LMFDB - Embedded newform 825.2.c.e.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.c (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:	$6.58765816676$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.e.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.82843i q^{12} +5.65685i q^{13} +2.00000 q^{14} +3.00000 q^{16} +6.82843i q^{17} +0.414214i q^{18} +1.17157 q^{19} +4.82843 q^{21} +0.414214i q^{22} -4.00000i q^{23} -1.58579 q^{24} +2.34315 q^{26} +1.00000i q^{27} +8.82843i q^{28} -0.828427 q^{29} -4.41421i q^{32} +1.00000i q^{33} +2.82843 q^{34} -1.82843 q^{36} -0.343146i q^{37} -0.485281i q^{38} +5.65685 q^{39} -0.828427 q^{41} -2.00000i q^{42} -3.17157i q^{43} -1.82843 q^{44} -1.65685 q^{46} +4.00000i q^{47} -3.00000i q^{48} -16.3137 q^{49} +6.82843 q^{51} +10.3431i q^{52} -13.3137i q^{53} +0.414214 q^{54} +7.65685 q^{56} -1.17157i q^{57} +0.343146i q^{58} +4.00000 q^{59} -0.343146 q^{61} -4.82843i q^{63} +4.17157 q^{64} +0.414214 q^{66} -5.65685i q^{67} +12.4853i q^{68} -4.00000 q^{69} +13.6569 q^{71} +1.58579i q^{72} -11.3137i q^{73} -0.142136 q^{74} +2.14214 q^{76} -4.82843i q^{77} -2.34315i q^{78} +8.48528 q^{79} +1.00000 q^{81} +0.343146i q^{82} -10.0000i q^{83} +8.82843 q^{84} -1.31371 q^{86} +0.828427i q^{87} +1.58579i q^{88} +7.65685 q^{89} -27.3137 q^{91} -7.31371i q^{92} +1.65685 q^{94} -4.41421 q^{96} -0.343146i q^{97} +6.75736i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{19} + 8 q^{21} - 12 q^{24} + 32 q^{26} + 8 q^{29} + 4 q^{36} + 8 q^{41} + 4 q^{44} + 16 q^{46} - 20 q^{49} + 16 q^{51} - 4 q^{54} + 8 q^{56} + 16 q^{59} - 24 q^{61} + 28 q^{64} - 4 q^{66} - 16 q^{69} + 32 q^{71} + 56 q^{74} - 48 q^{76} + 4 q^{81} + 24 q^{84} + 40 q^{86} + 8 q^{89} - 64 q^{91} - 16 q^{94} - 12 q^{96} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 - 4 * q^11 + 8 * q^14 + 12 * q^16 + 16 * q^19 + 8 * q^21 - 12 * q^24 + 32 * q^26 + 8 * q^29 + 4 * q^36 + 8 * q^41 + 4 * q^44 + 16 * q^46 - 20 * q^49 + 16 * q^51 - 4 * q^54 + 8 * q^56 + 16 * q^59 - 24 * q^61 + 28 * q^64 - 4 * q^66 - 16 * q^69 + 32 * q^71 + 56 * q^74 - 48 * q^76 + 4 * q^81 + 24 * q^84 + 40 * q^86 + 8 * q^89 - 64 * q^91 - 16 * q^94 - 12 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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	$-0.953216\pi$
$2$	− 0.414214i	− 0.292893i	0.989219	−	0.146447i	$-0.0467837\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	−0.414214	−0.169102
$7$	4.82843i	1.82497i	0.409106	+	0.912487i	$0.365841\pi$
$7$	4.82843i	1.82497i	−0.409106	+	0.912487i	$0.634159\pi$
$8$	− 1.58579i	− 0.560660i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 1.82843i	− 0.527821i
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	2.00000	0.534522
$15$	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	$0.689440\pi$
$18$	0.414214i	0.0976311i
$19$	1.17157	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$19$	1.17157	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$20$	0	0
$21$	4.82843	1.05365
$22$	0.414214i	0.0883106i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.58579	−0.323697
$25$	0	0
$26$	2.34315	0.459529
$27$	1.00000i	0.192450i
$28$	8.82843i	1.66842i
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 4.41421i	− 0.780330i
$33$	1.00000i	0.174078i
$34$	2.82843	0.485071
$35$	0	0
$36$	−1.82843	−0.304738
$37$	− 0.343146i	− 0.0564128i	−0.999602	−	0.0282064i	$-0.991020\pi$
$37$	− 0.343146i	− 0.0564128i	0.999602	−	0.0282064i	$-0.00897957\pi$
$38$	− 0.485281i	− 0.0787230i
$39$	5.65685	0.905822
$40$	0	0
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 3.17157i	− 0.483660i	−0.970319	−	0.241830i	$-0.922252\pi$
$43$	− 3.17157i	− 0.483660i	0.970319	−	0.241830i	$-0.0777477\pi$
$44$	−1.82843	−0.275646
$45$	0	0
$46$	−1.65685	−0.244290
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	− 3.00000i	− 0.433013i
$49$	−16.3137	−2.33053
$50$	0	0
$51$	6.82843	0.956171
$52$	10.3431i	1.43434i
$53$	− 13.3137i	− 1.82878i	−0.404836	−	0.914389i	$-0.632671\pi$
$53$	− 13.3137i	− 1.82878i	0.404836	−	0.914389i	$-0.367329\pi$
$54$	0.414214	0.0563673
$55$	0	0
$56$	7.65685	1.02319
$57$	− 1.17157i	− 0.155179i
$58$	0.343146i	0.0450572i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−0.343146	−0.0439353	−0.0219677	−	0.999759i	$-0.506993\pi$
$61$	−0.343146	−0.0439353	−0.0219677	+	0.999759i	$0.506993\pi$
$62$	0	0
$63$	− 4.82843i	− 0.608325i
$64$	4.17157	0.521447
$65$	0	0
$66$	0.414214	0.0509862
$67$	− 5.65685i	− 0.691095i	−0.938401	−	0.345547i	$-0.887693\pi$
$67$	− 5.65685i	− 0.691095i	0.938401	−	0.345547i	$-0.112307\pi$
$68$	12.4853i	1.51406i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	13.6569	1.62077	0.810385	−	0.585897i	$-0.199258\pi$
$71$	13.6569	1.62077	0.810385	+	0.585897i	$0.199258\pi$
$72$	1.58579i	0.186887i
$73$	− 11.3137i	− 1.32417i	−0.749429	−	0.662085i	$-0.769672\pi$
$73$	− 11.3137i	− 1.32417i	0.749429	−	0.662085i	$-0.230328\pi$
$74$	−0.142136	−0.0165229
$75$	0	0
$76$	2.14214	0.245720
$77$	− 4.82843i	− 0.550250i
$78$	− 2.34315i	− 0.265309i
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.343146i	0.0378941i
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	8.82843	0.963260
$85$	0	0
$86$	−1.31371	−0.141661
