LMFDB - Embedded newform 684.5.y.e.145.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 2358 x^{12} + 15572 x^{11} + 4050518 x^{10} + 21628620 x^{9} + 2974230644 x^{8} + \cdots + 96\!\cdots\!24 $ x^14 - 2x^13 + 2358x^12 + 15572x^11 + 4050518x^10 + 21628620x^9 + 2974230644x^8 + 5952397856x^7 + 1565987836020x^6 + 2094356466136x^5 + 410050182894688x^4 + 121654699639392x^3 + 75130751452572288x^2 + 84471822440408448*x + 96133502759457024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.6
Root			$9.98915 - 17.3017i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.5.y.e.217.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.9892 + 20.7658i) q^{5} -2.95794 q^{7} +O(q^{10})$	$q+(11.9892 + 20.7658i) q^{5} -2.95794 q^{7} -190.065 q^{11} +(-22.7900 - 13.1578i) q^{13} +(57.0672 + 98.8433i) q^{17} +(-360.990 - 2.68653i) q^{19} +(496.822 - 860.522i) q^{23} +(25.0203 - 43.3365i) q^{25} +(472.196 + 272.622i) q^{29} -512.283i q^{31} +(-35.4632 - 61.4240i) q^{35} -459.578i q^{37} +(2722.46 - 1571.81i) q^{41} +(-551.453 - 955.145i) q^{43} +(-526.277 + 911.539i) q^{47} -2392.25 q^{49} +(2495.00 + 1440.49i) q^{53} +(-2278.72 - 3946.86i) q^{55} +(2794.62 - 1613.47i) q^{59} +(-2894.62 + 5013.63i) q^{61} -631.005i q^{65} +(3584.82 + 2069.70i) q^{67} +(2387.05 - 1378.16i) q^{71} +(3134.42 + 5428.97i) q^{73} +562.200 q^{77} +(-867.616 + 500.918i) q^{79} +1987.92 q^{83} +(-1368.38 + 2370.10i) q^{85} +(1643.57 + 948.915i) q^{89} +(67.4115 + 38.9200i) q^{91} +(-4272.18 - 7528.46i) q^{95} +(2572.37 - 1485.16i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 30 q^{5} + 106 q^{7}+O(q^{10}) $ 14 * q + 30 * q^5 + 106 * q^7	$ 14 q + 30 q^{5} + 106 q^{7} + 264 q^{11} + 57 q^{13} - 282 q^{17} + 2 q^{19} - 96 q^{23} - 465 q^{25} + 630 q^{29} - 1434 q^{35} + 228 q^{41} - 2093 q^{43} + 4710 q^{47} + 4440 q^{49} - 2364 q^{53} + 6368 q^{55} - 11838 q^{59} - 1661 q^{61} - 20319 q^{67} - 624 q^{71} + 5851 q^{73} + 1080 q^{77} - 13299 q^{79} - 12252 q^{83} - 5740 q^{85} + 20010 q^{89} + 15951 q^{91} - 7770 q^{95} + 44904 q^{97}+O(q^{100}) $ 14 * q + 30 * q^5 + 106 * q^7 + 264 * q^11 + 57 * q^13 - 282 * q^17 + 2 * q^19 - 96 * q^23 - 465 * q^25 + 630 * q^29 - 1434 * q^35 + 228 * q^41 - 2093 * q^43 + 4710 * q^47 + 4440 * q^49 - 2364 * q^53 + 6368 * q^55 - 11838 * q^59 - 1661 * q^61 - 20319 * q^67 - 624 * q^71 + 5851 * q^73 + 1080 * q^77 - 13299 * q^79 - 12252 * q^83 - 5740 * q^85 + 20010 * q^89 + 15951 * q^91 - 7770 * q^95 + 44904 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	11.9892	+	20.7658i	0.479566	+	0.830633i	0.999725	−	0.0234364i	$-0.00746071\pi$
$5$	11.9892	+	20.7658i	0.479566	+	0.830633i	−0.520159	+	0.854069i	$0.674127\pi$
$6$		0			0
$7$	−2.95794			−0.0603661			−0.0301830	−	0.999544i	$-0.509609\pi$
$7$	−2.95794			−0.0603661			−0.0301830	+	0.999544i	$0.509609\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−190.065			−1.57078			−0.785392	−	0.618998i	$-0.787539\pi$
$11$	−190.065			−1.57078			−0.785392	+	0.618998i	$0.787539\pi$
$12$		0			0
$13$	−22.7900	−	13.1578i	−0.134852	−	0.0778570i	0.431056	−	0.902325i	$-0.358141\pi$
$13$	−22.7900	−	13.1578i	−0.134852	−	0.0778570i	−0.565908	+	0.824468i	$0.691474\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	57.0672	+	98.8433i	0.197464	+	0.342018i	0.947706	−	0.319146i	$-0.103396\pi$
$17$	57.0672	+	98.8433i	0.197464	+	0.342018i	−0.750241	+	0.661164i	$0.770063\pi$
$18$		0			0
$19$	−360.990	−	2.68653i	−0.999972	−	0.00744190i
$19$	−360.990	−	2.68653i	−0.999972	−	0.00744190i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	496.822	−	860.522i	0.939173	−	1.62669i	0.172154	−	0.985070i	$-0.444927\pi$
$23$	496.822	−	860.522i	0.939173	−	1.62669i	0.767019	−	0.641625i	$-0.221739\pi$
$24$		0			0
$25$	25.0203	−	43.3365i	0.0400325	−	0.0693384i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	472.196	+	272.622i	0.561469	+	0.324164i	0.753735	−	0.657179i	$-0.228250\pi$
$29$	472.196	+	272.622i	0.561469	+	0.324164i	−0.192266	+	0.981343i	$0.561584\pi$
$30$		0			0
$31$		−	512.283i		−	0.533073i	−0.963825	−	0.266537i	$-0.914121\pi$
$31$		−	512.283i		−	0.533073i	0.963825	−	0.266537i	$-0.0858793\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−35.4632	−	61.4240i	−0.0289495	−	0.0501420i
$36$		0			0
$37$		−	459.578i		−	0.335703i	−0.985812	−	0.167852i	$-0.946317\pi$
$37$		−	459.578i		−	0.335703i	0.985812	−	0.167852i	$-0.0536829\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2722.46	−	1571.81i	1.61955	−	0.935048i	0.632514	−	0.774549i	$-0.282023\pi$
$41$	2722.46	−	1571.81i	1.61955	−	0.935048i	0.987036	−	0.160498i	$-0.0513102\pi$
$42$		0			0
$43$	−551.453	−	955.145i	−0.298244	−	0.516574i	0.677490	−	0.735532i	$-0.263068\pi$
$43$	−551.453	−	955.145i	−0.298244	−	0.516574i	−0.975734	+	0.218958i	$0.929734\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−526.277	+	911.539i	−0.238242	+	0.412648i	−0.960210	−	0.279279i	$-0.909905\pi$
$47$	−526.277	+	911.539i	−0.238242	+	0.412648i	0.721968	+	0.691927i	$0.243238\pi$
$48$		0			0
$49$	−2392.25			−0.996356
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2495.00	+	1440.49i	0.888215	+	0.512811i	0.873358	−	0.487078i	$-0.161937\pi$
$53$	2495.00	+	1440.49i	0.888215	+	0.512811i	0.0148570	+	0.999890i	$0.495271\pi$
$54$		0			0
$55$	−2278.72	−	3946.86i	−0.753295	−	1.30475i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2794.62	−	1613.47i	0.802820	−	0.463508i	−0.0416364	−	0.999133i	$-0.513257\pi$
$59$	2794.62	−	1613.47i	0.802820	−	0.463508i	0.844456	+	0.535625i	$0.179924\pi$
$60$		0			0
$61$	−2894.62	+	5013.63i	−0.777915	+	1.34739i	0.155226	+	0.987879i	$0.450389\pi$
$61$	−2894.62	+	5013.63i	−0.777915	+	1.34739i	−0.933141	+	0.359509i	$0.882944\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	631.005i		−	0.149350i
$66$		0			0
$67$	3584.82	+	2069.70i	0.798578	+	0.461059i	0.842974	−	0.537955i	$-0.180803\pi$
