0.845191
$937$	− 1.45584i	− 0.0475604i	−0.999717	−	0.0237802i	$-0.992430\pi$
$937$	− 1.45584i	− 0.0475604i	0.999717	−	0.0237802i	$-0.00757018\pi$
$938$	60.2843i	1.96835i
$939$	−3.71573	−0.121258
$940$	0	0
$941$	−6.68629	−0.217967	−0.108983	−	0.994044i	$-0.534760\pi$
$941$	−6.68629	−0.217967	−0.108983	+	0.994044i	$0.534760\pi$
$942$	95.5980i	3.11475i
$943$	16.9706i	0.552638i
$944$	−4.97056	−0.161778
$945$	0	0
$946$	−14.4853	−0.470957
$947$	41.1716i	1.33790i	0.743309	+	0.668948i	$0.233255\pi$
$947$	41.1716i	1.33790i	−0.743309	+	0.668948i	$0.766745\pi$
$948$	43.3137i	1.40676i
$949$	−1.37258	−0.0445559
$950$	0	0
$951$	3.71573	0.120491
$952$	− 60.2843i	− 1.95382i
$953$	− 53.1716i	− 1.72240i	−0.508269	−	0.861198i	$-0.669715\pi$
$953$	− 53.1716i	− 1.72240i	0.508269	−	0.861198i	$-0.330285\pi$
$954$	−140.711	−4.55568
$955$	0	0
$956$	−89.2548	−2.88671
$957$	10.3431i	0.334346i
$958$	86.9117i	2.80799i
$959$	−21.9411	−0.708516
$960$	0	0
$961$	−31.0000	−1.00000
$962$	21.6569i	0.698245i
$963$	− 38.2843i	− 1.23369i
$964$	−22.9706	−0.739832
$965$	0	0
$966$	−38.6274	−1.24282
$967$	− 14.9706i	− 0.481421i	−0.970597	−	0.240710i	$-0.922620\pi$
$967$	− 14.9706i	− 0.481421i	0.970597	−	0.240710i	$-0.0773804\pi$
$968$	− 4.41421i	− 0.141878i
$969$	0	0
$970$	0	0
$971$	−8.68629	−0.278756	−0.139378	−	0.990239i	$-0.544510\pi$
$971$	−8.68629	−0.278756	−0.139378	+	0.990239i	$0.544510\pi$
$972$	54.1421i	1.73661i
$973$	− 8.00000i	− 0.256468i
$974$	59.1127	1.89409
$975$	0	0
$976$	−27.9411	−0.894374
$977$	32.3431i	1.03475i	0.855759	+	0.517374i	$0.173091\pi$
$977$	32.3431i	1.03475i	−0.855759	+	0.517374i	$0.826909\pi$
$978$	112.569i	3.59955i
$979$	13.3137	0.425508
$980$	0	0
$981$	−38.2843	−1.22232
$982$	− 1.65685i	− 0.0528723i
$983$	21.8579i	0.697158i	0.937279	+	0.348579i	$0.113336\pi$
$983$	21.8579i	0.697158i	−0.937279	+	0.348579i	$0.886664\pi$
$984$	74.9117	2.38810
$985$	0	0
$986$	−60.2843	−1.91984
$987$	16.0000i	0.509286i
$988$	0	0
$989$	−16.9706	−0.539633
$990$	0	0
$991$	−57.9411	−1.84056	−0.920280	−	0.391260i	$-0.872039\pi$
$991$	−57.9411	−1.84056	−0.920280	+	0.391260i	$0.872039\pi$
$992$	0	0
$993$	− 20.6863i	− 0.656460i
$994$	54.6274	1.73268
$995$	0	0
$996$	64.9706	2.05867
$997$	− 41.4558i	− 1.31292i	−0.754361	−	0.656460i	$-0.772053\pi$
$997$	− 41.4558i	− 1.31292i	0.754361	−	0.656460i	$-0.227947\pi$
$998$	− 23.3137i	− 0.737983i
$999$	−43.3137	−1.37039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.b.d.199.4		4
3.2	odd	2		2475.2.c.l.199.1		4
4.3	odd	2		4400.2.b.q.4049.2		4
5.2	odd	4		275.2.a.c.1.1		2
5.3	odd	4		55.2.a.b.1.2	✓	2
5.4	even	2	inner	275.2.b.d.199.1		4
15.2	even	4		2475.2.a.x.1.2		2
15.8	even	4		495.2.a.b.1.1		2
15.14	odd	2		2475.2.c.l.199.4		4
20.3	even	4		880.2.a.m.1.2		2
20.7	even	4		4400.2.a.bn.1.1		2
20.19	odd	2		4400.2.b.q.4049.3		4
35.13	even	4		2695.2.a.f.1.2		2
40.3	even	4		3520.2.a.bo.1.1		2
40.13	odd	4		3520.2.a.bn.1.2		2
55.3	odd	20		605.2.g.f.251.2		8