$87$	0.828427i	0.0888167i
$88$	1.58579i	0.169045i
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	−27.3137	−2.86325
$92$	− 7.31371i	− 0.762507i
$93$	0	0
$94$	1.65685	0.170891
$95$	0	0
$96$	−4.41421	−0.450524
$97$	− 0.343146i	− 0.0348412i	−0.999848	−	0.0174206i	$-0.994455\pi$
$97$	− 0.343146i	− 0.0348412i	0.999848	−	0.0174206i	$-0.00554543\pi$
$98$	6.75736i	0.682596i
$99$	1.00000	0.100504
$100$	0	0
$101$	4.82843	0.480446	0.240223	−	0.970718i	$-0.422779\pi$
$101$	4.82843	0.480446	0.240223	+	0.970718i	$0.422779\pi$
$102$	− 2.82843i	− 0.280056i
$103$	19.3137i	1.90304i	0.307593	+	0.951518i	$0.400477\pi$
$103$	19.3137i	1.90304i	−0.307593	+	0.951518i	$0.599523\pi$
$104$	8.97056	0.879636
$105$	0	0
$106$	−5.51472	−0.535637
$107$	5.31371i	0.513696i	0.966452	+	0.256848i	$0.0826839\pi$
$107$	5.31371i	0.513696i	−0.966452	+	0.256848i	$0.917316\pi$
$108$	1.82843i	0.175940i
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	0	0
$111$	−0.343146	−0.0325700
$112$	14.4853i	1.36873i
$113$	14.9706i	1.40831i	0.710045	+	0.704156i	$0.248674\pi$
$113$	14.9706i	1.40831i	−0.710045	+	0.704156i	$0.751326\pi$
$114$	−0.485281	−0.0454508
$115$	0	0
$116$	−1.51472	−0.140638
$117$	− 5.65685i	− 0.522976i
$118$	− 1.65685i	− 0.152526i
$119$	−32.9706	−3.02241
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.142136i	0.0128684i
$123$	0.828427i	0.0746968i
$124$	0	0
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 2.48528i	− 0.220533i	−0.993902	−	0.110267i	$-0.964830\pi$
$127$	− 2.48528i	− 0.220533i	0.993902	−	0.110267i	$-0.0351704\pi$
$128$	− 10.5563i	− 0.933058i
$129$	−3.17157	−0.279241
$130$	0	0
$131$	−19.3137	−1.68745	−0.843723	−	0.536778i	$-0.819641\pi$
$131$	−19.3137	−1.68745	−0.843723	+	0.536778i	$0.819641\pi$
$132$	1.82843i	0.159144i
$133$	5.65685i	0.490511i
$134$	−2.34315	−0.202417
$135$	0	0
$136$	10.8284	0.928530
$137$	− 9.31371i	− 0.795724i	−0.917445	−	0.397862i	$-0.869752\pi$
$137$	− 9.31371i	− 0.795724i	0.917445	−	0.397862i	$-0.130248\pi$
$138$	1.65685i	0.141041i
$139$	16.4853	1.39826	0.699132	−	0.714993i	$-0.253570\pi$
$139$	16.4853	1.39826	0.699132	+	0.714993i	$0.253570\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	− 5.65685i	− 0.474713i
$143$	− 5.65685i	− 0.473050i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	−4.68629	−0.387840
$147$	16.3137i	1.34553i
$148$	− 0.627417i	− 0.0515734i
$149$	−18.4853	−1.51437	−0.757187	−	0.653199i	$-0.773427\pi$
$149$	−18.4853	−1.51437	−0.757187	+	0.653199i	$0.773427\pi$
$150$	0	0
$151$	−0.485281	−0.0394916	−0.0197458	−	0.999805i	$-0.506286\pi$
$151$	−0.485281	−0.0394916	−0.0197458	+	0.999805i	$0.506286\pi$
$152$	− 1.85786i	− 0.150693i
$153$	− 6.82843i	− 0.552046i
$154$	−2.00000	−0.161165
$155$	0	0
$156$	10.3431	0.828114
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	− 3.51472i	− 0.279616i
$159$	−13.3137	−1.05585
$160$	0	0
$161$	19.3137	1.52213
$162$	− 0.414214i	− 0.0325437i
$163$	15.3137i	1.19946i	0.800202	+	0.599731i	$0.204726\pi$
$163$	15.3137i	1.19946i	−0.800202	+	0.599731i	$0.795274\pi$
$164$	−1.51472	−0.118280
$165$	0	0
$166$	−4.14214	−0.321492
$167$	− 9.31371i	− 0.720716i	−0.932814	−	0.360358i	$-0.882654\pi$
$167$	− 9.31371i	− 0.720716i	0.932814	−	0.360358i	$-0.117346\pi$
$168$	− 7.65685i	− 0.590739i
$169$	−19.0000	−1.46154
$170$	0	0
$171$	−1.17157	−0.0895924
$172$	− 5.79899i	− 0.442169i
$173$	− 2.82843i	− 0.215041i	−0.994203	−	0.107521i	$-0.965709\pi$
$173$	− 2.82843i	− 0.215041i	0.994203	−	0.107521i	$-0.0342912\pi$
$174$	0.343146	0.0260138
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 4.00000i	− 0.300658i
$178$	− 3.17157i	− 0.237719i
$179$	6.34315	0.474109	0.237054	−	0.971496i	$-0.423818\pi$
$179$	6.34315	0.474109	0.237054	+	0.971496i	$0.423818\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	11.3137i	0.838628i
$183$	0.343146i	0.0253661i
$184$	−6.34315	−0.467623
$185$	0	0
$186$	0	0
$187$	− 6.82843i	− 0.499344i
$188$	7.31371i	0.533407i
$189$	−4.82843	−0.351216
$190$	0	0
$191$	−5.65685	−0.409316	−0.204658	−	0.978834i	$-0.565608\pi$
$191$	−5.65685	−0.409316	−0.204658	+	0.978834i	$0.565608\pi$
$192$	− 4.17157i	− 0.301057i
$193$	2.34315i	0.168663i	0.996438	+	0.0843317i	$0.0268755\pi$
$193$	2.34315i	0.168663i	−0.996438	+	0.0843317i	$0.973124\pi$
$194$	−0.142136	−0.0102047
$195$	0	0
$196$	−29.8284	−2.13060
$197$	− 8.48528i	− 0.604551i	−0.953221	−	0.302276i	$-0.902254\pi$
$197$	− 8.48528i	− 0.604551i	0.953221	−	0.302276i	$-0.0977463\pi$
$198$	− 0.414214i	− 0.0294369i
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	0	0
$201$	−5.65685	−0.399004
$202$	− 2.00000i	− 0.140720i
$203$	− 4.00000i	− 0.280745i
$204$	12.4853	0.874145
$205$	0	0
$206$	8.00000	0.557386
$207$	4.00000i	0.278019i
$208$	16.9706i	1.17670i
$209$	−1.17157	−0.0810394
$210$	0	0
$211$	6.82843	0.470088	0.235044	−	0.971985i	$-0.424477\pi$
$211$	6.82843	0.470088	0.235044	+	0.971985i	$0.424477\pi$
$212$	− 24.3431i	− 1.67189i
$213$	− 13.6569i	− 0.935752i
$214$	2.20101	0.150458
$215$	0	0
$216$	1.58579	0.107899
$217$	0	0
$218$	− 2.20101i	− 0.149071i
$219$	−11.3137	−0.764510
$220$	0	0
$221$	−38.6274	−2.59836
$222$	0.142136i	0.00953952i
$223$	− 17.6569i	− 1.18239i	−0.806529	−	0.591195i	$-0.798656\pi$
