LMFDB - Embedded newform 1600.2.n.o.1343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1343.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1600.1343
Dual form			1600.2.n.o.1407.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.22474 + 2.22474i) q^{3} +(-2.00000 - 2.00000i) q^{7} -6.89898i q^{9} +O(q^{10})$	$q+(-2.22474 + 2.22474i) q^{3} +(-2.00000 - 2.00000i) q^{7} -6.89898i q^{9} +3.44949i q^{11} +(-4.44949 - 4.44949i) q^{13} +(-0.775255 + 0.775255i) q^{17} +2.55051 q^{19} +8.89898 q^{21} +(-0.449490 + 0.449490i) q^{23} +(8.67423 + 8.67423i) q^{27} +2.89898i q^{29} -2.89898i q^{31} +(-7.67423 - 7.67423i) q^{33} +(6.00000 - 6.00000i) q^{37} +19.7980 q^{39} +5.00000 q^{41} +(-2.89898 + 2.89898i) q^{43} +(7.34847 + 7.34847i) q^{47} +1.00000i q^{49} -3.44949i q^{51} +(6.44949 + 6.44949i) q^{53} +(-5.67423 + 5.67423i) q^{57} -7.79796 q^{59} +0.898979 q^{61} +(-13.7980 + 13.7980i) q^{63} +(-4.22474 - 4.22474i) q^{67} -2.00000i q^{69} -2.00000i q^{71} +(5.67423 + 5.67423i) q^{73} +(6.89898 - 6.89898i) q^{77} -0.898979 q^{79} -17.8990 q^{81} +(-4.67423 + 4.67423i) q^{83} +(-6.44949 - 6.44949i) q^{87} +11.8990i q^{89} +17.7980i q^{91} +(6.44949 + 6.44949i) q^{93} +(4.89898 - 4.89898i) q^{97} +23.7980 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7}+O(q^{10}) $ 4 * q - 4 * q^3 - 8 * q^7	$ 4 q - 4 q^{3} - 8 q^{7} - 8 q^{13} - 8 q^{17} + 20 q^{19} + 16 q^{21} + 8 q^{23} + 20 q^{27} - 16 q^{33} + 24 q^{37} + 40 q^{39} + 20 q^{41} + 8 q^{43} + 16 q^{53} - 8 q^{57} + 8 q^{59} - 16 q^{61} - 16 q^{63} - 12 q^{67} + 8 q^{73} + 8 q^{77} + 16 q^{79} - 52 q^{81} - 4 q^{83} - 16 q^{87} + 16 q^{93} + 56 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 - 8 * q^13 - 8 * q^17 + 20 * q^19 + 16 * q^21 + 8 * q^23 + 20 * q^27 - 16 * q^33 + 24 * q^37 + 40 * q^39 + 20 * q^41 + 8 * q^43 + 16 * q^53 - 8 * q^57 + 8 * q^59 - 16 * q^61 - 16 * q^63 - 12 * q^67 + 8 * q^73 + 8 * q^77 + 16 * q^79 - 52 * q^81 - 4 * q^83 - 16 * q^87 + 16 * q^93 + 56 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$	−2.22474	+	2.22474i	−1.28446	+	1.28446i	−0.346353	+	0.938104i	$0.612580\pi$
$3$	−2.22474	+	2.22474i	−1.28446	+	1.28446i	−0.938104	+	0.346353i	$0.887420\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	0.219650	−	0.975579i	$-0.429509\pi$
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	−0.975579	+	0.219650i	$0.929509\pi$
$8$		0			0
$9$		−	6.89898i		−	2.29966i
$10$		0			0
$11$			3.44949i			1.04006i	0.854148	+	0.520030i	$0.174079\pi$
$11$			3.44949i			1.04006i	−0.854148	+	0.520030i	$0.825921\pi$
$12$		0			0
$13$	−4.44949	−	4.44949i	−1.23407	−	1.23407i	−0.962388	−	0.271678i	$-0.912421\pi$
$13$	−4.44949	−	4.44949i	−1.23407	−	1.23407i	−0.271678	−	0.962388i	$-0.587579\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.775255	+	0.775255i	−0.188027	+	0.188027i	−0.794843	−	0.606816i	$-0.792447\pi$
$17$	−0.775255	+	0.775255i	−0.188027	+	0.188027i	0.606816	+	0.794843i	$0.292447\pi$
$18$		0			0