$67$	3584.82	+	2069.70i	0.798578	+	0.461059i	−0.0443956	+	0.999014i	$0.514136\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2387.05	−	1378.16i	0.473527	−	0.273391i	−0.244188	−	0.969728i	$-0.578521\pi$
$71$	2387.05	−	1378.16i	0.473527	−	0.273391i	0.717715	+	0.696337i	$0.245188\pi$
$72$		0			0
$73$	3134.42	+	5428.97i	0.588182	+	1.01876i	0.994471	+	0.105016i	$0.0334893\pi$
$73$	3134.42	+	5428.97i	0.588182	+	1.01876i	−0.406289	+	0.913745i	$0.633177\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	562.200			0.0948221
$78$		0			0
$79$	−867.616	+	500.918i	−0.139019	+	0.0802625i	−0.567896	−	0.823100i	$-0.692243\pi$
$79$	−867.616	+	500.918i	−0.139019	+	0.0802625i	0.428878	+	0.903363i	$0.358909\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1987.92			0.288565			0.144282	−	0.989537i	$-0.453913\pi$
$83$	1987.92			0.288565			0.144282	+	0.989537i	$0.453913\pi$
$84$		0			0
$85$	−1368.38	+	2370.10i	−0.189395	+	0.328041i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1643.57	+	948.915i	0.207495	+	0.119797i	0.600147	−	0.799890i	$-0.295109\pi$
$89$	1643.57	+	948.915i	0.207495	+	0.119797i	−0.392652	+	0.919687i	$0.628442\pi$
$90$		0			0
$91$	67.4115	+	38.9200i	0.00814050	+	0.00469992i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4272.18	−	7528.46i	−0.473371	−	0.834179i
$96$		0			0
$97$	2572.37	−	1485.16i	0.273394	−	0.157844i	−0.357035	−	0.934091i	$-0.616212\pi$
$97$	2572.37	−	1485.16i	0.273394	−	0.157844i	0.630429	+	0.776247i	$0.282879\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7158.85	−	12399.5i	0.701779	−	1.21552i	−0.266063	−	0.963956i	$-0.585723\pi$
$101$	7158.85	−	12399.5i	0.701779	−	1.21552i	0.967841	−	0.251561i	$-0.0809439\pi$
$102$		0			0
$103$		−	14617.6i		−	1.37785i	−0.724832	−	0.688925i	$-0.758083\pi$
$103$		−	14617.6i		−	1.37785i	0.724832	−	0.688925i	$-0.241917\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			20437.4i			1.78508i	0.450967	+	0.892540i	$0.351079\pi$
$107$			20437.4i			1.78508i	−0.450967	+	0.892540i	$0.648921\pi$
$108$		0			0
$109$	14602.2	−	8430.57i	1.22904	−	0.709584i	0.262207	−	0.965012i	$-0.415550\pi$
$109$	14602.2	−	8430.57i	1.22904	−	0.709584i	0.966828	+	0.255428i	$0.0822162\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11503.0i		−	0.900857i	−0.892812	−	0.450429i	$-0.851271\pi$
$113$		−	11503.0i		−	0.900857i	0.892812	−	0.450429i	$-0.148729\pi$
$114$		0			0
$115$	23825.9			1.80158
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−168.801	−	292.372i	−0.0119202	−	0.0206463i
$120$		0			0
$121$	21483.7			1.46737
$122$		0			0
$123$		0			0
$124$		0			0
$125$	16186.3			1.03593
$126$		0			0
$127$	−21453.1	−	12385.9i	−1.33009	−	0.767930i	−0.344780	−	0.938683i	$-0.612047\pi$
$127$	−21453.1	−	12385.9i	−1.33009	−	0.767930i	−0.985314	+	0.170753i	$0.945380\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6519.25	−	11291.7i	−0.379888	−	0.657985i	0.611158	−	0.791509i	$-0.290704\pi$
$131$	−6519.25	−	11291.7i	−0.379888	−	0.657985i	−0.991046	+	0.133524i	$0.957371\pi$
$132$		0			0
$133$	1067.79	+	7.94658i	0.0603644	+	0.000449238i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2463.80	−	4267.43i	0.131270	−	0.227366i	−0.792896	−	0.609356i	$-0.791428\pi$
$137$	2463.80	−	4267.43i	0.131270	−	0.227366i	0.924166	+	0.381990i	$0.124761\pi$
$138$		0			0
$139$	−6728.47	+	11654.1i	−0.348247	+	0.603181i	−0.985938	−	0.167111i	$-0.946556\pi$
$139$	−6728.47	+	11654.1i	−0.348247	+	0.603181i	0.637691	+	0.770292i	$0.279890\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4331.59	+	2500.84i	0.211824	+	0.122297i
$144$		0			0
$145$			13074.0i			0.621833i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10183.9	−	17639.0i	−0.458712	−	0.794513i	0.540181	−	0.841549i	$-0.318356\pi$
$149$	−10183.9	−	17639.0i	−0.458712	−	0.794513i	−0.998893	+	0.0470357i	$0.985023\pi$
$150$		0			0
$151$		−	27507.3i		−	1.20641i	−0.797587	−	0.603204i	$-0.793890\pi$
$151$		−	27507.3i		−	1.20641i	0.797587	−	0.603204i	$-0.206110\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	10638.0	−	6141.84i	0.442788	−	0.255644i
$156$		0			0
$157$	−2035.67	−	3525.88i	−0.0825862	−	0.143044i	0.821774	−	0.569814i	$-0.192985\pi$
$157$	−2035.67	−	3525.88i	−0.0825862	−	0.143044i	−0.904360	+	0.426770i	$0.859651\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1469.57	+	2545.37i	−0.0566942	+	0.0981972i
$162$		0			0
$163$	50013.8			1.88241			0.941206	−	0.337833i	$-0.109694\pi$
$163$	50013.8			1.88241			0.941206	+	0.337833i	$0.109694\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−28508.9	−	16459.6i	−1.02223	−	0.590184i	−0.107480	−	0.994207i	$-0.534278\pi$
$167$	−28508.9	−	16459.6i	−1.02223	−	0.590184i	−0.914749	+	0.404024i	$0.867611\pi$
$168$		0			0
$169$	−13934.2	−	24134.8i	−0.487877	−	0.845027i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10626.2	−	6135.02i	0.355046	−	0.204986i	−0.311860	−	0.950128i	$-0.600952\pi$
$173$	10626.2	−	6135.02i	0.355046	−	0.204986i	0.666905	+	0.745142i	$0.267619\pi$
$174$		0			0
$175$	−74.0086	+	128.187i	−0.00241661	+	0.00418569i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	10753.2i		−	0.335607i	−0.985820	−	0.167804i	$-0.946333\pi$
$179$		−	10753.2i		−	0.335607i	0.985820	−	0.167804i	$-0.0536674\pi$
$180$		0			0
$181$	11065.3	+	6388.57i	0.337759	+	0.195005i	0.659281	−	0.751897i	$-0.270861\pi$
$181$	11065.3	+	6388.57i	0.337759	+	0.195005i	−0.321522	+	0.946902i	$0.604194\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9543.51	−	5509.95i	0.278846	−	0.160992i
$186$		0			0
$187$	−10846.5	−	18786.7i	−0.310174	−	0.537237i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	57497.3			1.57609			0.788044	−	0.615619i	$-0.211094\pi$
$191$	57497.3			1.57609			0.788044	+	0.615619i	$0.211094\pi$
$192$		0			0
$193$	9359.51	−	5403.72i	0.251269	−	0.145070i	−0.369076	−	0.929399i	$-0.620326\pi$
$193$	9359.51	−	5403.72i	0.251269	−	0.145070i	0.620345	+	0.784329i	$0.286993\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3066.70			0.0790203			0.0395102	−	0.999219i	$-0.487420\pi$