$223$	− 17.6569i	− 1.18239i	0.806529	−	0.591195i	$-0.201344\pi$
$224$	21.3137	1.42408
$225$	0	0
$226$	6.20101	0.412485
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	− 2.14214i	− 0.141866i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−4.82843	−0.317687
$232$	1.31371i	0.0862492i
$233$	− 13.1716i	− 0.862898i	−0.902137	−	0.431449i	$-0.858002\pi$
$233$	− 13.1716i	− 0.862898i	0.902137	−	0.431449i	$-0.141998\pi$
$234$	−2.34315	−0.153176
$235$	0	0
$236$	7.31371	0.476082
$237$	− 8.48528i	− 0.551178i
$238$	13.6569i	0.885242i
$239$	−6.34315	−0.410304	−0.205152	−	0.978730i	$-0.565769\pi$
$239$	−6.34315	−0.410304	−0.205152	+	0.978730i	$0.565769\pi$
$240$	0	0
$241$	−23.6569	−1.52387	−0.761936	−	0.647652i	$-0.775751\pi$
$241$	−23.6569	−1.52387	−0.761936	+	0.647652i	$0.775751\pi$
$242$	− 0.414214i	− 0.0266267i
$243$	− 1.00000i	− 0.0641500i
$244$	−0.627417	−0.0401663
$245$	0	0
$246$	0.343146	0.0218782
$247$	6.62742i	0.421692i
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	−12.9706	−0.818695	−0.409347	−	0.912379i	$-0.634244\pi$
$251$	−12.9706	−0.818695	−0.409347	+	0.912379i	$0.634244\pi$
$252$	− 8.82843i	− 0.556139i
$253$	4.00000i	0.251478i
$254$	−1.02944	−0.0645926
$255$	0	0
$256$	3.97056	0.248160
$257$	27.6569i	1.72519i	0.505898	+	0.862594i	$0.331161\pi$
$257$	27.6569i	1.72519i	−0.505898	+	0.862594i	$0.668839\pi$
$258$	1.31371i	0.0817879i
$259$	1.65685	0.102952
$260$	0	0
$261$	0.828427	0.0512784
$262$	8.00000i	0.494242i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	1.58579	0.0975984
$265$	0	0
$266$	2.34315	0.143667
$267$	− 7.65685i	− 0.468592i
$268$	− 10.3431i	− 0.631808i
$269$	24.6274	1.50156	0.750780	−	0.660552i	$-0.229678\pi$
$269$	24.6274	1.50156	0.750780	+	0.660552i	$0.229678\pi$
$270$	0	0
$271$	27.7990	1.68867	0.844334	−	0.535817i	$-0.179996\pi$
$271$	27.7990	1.68867	0.844334	+	0.535817i	$0.179996\pi$
$272$	20.4853i	1.24210i
$273$	27.3137i	1.65310i
$274$	−3.85786	−0.233062
$275$	0	0
$276$	−7.31371	−0.440234
$277$	13.6569i	0.820561i	0.911959	+	0.410280i	$0.134569\pi$
$277$	13.6569i	0.820561i	−0.911959	+	0.410280i	$0.865431\pi$
$278$	− 6.82843i	− 0.409542i
$279$	0	0
$280$	0	0
$281$	−16.8284	−1.00390	−0.501950	−	0.864897i	$-0.667384\pi$
$281$	−16.8284	−1.00390	−0.501950	+	0.864897i	$0.667384\pi$
$282$	− 1.65685i	− 0.0986642i
$283$	− 3.17157i	− 0.188530i	−0.995547	−	0.0942652i	$-0.969950\pi$
$283$	− 3.17157i	− 0.188530i	0.995547	−	0.0942652i	$-0.0300502\pi$
$284$	24.9706	1.48173
$285$	0	0
$286$	−2.34315	−0.138553
$287$	− 4.00000i	− 0.236113i
$288$	4.41421i	0.260110i
$289$	−29.6274	−1.74279
$290$	0	0
$291$	−0.343146	−0.0201156
$292$	− 20.6863i	− 1.21057i
$293$	− 1.17157i	− 0.0684440i	−0.999414	−	0.0342220i	$-0.989105\pi$
$293$	− 1.17157i	− 0.0684440i	0.999414	−	0.0342220i	$-0.0108953\pi$
$294$	6.75736	0.394097
$295$	0	0
$296$	−0.544156	−0.0316284
$297$	− 1.00000i	− 0.0580259i
$298$	7.65685i	0.443550i
$299$	22.6274	1.30858
$300$	0	0
$301$	15.3137	0.882667
$302$	0.201010i	0.0115668i
$303$	− 4.82843i	− 0.277386i
$304$	3.51472	0.201583
$305$	0	0
$306$	−2.82843	−0.161690
$307$	8.82843i	0.503865i	0.967745	+	0.251932i	$0.0810661\pi$
$307$	8.82843i	0.503865i	−0.967745	+	0.251932i	$0.918934\pi$
$308$	− 8.82843i	− 0.503046i
$309$	19.3137	1.09872
$310$	0	0
$311$	19.3137	1.09518	0.547590	−	0.836747i	$-0.315545\pi$
$311$	19.3137	1.09518	0.547590	+	0.836747i	$0.315545\pi$
$312$	− 8.97056i	− 0.507858i
$313$	4.34315i	0.245489i	0.992438	+	0.122745i	$0.0391696\pi$
$313$	4.34315i	0.245489i	−0.992438	+	0.122745i	$0.960830\pi$
$314$	−7.45584	−0.420758
$315$	0	0
$316$	15.5147	0.872771
$317$	− 30.2843i	− 1.70093i	−0.526028	−	0.850467i	$-0.676319\pi$
$317$	− 30.2843i	− 1.70093i	0.526028	−	0.850467i	$-0.323681\pi$
$318$	5.51472i	0.309250i
$319$	0.828427	0.0463830
$320$	0	0
$321$	5.31371	0.296582
$322$	− 8.00000i	− 0.445823i
$323$	8.00000i	0.445132i
$324$	1.82843	0.101579
$325$	0	0
$326$	6.34315	0.351314
$327$	− 5.31371i	− 0.293849i
$328$	1.31371i	0.0725374i
$329$	−19.3137	−1.06480
$330$	0	0
$331$	17.6569	0.970508	0.485254	−	0.874373i	$-0.338727\pi$
$331$	17.6569	0.970508	0.485254	+	0.874373i	$0.338727\pi$
$332$	− 18.2843i	− 1.00348i
$333$	0.343146i	0.0188043i
$334$	−3.85786	−0.211093
$335$	0	0
$336$	14.4853	0.790237
$337$	19.3137i	1.05208i	0.850458	+	0.526042i	$0.176325\pi$
$337$	19.3137i	1.05208i	−0.850458	+	0.526042i	$0.823675\pi$
$338$	7.87006i	0.428075i
$339$	14.9706	0.813089
$340$	0	0
$341$	0	0
$342$	0.485281i	0.0262410i
$343$	− 44.9706i	− 2.42818i
$344$	−5.02944	−0.271169
$345$	0	0
$346$	−1.17157	−0.0629841
$347$	6.68629i	0.358939i	0.983764	+	0.179469i	$0.0574381\pi$
$347$	6.68629i	0.358939i	−0.983764	+	0.179469i	$0.942562\pi$
$348$	1.51472i	0.0811974i
$349$	22.9706	1.22959	0.614793	−	0.788688i	$-0.289240\pi$
$349$	22.9706	1.22959	0.614793	+	0.788688i	$0.289240\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	4.41421i	0.235278i
$353$	26.0000i	1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$	26.0000i	1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$	−1.65685	−0.0880608
$355$	0	0
$356$	14.0000	0.741999
$357$	32.9706i	1.74499i
$358$	− 2.62742i	− 0.138863i
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	5.79899i	0.304788i
$363$	− 1.00000i	− 0.0524864i