$19$	2.55051			0.585127			0.292564	−	0.956246i	$-0.405492\pi$
$19$	2.55051			0.585127			0.292564	+	0.956246i	$0.405492\pi$
$20$		0			0
$21$	8.89898			1.94192
$22$		0			0
$23$	−0.449490	+	0.449490i	−0.0937251	+	0.0937251i	−0.752415	−	0.658690i	$-0.771111\pi$
$23$	−0.449490	+	0.449490i	−0.0937251	+	0.0937251i	0.658690	+	0.752415i	$0.271111\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	8.67423	+	8.67423i	1.66936	+	1.66936i
$28$		0			0
$29$			2.89898i			0.538327i	0.963095	+	0.269163i	$0.0867472\pi$
$29$			2.89898i			0.538327i	−0.963095	+	0.269163i	$0.913253\pi$
$30$		0			0
$31$		−	2.89898i		−	0.520672i	−0.965518	−	0.260336i	$-0.916167\pi$
$31$		−	2.89898i		−	0.520672i	0.965518	−	0.260336i	$-0.0838333\pi$
$32$		0			0
$33$	−7.67423	−	7.67423i	−1.33591	−	1.33591i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.00000	−	6.00000i	0.986394	−	0.986394i	−0.0135147	−	0.999909i	$-0.504302\pi$
$37$	6.00000	−	6.00000i	0.986394	−	0.986394i	0.999909	+	0.0135147i	$0.00430201\pi$
$38$		0			0
$39$	19.7980			3.17021
$40$		0			0
$41$	5.00000			0.780869			0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000			0.780869			0.390434	+	0.920631i	$0.372325\pi$
$42$		0			0
$43$	−2.89898	+	2.89898i	−0.442090	+	0.442090i	−0.892714	−	0.450624i	$-0.851202\pi$
$43$	−2.89898	+	2.89898i	−0.442090	+	0.442090i	0.450624	+	0.892714i	$0.351202\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.34847	+	7.34847i	1.07188	+	1.07188i	0.997208	+	0.0746766i	$0.0237924\pi$
$47$	7.34847	+	7.34847i	1.07188	+	1.07188i	0.0746766	+	0.997208i	$0.476208\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$		−	3.44949i		−	0.483025i
$52$		0			0
$53$	6.44949	+	6.44949i	0.885906	+	0.885906i	0.994127	−	0.108221i	$-0.0345155\pi$
$53$	6.44949	+	6.44949i	0.885906	+	0.885906i	−0.108221	+	0.994127i	$0.534516\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.67423	+	5.67423i	−0.751571	+	0.751571i
$58$		0			0
$59$	−7.79796			−1.01521			−0.507604	−	0.861591i	$-0.669469\pi$
$59$	−7.79796			−1.01521			−0.507604	+	0.861591i	$0.669469\pi$
$60$		0			0
$61$	0.898979			0.115103			0.0575513	−	0.998343i	$-0.481671\pi$
$61$	0.898979			0.115103			0.0575513	+	0.998343i	$0.481671\pi$
$62$		0			0
$63$	−13.7980	+	13.7980i	−1.73838	+	1.73838i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.22474	−	4.22474i	−0.516135	−	0.516135i	0.400265	−	0.916400i	$-0.368918\pi$
$67$	−4.22474	−	4.22474i	−0.516135	−	0.516135i	−0.916400	+	0.400265i	$0.868918\pi$
$68$		0			0
$69$		−	2.00000i		−	0.240772i
$70$		0			0
$71$		−	2.00000i		−	0.237356i	−0.992933	−	0.118678i	$-0.962134\pi$
$71$		−	2.00000i		−	0.237356i	0.992933	−	0.118678i	$-0.0378657\pi$
$72$		0			0
$73$	5.67423	+	5.67423i	0.664119	+	0.664119i	0.956348	−	0.292229i	$-0.0943971\pi$
$73$	5.67423	+	5.67423i	0.664119	+	0.664119i	−0.292229	+	0.956348i	$0.594397\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.89898	−	6.89898i	0.786212	−	0.786212i
$78$		0			0
$79$	−0.898979			−0.101143			−0.0505715	−	0.998720i	$-0.516104\pi$
