LMFDB - Embedded newform 3468.2.j.f.829.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3468,2,Mod(829,3468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3468, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("3468.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3468 = 2^{2} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3468.j (of order $4$, degree $2$, 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:	$27.6921194210$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 204)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			829.1
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3468.829
Dual form			3468.2.j.f.3217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{9} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{9} +(-3.53553 + 3.53553i) q^{11} +5.00000 q^{13} -1.00000i q^{15} -1.00000i q^{19} +(2.12132 - 2.12132i) q^{23} -4.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(1.41421 + 1.41421i) q^{29} +(-1.41421 - 1.41421i) q^{31} +5.00000 q^{33} +(5.65685 + 5.65685i) q^{37} +(-3.53553 - 3.53553i) q^{39} +(-3.53553 + 3.53553i) q^{41} -9.00000i q^{43} +(-0.707107 + 0.707107i) q^{45} -6.00000 q^{47} +7.00000i q^{49} +6.00000i q^{53} -5.00000 q^{55} +(-0.707107 + 0.707107i) q^{57} +6.00000i q^{59} +(-2.82843 + 2.82843i) q^{61} +(3.53553 + 3.53553i) q^{65} +12.0000 q^{67} -3.00000 q^{69} +(8.48528 + 8.48528i) q^{71} +(-1.41421 - 1.41421i) q^{73} +(-2.82843 + 2.82843i) q^{75} +(-7.07107 + 7.07107i) q^{79} -1.00000 q^{81} +2.00000i q^{83} -2.00000i q^{87} -12.0000 q^{89} +2.00000i q^{93} +(0.707107 - 0.707107i) q^{95} +(11.3137 + 11.3137i) q^{97} +(-3.53553 - 3.53553i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 20 q^{13} + 20 q^{33} - 24 q^{47} - 20 q^{55} + 48 q^{67} - 12 q^{69} - 4 q^{81} - 48 q^{89}+O(q^{100}) $ 4 * q + 20 * q^13 + 20 * q^33 - 24 * q^47 - 20 * q^55 + 48 * q^67 - 12 * q^69 - 4 * q^81 - 48 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3468\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$2893$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$2$		0			0
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i	0.847316	−	0.531089i	$-0.178217\pi$
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i	−0.531089	+	0.847316i	$0.678217\pi$
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−3.53553	+	3.53553i	−1.06600	+	1.06600i	−0.0683416	+	0.997662i	$0.521771\pi$
$11$	−3.53553	+	3.53553i	−1.06600	+	1.06600i	−0.997662	+	0.0683416i	$0.978229\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$		0			0
$17$		0			0
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.12132	−	2.12132i	0.442326	−	0.442326i	−0.450467	−	0.892793i	$-0.648743\pi$
$23$	2.12132	−	2.12132i	0.442326	−	0.442326i	0.892793	+	0.450467i	$0.148743\pi$
$24$		0			0
$25$		−	4.00000i		−	0.800000i
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$	1.41421	+	1.41421i	0.262613	+	0.262613i	0.826115	−	0.563502i	$-0.190546\pi$
$29$	1.41421	+	1.41421i	0.262613	+	0.262613i	−0.563502	+	0.826115i	$0.690546\pi$
$30$		0			0
$31$	−1.41421	−	1.41421i	−0.254000	−	0.254000i	0.568608	−	0.822608i	$-0.307482\pi$
$31$	−1.41421	−	1.41421i	−0.254000	−	0.254000i	−0.822608	+	0.568608i	$0.807482\pi$
$32$		0			0
$33$	5.00000			0.870388
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.65685	+	5.65685i	0.929981	+	0.929981i	0.997704	−	0.0677230i	$-0.0215734\pi$
$37$	5.65685	+	5.65685i	0.929981	+	0.929981i	−0.0677230	+	0.997704i	$0.521573\pi$
$38$		0			0
$39$	−3.53553	−	3.53553i	−0.566139	−	0.566139i
$40$		0			0
$41$	−3.53553	+	3.53553i	−0.552158	+	0.552158i	−0.927063	−	0.374905i	$-0.877675\pi$
$41$	−3.53553	+	3.53553i	−0.552158	+	0.552158i	0.374905	+	0.927063i	$0.377675\pi$
$42$		0			0
$43$		−	9.00000i		−	1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$		−	9.00000i		−	1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$		0			0
$45$	−0.707107	+	0.707107i	−0.105409	+	0.105409i
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$		0			0
$57$	−0.707107	+	0.707107i	−0.0936586	+	0.0936586i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	−2.82843	+	2.82843i	−0.362143	+	0.362143i	−0.864601	−	0.502458i	$-0.832429\pi$
$61$	−2.82843	+	2.82843i	−0.362143	+	0.362143i	0.502458	+	0.864601i	$0.332429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.53553	+	3.53553i	0.438529	+	0.438529i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$	−3.00000			−0.361158
$70$		0			0
$71$	8.48528	+	8.48528i	1.00702	+	1.00702i	0.999975	+	0.00704243i	$0.00224169\pi$
$71$	8.48528	+	8.48528i	1.00702	+	1.00702i	0.00704243	+	0.999975i	$0.497758\pi$
$72$		0			0
$73$	−1.41421	−	1.41421i	−0.165521	−	0.165521i	0.619486	−	0.785007i	$-0.287341\pi$
$73$	−1.41421	−	1.41421i	−0.165521	−	0.165521i	−0.785007	+	0.619486i	$0.787341\pi$
$74$		0			0
$75$	−2.82843	+	2.82843i	−0.326599	+	0.326599i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.07107	+	7.07107i	−0.795557	+	0.795557i	−0.982391	−	0.186834i	$-0.940177\pi$
$79$	−7.07107	+	7.07107i	−0.795557	+	0.795557i	0.186834	+	0.982391i	$0.440177\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$	−12.0000			−1.27200			−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000			−1.27200			−0.635999	+	0.771690i	$0.719412\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	0.707107	−	0.707107i	0.0725476	−	0.0725476i
$96$		0			0