$364$	−49.9411	−2.61763
$365$	0	0
$366$	0.142136	0.00742955
$367$	1.65685i	0.0864871i	0.999065	+	0.0432435i	$0.0137691\pi$
$367$	1.65685i	0.0864871i	−0.999065	+	0.0432435i	$0.986231\pi$
$368$	− 12.0000i	− 0.625543i
$369$	0.828427	0.0431262
$370$	0	0
$371$	64.2843	3.33747
$372$	0	0
$373$	− 34.6274i	− 1.79294i	−0.443105	−	0.896470i	$-0.646123\pi$
$373$	− 34.6274i	− 1.79294i	0.443105	−	0.896470i	$-0.353877\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	6.34315	0.327123
$377$	− 4.68629i	− 0.241356i
$378$	2.00000i	0.102869i
$379$	0.686292	0.0352524	0.0176262	−	0.999845i	$-0.494389\pi$
$379$	0.686292	0.0352524	0.0176262	+	0.999845i	$0.494389\pi$
$380$	0	0
$381$	−2.48528	−0.127325
$382$	2.34315i	0.119886i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	−10.5563	−0.538701
$385$	0	0
$386$	0.970563	0.0494003
$387$	3.17157i	0.161220i
$388$	− 0.627417i	− 0.0318523i
$389$	12.3431	0.625822	0.312911	−	0.949782i	$-0.398696\pi$
$389$	12.3431	0.625822	0.312911	+	0.949782i	$0.398696\pi$
$390$	0	0
$391$	27.3137	1.38131
$392$	25.8701i	1.30664i
$393$	19.3137i	0.974248i
$394$	−3.51472	−0.177069
$395$	0	0
$396$	1.82843	0.0918819
$397$	− 18.9706i	− 0.952105i	−0.879417	−	0.476053i	$-0.842067\pi$
$397$	− 18.9706i	− 0.952105i	0.879417	−	0.476053i	$-0.157933\pi$
$398$	− 4.28427i	− 0.214751i
$399$	5.65685	0.283197
$400$	0	0
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	2.34315i	0.116865i
$403$	0	0
$404$	8.82843	0.439231
$405$	0	0
$406$	−1.65685	−0.0822283
$407$	0.343146i	0.0170091i
$408$	− 10.8284i	− 0.536087i
$409$	8.34315	0.412542	0.206271	−	0.978495i	$-0.433867\pi$
$409$	8.34315	0.412542	0.206271	+	0.978495i	$0.433867\pi$
$410$	0	0
$411$	−9.31371	−0.459411
$412$	35.3137i	1.73978i
$413$	19.3137i	0.950365i
$414$	1.65685	0.0814299
$415$	0	0
$416$	24.9706	1.22428
$417$	− 16.4853i	− 0.807288i
$418$	0.485281i	0.0237359i
$419$	3.02944	0.147998	0.0739988	−	0.997258i	$-0.476424\pi$
$419$	3.02944	0.147998	0.0739988	+	0.997258i	$0.476424\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	− 2.82843i	− 0.137686i
$423$	− 4.00000i	− 0.194487i
$424$	−21.1127	−1.02532
$425$	0	0
$426$	−5.65685	−0.274075
$427$	− 1.65685i	− 0.0801808i
$428$	9.71573i	0.469627i
$429$	−5.65685	−0.273115
$430$	0	0
$431$	10.3431	0.498212	0.249106	−	0.968476i	$-0.419863\pi$
$431$	10.3431	0.498212	0.249106	+	0.968476i	$0.419863\pi$
$432$	3.00000i	0.144338i
$433$	− 4.34315i	− 0.208718i	−0.994540	−	0.104359i	$-0.966721\pi$
$433$	− 4.34315i	− 0.208718i	0.994540	−	0.104359i	$-0.0332791\pi$
$434$	0	0
$435$	0	0
$436$	9.71573	0.465299
$437$	− 4.68629i	− 0.224176i
$438$	4.68629i	0.223920i
$439$	3.51472	0.167748	0.0838742	−	0.996476i	$-0.473271\pi$
$439$	3.51472	0.167748	0.0838742	+	0.996476i	$0.473271\pi$
$440$	0	0
$441$	16.3137	0.776843
$442$	16.0000i	0.761042i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−0.627417	−0.0297759
$445$	0	0
$446$	−7.31371	−0.346314
$447$	18.4853i	0.874324i
$448$	20.1421i	0.951626i
$449$	2.97056	0.140190	0.0700948	−	0.997540i	$-0.477670\pi$
$449$	2.97056	0.140190	0.0700948	+	0.997540i	$0.477670\pi$
$450$	0	0
$451$	0.828427	0.0390091
$452$	27.3726i	1.28750i
$453$	0.485281i	0.0228005i
$454$	5.79899	0.272160
$455$	0	0
$456$	−1.85786	−0.0870025
$457$	0.686292i	0.0321034i	0.999871	+	0.0160517i	$0.00510963\pi$
$457$	0.686292i	0.0321034i	−0.999871	+	0.0160517i	$0.994890\pi$
$458$	− 0.828427i	− 0.0387099i
$459$	−6.82843	−0.318724
$460$	0	0
$461$	−28.1421	−1.31071	−0.655355	−	0.755321i	$-0.727481\pi$
$461$	−28.1421	−1.31071	−0.655355	+	0.755321i	$0.727481\pi$
$462$	2.00000i	0.0930484i
$463$	− 28.9706i	− 1.34638i	−0.739471	−	0.673188i	$-0.764924\pi$
$463$	− 28.9706i	− 1.34638i	0.739471	−	0.673188i	$-0.235076\pi$
$464$	−2.48528	−0.115376
$465$	0	0
$466$	−5.45584	−0.252737
$467$	− 22.6274i	− 1.04707i	−0.852004	−	0.523536i	$-0.824613\pi$
$467$	− 22.6274i	− 1.04707i	0.852004	−	0.523536i	$-0.175387\pi$
$468$	− 10.3431i	− 0.478112i
$469$	27.3137	1.26123
$470$	0	0
$471$	−18.0000	−0.829396
$472$	− 6.34315i	− 0.291967i
$473$	3.17157i	0.145829i
$474$	−3.51472	−0.161436
$475$	0	0
$476$	−60.2843	−2.76313
$477$	13.3137i	0.609593i
$478$	2.62742i	0.120175i
$479$	−3.02944	−0.138419	−0.0692093	−	0.997602i	$-0.522048\pi$
$479$	−3.02944	−0.138419	−0.0692093	+	0.997602i	$0.522048\pi$
$480$	0	0
$481$	1.94113	0.0885077
$482$	9.79899i	0.446332i
$483$	− 19.3137i	− 0.878804i
$484$	1.82843	0.0831103
$485$	0	0
$486$	−0.414214	−0.0187891
$487$	− 20.9706i	− 0.950267i	−0.879914	−	0.475133i	$-0.842400\pi$
$487$	− 20.9706i	− 0.950267i	0.879914	−	0.475133i	$-0.157600\pi$
$488$	0.544156i	0.0246328i
$489$	15.3137	0.692510
$490$	0	0
$491$	25.6569	1.15788	0.578939	−	0.815371i	$-0.303467\pi$
$491$	25.6569	1.15788	0.578939	+	0.815371i	$0.303467\pi$
$492$	1.51472i	0.0682888i
$493$	− 5.65685i	− 0.254772i
$494$	2.74517	0.123511
$495$	0	0
$496$	0	0
$497$	65.9411i	2.95786i
$498$	4.14214i	0.185614i
$499$	33.6569	1.50669	0.753344	−	0.657627i	$-0.228440\pi$
$499$	33.6569	1.50669	0.753344	+	0.657627i	$0.228440\pi$
$500$	0	0
$501$	−9.31371	−0.416106
$502$	5.37258i	0.239790i
$503$	− 5.31371i	− 0.236927i	−0.992958	−	0.118463i	$-0.962203\pi$
$503$	− 5.31371i	− 0.236927i	0.992958	−	0.118463i	$-0.0377968\pi$