$197$	3066.70			0.0790203			0.0395102	+	0.999219i	$0.487420\pi$
$198$		0			0
$199$	−2700.89	+	4678.09i	−0.0682027	+	0.118130i	−0.898110	−	0.439771i	$-0.855060\pi$
$199$	−2700.89	+	4678.09i	−0.0682027	+	0.118130i	0.829908	+	0.557901i	$0.188393\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1396.72	−	806.399i	−0.0338937	−	0.0195685i
$204$		0			0
$205$	65280.1	+	37689.5i	1.55336	+	0.896834i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	68611.6	+	510.615i	1.57074	+	0.0116896i
$210$		0			0
$211$	52467.7	−	30292.3i	1.17849	−	0.680404i	0.222829	−	0.974858i	$-0.428471\pi$
$211$	52467.7	−	30292.3i	1.17849	−	0.680404i	0.955666	+	0.294454i	$0.0951376\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13222.9	−	22902.8i	0.286056	−	0.495463i
$216$		0			0
$217$			1515.30i			0.0321795i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3003.52i		−	0.0614959i
$222$		0			0
$223$	−43123.5	+	24897.4i	−0.867171	+	0.500661i	−0.866407	−	0.499338i	$-0.833576\pi$
$223$	−43123.5	+	24897.4i	−0.867171	+	0.500661i	−0.000763916		1.00000i	$0.500243\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			37686.4i			0.731362i	0.930740	+	0.365681i	$0.119164\pi$
$227$			37686.4i			0.731362i	−0.930740	+	0.365681i	$0.880836\pi$
$228$		0			0
$229$	−74757.6			−1.42556			−0.712778	−	0.701390i	$-0.752563\pi$
$229$	−74757.6			−1.42556			−0.712778	+	0.701390i	$0.752563\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13577.2	+	23516.4i	0.250091	+	0.433171i	0.963551	−	0.267526i	$-0.0862061\pi$
$233$	13577.2	+	23516.4i	0.250091	+	0.433171i	−0.713460	+	0.700696i	$0.752873\pi$
$234$		0			0
$235$	−25238.5			−0.457012
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−25577.1			−0.447771			−0.223886	−	0.974615i	$-0.571874\pi$
$239$	−25577.1			−0.447771			−0.223886	+	0.974615i	$0.571874\pi$
$240$		0			0
$241$	−72690.8	−	41968.1i	−1.25154	−	0.722578i	−0.280126	−	0.959963i	$-0.590376\pi$
$241$	−72690.8	−	41968.1i	−1.25154	−	0.722578i	−0.971416	+	0.237385i	$0.923710\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−28681.1	−	49677.1i	−0.477819	−	0.827606i
$246$		0			0
$247$	8191.62	+	4811.07i	0.134269	+	0.0788584i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	32484.1	−	56264.0i	0.515612	−	0.893066i	−0.484224	−	0.874944i	$-0.660898\pi$
$251$	32484.1	−	56264.0i	0.515612	−	0.893066i	0.999836	−	0.0181218i	$-0.00576867\pi$
$252$		0			0
$253$	−94428.5	+	163555.i	−1.47524	+	2.55519i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	66055.3	+	38137.0i	1.00009	+	0.577405i	0.908276	−	0.418371i	$-0.137399\pi$
$257$	66055.3	+	38137.0i	1.00009	+	0.577405i	0.0918186	+	0.995776i	$0.470732\pi$
$258$		0			0
$259$			1359.40i			0.0202651i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	52563.6	+	91042.8i	0.759930	+	1.31624i	0.942886	+	0.333116i	$0.108100\pi$
$263$	52563.6	+	91042.8i	0.759930	+	1.31624i	−0.182956	+	0.983121i	$0.558567\pi$
$264$		0			0
$265$			69080.9i			0.983708i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	57879.3	−	33416.6i	0.799869	−	0.461804i	−0.0435566	−	0.999051i	$-0.513869\pi$
$269$	57879.3	−	33416.6i	0.799869	−	0.461804i	0.843425	+	0.537247i	$0.180536\pi$
$270$		0			0
$271$	−57130.9	−	98953.6i	−0.777916	−	1.34739i	−0.933141	−	0.359511i	$-0.882944\pi$
$271$	−57130.9	−	98953.6i	−0.777916	−	1.34739i	0.155225	−	0.987879i	$-0.450390\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4755.49	+	8236.75i	−0.0628825	+	0.108916i
$276$		0			0
$277$	13914.8			0.181350			0.0906752	−	0.995881i	$-0.471097\pi$
$277$	13914.8			0.181350			0.0906752	+	0.995881i	$0.471097\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18142.1	−	10474.3i	−0.229760	−	0.132652i	0.380701	−	0.924698i	$-0.375683\pi$
$281$	−18142.1	−	10474.3i	−0.229760	−	0.132652i	−0.610461	+	0.792046i	$0.709016\pi$
$282$		0			0
$283$	−9109.27	−	15777.7i	−0.113739	−	0.197002i	0.803536	−	0.595256i	$-0.202950\pi$
$283$	−9109.27	−	15777.7i	−0.113739	−	0.197002i	−0.917275	+	0.398254i	$0.869616\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8052.87	+	4649.33i	−0.0977658	+	0.0564451i
$288$		0			0
$289$	35247.2	−	61049.9i	0.422016	−	0.730952i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			7841.64i			0.0913422i	0.998957	+	0.0456711i	$0.0145426\pi$
$293$			7841.64i			0.0913422i	−0.998957	+	0.0456711i	$0.985457\pi$
$294$		0			0
$295$	67010.2	+	38688.3i	0.770010	+	0.444566i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−22645.2	+	13074.2i	−0.253299	+	0.146242i
$300$		0			0
$301$	1631.16	+	2825.26i	0.0180038	+	0.0311835i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−138816.			−1.49225
$306$		0			0
$307$	96956.9	−	55978.1i	1.02873	−	0.593938i	0.112110	−	0.993696i	$-0.464239\pi$
$307$	96956.9	−	55978.1i	1.02873	−	0.593938i	0.916621	+	0.399758i	$0.130906\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−66063.3			−0.683029			−0.341515	−	0.939876i	$-0.610940\pi$
$311$	−66063.3			−0.683029			−0.341515	+	0.939876i	$0.610940\pi$
$312$		0			0
$313$	−15394.0	+	26663.2i	−0.157131	+	0.272159i	−0.933833	−	0.357709i	$-0.883558\pi$
$313$	−15394.0	+	26663.2i	−0.157131	+	0.272159i	0.776702	+	0.629869i	$0.216891\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7299.00	−	4214.08i	−0.0726348	−	0.0419357i	0.463243	−	0.886231i	$-0.346686\pi$
$317$	−7299.00	−	4214.08i	−0.0726348	−	0.0419357i	−0.535878	+	0.844296i	$0.680019\pi$
$318$		0			0
$319$	−89747.8	−	51815.9i	−0.881947	−	0.509192i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20335.2	−	35834.8i	−0.194914	−	0.343479i
$324$		0			0
$325$	−1140.43	+	658.427i	−0.0107970	+	0.00623363i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1556.70	−	2696.28i	0.0143818	−	0.0249099i
$330$		0			0
$331$		−	98871.8i		−	0.902436i	−0.892414	−	0.451218i	$-0.850990\pi$
$331$		−	98871.8i		−	0.902436i	0.892414	−	0.451218i	$-0.149010\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			99255.6i			0.884434i
$336$		0			0