$97$	11.3137	+	11.3137i	1.14873	+	1.14873i	0.986802	+	0.161931i	$0.0517722\pi$
$97$	11.3137	+	11.3137i	1.14873	+	1.14873i	0.161931	+	0.986802i	$0.448228\pi$
$98$		0			0
$99$	−3.53553	−	3.53553i	−0.355335	−	0.355335i
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$	7.00000			0.689730			0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000			0.689730			0.344865	+	0.938652i	$0.387925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.94975	−	4.94975i	−0.478510	−	0.478510i	0.426145	−	0.904655i	$-0.359871\pi$
$107$	−4.94975	−	4.94975i	−0.478510	−	0.478510i	−0.904655	+	0.426145i	$0.859871\pi$
$108$		0			0
$109$	−2.82843	+	2.82843i	−0.270914	+	0.270914i	−0.829468	−	0.558554i	$-0.811356\pi$
$109$	−2.82843	+	2.82843i	−0.270914	+	0.270914i	0.558554	+	0.829468i	$0.311356\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$		0			0
$113$	−12.0208	+	12.0208i	−1.13082	+	1.13082i	−0.140783	+	0.990041i	$0.544962\pi$
$113$	−12.0208	+	12.0208i	−1.13082	+	1.13082i	−0.990041	+	0.140783i	$0.955038\pi$
$114$		0			0
$115$	3.00000			0.279751
$116$		0			0
$117$			5.00000i			0.462250i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	14.0000i		−	1.27273i
$122$		0			0
$123$	5.00000			0.450835
$124$		0			0
$125$	6.36396	−	6.36396i	0.569210	−	0.569210i
$126$		0			0
$127$			9.00000i			0.798621i	0.916816	+	0.399310i	$0.130750\pi$
$127$			9.00000i			0.798621i	−0.916816	+	0.399310i	$0.869250\pi$
$128$		0			0
$129$	−6.36396	+	6.36396i	−0.560316	+	0.560316i
$130$		0			0
$131$	2.12132	+	2.12132i	0.185341	+	0.185341i	0.793678	−	0.608338i	$-0.208163\pi$
$131$	2.12132	+	2.12132i	0.185341	+	0.185341i	−0.608338	+	0.793678i	$0.708163\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	22.0000			1.87959			0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000			1.87959			0.939793	+	0.341743i	$0.111017\pi$
$138$		0			0
$139$	−9.89949	−	9.89949i	−0.839664	−	0.839664i	0.149150	−	0.988815i	$-0.452346\pi$
$139$	−9.89949	−	9.89949i	−0.839664	−	0.839664i	−0.988815	+	0.149150i	$0.952346\pi$
$140$		0			0
$141$	4.24264	+	4.24264i	0.357295	+	0.357295i
$142$		0			0
$143$	−17.6777	+	17.6777i	−1.47828	+	1.47828i
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	4.94975	−	4.94975i	0.408248	−	0.408248i
$148$		0			0
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.00000i		−	0.160644i
$156$		0			0
$157$	17.0000			1.35675			0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000			1.35675			0.678374	+	0.734717i	$0.262685\pi$
$158$		0			0
$159$	4.24264	−	4.24264i	0.336463	−	0.336463i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−9.89949	+	9.89949i	−0.775388	+	0.775388i	−0.979043	−	0.203655i	$-0.934718\pi$
$163$	−9.89949	+	9.89949i	−0.775388	+	0.775388i	0.203655	+	0.979043i	$0.434718\pi$
$164$		0			0
$165$	3.53553	+	3.53553i	0.275241	+	0.275241i
$166$		0			0
$167$	16.2635	+	16.2635i	1.25850	+	1.25850i	0.951807	+	0.306697i	$0.0992237\pi$
$167$	16.2635	+	16.2635i	1.25850	+	1.25850i	0.306697	+	0.951807i	$0.400776\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	1.00000			0.0764719
$172$		0			0
$173$	2.12132	+	2.12132i	0.161281	+	0.161281i	0.783134	−	0.621853i	$-0.213620\pi$
$173$	2.12132	+	2.12132i	0.161281	+	0.161281i	−0.621853	+	0.783134i	$0.713620\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.24264	−	4.24264i	0.318896	−	0.318896i
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$	−9.89949	+	9.89949i	−0.735824	+	0.735824i	−0.971767	−	0.235943i	$-0.924182\pi$
$181$	−9.89949	+	9.89949i	−0.735824	+	0.735824i	0.235943	+	0.971767i	$0.424182\pi$
$182$		0			0
$183$	4.00000			0.295689
$184$		0			0
$185$			8.00000i			0.588172i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−26.0000			−1.88129			−0.940647	−	0.339387i	$-0.889781\pi$
$191$	−26.0000			−1.88129			−0.940647	+	0.339387i	$0.889781\pi$
$192$		0			0
$193$	12.7279	−	12.7279i	0.916176	−	0.916176i	−0.0805728	−	0.996749i	$-0.525675\pi$
$193$	12.7279	−	12.7279i	0.916176	−	0.916176i	0.996749	+	0.0805728i	$0.0256750\pi$
$194$		0			0
$195$		−	5.00000i		−	0.358057i
$196$		0			0
$197$	12.0208	−	12.0208i	0.856448	−	0.856448i	−0.134470	−	0.990918i	$-0.542933\pi$
$197$	12.0208	−	12.0208i	0.856448	−	0.856448i	0.990918	+	0.134470i	$0.0429332\pi$
$198$		0			0
$199$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$199$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$200$		0			0
$201$	−8.48528	−	8.48528i	−0.598506	−	0.598506i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−5.00000			−0.349215
$206$		0			0
$207$	2.12132	+	2.12132i	0.147442	+	0.147442i
$208$		0			0
$209$	3.53553	+	3.53553i	0.244558	+	0.244558i
$210$		0			0
$211$	−15.5563	+	15.5563i	−1.07094	+	1.07094i	−0.0736598	+	0.997283i	$0.523468\pi$
$211$	−15.5563	+	15.5563i	−1.07094	+	1.07094i	−0.997283	+	0.0736598i	$0.976532\pi$
$212$		0			0
$213$		−	12.0000i		−	0.822226i
$214$		0			0
$215$	6.36396	−	6.36396i	0.434019	−	0.434019i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			2.00000i			0.135147i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			11.0000i			0.736614i	0.929704	+	0.368307i	$0.120063\pi$
$223$			11.0000i			0.736614i	−0.929704	+	0.368307i	$0.879937\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$	3.53553	−	3.53553i	0.234662	−	0.234662i	−0.579974	−	0.814635i	$-0.696937\pi$
$227$	3.53553	−	3.53553i	0.234662	−	0.234662i	0.814635	+	0.579974i	$0.196937\pi$
$228$		0			0