$504$	−7.65685	−0.341063
$505$	0	0
$506$	1.65685	0.0736562
$507$	19.0000i	0.843820i
$508$	− 4.54416i	− 0.201614i
$509$	−41.3137	−1.83120	−0.915599	−	0.402093i	$-0.868283\pi$
$509$	−41.3137	−1.83120	−0.915599	+	0.402093i	$0.868283\pi$
$510$	0	0
$511$	54.6274	2.41657
$512$	− 22.7574i	− 1.00574i
$513$	1.17157i	0.0517262i
$514$	11.4558	0.505296
$515$	0	0
$516$	−5.79899	−0.255286
$517$	− 4.00000i	− 0.175920i
$518$	− 0.686292i	− 0.0301539i
$519$	−2.82843	−0.124154
$520$	0	0
$521$	12.6274	0.553217	0.276609	−	0.960983i	$-0.410789\pi$
$521$	12.6274	0.553217	0.276609	+	0.960983i	$0.410789\pi$
$522$	− 0.343146i	− 0.0150191i
$523$	− 26.4853i	− 1.15812i	−0.815285	−	0.579060i	$-0.803420\pi$
$523$	− 26.4853i	− 1.15812i	0.815285	−	0.579060i	$-0.196580\pi$
$524$	−35.3137	−1.54269
$525$	0	0
$526$	7.45584	0.325090
$527$	0	0
$528$	3.00000i	0.130558i
$529$	7.00000	0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	10.3431i	0.448432i
$533$	− 4.68629i	− 0.202986i
$534$	−3.17157	−0.137247
$535$	0	0
$536$	−8.97056	−0.387469
$537$	− 6.34315i	− 0.273727i
$538$	− 10.2010i	− 0.439797i
$539$	16.3137	0.702681
$540$	0	0
$541$	−5.31371	−0.228454	−0.114227	−	0.993455i	$-0.536439\pi$
$541$	−5.31371	−0.228454	−0.114227	+	0.993455i	$0.536439\pi$
$542$	− 11.5147i	− 0.494600i
$543$	14.0000i	0.600798i
$544$	30.1421	1.29233
$545$	0	0
$546$	11.3137	0.484182
$547$	− 20.1421i	− 0.861216i	−0.902539	−	0.430608i	$-0.858299\pi$
$547$	− 20.1421i	− 0.861216i	0.902539	−	0.430608i	$-0.141701\pi$
$548$	− 17.0294i	− 0.727462i
$549$	0.343146	0.0146451
$550$	0	0
$551$	−0.970563	−0.0413474
$552$	6.34315i	0.269982i
$553$	40.9706i	1.74225i
$554$	5.65685	0.240337
$555$	0	0
$556$	30.1421	1.27831
$557$	− 10.8284i	− 0.458815i	−0.973330	−	0.229408i	$-0.926321\pi$
$557$	− 10.8284i	− 0.458815i	0.973330	−	0.229408i	$-0.0736789\pi$
$558$	0	0
$559$	17.9411	0.758829
$560$	0	0
$561$	−6.82843	−0.288296
$562$	6.97056i	0.294035i
$563$	20.3431i	0.857361i	0.903456	+	0.428681i	$0.141021\pi$
$563$	20.3431i	0.857361i	−0.903456	+	0.428681i	$0.858979\pi$
$564$	7.31371	0.307963
$565$	0	0
$566$	−1.31371	−0.0552193
$567$	4.82843i	0.202775i
$568$	− 21.6569i	− 0.908701i
$569$	−15.4558	−0.647943	−0.323971	−	0.946067i	$-0.605018\pi$
$569$	−15.4558	−0.647943	−0.323971	+	0.946067i	$0.605018\pi$
$570$	0	0
$571$	0.485281	0.0203084	0.0101542	−	0.999948i	$-0.496768\pi$
$571$	0.485281	0.0203084	0.0101542	+	0.999948i	$0.496768\pi$
$572$	− 10.3431i	− 0.432469i
$573$	5.65685i	0.236318i
$574$	−1.65685	−0.0691558
$575$	0	0
$576$	−4.17157	−0.173816
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	12.2721i	0.510451i
$579$	2.34315	0.0973778
$580$	0	0
$581$	48.2843	2.00317
$582$	0.142136i	0.00589171i
$583$	13.3137i	0.551397i
$584$	−17.9411	−0.742409
$585$	0	0
$586$	−0.485281	−0.0200468
$587$	30.6274i	1.26413i	0.774916	+	0.632064i	$0.217792\pi$
$587$	30.6274i	1.26413i	−0.774916	+	0.632064i	$0.782208\pi$
$588$	29.8284i	1.23010i
$589$	0	0
$590$	0	0
$591$	−8.48528	−0.349038
$592$	− 1.02944i	− 0.0423096i
$593$	− 17.1716i	− 0.705152i	−0.935783	−	0.352576i	$-0.885306\pi$
$593$	− 17.1716i	− 0.705152i	0.935783	−	0.352576i	$-0.114694\pi$
$594$	−0.414214	−0.0169954
$595$	0	0
$596$	−33.7990	−1.38446
$597$	− 10.3431i	− 0.423317i
$598$	− 9.37258i	− 0.383273i
$599$	−4.68629	−0.191477	−0.0957383	−	0.995407i	$-0.530521\pi$
$599$	−4.68629	−0.191477	−0.0957383	+	0.995407i	$0.530521\pi$
$600$	0	0
$601$	17.3137	0.706241	0.353120	−	0.935578i	$-0.385121\pi$
$601$	17.3137	0.706241	0.353120	+	0.935578i	$0.385121\pi$
$602$	− 6.34315i	− 0.258527i
$603$	5.65685i	0.230365i
$604$	−0.887302	−0.0361038
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	− 18.4853i	− 0.750294i	−0.926965	−	0.375147i	$-0.877592\pi$
$607$	− 18.4853i	− 0.750294i	0.926965	−	0.375147i	$-0.122408\pi$
$608$	− 5.17157i	− 0.209735i
$609$	−4.00000	−0.162088
$610$	0	0
$611$	−22.6274	−0.915407
$612$	− 12.4853i	− 0.504688i
$613$	21.9411i	0.886194i	0.896474	+	0.443097i	$0.146120\pi$
$613$	21.9411i	0.886194i	−0.896474	+	0.443097i	$0.853880\pi$
$614$	3.65685	0.147579
$615$	0	0
$616$	−7.65685	−0.308503
$617$	− 11.6569i	− 0.469287i	−0.972081	−	0.234644i	$-0.924608\pi$
$617$	− 11.6569i	− 0.469287i	0.972081	−	0.234644i	$-0.0753923\pi$
$618$	− 8.00000i	− 0.321807i
$619$	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
$621$	4.00000	0.160514
$622$	− 8.00000i	− 0.320771i
$623$	36.9706i	1.48119i
$624$	16.9706	0.679366
$625$	0	0
$626$	1.79899	0.0719021
$627$	1.17157i	0.0467881i
$628$	− 32.9117i	− 1.31332i
$629$	2.34315	0.0934273
$630$	0	0
$631$	34.3431	1.36718	0.683590	−	0.729867i	$-0.260418\pi$
$631$	34.3431	1.36718	0.683590	+	0.729867i	$0.260418\pi$
$632$	− 13.4558i	− 0.535245i
$633$	− 6.82843i	− 0.271406i
$634$	−12.5442	−0.498192
$635$	0	0
$636$	−24.3431	−0.965269
$637$	− 92.2843i	− 3.65644i
$638$	− 0.343146i	− 0.0135853i
$639$	−13.6569	−0.540257
$640$	0	0
$641$	−26.9706	−1.06527	−0.532637	−	0.846344i	$-0.678799\pi$
$641$	−26.9706	−1.06527	−0.532637	+	0.846344i	$0.678799\pi$
$642$	− 2.20101i	− 0.0868669i
$643$	− 29.9411i	− 1.18076i	−0.807124	−	0.590381i	$-0.798977\pi$
$643$	− 29.9411i	− 1.18076i	0.807124	−	0.590381i	$-0.201023\pi$
$644$	35.3137	1.39156
$645$	0	0
$646$	3.31371	0.130376