$337$	−24617.7	+	14213.1i	−0.216765	+	0.125149i	−0.604451	−	0.796642i	$-0.706608\pi$
$337$	−24617.7	+	14213.1i	−0.216765	+	0.125149i	0.387687	+	0.921791i	$0.373274\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			97367.1i			0.837343i
$342$		0			0
$343$	14178.1			0.120512
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−79485.8	−	137673.i	−0.660132	−	1.14338i	−0.980581	−	0.196116i	$-0.937167\pi$
$347$	−79485.8	−	137673.i	−0.660132	−	1.14338i	0.320449	−	0.947266i	$-0.396166\pi$
$348$		0			0
$349$	−10494.7			−0.0861624			−0.0430812	−	0.999072i	$-0.513717\pi$
$349$	−10494.7			−0.0861624			−0.0430812	+	0.999072i	$0.513717\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−98601.5			−0.791287			−0.395644	−	0.918404i	$-0.629478\pi$
$353$	−98601.5			−0.791287			−0.395644	+	0.918404i	$0.629478\pi$
$354$		0			0
$355$	57237.4	+	33046.0i	0.454175	+	0.262218i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−66427.6	−	115056.i	−0.515418	−	0.892731i	−0.999840	−	0.0178959i	$-0.994303\pi$
$359$	−66427.6	−	115056.i	−0.515418	−	0.892731i	0.484422	−	0.874835i	$-0.339030\pi$
$360$		0			0
$361$	130307.	+	1939.62i	0.999889	+	0.0148834i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−75158.1	+	130178.i	−0.564144	+	0.977126i
$366$		0			0
$367$	−2509.04	+	4345.79i	−0.0186284	+	0.0322654i	−0.875189	−	0.483781i	$-0.839263\pi$
$367$	−2509.04	+	4345.79i	−0.0186284	+	0.0322654i	0.856561	+	0.516046i	$0.172597\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7380.04	−	4260.87i	−0.0536181	−	0.0309564i
$372$		0			0
$373$			6806.82i			0.0489245i	0.999701	+	0.0244623i	$0.00778736\pi$
$373$			6806.82i			0.0489245i	−0.999701	+	0.0244623i	$0.992213\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7174.23	−	12426.1i	−0.0504769	−	0.0874286i
$378$		0			0
$379$			209091.i			1.45565i	0.685762	+	0.727825i	$0.259469\pi$
$379$			209091.i			1.45565i	−0.685762	+	0.727825i	$0.740531\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−60283.4	+	34804.6i	−0.410961	+	0.237268i	−0.691202	−	0.722661i	$-0.742919\pi$
$383$	−60283.4	+	34804.6i	−0.410961	+	0.237268i	0.280242	+	0.959929i	$0.409585\pi$
$384$		0			0
$385$	6740.30	+	11674.6i	0.0454735	+	0.0787624i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−94417.4	+	163536.i	−0.623955	+	1.08072i	0.364787	+	0.931091i	$0.381142\pi$
$389$	−94417.4	+	163536.i	−0.623955	+	1.08072i	−0.988742	+	0.149630i	$0.952192\pi$
$390$		0			0
$391$	113409.			0.741813
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20804.0	−	12011.2i	−0.133337	−	0.0769824i
$396$		0			0
$397$	−73958.9	−	128101.i	−0.469256	−	0.812775i	0.530127	−	0.847918i	$-0.322144\pi$
$397$	−73958.9	−	128101.i	−0.469256	−	0.812775i	−0.999382	+	0.0351439i	$0.988811\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−77094.6	+	44510.6i	−0.479441	+	0.276805i	−0.720183	−	0.693784i	$-0.755942\pi$
$401$	−77094.6	+	44510.6i	−0.479441	+	0.276805i	0.240743	+	0.970589i	$0.422609\pi$
$402$		0			0
$403$	−6740.53	+	11674.9i	−0.0415035	+	0.0718861i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			87349.6i			0.527317i
$408$		0			0
$409$	−93581.9	−	54029.6i	−0.559430	−	0.322987i	0.193487	−	0.981103i	$-0.438020\pi$
$409$	−93581.9	−	54029.6i	−0.559430	−	0.322987i	−0.752917	+	0.658116i	$0.771354\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8266.30	+	4772.55i	−0.0484631	+	0.0279802i
$414$		0			0
$415$	23833.5	+	41280.9i	0.138386	+	0.239691i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−225839.			−1.28638			−0.643191	−	0.765706i	$-0.722390\pi$
$419$	−225839.			−1.28638			−0.643191	+	0.765706i	$0.722390\pi$
$420$		0			0
$421$	−266565.	+	153901.i	−1.50397	+	0.868316i	−0.503978	+	0.863717i	$0.668131\pi$
$421$	−266565.	+	153901.i	−1.50397	+	0.868316i	−0.999989	+	0.00459947i	$0.998536\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5711.37			0.0316200
$426$		0			0
$427$	8562.11	−	14830.0i	0.0469597	−	0.0813365i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9580.59	+	5531.36i	0.0515748	+	0.0297767i	0.525566	−	0.850753i	$-0.323854\pi$
$431$	9580.59	+	5531.36i	0.0515748	+	0.0297767i	−0.473991	+	0.880530i	$0.657187\pi$
$432$		0			0
$433$	61894.5	+	35734.8i	0.330124	+	0.190597i	0.655896	−	0.754851i	$-0.272291\pi$
$433$	61894.5	+	35734.8i	0.330124	+	0.190597i	−0.325772	+	0.945448i	$0.605624\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−181660.	+	309305.i	−0.951252	+	1.61966i
$438$		0			0
$439$	163282.	−	94270.9i	0.847245	−	0.489157i	−0.0124752	−	0.999922i	$-0.503971\pi$
$439$	163282.	−	94270.9i	0.847245	−	0.489157i	0.859720	+	0.510765i	$0.170638\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1593.29	+	2759.66i	−0.00811871	+	0.0140620i	−0.870056	−	0.492953i	$-0.835918\pi$
$443$	−1593.29	+	2759.66i	−0.00811871	+	0.0140620i	0.861938	+	0.507015i	$0.169251\pi$
$444$		0			0
$445$			45506.8i			0.229803i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			232278.i			1.15217i	0.817391	+	0.576084i	$0.195420\pi$
$449$			232278.i			1.15217i	−0.817391	+	0.576084i	$0.804580\pi$
$450$		0			0
$451$	−517445.	+	298747.i	−2.54396	+	1.46876i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1866.47i			0.00901569i
$456$		0			0
$457$	283411.			1.35701			0.678507	−	0.734594i	$-0.262627\pi$
$457$	283411.			1.35701			0.678507	+	0.734594i	$0.262627\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−49277.8	−	85351.6i	−0.231873	−	0.401615i	0.726487	−	0.687181i	$-0.241152\pi$
$461$	−49277.8	−	85351.6i	−0.231873	−	0.401615i	−0.958359	+	0.285566i	$0.907819\pi$
$462$		0			0
$463$	−7906.98			−0.0368849			−0.0184424	−	0.999830i	$-0.505871\pi$
$463$	−7906.98			−0.0368849			−0.0184424	+	0.999830i	$0.505871\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−246785.			−1.13158			−0.565790	−	0.824550i	$-0.691429\pi$
$467$	−246785.			−1.13158			−0.565790	+	0.824550i	$0.691429\pi$
$468$		0			0