$229$			22.0000i			1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$			22.0000i			1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.19239	−	9.19239i	−0.602213	−	0.602213i	0.338686	−	0.940899i	$-0.390018\pi$
$233$	−9.19239	−	9.19239i	−0.602213	−	0.602213i	−0.940899	+	0.338686i	$0.890018\pi$
$234$		0			0
$235$	−4.24264	−	4.24264i	−0.276759	−	0.276759i
$236$		0			0
$237$	10.0000			0.649570
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	2.82843	+	2.82843i	0.182195	+	0.182195i	0.792312	−	0.610117i	$-0.208877\pi$
$241$	2.82843	+	2.82843i	0.182195	+	0.182195i	−0.610117	+	0.792312i	$0.708877\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$	−4.94975	+	4.94975i	−0.316228	+	0.316228i
$246$		0			0
$247$		−	5.00000i		−	0.318142i
$248$		0			0
$249$	1.41421	−	1.41421i	0.0896221	−	0.0896221i
$250$		0			0
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		0			0
$253$			15.0000i			0.943042i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	8.00000i		−	0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$		−	8.00000i		−	0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.41421	+	1.41421i	−0.0875376	+	0.0875376i
$262$		0			0
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		0			0
$265$	−4.24264	+	4.24264i	−0.260623	+	0.260623i
$266$		0			0
$267$	8.48528	+	8.48528i	0.519291	+	0.519291i
$268$		0			0
$269$	14.8492	+	14.8492i	0.905374	+	0.905374i	0.995895	−	0.0905203i	$-0.0288530\pi$
$269$	14.8492	+	14.8492i	0.905374	+	0.905374i	−0.0905203	+	0.995895i	$0.528853\pi$
$270$		0			0
$271$	25.0000			1.51864			0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000			1.51864			0.759321	+	0.650716i	$0.225531\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.1421	+	14.1421i	0.852803	+	0.852803i
$276$		0			0
$277$	−12.7279	−	12.7279i	−0.764747	−	0.764747i	0.212430	−	0.977176i	$-0.431862\pi$
$277$	−12.7279	−	12.7279i	−0.764747	−	0.764747i	−0.977176	+	0.212430i	$0.931862\pi$
$278$		0			0
$279$	1.41421	−	1.41421i	0.0846668	−	0.0846668i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$	21.2132	−	21.2132i	1.26099	−	1.26099i	0.310382	−	0.950612i	$-0.399543\pi$
$283$	21.2132	−	21.2132i	1.26099	−	1.26099i	0.950612	−	0.310382i	$-0.100457\pi$
$284$		0			0
$285$	−1.00000			−0.0592349
$286$		0			0
$287$		0			0
$288$		0			0
$289$		0			0
$290$		0			0
$291$		−	16.0000i		−	0.937937i
$292$		0			0
$293$	−4.00000			−0.233682			−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000			−0.233682			−0.116841	+	0.993151i	$0.537277\pi$
$294$		0			0
$295$	−4.24264	+	4.24264i	−0.247016	+	0.247016i
$296$		0			0
$297$			5.00000i			0.290129i
$298$		0			0
$299$	10.6066	−	10.6066i	0.613396	−	0.613396i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−5.65685	−	5.65685i	−0.324978	−	0.324978i
$304$		0			0
$305$	−4.00000			−0.229039
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	−4.94975	−	4.94975i	−0.281581	−	0.281581i
$310$		0			0
$311$	16.9706	+	16.9706i	0.962312	+	0.962312i	0.999315	−	0.0370028i	$-0.0117811\pi$
$311$	16.9706	+	16.9706i	0.962312	+	0.962312i	−0.0370028	+	0.999315i	$0.511781\pi$
$312$		0			0
$313$	11.3137	−	11.3137i	0.639489	−	0.639489i	−0.310941	−	0.950429i	$-0.600644\pi$
$313$	11.3137	−	11.3137i	0.639489	−	0.639489i	0.950429	+	0.310941i	$0.100644\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.2132	+	21.2132i	−1.19145	+	1.19145i	−0.214792	+	0.976660i	$0.568908\pi$
$317$	−21.2132	+	21.2132i	−1.19145	+	1.19145i	−0.976660	+	0.214792i	$0.931092\pi$
$318$		0			0
$319$	−10.0000			−0.559893
$320$		0			0
$321$			7.00000i			0.390702i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	20.0000i		−	1.10940i
$326$		0			0
$327$	4.00000			0.221201
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	11.0000i		−	0.604615i	−0.953211	−	0.302307i	$-0.902243\pi$
$331$		−	11.0000i		−	0.604615i	0.953211	−	0.302307i	$-0.0977569\pi$
$332$		0			0
$333$	−5.65685	+	5.65685i	−0.309994	+	0.309994i
$334$		0			0
$335$	8.48528	+	8.48528i	0.463600	+	0.463600i
$336$		0			0
$337$	1.41421	+	1.41421i	0.0770371	+	0.0770371i	0.744575	−	0.667538i	$-0.232652\pi$
$337$	1.41421	+	1.41421i	0.0770371	+	0.0770371i	−0.667538	+	0.744575i	$0.732652\pi$
$338$		0			0
$339$	17.0000			0.923313
$340$		0			0
$341$	10.0000			0.541530
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.12132	−	2.12132i	−0.114208	−	0.114208i
$346$		0			0
$347$	2.82843	−	2.82843i	0.151838	−	0.151838i	−0.627100	−	0.778938i	$-0.715758\pi$
$347$	2.82843	−	2.82843i	0.151838	−	0.151838i	0.778938	+	0.627100i	$0.215758\pi$
$348$		0			0
$349$		−	7.00000i		−	0.374701i	−0.982293	−	0.187351i	$-0.940010\pi$
$349$		−	7.00000i		−	0.374701i	0.982293	−	0.187351i	$-0.0599901\pi$
$350$		0			0
$351$	3.53553	−	3.53553i	0.188713	−	0.188713i
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$			12.0000i			0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			36.0000i			1.90001i	0.312239	+	0.950004i	$0.398921\pi$
$359$			36.0000i			1.90001i	−0.312239	+	0.950004i	$0.601079\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$	−9.89949	+	9.89949i	−0.519589	+	0.519589i
$364$		0			0
$365$		−	2.00000i		−	0.104685i
$366$		0			0
$367$	8.48528	−	8.48528i	0.442928	−	0.442928i	−0.450067	−	0.892995i	$-0.648600\pi$
$367$	8.48528	−	8.48528i	0.442928	−	0.442928i	0.892995	+	0.450067i	$0.148600\pi$
$368$		0			0
$369$	−3.53553	−	3.53553i	−0.184053	−	0.184053i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$	−9.00000			−0.464758
$376$		0			0