55.8	even	20		605.2.g.l.251.1		8
55.13	even	20		605.2.g.l.81.2		8
55.18	even	20		605.2.g.l.511.1		8
55.28	even	20		605.2.g.l.366.2		8
55.32	even	4		3025.2.a.o.1.2		2
55.38	odd	20		605.2.g.f.366.1		8
55.43	even	4		605.2.a.d.1.1		2
55.48	odd	20		605.2.g.f.511.2		8
55.53	odd	20		605.2.g.f.81.1		8
60.23	odd	4		7920.2.a.ch.1.2		2
65.38	odd	4		9295.2.a.g.1.1		2
165.98	odd	4		5445.2.a.y.1.2		2
220.43	odd	4		9680.2.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.a.b.1.2	✓	2	5.3	odd	4
275.2.a.c.1.1		2	5.2	odd	4
275.2.b.d.199.1		4	5.4	even	2	inner
275.2.b.d.199.4		4	1.1	even	1	trivial
495.2.a.b.1.1		2	15.8	even	4
605.2.a.d.1.1		2	55.43	even	4
605.2.g.f.81.1		8	55.53	odd	20
605.2.g.f.251.2		8	55.3	odd	20
605.2.g.f.366.1		8	55.38	odd	20
605.2.g.f.511.2		8	55.48	odd	20
605.2.g.l.81.2		8	55.13	even	20
605.2.g.l.251.1		8	55.8	even	20
605.2.g.l.366.2		8	55.28	even	20
605.2.g.l.511.1		8	55.18	even	20
880.2.a.m.1.2		2	20.3	even	4
2475.2.a.x.1.2		2	15.2	even	4
2475.2.c.l.199.1		4	3.2	odd	2
2475.2.c.l.199.4		4	15.14	odd	2
2695.2.a.f.1.2		2	35.13	even	4
3025.2.a.o.1.2		2	55.32	even	4
3520.2.a.bn.1.2		2	40.13	odd	4
3520.2.a.bo.1.1		2	40.3	even	4
4400.2.a.bn.1.1		2	20.7	even	4
4400.2.b.q.4049.2		4	4.3	odd	2
4400.2.b.q.4049.3		4	20.19	odd	2
5445.2.a.y.1.2		2	165.98	odd	4
7920.2.a.ch.1.2		2	60.23	odd	4
9295.2.a.g.1.1		2	65.38	odd	4
9680.2.a.bn.1.2		2	220.43	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)

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} +2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} -2.00000i q^{7} -4.41421i q^{8} -5.00000 q^{9} +O(q^{10})\)	\(q+2.41421i q^{2} +2.82843i q^{3} -3.82843 q^{4} -6.82843 q^{6} -2.00000i q^{7} -4.41421i q^{8} -5.00000 q^{9} +1.00000 q^{11} -10.8284i q^{12} +1.17157i q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843i q^{17} -12.0711i q^{18} +5.65685 q^{21} +2.41421i q^{22} +2.82843i q^{23} +12.4853 q^{24} -2.82843 q^{26} -5.65685i q^{27} +7.65685i q^{28} +3.65685 q^{29} -1.58579i q^{32} +2.82843i q^{33} -16.4853 q^{34} +19.1421 q^{36} -7.65685i q^{37} -3.31371 q^{39} +6.00000 q^{41} +13.6569i q^{42} +6.00000i q^{43} -3.82843 q^{44} -6.82843 q^{46} +2.82843i q^{47} +8.48528i q^{48} +3.00000 q^{49} -19.3137 q^{51} -4.48528i q^{52} -11.6569i q^{53} +13.6569 q^{54} -8.82843 q^{56} +8.82843i q^{58} -1.65685 q^{59} -9.31371 q^{61} +10.0000i q^{63} +9.82843 q^{64} -6.82843 q^{66} +12.4853i q^{67} -26.1421i q^{68} -8.00000 q^{69} +11.3137 q^{71} +22.0711i q^{72} +1.17157i q^{73} +18.4853 q^{74} -2.00000i q^{77} -8.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +14.4853i q^{82} +6.00000i q^{83} -21.6569 q^{84} -14.4853 q^{86} +10.3431i q^{87} -4.41421i q^{88} +13.3137 q^{89} +2.34315 q^{91} -10.8284i q^{92} -6.82843 q^{94} +4.48528 q^{96} +3.65685i q^{97} +7.24264i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10}) \) 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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