$469$	−10603.7	−	6122.03i	−0.0482070	−	0.0278323i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	104812.	+	181540.i	0.468477	+	0.811427i
$474$		0			0
$475$	−9148.52	+	15576.8i	−0.0405474	+	0.0690386i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	127554.	−	220930.i	0.555933	−	0.962904i	−0.441897	−	0.897066i	$-0.645694\pi$
$479$	127554.	−	220930.i	0.555933	−	0.962904i	0.997830	−	0.0658386i	$-0.0209722\pi$
$480$		0			0
$481$	−6047.04	+	10473.8i	−0.0261368	+	0.0452703i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	61681.0	+	35611.5i	0.262221	+	0.151393i
$486$		0			0
$487$		−	143792.i		−	0.606287i	−0.952945	−	0.303143i	$-0.901964\pi$
$487$		−	143792.i		−	0.606287i	0.952945	−	0.303143i	$-0.0980361\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−42706.5	−	73969.9i	−0.177146	−	0.306826i	0.763756	−	0.645505i	$-0.223353\pi$
$491$	−42706.5	−	73969.9i	−0.177146	−	0.306826i	−0.940902	+	0.338679i	$0.890020\pi$
$492$		0			0
$493$			62231.2i			0.256044i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7060.74	+	4076.52i	−0.0285850	+	0.0165035i
$498$		0			0
$499$	241908.	+	418996.i	0.971512	+	1.68271i	0.690994	+	0.722860i	$0.257173\pi$
$499$	241908.	+	418996.i	0.971512	+	1.68271i	0.280518	+	0.959849i	$0.409494\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	46410.2	−	80384.8i	0.183433	−	0.317715i	−0.759614	−	0.650374i	$-0.774612\pi$
$503$	46410.2	−	80384.8i	0.183433	−	0.317715i	0.943047	+	0.332658i	$0.107946\pi$
$504$		0			0
$505$	343314.			1.34620
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−65210.7	−	37649.4i	−0.251700	−	0.145319i	0.368842	−	0.929492i	$-0.379754\pi$
$509$	−65210.7	−	37649.4i	−0.251700	−	0.145319i	−0.620542	+	0.784173i	$0.713088\pi$
$510$		0			0
$511$	−9271.42	−	16058.6i	−0.0355062	−	0.0614985i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	303547.	−	175253.i	1.14449	−	0.660771i
$516$		0			0
$517$	100027.	−	173252.i	0.374228	−	0.648181i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	55779.7i		−	0.205495i	−0.994707	−	0.102747i	$-0.967237\pi$
$521$		−	55779.7i		−	0.205495i	0.994707	−	0.102747i	$-0.0327633\pi$
$522$		0			0
$523$	20549.7	+	11864.4i	0.0751281	+	0.0433752i	0.537093	−	0.843523i	$-0.319522\pi$
$523$	20549.7	+	11864.4i	0.0751281	+	0.0433752i	−0.461965	+	0.886898i	$0.652856\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	50635.8	−	29234.6i	0.182321	−	0.105263i
$528$		0			0
$529$	−353744.	−	612703.i	−1.26409	−	2.18947i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−82726.7			−0.291200
$534$		0			0
$535$	−424399.	+	245027.i	−1.48275	+	0.856064i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	454683.			1.56506
$540$		0			0
$541$	−192553.	+	333512.i	−0.657895	+	1.13951i	0.323264	+	0.946309i	$0.395220\pi$
$541$	−192553.	+	333512.i	−0.657895	+	1.13951i	−0.981159	+	0.193200i	$0.938114\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	350135.	+	202151.i	1.17881	+	0.680585i
$546$		0			0
$547$	−467847.	−	270111.i	−1.56361	−	0.902751i	−0.996887	−	0.0788437i	$-0.974877\pi$
$547$	−467847.	−	270111.i	−1.56361	−	0.902751i	−0.566724	−	0.823908i	$-0.691789\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−169725.	−	99682.5i	−0.559041	−	0.328334i
$552$		0			0
$553$	2566.35	−	1481.68i	0.00839201	−	0.00484513i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−119750.	+	207414.i	−0.385981	+	0.668539i	−0.991905	−	0.126984i	$-0.959470\pi$
$557$	−119750.	+	207414.i	−0.385981	+	0.668539i	0.605924	+	0.795523i	$0.292804\pi$
$558$		0			0
$559$			29023.7i			0.0928815i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	580908.i		−	1.83270i	−0.400382	−	0.916348i	$-0.631122\pi$
$563$		−	580908.i		−	1.83270i	0.400382	−	0.916348i	$-0.368878\pi$
$564$		0			0
$565$	238870.	−	137912.i	0.748282	−	0.432021i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			207938.i			0.642259i	0.947035	+	0.321130i	$0.104062\pi$
$569$			207938.i			0.642259i	−0.947035	+	0.321130i	$0.895938\pi$
$570$		0			0
$571$	−232795.			−0.714005			−0.357003	−	0.934103i	$-0.616201\pi$
$571$	−232795.			−0.714005			−0.357003	+	0.934103i	$0.616201\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24861.3	−	43061.1i	−0.0751949	−	0.130241i
$576$		0			0
$577$	115367.			0.346522			0.173261	−	0.984876i	$-0.444570\pi$
$577$	115367.			0.346522			0.173261	+	0.984876i	$0.444570\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5880.15			−0.0174195
$582$		0			0
$583$	−474211.	−	273786.i	−1.39520	−	0.805516i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−193520.	−	335186.i	−0.561629	−	0.972770i	−0.997355	−	0.0726906i	$-0.976841\pi$
$587$	−193520.	−	335186.i	−0.561629	−	0.972770i	0.435725	−	0.900080i	$-0.356492\pi$
$588$		0			0
$589$	−1376.26	+	184929.i	−0.00396708	+	0.533058i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−243339.	+	421475.i	−0.691993	+	1.19857i	0.279191	+	0.960236i	$0.409934\pi$
$593$	−243339.	+	421475.i	−0.691993	+	1.19857i	−0.971184	+	0.238331i	$0.923400\pi$
$594$		0			0
$595$	4047.57	−	7010.60i	0.0114330	−	0.0198025i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	422678.	+	244033.i	1.17803	+	0.680135i	0.955558	−	0.294804i	$-0.0952543\pi$
$599$	422678.	+	244033.i	1.17803	+	0.680135i	0.222471	+	0.974939i	$0.428588\pi$
$600$		0			0
$601$			86130.8i			0.238457i	0.992867	+	0.119228i	$0.0380421\pi$
$601$			86130.8i			0.238457i	−0.992867	+	0.119228i	$0.961958\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	257571.	+	446127.i	0.703699	+	1.21884i
$606$		0			0
$607$		−	676987.i		−	1.83740i	−0.394960	−	0.918698i	$-0.629241\pi$
$607$		−	676987.i		−	1.83740i	0.394960	−	0.918698i	$-0.370759\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	23987.8	−	13849.3i	0.0642550	−	0.0370977i
$612$		0			0
$613$	184792.	+	320069.i	0.491771	+	0.851772i	0.999955	−	0.00947643i	$-0.00301649\pi$
$613$	184792.	+	320069.i	0.491771	+	0.851772i	−0.508184	+	0.861248i	$0.669683\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	180024.	−	311812.i	0.472891	−	0.819072i	−0.526627	−	0.850096i	$-0.676544\pi$