$647$	− 27.3137i	− 1.07381i	−0.843642	−	0.536906i	$-0.819593\pi$
$647$	− 27.3137i	− 1.07381i	0.843642	−	0.536906i	$-0.180407\pi$
$648$	− 1.58579i	− 0.0622956i
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	28.0000i	1.09656i
$653$	− 26.9706i	− 1.05544i	−0.849418	−	0.527720i	$-0.823047\pi$
$653$	− 26.9706i	− 1.05544i	0.849418	−	0.527720i	$-0.176953\pi$
$654$	−2.20101	−0.0860663
$655$	0	0
$656$	−2.48528	−0.0970339
$657$	11.3137i	0.441390i
$658$	8.00000i	0.311872i
$659$	−7.31371	−0.284902	−0.142451	−	0.989802i	$-0.545498\pi$
$659$	−7.31371	−0.284902	−0.142451	+	0.989802i	$0.545498\pi$
$660$	0	0
$661$	−13.3137	−0.517843	−0.258922	−	0.965898i	$-0.583367\pi$
$661$	−13.3137	−0.517843	−0.258922	+	0.965898i	$0.583367\pi$
$662$	− 7.31371i	− 0.284255i
$663$	38.6274i	1.50016i
$664$	−15.8579	−0.615404
$665$	0	0
$666$	0.142136	0.00550764
$667$	3.31371i	0.128307i
$668$	− 17.0294i	− 0.658889i
$669$	−17.6569	−0.682653
$670$	0	0
$671$	0.343146	0.0132470
$672$	− 21.3137i	− 0.822194i
$673$	29.6569i	1.14319i	0.820537	+	0.571594i	$0.193675\pi$
$673$	29.6569i	1.14319i	−0.820537	+	0.571594i	$0.806325\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−34.7401	−1.33616
$677$	21.4558i	0.824615i	0.911045	+	0.412308i	$0.135277\pi$
$677$	21.4558i	0.824615i	−0.911045	+	0.412308i	$0.864723\pi$
$678$	− 6.20101i	− 0.238148i
$679$	1.65685	0.0635842
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	−2.14214	−0.0819066
$685$	0	0
$686$	−18.6274	−0.711198
$687$	− 2.00000i	− 0.0763048i
$688$	− 9.51472i	− 0.362745i
$689$	75.3137	2.86922
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 5.17157i	− 0.196594i
$693$	4.82843i	0.183417i
$694$	2.76955	0.105131
$695$	0	0
$696$	1.31371	0.0497960
$697$	− 5.65685i	− 0.214269i
$698$	− 9.51472i	− 0.360137i
$699$	−13.1716	−0.498195
$700$	0	0
$701$	7.85786	0.296787	0.148394	−	0.988928i	$-0.452590\pi$
$701$	7.85786	0.296787	0.148394	+	0.988928i	$0.452590\pi$
$702$	2.34315i	0.0884363i
$703$	− 0.402020i	− 0.0151625i
$704$	−4.17157	−0.157222
$705$	0	0
$706$	10.7696	0.405317
$707$	23.3137i	0.876802i
$708$	− 7.31371i	− 0.274866i
$709$	−29.3137	−1.10090	−0.550450	−	0.834868i	$-0.685544\pi$
$709$	−29.3137	−1.10090	−0.550450	+	0.834868i	$0.685544\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	− 12.1421i	− 0.455046i
$713$	0	0
$714$	13.6569	0.511095
$715$	0	0
$716$	11.5980	0.433437
$717$	6.34315i	0.236889i
$718$	4.97056i	0.185500i
$719$	−31.5980	−1.17841	−0.589203	−	0.807985i	$-0.700558\pi$
$719$	−31.5980	−1.17841	−0.589203	+	0.807985i	$0.700558\pi$
$720$	0	0
$721$	−93.2548	−3.47299
$722$	7.30152i	0.271734i
$723$	23.6569i	0.879808i
$724$	−25.5980	−0.951341
$725$	0	0
$726$	−0.414214	−0.0153729
$727$	33.9411i	1.25881i	0.777079	+	0.629403i	$0.216701\pi$
$727$	33.9411i	1.25881i	−0.777079	+	0.629403i	$0.783299\pi$
$728$	43.3137i	1.60531i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	21.6569	0.801008
$732$	0.627417i	0.0231900i
$733$	− 17.6569i	− 0.652171i	−0.945340	−	0.326085i	$-0.894270\pi$
$733$	− 17.6569i	− 0.652171i	0.945340	−	0.326085i	$-0.105730\pi$
$734$	0.686292	0.0253315
$735$	0	0
$736$	−17.6569	−0.650840
$737$	5.65685i	0.208373i
$738$	− 0.343146i	− 0.0126314i
$739$	−47.1127	−1.73307	−0.866534	−	0.499118i	$-0.833658\pi$
$739$	−47.1127	−1.73307	−0.866534	+	0.499118i	$0.833658\pi$
$740$	0	0
$741$	6.62742	0.243464
$742$	− 26.6274i	− 0.977523i
$743$	47.6569i	1.74836i	0.485602	+	0.874180i	$0.338601\pi$
$743$	47.6569i	1.74836i	−0.485602	+	0.874180i	$0.661399\pi$
$744$	0	0
$745$	0	0
$746$	−14.3431	−0.525140
$747$	10.0000i	0.365881i
$748$	− 12.4853i	− 0.456507i
$749$	−25.6569	−0.937481
$750$	0	0
$751$	−36.2843	−1.32403	−0.662016	−	0.749490i	$-0.730299\pi$
$751$	−36.2843	−1.32403	−0.662016	+	0.749490i	$0.730299\pi$
$752$	12.0000i	0.437595i
$753$	12.9706i	0.472674i
$754$	−1.94113	−0.0706916
$755$	0	0
$756$	−8.82843	−0.321087
$757$	− 8.62742i	− 0.313569i	−0.987633	−	0.156784i	$-0.949887\pi$
$757$	− 8.62742i	− 0.313569i	0.987633	−	0.156784i	$-0.0501128\pi$
$758$	− 0.284271i	− 0.0103252i
$759$	4.00000	0.145191
$760$	0	0
$761$	−23.1716	−0.839969	−0.419984	−	0.907531i	$-0.637964\pi$
$761$	−23.1716	−0.839969	−0.419984	+	0.907531i	$0.637964\pi$
$762$	1.02944i	0.0372926i
$763$	25.6569i	0.928840i
$764$	−10.3431	−0.374202
$765$	0	0
$766$	−3.31371	−0.119729
$767$	22.6274i	0.817029i
$768$	− 3.97056i	− 0.143275i
$769$	−33.3137	−1.20132	−0.600662	−	0.799503i	$-0.705096\pi$
$769$	−33.3137	−1.20132	−0.600662	+	0.799503i	$0.705096\pi$
$770$	0	0
$771$	27.6569	0.996037
$772$	4.28427i	0.154194i
$773$	− 7.65685i	− 0.275398i	−0.990474	−	0.137699i	$-0.956029\pi$
$773$	− 7.65685i	− 0.275398i	0.990474	−	0.137699i	$-0.0439706\pi$
$774$	1.31371	0.0472203
$775$	0	0
$776$	−0.544156	−0.0195341
$777$	− 1.65685i	− 0.0594393i
$778$	− 5.11270i	− 0.183299i
$779$	−0.970563	−0.0347740
$780$	0	0
$781$	−13.6569	−0.488681
$782$	− 11.3137i	− 0.404577i
$783$	− 0.828427i	− 0.0296056i
$784$	−48.9411	−1.74790
$785$	0	0
$786$	8.00000	0.285351
$787$	− 8.14214i	− 0.290236i	−0.989414	−	0.145118i	$-0.953644\pi$
$787$	− 8.14214i	− 0.290236i	0.989414	−	0.145118i	$-0.0463561\pi$
$788$	− 15.5147i	− 0.552689i
$789$	18.0000	0.640817
$790$	0	0
$791$	−72.2843	−2.57013
$792$	− 1.58579i	− 0.0563485i