$377$	7.07107	+	7.07107i	0.364179	+	0.364179i
$378$		0			0
$379$	−2.82843	−	2.82843i	−0.145287	−	0.145287i	0.630722	−	0.776009i	$-0.282759\pi$
$379$	−2.82843	−	2.82843i	−0.145287	−	0.145287i	−0.776009	+	0.630722i	$0.782759\pi$
$380$		0			0
$381$	6.36396	−	6.36396i	0.326036	−	0.326036i
$382$		0			0
$383$			20.0000i			1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$			20.0000i			1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.00000			0.457496
$388$		0			0
$389$			8.00000i			0.405616i	0.979219	+	0.202808i	$0.0650067\pi$
$389$			8.00000i			0.405616i	−0.979219	+	0.202808i	$0.934993\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	3.00000i		−	0.151330i
$394$		0			0
$395$	−10.0000			−0.503155
$396$		0			0
$397$	−2.82843	+	2.82843i	−0.141955	+	0.141955i	−0.774513	−	0.632558i	$-0.782005\pi$
$397$	−2.82843	+	2.82843i	−0.141955	+	0.141955i	0.632558	+	0.774513i	$0.282005\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0208	+	12.0208i	−0.600291	+	0.600291i	−0.940390	−	0.340099i	$-0.889539\pi$
$401$	−12.0208	+	12.0208i	−0.600291	+	0.600291i	0.340099	+	0.940390i	$0.389539\pi$
$402$		0			0
$403$	−7.07107	−	7.07107i	−0.352235	−	0.352235i
$404$		0			0
$405$	−0.707107	−	0.707107i	−0.0351364	−	0.0351364i
$406$		0			0
$407$	−40.0000			−1.98273
$408$		0			0
$409$	5.00000			0.247234			0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000			0.247234			0.123617	+	0.992330i	$0.460551\pi$
$410$		0			0
$411$	−15.5563	−	15.5563i	−0.767338	−	0.767338i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.41421	+	1.41421i	−0.0694210	+	0.0694210i
$416$		0			0
$417$			14.0000i			0.685583i
$418$		0			0
$419$	−19.7990	+	19.7990i	−0.967244	+	0.967244i	−0.999480	−	0.0322363i	$-0.989737\pi$
$419$	−19.7990	+	19.7990i	−0.967244	+	0.967244i	0.0322363	+	0.999480i	$0.489737\pi$
$420$		0			0
$421$	29.0000			1.41337			0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000			1.41337			0.706687	+	0.707527i	$0.250189\pi$
$422$		0			0
$423$		−	6.00000i		−	0.291730i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	25.0000			1.20701
$430$		0			0
$431$	22.6274	−	22.6274i	1.08992	−	1.08992i	0.0943889	−	0.995535i	$-0.469910\pi$
$431$	22.6274	−	22.6274i	1.08992	−	1.08992i	0.995535	−	0.0943889i	$-0.0300897\pi$
$432$		0			0
$433$		−	41.0000i		−	1.97033i	−0.171598	−	0.985167i	$-0.554893\pi$
$433$		−	41.0000i		−	1.97033i	0.171598	−	0.985167i	$-0.445107\pi$
$434$		0			0
$435$	1.41421	−	1.41421i	0.0678064	−	0.0678064i
$436$		0			0
$437$	−2.12132	−	2.12132i	−0.101477	−	0.101477i
$438$		0			0
$439$	−19.7990	−	19.7990i	−0.944954	−	0.944954i	0.0536078	−	0.998562i	$-0.482928\pi$
$439$	−19.7990	−	19.7990i	−0.944954	−	0.944954i	−0.998562	+	0.0536078i	$0.982928\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	2.00000			0.0950229			0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000			0.0950229			0.0475114	+	0.998871i	$0.484871\pi$
$444$		0			0
$445$	−8.48528	−	8.48528i	−0.402241	−	0.402241i
$446$		0			0
$447$	−15.5563	−	15.5563i	−0.735790	−	0.735790i
$448$		0			0
$449$	−7.07107	+	7.07107i	−0.333704	+	0.333704i	−0.853991	−	0.520287i	$-0.825825\pi$
$449$	−7.07107	+	7.07107i	−0.333704	+	0.333704i	0.520287	+	0.853991i	$0.325825\pi$
$450$		0			0
$451$		−	25.0000i		−	1.17720i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	13.0000i		−	0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$457$		−	13.0000i		−	0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$	−1.41421	+	1.41421i	−0.0655826	+	0.0655826i
$466$		0			0
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−12.0208	−	12.0208i	−0.553890	−	0.553890i
$472$		0			0
$473$	31.8198	+	31.8198i	1.46308	+	1.46308i
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−6.36396	−	6.36396i	−0.290777	−	0.290777i	0.546610	−	0.837387i	$-0.315918\pi$
$479$	−6.36396	−	6.36396i	−0.290777	−	0.290777i	−0.837387	+	0.546610i	$0.815918\pi$
$480$		0			0
$481$	28.2843	+	28.2843i	1.28965	+	1.28965i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			16.0000i			0.726523i
$486$		0			0
$487$	24.0416	−	24.0416i	1.08943	−	1.08943i	0.0938433	−	0.995587i	$-0.470085\pi$
$487$	24.0416	−	24.0416i	1.08943	−	1.08943i	0.995587	−	0.0938433i	$-0.0299153\pi$
$488$		0			0
$489$	14.0000			0.633102
$490$		0			0
$491$		−	30.0000i		−	1.35388i	−0.736038	−	0.676941i	$-0.763305\pi$
$491$		−	30.0000i		−	1.35388i	0.736038	−	0.676941i	$-0.236695\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	5.00000i		−	0.224733i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7.07107	+	7.07107i	−0.316544	+	0.316544i	−0.847438	−	0.530894i	$-0.821856\pi$
$499$	−7.07107	+	7.07107i	−0.316544	+	0.316544i	0.530894	+	0.847438i	$0.321856\pi$
$500$		0			0
$501$		−	23.0000i		−	1.02756i
$502$		0			0
$503$	19.0919	−	19.0919i	0.851265	−	0.851265i	−0.139024	−	0.990289i	$-0.544397\pi$
$503$	19.0919	−	19.0919i	0.851265	−	0.851265i	0.990289	+	0.139024i	$0.0443965\pi$
$504$		0			0
$505$	5.65685	+	5.65685i	0.251727	+	0.251727i
$506$		0			0
$507$	−8.48528	−	8.48528i	−0.376845	−	0.376845i
$508$		0			0
$509$	−36.0000			−1.59567			−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000			−1.59567			−0.797836	+	0.602875i	$0.794022\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−0.707107	−	0.707107i	−0.0312195	−	0.0312195i
$514$		0			0
$515$	4.94975	+	4.94975i	0.218112	+	0.218112i
$516$		0			0
$517$	21.2132	−	21.2132i	0.932956	−	0.932956i
$518$		0			0