$617$	180024.	−	311812.i	0.472891	−	0.819072i	0.999519	+	0.0310246i	$0.00987703\pi$
$618$		0			0
$619$	166422.			0.434340			0.217170	−	0.976134i	$-0.430317\pi$
$619$	166422.			0.434340			0.217170	+	0.976134i	$0.430317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4861.57	−	2806.83i	−0.0125257	−	0.00723170i
$624$		0			0
$625$	178423.	+	309037.i	0.456762	+	0.791135i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	45426.2	−	26226.8i	0.114817	−	0.0662894i
$630$		0			0
$631$	51126.7	−	88554.0i	0.128407	−	0.222408i	−0.794653	−	0.607065i	$-0.792347\pi$
$631$	51126.7	−	88554.0i	0.128407	−	0.222408i	0.923060	+	0.384657i	$0.125680\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	593988.i		−	1.47309i
$636$		0			0
$637$	54519.5	+	31476.8i	0.134361	+	0.0775733i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	225748.	−	130335.i	0.549423	−	0.317210i	−0.199466	−	0.979905i	$-0.563921\pi$
$641$	225748.	−	130335.i	0.549423	−	0.317210i	0.748889	+	0.662695i	$0.230587\pi$
$642$		0			0
$643$	−265841.	−	460451.i	−0.642985	−	1.11368i	−0.984763	−	0.173901i	$-0.944363\pi$
$643$	−265841.	−	460451.i	−0.642985	−	1.11368i	0.341779	−	0.939781i	$-0.388971\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	202465.			0.483662			0.241831	−	0.970318i	$-0.422252\pi$
$647$	202465.			0.483662			0.241831	+	0.970318i	$0.422252\pi$
$648$		0			0
$649$	−531159.	+	306665.i	−1.26106	+	0.728072i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	665316.			1.56028			0.780139	−	0.625607i	$-0.215149\pi$
$653$	665316.			1.56028			0.780139	+	0.625607i	$0.215149\pi$
$654$		0			0
$655$	156321.	−	270755.i	0.364362	−	0.631094i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−525310.	−	303288.i	−1.20961	−	0.698368i	−0.246935	−	0.969032i	$-0.579423\pi$
$659$	−525310.	−	303288.i	−1.20961	−	0.698368i	−0.962674	+	0.270664i	$0.912757\pi$
$660$		0			0
$661$	45772.3	+	26426.7i	0.104761	+	0.0604838i	0.551465	−	0.834198i	$-0.314069\pi$
$661$	45772.3	+	26426.7i	0.104761	+	0.0604838i	−0.446704	+	0.894682i	$0.647402\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12636.8	+	22268.7i	0.0285756	+	0.0503561i
$666$		0			0
$667$	469195.	−	270890.i	1.05463	−	0.608893i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	550166.	−	952916.i	1.22194	−	2.11646i
$672$		0			0
$673$			246375.i			0.543960i	0.962303	+	0.271980i	$0.0876785\pi$
$673$			246375.i			0.543960i	−0.962303	+	0.271980i	$0.912321\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			631064.i			1.37688i	0.725293	+	0.688440i	$0.241704\pi$
$677$			631064.i			1.37688i	−0.725293	+	0.688440i	$0.758296\pi$
$678$		0			0
$679$	−7608.90	+	4393.00i	−0.0165037	+	0.00952843i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			201601.i			0.432168i	0.976375	+	0.216084i	$0.0693285\pi$
$683$			201601.i			0.432168i	−0.976375	+	0.216084i	$0.930672\pi$
$684$		0			0
$685$	118156.			0.251810
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−37907.4	−	65657.5i	−0.0798519	−	0.138307i
$690$		0			0
$691$	860170.			1.80148			0.900738	−	0.434363i	$-0.143027\pi$
$691$	860170.			1.80148			0.900738	+	0.434363i	$0.143027\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−322675.			−0.668029
$696$		0			0
$697$	310727.	+	179398.i	0.639607	+	0.369277i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−131501.	−	227766.i	−0.267603	−	0.463503i	0.700639	−	0.713516i	$-0.252898\pi$
$701$	−131501.	−	227766.i	−0.267603	−	0.463503i	−0.968242	+	0.250013i	$0.919565\pi$
$702$		0			0
$703$	−1234.67	+	165903.i	−0.00249827	+	0.335694i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21175.4	+	36676.9i	−0.0423636	+	0.0733759i
$708$		0			0
$709$	389759.	−	675082.i	0.775360	−	1.34296i	−0.159233	−	0.987241i	$-0.550902\pi$
$709$	389759.	−	675082.i	0.775360	−	1.34296i	0.934592	−	0.355721i	$-0.115765\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−440831.	−	254514.i	−0.867147	−	0.500648i
$714$		0			0
$715$			119932.i			0.234597i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	228362.	+	395534.i	0.441739	+	0.765114i	0.997819	−	0.0660151i	$-0.0210285\pi$
$719$	228362.	+	395534.i	0.441739	+	0.765114i	−0.556080	+	0.831129i	$0.687695\pi$
$720$		0			0
$721$			43238.0i			0.0831754i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	23629.0	−	13642.2i	0.0449541	−	0.0259542i
$726$		0			0
$727$	66442.3	+	115081.i	0.125712	+	0.217739i	0.922011	−	0.387164i	$-0.126545\pi$
$727$	66442.3	+	115081.i	0.125712	+	0.217739i	−0.796299	+	0.604903i	$0.793212\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	62939.8	−	109015.i	0.117785	−	0.204010i
$732$		0			0
$733$	775961.			1.44422			0.722108	−	0.691780i	$-0.243173\pi$
$733$	775961.			1.44422			0.722108	+	0.691780i	$0.243173\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−681348.	−	393377.i	−1.25439	−	0.724225i
$738$		0			0
$739$	367561.	+	636635.i	0.673040	+	1.16574i	0.977038	+	0.213067i	$0.0683452\pi$
$739$	367561.	+	636635.i	0.673040	+	1.16574i	−0.303997	+	0.952673i	$0.598321\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−303308.	+	175115.i	−0.549423	+	0.317209i	−0.748889	−	0.662695i	$-0.769412\pi$
$743$	−303308.	+	175115.i	−0.549423	+	0.317209i	0.199467	+	0.979905i	$0.436079\pi$
$744$		0			0
$745$	244192.	−	422953.i	0.439966	−	0.762043i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	60452.5i		−	0.107758i
$750$		0			0
$751$	220009.	+	127022.i	0.390086	+	0.225216i	0.682198	−	0.731168i	$-0.261024\pi$
$751$	220009.	+	127022.i	0.390086	+	0.225216i	−0.292111	+	0.956384i	$0.594358\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	571212.	−	329789.i	1.00208	−	0.578553i
$756$		0			0
$757$	−187464.	−	324697.i	−0.327134	−	0.566612i	0.654808	−	0.755795i	$-0.272749\pi$
$757$	−187464.	−	324697.i	−0.327134	−	0.566612i	−0.981942	+	0.189183i	$0.939416\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	748073.			1.29174			0.645869	−	0.763448i	$-0.276495\pi$