$793$	− 1.94113i	− 0.0689314i
$794$	−7.85786	−0.278865
$795$	0	0
$796$	18.9117	0.670307
$797$	− 1.02944i	− 0.0364645i	−0.999834	−	0.0182323i	$-0.994196\pi$
$797$	− 1.02944i	− 0.0364645i	0.999834	−	0.0182323i	$-0.00580383\pi$
$798$	− 2.34315i	− 0.0829465i
$799$	−27.3137	−0.966290
$800$	0	0
$801$	−7.65685	−0.270542
$802$	12.1421i	0.428754i
$803$	11.3137i	0.399252i
$804$	−10.3431	−0.364775
$805$	0	0
$806$	0	0
$807$	− 24.6274i	− 0.866926i
$808$	− 7.65685i	− 0.269367i
$809$	56.4264	1.98385	0.991923	−	0.126838i	$-0.0404829\pi$
$809$	56.4264	1.98385	0.991923	+	0.126838i	$0.0404829\pi$
$810$	0	0
$811$	−16.4853	−0.578877	−0.289438	−	0.957197i	$-0.593468\pi$
$811$	−16.4853	−0.578877	−0.289438	+	0.957197i	$0.593468\pi$
$812$	− 7.31371i	− 0.256661i
$813$	− 27.7990i	− 0.974953i
$814$	0.142136	0.00498185
$815$	0	0
$816$	20.4853	0.717128
$817$	− 3.71573i	− 0.129997i
$818$	− 3.45584i	− 0.120831i
$819$	27.3137	0.954418
$820$	0	0
$821$	−7.17157	−0.250290	−0.125145	−	0.992138i	$-0.539940\pi$
$821$	−7.17157	−0.250290	−0.125145	+	0.992138i	$0.539940\pi$
$822$	3.85786i	0.134558i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	30.6274	1.06696
$825$	0	0
$826$	8.00000	0.278356
$827$	− 18.6863i	− 0.649786i	−0.945751	−	0.324893i	$-0.894672\pi$
$827$	− 18.6863i	− 0.649786i	0.945751	−	0.324893i	$-0.105328\pi$
$828$	7.31371i	0.254169i
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	13.6569	0.473751
$832$	23.5980i	0.818113i
$833$	− 111.397i	− 3.85968i
$834$	−6.82843	−0.236449
$835$	0	0
$836$	−2.14214	−0.0740873
$837$	0	0
$838$	− 1.25483i	− 0.0433475i
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	2.48528i	0.0856485i
$843$	16.8284i	0.579602i
$844$	12.4853	0.429761
$845$	0	0
$846$	−1.65685	−0.0569638
$847$	4.82843i	0.165907i
$848$	− 39.9411i	− 1.37158i
$849$	−3.17157	−0.108848
$850$	0	0
$851$	−1.37258	−0.0470515
$852$	− 24.9706i	− 0.855477i
$853$	− 31.3137i	− 1.07216i	−0.844167	−	0.536080i	$-0.819904\pi$
$853$	− 31.3137i	− 1.07216i	0.844167	−	0.536080i	$-0.180096\pi$
$854$	−0.686292	−0.0234844
$855$	0	0
$856$	8.42641	0.288009
$857$	11.5147i	0.393335i	0.980470	+	0.196668i	$0.0630120\pi$
$857$	11.5147i	0.393335i	−0.980470	+	0.196668i	$0.936988\pi$
$858$	2.34315i	0.0799937i
$859$	19.0294	0.649276	0.324638	−	0.945838i	$-0.394758\pi$
$859$	19.0294	0.649276	0.324638	+	0.945838i	$0.394758\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	− 4.28427i	− 0.145923i
$863$	− 43.3137i	− 1.47442i	−0.675666	−	0.737208i	$-0.736144\pi$
$863$	− 43.3137i	− 1.47442i	0.675666	−	0.737208i	$-0.263856\pi$
$864$	4.41421	0.150175
$865$	0	0
$866$	−1.79899	−0.0611322
$867$	29.6274i	1.00620i
$868$	0	0
$869$	−8.48528	−0.287843
$870$	0	0
$871$	32.0000	1.08428
$872$	− 8.42641i	− 0.285354i
$873$	0.343146i	0.0116137i
$874$	−1.94113	−0.0656595
$875$	0	0
$876$	−20.6863	−0.698925
$877$	42.6274i	1.43943i	0.694272	+	0.719713i	$0.255727\pi$
$877$	42.6274i	1.43943i	−0.694272	+	0.719713i	$0.744273\pi$
$878$	− 1.45584i	− 0.0491324i
$879$	−1.17157	−0.0395162
$880$	0	0
$881$	−13.0294	−0.438973	−0.219486	−	0.975616i	$-0.570438\pi$
$881$	−13.0294	−0.438973	−0.219486	+	0.975616i	$0.570438\pi$
$882$	− 6.75736i	− 0.227532i
$883$	− 50.6274i	− 1.70375i	−0.523747	−	0.851874i	$-0.675466\pi$
$883$	− 50.6274i	− 1.70375i	0.523747	−	0.851874i	$-0.324534\pi$
$884$	−70.6274	−2.37546
$885$	0	0
$886$	4.97056	0.166989
$887$	4.34315i	0.145829i	0.997338	+	0.0729143i	$0.0232300\pi$
$887$	4.34315i	0.145829i	−0.997338	+	0.0729143i	$0.976770\pi$
$888$	0.544156i	0.0182607i
$889$	12.0000	0.402467
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 32.2843i	− 1.08096i
$893$	4.68629i	0.156821i
$894$	7.65685	0.256084
$895$	0	0
$896$	50.9706	1.70281
$897$	− 22.6274i	− 0.755507i
$898$	− 1.23045i	− 0.0410606i
$899$	0	0
$900$	0	0
$901$	90.9117	3.02871
$902$	− 0.343146i	− 0.0114255i
$903$	− 15.3137i	− 0.509608i
$904$	23.7401	0.789584
$905$	0	0
$906$	0.201010	0.00667811
$907$	− 7.02944i	− 0.233409i	−0.993167	−	0.116704i	$-0.962767\pi$
$907$	− 7.02944i	− 0.233409i	0.993167	−	0.116704i	$-0.0372330\pi$
$908$	25.5980i	0.849499i
$909$	−4.82843	−0.160149
$910$	0	0
$911$	−15.0294	−0.497947	−0.248974	−	0.968510i	$-0.580093\pi$
$911$	−15.0294	−0.497947	−0.248974	+	0.968510i	$0.580093\pi$
$912$	− 3.51472i	− 0.116384i
$913$	10.0000i	0.330952i
$914$	0.284271	0.00940286
$915$	0	0
$916$	3.65685	0.120826
$917$	− 93.2548i	− 3.07955i
$918$	2.82843i	0.0933520i
$919$	−28.4853	−0.939643	−0.469821	−	0.882762i	$-0.655682\pi$
$919$	−28.4853	−0.939643	−0.469821	+	0.882762i	$0.655682\pi$
$920$	0	0
$921$	8.82843	0.290907
$922$	11.6569i	0.383898i
$923$	77.2548i	2.54287i
$924$	−8.82843	−0.290434
$925$	0	0
$926$	−12.0000	−0.394344
$927$	− 19.3137i	− 0.634345i
$928$	3.65685i	0.120042i
$929$	−33.5980	−1.10231	−0.551157	−	0.834402i	$-0.685813\pi$
$929$	−33.5980	−1.10231	−0.551157	+	0.834402i	$0.685813\pi$
$930$	0	0
$931$	−19.1127	−0.626393
$932$	− 24.0833i	− 0.788873i
$933$	− 19.3137i	− 0.632302i
$934$	−9.37258	−0.306680
$935$	0	0
$936$	−8.97056	−0.293212
$937$	− 44.9706i	− 1.46912i	−0.678541	−	0.734562i	$-0.737388\pi$
$937$	− 44.9706i	− 1.46912i	0.678541	−	0.734562i	$-0.262612\pi$
$938$	− 11.3137i	− 0.369406i
$939$	4.34315	0.141733
$940$	0	0