$519$		−	3.00000i		−	0.131685i
$520$		0			0
$521$	9.19239	−	9.19239i	0.402726	−	0.402726i	−0.476467	−	0.879193i	$-0.658083\pi$
$521$	9.19239	−	9.19239i	0.402726	−	0.402726i	0.879193	+	0.476467i	$0.158083\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			14.0000i			0.608696i
$530$		0			0
$531$	−6.00000			−0.260378
$532$		0			0
$533$	−17.6777	+	17.6777i	−0.765705	+	0.765705i
$534$		0			0
$535$		−	7.00000i		−	0.302636i
$536$		0			0
$537$	−12.7279	+	12.7279i	−0.549250	+	0.549250i
$538$		0			0
$539$	−24.7487	−	24.7487i	−1.06600	−	1.06600i
$540$		0			0
$541$	−31.1127	−	31.1127i	−1.33764	−	1.33764i	−0.898341	−	0.439298i	$-0.855227\pi$
$541$	−31.1127	−	31.1127i	−1.33764	−	1.33764i	−0.439298	−	0.898341i	$-0.644773\pi$
$542$		0			0
$543$	14.0000			0.600798
$544$		0			0
$545$	−4.00000			−0.171341
$546$		0			0
$547$	8.48528	+	8.48528i	0.362804	+	0.362804i	0.864844	−	0.502040i	$-0.167417\pi$
$547$	8.48528	+	8.48528i	0.362804	+	0.362804i	−0.502040	+	0.864844i	$0.667417\pi$
$548$		0			0
$549$	−2.82843	−	2.82843i	−0.120714	−	0.120714i
$550$		0			0
$551$	1.41421	−	1.41421i	0.0602475	−	0.0602475i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	5.65685	−	5.65685i	0.240120	−	0.240120i
$556$		0			0
$557$	−14.0000			−0.593199			−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000			−0.593199			−0.296600	+	0.955002i	$0.595853\pi$
$558$		0			0
$559$		−	45.0000i		−	1.90330i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.0000i			0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$			18.0000i			0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$		0			0
$565$	−17.0000			−0.715195
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.0000i		−	1.00613i	−0.864248	−	0.503066i	$-0.832205\pi$
$569$		−	24.0000i		−	1.00613i	0.864248	−	0.503066i	$-0.167795\pi$
$570$		0			0
$571$	11.3137	−	11.3137i	0.473464	−	0.473464i	−0.429570	−	0.903034i	$-0.641335\pi$
$571$	11.3137	−	11.3137i	0.473464	−	0.473464i	0.903034	+	0.429570i	$0.141335\pi$
$572$		0			0
$573$	18.3848	+	18.3848i	0.768035	+	0.768035i
$574$		0			0
$575$	−8.48528	−	8.48528i	−0.353861	−	0.353861i
$576$		0			0
$577$	−7.00000			−0.291414			−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000			−0.291414			−0.145707	+	0.989328i	$0.546546\pi$
$578$		0			0
$579$	−18.0000			−0.748054
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−21.2132	−	21.2132i	−0.878561	−	0.878561i
$584$		0			0
$585$	−3.53553	+	3.53553i	−0.146176	+	0.146176i
$586$		0			0
$587$			4.00000i			0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$			4.00000i			0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$		0			0
$589$	−1.41421	+	1.41421i	−0.0582717	+	0.0582717i
$590$		0			0
$591$	−17.0000			−0.699287
$592$		0			0
$593$		−	22.0000i		−	0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$		−	22.0000i		−	0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.0000			−0.408589			−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000			−0.408589			−0.204294	+	0.978909i	$0.565490\pi$
$600$		0			0
$601$	−29.6985	+	29.6985i	−1.21143	+	1.21143i	−0.240869	+	0.970558i	$0.577432\pi$
$601$	−29.6985	+	29.6985i	−1.21143	+	1.21143i	−0.970558	+	0.240869i	$0.922568\pi$
$602$		0			0
$603$			12.0000i			0.488678i
$604$		0			0
$605$	9.89949	−	9.89949i	0.402472	−	0.402472i
$606$		0			0
$607$	−4.24264	−	4.24264i	−0.172203	−	0.172203i	0.615743	−	0.787947i	$-0.288856\pi$
$607$	−4.24264	−	4.24264i	−0.172203	−	0.172203i	−0.787947	+	0.615743i	$0.788856\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−30.0000			−1.21367
$612$		0			0
$613$	−25.0000			−1.00974			−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000			−1.00974			−0.504870	+	0.863195i	$0.668460\pi$
$614$		0			0
$615$	3.53553	+	3.53553i	0.142566	+	0.142566i
$616$		0			0
$617$	−12.7279	−	12.7279i	−0.512407	−	0.512407i	0.402856	−	0.915263i	$-0.368017\pi$
$617$	−12.7279	−	12.7279i	−0.512407	−	0.512407i	−0.915263	+	0.402856i	$0.868017\pi$
$618$		0			0
$619$	24.0416	−	24.0416i	0.966315	−	0.966315i	−0.0331361	−	0.999451i	$-0.510549\pi$
$619$	24.0416	−	24.0416i	0.966315	−	0.966315i	0.999451	+	0.0331361i	$0.0105495\pi$
$620$		0			0
$621$		−	3.00000i		−	0.120386i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		−	5.00000i		−	0.199681i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	9.00000i		−	0.358284i	−0.983823	−	0.179142i	$-0.942668\pi$
$631$		−	9.00000i		−	0.358284i	0.983823	−	0.179142i	$-0.0573322\pi$
$632$		0			0
$633$	22.0000			0.874421
$634$		0			0
$635$	−6.36396	+	6.36396i	−0.252546	+	0.252546i
$636$		0			0
$637$			35.0000i			1.38675i
$638$		0			0
$639$	−8.48528	+	8.48528i	−0.335673	+	0.335673i
$640$		0			0
$641$	−4.94975	−	4.94975i	−0.195503	−	0.195503i	0.602566	−	0.798069i	$-0.294145\pi$
$641$	−4.94975	−	4.94975i	−0.195503	−	0.195503i	−0.798069	+	0.602566i	$0.794145\pi$
$642$		0			0
$643$	−5.65685	−	5.65685i	−0.223085	−	0.223085i	0.586711	−	0.809796i	$-0.300422\pi$
$643$	−5.65685	−	5.65685i	−0.223085	−	0.223085i	−0.809796	+	0.586711i	$0.800422\pi$
$644$		0			0
$645$	−9.00000			−0.354375
$646$		0			0
$647$	−22.0000			−0.864909			−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000			−0.864909			−0.432455	+	0.901656i	$0.642352\pi$
$648$		0			0
$649$	−21.2132	−	21.2132i	−0.832691	−	0.832691i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.3345	−	23.3345i	0.913150	−	0.913150i	−0.0833683	−	0.996519i	$-0.526568\pi$