$761$	748073.			1.29174			0.645869	+	0.763448i	$0.276495\pi$
$762$		0			0
$763$	−43192.3	+	24937.1i	−0.0741920	+	0.0428348i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−84919.1			−0.144349
$768$		0			0
$769$	152987.	−	264982.i	0.258704	−	0.448088i	−0.707191	−	0.707022i	$-0.750038\pi$
$769$	152987.	−	264982.i	0.258704	−	0.448088i	0.965895	+	0.258934i	$0.0833713\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	634560.	+	366363.i	1.06197	+	0.613131i	0.925977	−	0.377579i	$-0.123243\pi$
$773$	634560.	+	366363.i	1.06197	+	0.613131i	0.135996	+	0.990709i	$0.456577\pi$
$774$		0			0
$775$	−22200.6	−	12817.5i	−0.0369624	−	0.0213403i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−987005.	+	560096.i	−1.62646	+	0.922969i
$780$		0			0
$781$	−453695.	+	261941.i	−0.743809	+	0.429438i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	48811.9	−	84544.6i	0.0792111	−	0.137198i
$786$		0			0
$787$			521333.i			0.841716i	0.907127	+	0.420858i	$0.138271\pi$
$787$			521333.i			0.841716i	−0.907127	+	0.420858i	$0.861729\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			34025.3i			0.0543812i
$792$		0			0
$793$	131937.	−	76173.9i	0.209807	−	0.121132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			920317.i			1.44884i	0.689358	+	0.724421i	$0.257893\pi$
$797$			920317.i			1.44884i	−0.689358	+	0.724421i	$0.742107\pi$
$798$		0			0
$799$	−120133.			−0.188178
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−595743.	−	1.03186e6i	−0.923907	−	1.60025i
$804$		0			0
$805$	−70475.6			−0.108754
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−234113.			−0.357708			−0.178854	−	0.983876i	$-0.557239\pi$
$809$	−234113.			−0.357708			−0.178854	+	0.983876i	$0.557239\pi$
$810$		0			0
$811$	374589.	+	216269.i	0.569526	+	0.328816i	0.756960	−	0.653461i	$-0.226684\pi$
$811$	374589.	+	216269.i	0.569526	+	0.328816i	−0.187434	+	0.982277i	$0.560017\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	599623.	+	1.03858e6i	0.902741	+	1.56359i
$816$		0			0
$817$	196503.	+	346279.i	0.294392	+	0.518779i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	214949.	−	372302.i	0.318895	−	0.552343i	−0.661362	−	0.750066i	$-0.730021\pi$
$821$	214949.	−	372302.i	0.318895	−	0.552343i	0.980258	+	0.197723i	$0.0633548\pi$
$822$		0			0
$823$	−150965.	+	261479.i	−0.222883	+	0.386044i	−0.955682	−	0.294401i	$-0.904880\pi$
$823$	−150965.	+	261479.i	−0.222883	+	0.386044i	0.732799	+	0.680445i	$0.238213\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−360514.	−	208143.i	−0.527122	−	0.304334i	0.212722	−	0.977113i	$-0.431767\pi$
$827$	−360514.	−	208143.i	−0.527122	−	0.304334i	−0.739844	+	0.672779i	$0.765101\pi$
$828$		0			0
$829$			111236.i			0.161859i	0.996720	+	0.0809296i	$0.0257889\pi$
$829$			111236.i			0.161859i	−0.996720	+	0.0809296i	$0.974211\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−136519.	−	236458.i	−0.196745	−	0.340772i
$834$		0			0
$835$		−	789348.i		−	1.13213i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	890595.	−	514185.i	1.26519	−	0.730459i	0.291117	−	0.956687i	$-0.405973\pi$
$839$	890595.	−	514185.i	1.26519	−	0.730459i	0.974074	+	0.226229i	$0.0726397\pi$
$840$		0			0
$841$	−204995.	−	355061.i	−0.289835	−	0.502009i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	334120.	−	578712.i	0.467938	−	0.810493i
$846$		0			0
$847$	−63547.4			−0.0885790
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−395476.	−	228328.i	−0.546087	−	0.315283i
$852$		0			0
$853$	−540959.	−	936969.i	−0.743476	−	1.28774i	−0.950904	−	0.309487i	$-0.899843\pi$
$853$	−540959.	−	936969.i	−0.743476	−	1.28774i	0.207428	−	0.978250i	$-0.433491\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	730667.	−	421851.i	0.994850	−	0.574377i	0.0881296	−	0.996109i	$-0.471911\pi$
$857$	730667.	−	421851.i	0.994850	−	0.574377i	0.906721	+	0.421732i	$0.138578\pi$
$858$		0			0
$859$	499488.	−	865138.i	0.676922	−	1.17246i	−0.298981	−	0.954259i	$-0.596647\pi$
$859$	499488.	−	865138.i	0.676922	−	1.17246i	0.975903	−	0.218204i	$-0.0700199\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.00727e6i		−	1.35246i	−0.736690	−	0.676231i	$-0.763612\pi$
$863$		−	1.00727e6i		−	1.35246i	0.736690	−	0.676231i	$-0.236388\pi$
$864$		0			0
$865$	254797.	+	147107.i	0.340536	+	0.196608i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	164903.	−	95207.0i	0.218369	−	0.126075i
$870$		0			0
$871$	−54465.4	−	94336.8i	−0.0717934	−	0.124350i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−47878.2			−0.0625347
$876$		0			0
$877$	−1.08083e6	+	624020.i	−1.40527	+	0.811333i	−0.994927	−	0.100598i	$-0.967924\pi$
$877$	−1.08083e6	+	624020.i	−1.40527	+	0.811333i	−0.410343	+	0.911931i	$0.634591\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−500467.			−0.644798			−0.322399	−	0.946604i	$-0.604489\pi$
$881$	−500467.			−0.644798			−0.322399	+	0.946604i	$0.604489\pi$
$882$		0			0
$883$	317677.	−	550233.i	0.407441	−	0.705709i	−0.587161	−	0.809470i	$-0.699754\pi$
$883$	317677.	−	550233.i	0.407441	−	0.705709i	0.994602	+	0.103761i	$0.0330878\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.24371e6	+	718059.i	1.58079	+	0.912668i	0.994745	+	0.102387i	$0.0326480\pi$
$887$	1.24371e6	+	718059.i	1.58079	+	0.912668i	0.586042	+	0.810281i	$0.300685\pi$
$888$		0			0
$889$	63456.9	+	36636.8i	0.0802925	+	0.0463569i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	192430.	−	327643.i	0.241307	−	0.410864i
$894$		0			0
$895$	223299.	−	128922.i	0.278766	−	0.160946i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	139660.	−	241898.i	0.172803	−	0.299304i
$900$		0			0
$901$			328818.i			0.405048i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			306374.i			0.374072i
$906$		0			0
$907$	−724008.	+	418006.i	−0.880093	+	0.508122i	−0.870689	−	0.491834i	$-0.836327\pi$
$907$	−724008.	+	418006.i	−0.880093	+	0.508122i	−0.00940399	+	0.999956i	$0.502993\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1.36684e6i			1.64695i	0.567353	+	0.823475i	$0.307967\pi$