$941$	38.7696	1.26385	0.631926	−	0.775029i	$-0.282265\pi$
$941$	38.7696	1.26385	0.631926	+	0.775029i	$0.282265\pi$
$942$	7.45584i	0.242925i
$943$	3.31371i	0.107909i
$944$	12.0000	0.390567
$945$	0	0
$946$	1.31371	0.0427123
$947$	38.6274i	1.25522i	0.778527	+	0.627611i	$0.215967\pi$
$947$	38.6274i	1.25522i	−0.778527	+	0.627611i	$0.784033\pi$
$948$	− 15.5147i	− 0.503895i
$949$	64.0000	2.07753
$950$	0	0
$951$	−30.2843	−0.982035
$952$	52.2843i	1.69454i
$953$	27.7990i	0.900498i	0.892903	+	0.450249i	$0.148665\pi$
$953$	27.7990i	0.900498i	−0.892903	+	0.450249i	$0.851335\pi$
$954$	5.51472	0.178546
$955$	0	0
$956$	−11.5980	−0.375105
$957$	− 0.828427i	− 0.0267792i
$958$	1.25483i	0.0405418i
$959$	44.9706	1.45218
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 0.804041i	− 0.0259233i
$963$	− 5.31371i	− 0.171232i
$964$	−43.2548	−1.39314
$965$	0	0
$966$	−8.00000	−0.257396
$967$	39.4558i	1.26881i	0.772999	+	0.634407i	$0.218756\pi$
$967$	39.4558i	1.26881i	−0.772999	+	0.634407i	$0.781244\pi$
$968$	− 1.58579i	− 0.0509691i
$969$	8.00000	0.256997
$970$	0	0
$971$	10.6274	0.341050	0.170525	−	0.985353i	$-0.445454\pi$
$971$	10.6274	0.341050	0.170525	+	0.985353i	$0.445454\pi$
$972$	− 1.82843i	− 0.0586468i
$973$	79.5980i	2.55179i
$974$	−8.68629	−0.278327
$975$	0	0
$976$	−1.02944	−0.0329515
$977$	− 25.3137i	− 0.809857i	−0.914348	−	0.404929i	$-0.867296\pi$
$977$	− 25.3137i	− 0.809857i	0.914348	−	0.404929i	$-0.132704\pi$
$978$	− 6.34315i	− 0.202831i
$979$	−7.65685	−0.244714
$980$	0	0
$981$	−5.31371	−0.169654
$982$	− 10.6274i	− 0.339135i
$983$	14.6274i	0.466542i	0.972412	+	0.233271i	$0.0749429\pi$
$983$	14.6274i	0.466542i	−0.972412	+	0.233271i	$0.925057\pi$
$984$	1.31371	0.0418795
$985$	0	0
$986$	−2.34315	−0.0746210
$987$	19.3137i	0.614762i
$988$	12.1177i	0.385517i
$989$	−12.6863	−0.403401
$990$	0	0
$991$	14.6274	0.464655	0.232328	−	0.972638i	$-0.425366\pi$
$991$	14.6274	0.464655	0.232328	+	0.972638i	$0.425366\pi$
$992$	0	0
$993$	− 17.6569i	− 0.560323i
$994$	27.3137	0.866338
$995$	0	0
$996$	−18.2843	−0.579359
$997$	16.6863i	0.528460i	0.964460	+	0.264230i	$0.0851178\pi$
$997$	16.6863i	0.528460i	−0.964460	+	0.264230i	$0.914882\pi$
$998$	− 13.9411i	− 0.441299i
$999$	0.343146	0.0108567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.e.199.2		4
3.2	odd	2		2475.2.c.m.199.3		4
5.2	odd	4		165.2.a.a.1.2	✓	2
5.3	odd	4		825.2.a.g.1.1		2
5.4	even	2	inner	825.2.c.e.199.3		4
15.2	even	4		495.2.a.d.1.1		2
15.8	even	4		2475.2.a.m.1.2		2
15.14	odd	2		2475.2.c.m.199.2		4
20.7	even	4		2640.2.a.bb.1.2		2
35.27	even	4		8085.2.a.ba.1.2		2
55.32	even	4		1815.2.a.k.1.1		2
55.43	even	4		9075.2.a.v.1.2		2
60.47	odd	4		7920.2.a.cg.1.2		2
165.32	odd	4		5445.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.a.1.2	✓	2	5.2	odd	4
495.2.a.d.1.1		2	15.2	even	4
825.2.a.g.1.1		2	5.3	odd	4
825.2.c.e.199.2		4	1.1	even	1	trivial
825.2.c.e.199.3		4	5.4	even	2	inner
1815.2.a.k.1.1		2	55.32	even	4
2475.2.a.m.1.2		2	15.8	even	4
2475.2.c.m.199.2		4	15.14	odd	2
2475.2.c.m.199.3		4	3.2	odd	2
2640.2.a.bb.1.2		2	20.7	even	4
5445.2.a.m.1.2		2	165.32	odd	4
7920.2.a.cg.1.2		2	60.47	odd	4
8085.2.a.ba.1.2		2	35.27	even	4
9075.2.a.v.1.2		2	55.43	even	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.82843i q^{12} +5.65685i q^{13} +2.00000 q^{14} +3.00000 q^{16} +6.82843i q^{17} +0.414214i q^{18} +1.17157 q^{19} +4.82843 q^{21} +0.414214i q^{22} -4.00000i q^{23} -1.58579 q^{24} +2.34315 q^{26} +1.00000i q^{27} +8.82843i q^{28} -0.828427 q^{29} -4.41421i q^{32} +1.00000i q^{33} +2.82843 q^{34} -1.82843 q^{36} -0.343146i q^{37} -0.485281i q^{38} +5.65685 q^{39} -0.828427 q^{41} -2.00000i q^{42} -3.17157i q^{43} -1.82843 q^{44} -1.65685 q^{46} +4.00000i q^{47} -3.00000i q^{48} -16.3137 q^{49} +6.82843 q^{51} +10.3431i q^{52} -13.3137i q^{53} +0.414214 q^{54} +7.65685 q^{56} -1.17157i q^{57} +0.343146i q^{58} +4.00000 q^{59} -0.343146 q^{61} -4.82843i q^{63} +4.17157 q^{64} +0.414214 q^{66} -5.65685i q^{67} +12.4853i q^{68} -4.00000 q^{69} +13.6569 q^{71} +1.58579i q^{72} -11.3137i q^{73} -0.142136 q^{74} +2.14214 q^{76} -4.82843i q^{77} -2.34315i q^{78} +8.48528 q^{79} +1.00000 q^{81} +0.343146i q^{82} -10.0000i q^{83} +8.82843 q^{84} -1.31371 q^{86} +0.828427i q^{87} +1.58579i q^{88} +7.65685 q^{89} -27.3137 q^{91} -7.31371i q^{92} +1.65685 q^{94} -4.41421 q^{96} -0.343146i q^{97} +6.75736i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{19} + 8 q^{21} - 12 q^{24} + 32 q^{26} + 8 q^{29} + 4 q^{36} + 8 q^{41} + 4 q^{44} + 16 q^{46} - 20 q^{49} + 16 q^{51} - 4 q^{54} + 8 q^{56} + 16 q^{59} - 24 q^{61} + 28 q^{64} - 4 q^{66} - 16 q^{69} + 32 q^{71} + 56 q^{74} - 48 q^{76} + 4 q^{81} + 24 q^{84} + 40 q^{86} + 8 q^{89} - 64 q^{91} - 16 q^{94} - 12 q^{96} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 - 4 * q^11 + 8 * q^14 + 12 * q^16 + 16 * q^19 + 8 * q^21 - 12 * q^24 + 32 * q^26 + 8 * q^29 + 4 * q^36 + 8 * q^41 + 4 * q^44 + 16 * q^46 - 20 * q^49 + 16 * q^51 - 4 * q^54 + 8 * q^56 + 16 * q^59 - 24 * q^61 + 28 * q^64 - 4 * q^66 - 16 * q^69 + 32 * q^71 + 56 * q^74 - 48 * q^76 + 4 * q^81 + 24 * q^84 + 40 * q^86 + 8 * q^89 - 64 * q^91 - 16 * q^94 - 12 * q^96 + 4 * q^99

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