$653$	23.3345	−	23.3345i	0.913150	−	0.913150i	0.996519	+	0.0833683i	$0.0265678\pi$
$654$		0			0
$655$			3.00000i			0.117220i
$656$		0			0
$657$	1.41421	−	1.41421i	0.0551737	−	0.0551737i
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$			3.00000i			0.116686i	0.998297	+	0.0583432i	$0.0185818\pi$
$661$			3.00000i			0.116686i	−0.998297	+	0.0583432i	$0.981418\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.00000			0.232321
$668$		0			0
$669$	7.77817	−	7.77817i	0.300722	−	0.300722i
$670$		0			0
$671$		−	20.0000i		−	0.772091i
$672$		0			0
$673$	−24.0416	+	24.0416i	−0.926737	+	0.926737i	−0.997494	−	0.0707568i	$-0.977459\pi$
$673$	−24.0416	+	24.0416i	−0.926737	+	0.926737i	0.0707568	+	0.997494i	$0.477459\pi$
$674$		0			0
$675$	−2.82843	−	2.82843i	−0.108866	−	0.108866i
$676$		0			0
$677$	28.9914	+	28.9914i	1.11423	+	1.11423i	0.992572	+	0.121657i	$0.0388208\pi$
$677$	28.9914	+	28.9914i	1.11423	+	1.11423i	0.121657	+	0.992572i	$0.461179\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−5.00000			−0.191600
$682$		0			0
$683$	3.53553	+	3.53553i	0.135283	+	0.135283i	0.771506	−	0.636222i	$-0.219504\pi$
$683$	3.53553	+	3.53553i	0.135283	+	0.135283i	−0.636222	+	0.771506i	$0.719504\pi$
$684$		0			0
$685$	15.5563	+	15.5563i	0.594378	+	0.594378i
$686$		0			0
$687$	15.5563	−	15.5563i	0.593512	−	0.593512i
$688$		0			0
$689$			30.0000i			1.14291i
$690$		0			0
$691$	19.7990	−	19.7990i	0.753189	−	0.753189i	−0.221884	−	0.975073i	$-0.571221\pi$
$691$	19.7990	−	19.7990i	0.753189	−	0.753189i	0.975073	+	0.221884i	$0.0712206\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	14.0000i		−	0.531050i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			13.0000i			0.491705i
$700$		0			0
$701$	−40.0000			−1.51078			−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000			−1.51078			−0.755390	+	0.655276i	$0.772552\pi$
$702$		0			0
$703$	5.65685	−	5.65685i	0.213352	−	0.213352i
$704$		0			0
$705$			6.00000i			0.225973i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.07107	+	7.07107i	0.265560	+	0.265560i	0.827308	−	0.561749i	$-0.189871\pi$
$709$	7.07107	+	7.07107i	0.265560	+	0.265560i	−0.561749	+	0.827308i	$0.689871\pi$
$710$		0			0
$711$	−7.07107	−	7.07107i	−0.265186	−	0.265186i
$712$		0			0
$713$	−6.00000			−0.224702
$714$		0			0
$715$	−25.0000			−0.934947
$716$		0			0
$717$	−8.48528	−	8.48528i	−0.316889	−	0.316889i
$718$		0			0
$719$	4.94975	+	4.94975i	0.184594	+	0.184594i	0.793354	−	0.608760i	$-0.208333\pi$
$719$	4.94975	+	4.94975i	0.184594	+	0.184594i	−0.608760	+	0.793354i	$0.708333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	4.00000i		−	0.148762i
$724$		0			0
$725$	5.65685	−	5.65685i	0.210090	−	0.210090i
$726$		0			0
$727$	−16.0000			−0.593407			−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000			−0.593407			−0.296704	+	0.954970i	$0.595887\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	10.0000i		−	0.369358i	−0.982799	−	0.184679i	$-0.940875\pi$
$733$		−	10.0000i		−	0.369358i	0.982799	−	0.184679i	$-0.0591246\pi$
$734$		0			0
$735$	7.00000			0.258199
$736$		0			0
$737$	−42.4264	+	42.4264i	−1.56280	+	1.56280i
$738$		0			0
$739$		−	23.0000i		−	0.846069i	−0.906114	−	0.423034i	$-0.860965\pi$
$739$		−	23.0000i		−	0.846069i	0.906114	−	0.423034i	$-0.139035\pi$
$740$		0			0
$741$	−3.53553	+	3.53553i	−0.129881	+	0.129881i
$742$		0			0
$743$	−11.3137	−	11.3137i	−0.415060	−	0.415060i	0.468437	−	0.883497i	$-0.344817\pi$
$743$	−11.3137	−	11.3137i	−0.415060	−	0.415060i	−0.883497	+	0.468437i	$0.844817\pi$
$744$		0			0
$745$	15.5563	+	15.5563i	0.569941	+	0.569941i
$746$		0			0
$747$	−2.00000			−0.0731762
$748$		0			0
$749$		0			0
$750$		0			0
$751$	15.5563	+	15.5563i	0.567659	+	0.567659i	0.931472	−	0.363813i	$-0.118525\pi$
$751$	15.5563	+	15.5563i	0.567659	+	0.567659i	−0.363813	+	0.931472i	$0.618525\pi$
$752$		0			0
$753$	19.7990	+	19.7990i	0.721515	+	0.721515i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			41.0000i			1.49017i	0.666969	+	0.745085i	$0.267591\pi$
$757$			41.0000i			1.49017i	−0.666969	+	0.745085i	$0.732409\pi$
$758$		0			0
$759$	10.6066	−	10.6066i	0.384995	−	0.384995i
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			30.0000i			1.08324i
$768$		0			0
$769$	39.0000			1.40638			0.703188	−	0.711004i	$-0.251759\pi$
$769$	39.0000			1.40638			0.703188	+	0.711004i	$0.251759\pi$
$770$		0			0
$771$	−5.65685	+	5.65685i	−0.203727	+	0.203727i
$772$		0			0
$773$		−	16.0000i		−	0.575480i	−0.957709	−	0.287740i	$-0.907096\pi$
$773$		−	16.0000i		−	0.575480i	0.957709	−	0.287740i	$-0.0929039\pi$
$774$		0			0
$775$	−5.65685	+	5.65685i	−0.203200	+	0.203200i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.53553	+	3.53553i	0.126674	+	0.126674i
$780$		0			0
$781$	−60.0000			−2.14697
$782$		0			0
$783$	2.00000			0.0714742
$784$		0			0
$785$	12.0208	+	12.0208i	0.429041	+	0.429041i
$786$		0			0
$787$	−22.6274	−	22.6274i	−0.806580	−	0.806580i	0.177534	−	0.984115i	$-0.443188\pi$
$787$	−22.6274	−	22.6274i	−0.806580	−	0.806580i	−0.984115	+	0.177534i	$0.943188\pi$
$788$		0			0
$789$	11.3137	−	11.3137i	0.402779	−	0.402779i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−14.1421	+	14.1421i	−0.502202	+	0.502202i
$794$		0			0
$795$	6.00000			0.212798
$796$		0			0
$797$		−	8.00000i		−	0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$		−	8.00000i		−	0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	12.0000i		−	0.423999i