$911$			1.36684e6i			1.64695i	−0.567353	+	0.823475i	$0.692033\pi$
$912$		0			0
$913$	−377834.			−0.453273
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19283.5	+	33400.1i	0.0229323	+	0.0397199i
$918$		0			0
$919$	−19731.1			−0.0233625			−0.0116813	−	0.999932i	$-0.503718\pi$
$919$	−19731.1			−0.0233625			−0.0116813	+	0.999932i	$0.503718\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−72534.6			−0.0851416
$924$		0			0
$925$	−19916.5	−	11498.8i	−0.0232771	−	0.0134391i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−348332.	−	603329.i	−0.403610	−	0.699074i	0.590548	−	0.807002i	$-0.298912\pi$
$929$	−348332.	−	603329.i	−0.403610	−	0.699074i	−0.994159	+	0.107929i	$0.965578\pi$
$930$		0			0
$931$	863579.	+	6426.85i	0.996328	+	0.00741478i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	260080.	−	450472.i	0.297498	−	0.515282i
$936$		0			0
$937$	−586870.	+	1.01649e6i	−0.668441	+	1.15777i	0.309899	+	0.950769i	$0.399705\pi$
$937$	−586870.	+	1.01649e6i	−0.668441	+	1.15777i	−0.978340	+	0.207004i	$0.933629\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−451586.	−	260723.i	−0.509990	−	0.294443i	0.222840	−	0.974855i	$-0.428467\pi$
$941$	−451586.	−	260723.i	−0.509990	−	0.294443i	−0.732829	+	0.680412i	$0.761801\pi$
$942$		0			0
$943$		−	3.12365e6i		−	3.51268i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	436087.	+	755324.i	0.486265	+	0.842235i	0.999875	−	0.0157882i	$-0.00502575\pi$
$947$	436087.	+	755324.i	0.486265	+	0.842235i	−0.513611	+	0.858023i	$0.671692\pi$
$948$		0			0
$949$		−	164969.i		−	0.183176i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	425119.	−	245442.i	0.468085	−	0.270249i	−0.247353	−	0.968925i	$-0.579561\pi$
$953$	425119.	−	245442.i	0.468085	−	0.270249i	0.715438	+	0.698677i	$0.246227\pi$
$954$		0			0
$955$	689344.	+	1.19398e6i	0.755839	+	1.30915i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7287.78	+	12622.8i	−0.00792424	+	0.0137252i
$960$		0			0
$961$	661087.			0.715833
$962$		0			0
$963$		0			0
$964$		0			0
$965$	224425.	+	129572.i	0.241000	+	0.139141i
$966$		0			0
$967$	−17655.4	−	30580.1i	−0.0188810	−	0.0327028i	0.856431	−	0.516262i	$-0.172677\pi$
$967$	−17655.4	−	30580.1i	−0.0188810	−	0.0327028i	−0.875312	+	0.483559i	$0.839344\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.27208e6	−	734436.i	1.34920	−	0.778960i	0.361063	−	0.932541i	$-0.382414\pi$
$971$	1.27208e6	−	734436.i	1.34920	−	0.778960i	0.988136	+	0.153581i	$0.0490806\pi$
$972$		0			0
$973$	19902.4	−	34472.0i	0.0210223	−	0.0364117i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.40349e6i			1.47035i	0.677879	+	0.735173i	$0.262899\pi$
$977$			1.40349e6i			1.47035i	−0.677879	+	0.735173i	$0.737101\pi$
$978$		0			0
$979$	−312385.	−	180356.i	−0.325930	−	0.188176i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.37527e6	+	794013.i	−1.42325	+	0.821714i	−0.996575	−	0.0826913i	$-0.973648\pi$
$983$	−1.37527e6	+	794013.i	−1.42325	+	0.821714i	−0.426675	+	0.904405i	$0.640315\pi$
$984$		0			0
$985$	36767.1	+	63682.5i	0.0378955	+	0.0656369i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.09590e6			−1.12041
$990$		0			0
$991$	−1.22784e6	+	708893.i	−1.25024	+	0.721828i	−0.971158	−	0.238439i	$-0.923364\pi$
$991$	−1.22784e6	+	708893.i	−1.25024	+	0.721828i	−0.279085	+	0.960267i	$0.590031\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−129526.			−0.130831
$996$		0			0
$997$	436408.	−	755881.i	0.439038	−	0.760437i	−0.558577	−	0.829452i	$-0.688653\pi$
$997$	436408.	−	755881.i	0.439038	−	0.760437i	0.997616	+	0.0690159i	$0.0219859\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.y.e.145.6		14
3.2	odd	2		228.5.l.a.145.2	✓	14
19.8	odd	6	inner	684.5.y.e.217.6		14
57.8	even	6		228.5.l.a.217.2	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.5.l.a.145.2	✓	14	3.2	odd	2
228.5.l.a.217.2	yes	14	57.8	even	6
684.5.y.e.145.6		14	1.1	even	1	trivial
684.5.y.e.217.6		14	19.8	odd	6	inner

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(11.9892 + 20.7658i) q^{5} -2.95794 q^{7} +O(q^{10})\)	\(q+(11.9892 + 20.7658i) q^{5} -2.95794 q^{7} -190.065 q^{11} +(-22.7900 - 13.1578i) q^{13} +(57.0672 + 98.8433i) q^{17} +(-360.990 - 2.68653i) q^{19} +(496.822 - 860.522i) q^{23} +(25.0203 - 43.3365i) q^{25} +(472.196 + 272.622i) q^{29} -512.283i q^{31} +(-35.4632 - 61.4240i) q^{35} -459.578i q^{37} +(2722.46 - 1571.81i) q^{41} +(-551.453 - 955.145i) q^{43} +(-526.277 + 911.539i) q^{47} -2392.25 q^{49} +(2495.00 + 1440.49i) q^{53} +(-2278.72 - 3946.86i) q^{55} +(2794.62 - 1613.47i) q^{59} +(-2894.62 + 5013.63i) q^{61} -631.005i q^{65} +(3584.82 + 2069.70i) q^{67} +(2387.05 - 1378.16i) q^{71} +(3134.42 + 5428.97i) q^{73} +562.200 q^{77} +(-867.616 + 500.918i) q^{79} +1987.92 q^{83} +(-1368.38 + 2370.10i) q^{85} +(1643.57 + 948.915i) q^{89} +(67.4115 + 38.9200i) q^{91} +(-4272.18 - 7528.46i) q^{95} +(2572.37 - 1485.16i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 30 q^{5} + 106 q^{7}+O(q^{10}) \) 14 * q + 30 * q^5 + 106 * q^7	\( 14 q + 30 q^{5} + 106 q^{7} + 264 q^{11} + 57 q^{13} - 282 q^{17} + 2 q^{19} - 96 q^{23} - 465 q^{25} + 630 q^{29} - 1434 q^{35} + 228 q^{41} - 2093 q^{43} + 4710 q^{47} + 4440 q^{49} - 2364 q^{53} + 6368 q^{55} - 11838 q^{59} - 1661 q^{61} - 20319 q^{67} - 624 q^{71} + 5851 q^{73} + 1080 q^{77} - 13299 q^{79} - 12252 q^{83} - 5740 q^{85} + 20010 q^{89} + 15951 q^{91} - 7770 q^{95} + 44904 q^{97}+O(q^{100}) \) 14 * q + 30 * q^5 + 106 * q^7 + 264 * q^11 + 57 * q^13 - 282 * q^17 + 2 * q^19 - 96 * q^23 - 465 * q^25 + 630 * q^29 - 1434 * q^35 + 228 * q^41 - 2093 * q^43 + 4710 * q^47 + 4440 * q^49 - 2364 * q^53 + 6368 * q^55 - 11838 * q^59 - 1661 * q^61 - 20319 * q^67 - 624 * q^71 + 5851 * q^73 + 1080 * q^77 - 13299 * q^79 - 12252 * q^83 - 5740 * q^85 + 20010 * q^89 + 15951 * q^91 - 7770 * q^95 + 44904 * q^97

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