$802$		0			0
$803$	10.0000			0.352892
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	21.0000i		−	0.739235i
$808$		0			0
$809$	7.77817	−	7.77817i	0.273466	−	0.273466i	−0.557028	−	0.830494i	$-0.688058\pi$
$809$	7.77817	−	7.77817i	0.273466	−	0.273466i	0.830494	+	0.557028i	$0.188058\pi$
$810$		0			0
$811$	−18.3848	−	18.3848i	−0.645577	−	0.645577i	0.306344	−	0.951921i	$-0.400894\pi$
$811$	−18.3848	−	18.3848i	−0.645577	−	0.645577i	−0.951921	+	0.306344i	$0.900894\pi$
$812$		0			0
$813$	−17.6777	−	17.6777i	−0.619983	−	0.619983i
$814$		0			0
$815$	−14.0000			−0.490399
$816$		0			0
$817$	−9.00000			−0.314870
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.9203	−	21.9203i	−0.765024	−	0.765024i	0.212202	−	0.977226i	$-0.431937\pi$
$821$	−21.9203	−	21.9203i	−0.765024	−	0.765024i	−0.977226	+	0.212202i	$0.931937\pi$
$822$		0			0
$823$	5.65685	−	5.65685i	0.197186	−	0.197186i	−0.601607	−	0.798792i	$-0.705473\pi$
$823$	5.65685	−	5.65685i	0.197186	−	0.197186i	0.798792	+	0.601607i	$0.205473\pi$
$824$		0			0
$825$		−	20.0000i		−	0.696311i
$826$		0			0
$827$	3.53553	−	3.53553i	0.122943	−	0.122943i	−0.642958	−	0.765901i	$-0.722293\pi$
$827$	3.53553	−	3.53553i	0.122943	−	0.122943i	0.765901	+	0.642958i	$0.222293\pi$
$828$		0			0
$829$	10.0000			0.347314			0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000			0.347314			0.173657	+	0.984806i	$0.444442\pi$
$830$		0			0
$831$			18.0000i			0.624413i
$832$		0			0
$833$		0			0
$834$		0			0
$835$			23.0000i			0.795948i
$836$		0			0
$837$	−2.00000			−0.0691301
$838$		0			0
$839$	14.8492	−	14.8492i	0.512653	−	0.512653i	−0.402686	−	0.915338i	$-0.631923\pi$
$839$	14.8492	−	14.8492i	0.512653	−	0.512653i	0.915338	+	0.402686i	$0.131923\pi$
$840$		0			0
$841$		−	25.0000i		−	0.862069i
$842$		0			0
$843$	16.9706	−	16.9706i	0.584497	−	0.584497i
$844$		0			0
$845$	8.48528	+	8.48528i	0.291903	+	0.291903i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−30.0000			−1.02960
$850$		0			0
$851$	24.0000			0.822709
$852$		0			0
$853$	−41.0122	−	41.0122i	−1.40423	−	1.40423i	−0.785984	−	0.618246i	$-0.787843\pi$
$853$	−41.0122	−	41.0122i	−1.40423	−	1.40423i	−0.618246	−	0.785984i	$-0.712157\pi$
$854$		0			0
$855$	0.707107	+	0.707107i	0.0241825	+	0.0241825i
$856$		0			0
$857$	−4.24264	+	4.24264i	−0.144926	+	0.144926i	−0.775847	−	0.630921i	$-0.782677\pi$
$857$	−4.24264	+	4.24264i	−0.144926	+	0.144926i	0.630921	+	0.775847i	$0.282677\pi$
$858$		0			0
$859$			36.0000i			1.22830i	0.789188	+	0.614152i	$0.210502\pi$
$859$			36.0000i			1.22830i	−0.789188	+	0.614152i	$0.789498\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.00000			−0.136162			−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000			−0.136162			−0.0680808	+	0.997680i	$0.521688\pi$
$864$		0			0
$865$			3.00000i			0.102003i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	50.0000i		−	1.69613i
$870$		0			0
$871$	60.0000			2.03302
$872$		0			0
$873$	−11.3137	+	11.3137i	−0.382911	+	0.382911i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−18.3848	+	18.3848i	−0.620810	+	0.620810i	−0.945739	−	0.324929i	$-0.894660\pi$
$877$	−18.3848	+	18.3848i	−0.620810	+	0.620810i	0.324929	+	0.945739i	$0.394660\pi$
$878$		0			0
$879$	2.82843	+	2.82843i	0.0954005	+	0.0954005i
$880$		0			0
$881$	−9.89949	−	9.89949i	−0.333522	−	0.333522i	0.520400	−	0.853923i	$-0.325783\pi$
$881$	−9.89949	−	9.89949i	−0.333522	−	0.333522i	−0.853923	+	0.520400i	$0.825783\pi$
$882$		0			0
$883$	−27.0000			−0.908622			−0.454311	−	0.890843i	$-0.650115\pi$
$883$	−27.0000			−0.908622			−0.454311	+	0.890843i	$0.650115\pi$
$884$		0			0
$885$	6.00000			0.201688
$886$		0			0
$887$	−19.0919	−	19.0919i	−0.641043	−	0.641043i	0.309769	−	0.950812i	$-0.399748\pi$
$887$	−19.0919	−	19.0919i	−0.641043	−	0.641043i	−0.950812	+	0.309769i	$0.899748\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	3.53553	−	3.53553i	0.118445	−	0.118445i
$892$		0			0
$893$			6.00000i			0.200782i
$894$		0			0
$895$	12.7279	−	12.7279i	0.425448	−	0.425448i
$896$		0			0
$897$	−15.0000			−0.500835
$898$		0			0
$899$		−	4.00000i		−	0.133407i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.0000			−0.465376
$906$		0			0
$907$	1.41421	−	1.41421i	0.0469582	−	0.0469582i	−0.683238	−	0.730196i	$-0.739429\pi$
$907$	1.41421	−	1.41421i	0.0469582	−	0.0469582i	0.730196	+	0.683238i	$0.239429\pi$
$908$		0			0
$909$			8.00000i			0.265343i
$910$		0			0
$911$	12.0208	−	12.0208i	0.398267	−	0.398267i	−0.479354	−	0.877622i	$-0.659129\pi$
$911$	12.0208	−	12.0208i	0.398267	−	0.398267i	0.877622	+	0.479354i	$0.159129\pi$
$912$		0			0
$913$	−7.07107	−	7.07107i	−0.234018	−	0.234018i
$914$		0			0
$915$	2.82843	+	2.82843i	0.0935049	+	0.0935049i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−15.0000			−0.494804			−0.247402	−	0.968913i	$-0.579577\pi$
$919$	−15.0000			−0.494804			−0.247402	+	0.968913i	$0.579577\pi$
$920$		0			0
$921$	8.48528	+	8.48528i	0.279600	+	0.279600i
$922$		0			0
$923$	42.4264	+	42.4264i	1.39648	+	1.39648i
$924$		0			0
$925$	22.6274	−	22.6274i	0.743985	−	0.743985i
$926$		0			0
$927$			7.00000i			0.229910i
$928$		0			0
$929$	21.9203	−	21.9203i	0.719182	−	0.719182i	−0.249256	−	0.968438i	$-0.580186\pi$
$929$	21.9203	−	21.9203i	0.719182	−	0.719182i	0.968438	+	0.249256i	$0.0801859\pi$
$930$		0			0
$931$	7.00000			0.229416
$932$		0			0
$933$		−	24.0000i		−	0.785725i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			22.0000i			0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$			22.0000i			0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$		0			0
$939$	−16.0000			−0.522140
$940$		0			0
$941$	24.0416	−	24.0416i	0.783735	−	0.783735i	−0.196724	−	0.980459i	$-0.563030\pi$
$941$	24.0416	−	24.0416i	0.783735	−	0.783735i	0.980459	+	0.196724i	$0.0630303\pi$
$942$		0			0
$943$			15.0000i			0.488467i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.7990	−	19.7990i	−0.643381	−	0.643381i	0.308004	−	0.951385i	$-0.400339\pi$
$947$	−19.7990	−	19.7990i	−0.643381	−	0.643381i	−0.951385	+	0.308004i	$0.900339\pi$
$948$		0			0
$949$	−7.07107	−	7.07107i	−0.229537	−	0.229537i
$950$		0			0
$951$	30.0000			0.972817
$952$		0			0
$953$	36.0000			1.16615			0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000			1.16615			0.583077	+	0.812417i	$0.301849\pi$
$954$		0			0
$955$	−18.3848	−	18.3848i	−0.594917	−	0.594917i
$956$		0			0
$957$	7.07107	+	7.07107i	0.228575	+	0.228575i
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	27.0000i		−	0.870968i
$962$		0			0
$963$	4.94975	−	4.94975i	0.159503	−	0.159503i
$964$		0			0
$965$	18.0000			0.579441
$966$		0			0
$967$		−	19.0000i		−	0.610999i	−0.952192	−	0.305499i	$-0.901177\pi$
$967$		−	19.0000i		−	0.610999i	0.952192	−	0.305499i	$-0.0988234\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			22.0000i			0.706014i	0.935621	+	0.353007i	$0.114841\pi$
$971$			22.0000i			0.706014i	−0.935621	+	0.353007i	$0.885159\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−14.1421	+	14.1421i	−0.452911	+	0.452911i
$976$		0			0
$977$			14.0000i			0.447900i	0.974601	+	0.223950i	$0.0718952\pi$
$977$			14.0000i			0.447900i	−0.974601	+	0.223950i	$0.928105\pi$
$978$		0			0
$979$	42.4264	−	42.4264i	1.35595	−	1.35595i
$980$		0			0
$981$	−2.82843	−	2.82843i	−0.0903047	−	0.0903047i
$982$		0			0
$983$	−36.0624	−	36.0624i	−1.15021	−	1.15021i	−0.986510	−	0.163704i	$-0.947656\pi$
$983$	−36.0624	−	36.0624i	−1.15021	−	1.15021i	−0.163704	−	0.986510i	$-0.552344\pi$
$984$		0			0
$985$	17.0000			0.541665
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.0919	−	19.0919i	−0.607087	−	0.607087i
$990$		0			0
$991$	14.1421	+	14.1421i	0.449240	+	0.449240i	0.895102	−	0.445862i	$-0.147103\pi$
$991$	14.1421	+	14.1421i	0.449240	+	0.449240i	−0.445862	+	0.895102i	$0.647103\pi$
$992$		0			0
$993$	−7.77817	+	7.77817i	−0.246833	+	0.246833i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.41421	−	1.41421i	0.0447886	−	0.0447886i	−0.684358	−	0.729146i	$-0.739917\pi$
$997$	1.41421	−	1.41421i	0.0447886	−	0.0447886i	0.729146	+	0.684358i	$0.239917\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3468.2.j.f.829.1		4
17.2	even	8		3468.2.a.a.1.1		1
17.4	even	4	inner	3468.2.j.f.3217.1		4
17.8	even	8		3468.2.b.a.577.2		2
17.9	even	8		3468.2.b.a.577.1		2
17.13	even	4	inner	3468.2.j.f.3217.2		4
17.15	even	8		204.2.a.b.1.1	✓	1
17.16	even	2	inner	3468.2.j.f.829.2		4
51.32	odd	8		612.2.a.b.1.1		1
68.15	odd	8		816.2.a.e.1.1		1
85.32	odd	8		5100.2.g.l.2449.1		2
85.49	even	8		5100.2.a.f.1.1		1
85.83	odd	8		5100.2.g.l.2449.2		2
119.83	odd	8		9996.2.a.b.1.1		1
136.83	odd	8		3264.2.a.u.1.1		1
136.117	even	8		3264.2.a.f.1.1		1
204.83	even	8		2448.2.a.e.1.1		1
408.83	even	8		9792.2.a.bn.1.1		1
408.389	odd	8		9792.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
204.2.a.b.1.1	✓	1	17.15	even	8
612.2.a.b.1.1		1	51.32	odd	8
816.2.a.e.1.1		1	68.15	odd	8
2448.2.a.e.1.1		1	204.83	even	8
3264.2.a.f.1.1		1	136.117	even	8
3264.2.a.u.1.1		1	136.83	odd	8
3468.2.a.a.1.1		1	17.2	even	8
3468.2.b.a.577.1		2	17.9	even	8
3468.2.b.a.577.2		2	17.8	even	8
3468.2.j.f.829.1		4	1.1	even	1	trivial
3468.2.j.f.829.2		4	17.16	even	2	inner
3468.2.j.f.3217.1		4	17.4	even	4	inner
3468.2.j.f.3217.2		4	17.13	even	4	inner
5100.2.a.f.1.1		1	85.49	even	8
5100.2.g.l.2449.1		2	85.32	odd	8
5100.2.g.l.2449.2		2	85.83	odd	8
9792.2.a.bn.1.1		1	408.83	even	8
9792.2.a.bo.1.1		1	408.389	odd	8
9996.2.a.b.1.1		1	119.83	odd	8

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:	\( N \)	\(=\)	\( 3468 = 2^{2} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3468.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{9} +(-3.53553 + 3.53553i) q^{11} +5.00000 q^{13} -1.00000i q^{15} -1.00000i q^{19} +(2.12132 - 2.12132i) q^{23} -4.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(1.41421 + 1.41421i) q^{29} +(-1.41421 - 1.41421i) q^{31} +5.00000 q^{33} +(5.65685 + 5.65685i) q^{37} +(-3.53553 - 3.53553i) q^{39} +(-3.53553 + 3.53553i) q^{41} -9.00000i q^{43} +(-0.707107 + 0.707107i) q^{45} -6.00000 q^{47} +7.00000i q^{49} +6.00000i q^{53} -5.00000 q^{55} +(-0.707107 + 0.707107i) q^{57} +6.00000i q^{59} +(-2.82843 + 2.82843i) q^{61} +(3.53553 + 3.53553i) q^{65} +12.0000 q^{67} -3.00000 q^{69} +(8.48528 + 8.48528i) q^{71} +(-1.41421 - 1.41421i) q^{73} +(-2.82843 + 2.82843i) q^{75} +(-7.07107 + 7.07107i) q^{79} -1.00000 q^{81} +2.00000i q^{83} -2.00000i q^{87} -12.0000 q^{89} +2.00000i q^{93} +(0.707107 - 0.707107i) q^{95} +(11.3137 + 11.3137i) q^{97} +(-3.53553 - 3.53553i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 20 q^{13} + 20 q^{33} - 24 q^{47} - 20 q^{55} + 48 q^{67} - 12 q^{69} - 4 q^{81} - 48 q^{89}+O(q^{100}) \) 4 * q + 20 * q^13 + 20 * q^33 - 24 * q^47 - 20 * q^55 + 48 * q^67 - 12 * q^69 - 4 * q^81 - 48 * q^89

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