$79$	−0.898979			−0.101143			−0.0505715	+	0.998720i	$0.516104\pi$
$80$		0			0
$81$	−17.8990			−1.98878
$82$		0			0
$83$	−4.67423	+	4.67423i	−0.513064	+	0.513064i	−0.915464	−	0.402400i	$-0.868176\pi$
$83$	−4.67423	+	4.67423i	−0.513064	+	0.513064i	0.402400	+	0.915464i	$0.368176\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.44949	−	6.44949i	−0.691458	−	0.691458i
$88$		0			0
$89$			11.8990i			1.26129i	0.776072	+	0.630645i	$0.217209\pi$
$89$			11.8990i			1.26129i	−0.776072	+	0.630645i	$0.782791\pi$
$90$		0			0
$91$			17.7980i			1.86573i
$92$		0			0
$93$	6.44949	+	6.44949i	0.668781	+	0.668781i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.89898	−	4.89898i	0.497416	−	0.497416i	−0.413217	−	0.910633i	$-0.635595\pi$
$97$	4.89898	−	4.89898i	0.497416	−	0.497416i	0.910633	+	0.413217i	$0.135595\pi$
$98$		0			0
$99$	23.7980			2.39178
$100$		0			0
$101$	−14.6969			−1.46240			−0.731200	−	0.682163i	$-0.761039\pi$
$101$	−14.6969			−1.46240			−0.731200	+	0.682163i	$0.761039\pi$
$102$		0			0
$103$	−2.44949	+	2.44949i	−0.241355	+	0.241355i	−0.817411	−	0.576055i	$-0.804591\pi$
$103$	−2.44949	+	2.44949i	−0.241355	+	0.241355i	0.576055	+	0.817411i	$0.304591\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.57321	+	5.57321i	0.538783	+	0.538783i	0.923171	−	0.384388i	$-0.125588\pi$
$107$	5.57321	+	5.57321i	0.538783	+	0.538783i	−0.384388	+	0.923171i	$0.625588\pi$
$108$		0			0
$109$			3.10102i			0.297024i	0.988911	+	0.148512i	$0.0474483\pi$
$109$			3.10102i			0.297024i	−0.988911	+	0.148512i	$0.952552\pi$
$110$		0			0
$111$			26.6969i			2.53396i
$112$		0			0
$113$	11.6742	+	11.6742i	1.09822	+	1.09822i	0.994619	+	0.103601i	$0.0330364\pi$
$113$	11.6742	+	11.6742i	1.09822	+	1.09822i	0.103601	+	0.994619i	$0.466964\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−30.6969	+	30.6969i	−2.83793	+	2.83793i
$118$		0			0
$119$	3.10102			0.284270
$120$		0			0
$121$	−0.898979			−0.0817254
$122$		0			0
$123$	−11.1237	+	11.1237i	−1.00299	+	1.00299i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.44949	+	8.44949i	0.749771	+	0.749771i	0.974436	−	0.224665i	$-0.0721288\pi$
$127$	8.44949	+	8.44949i	0.749771	+	0.749771i	−0.224665	+	0.974436i	$0.572129\pi$
$128$		0			0
$129$		−	12.8990i		−	1.13569i
$130$		0			0
$131$		−	15.7980i		−	1.38027i	−0.723679	−	0.690137i	$-0.757550\pi$
$131$		−	15.7980i		−	1.38027i	0.723679	−	0.690137i	$-0.242450\pi$
$132$		0			0
$133$	−5.10102	−	5.10102i	−0.442315	−	0.442315i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.12372	+	2.12372i	−0.181442	+	0.181442i	−0.791984	−	0.610542i	$-0.790952\pi$
$137$	−2.12372	+	2.12372i	−0.181442	+	0.181442i	0.610542	+	0.791984i	$0.290952\pi$
$138$		0			0
$139$	9.24745			0.784358			0.392179	−	0.919889i	$-0.371721\pi$
$139$	9.24745			0.784358			0.392179	+	0.919889i	$0.371721\pi$
$140$		0			0
$141$	−32.6969			−2.75358
$142$		0			0
$143$	15.3485	−	15.3485i	1.28350	−	1.28350i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.22474	−	2.22474i	−0.183494	−	0.183494i
$148$		0			0
$149$			18.8990i			1.54826i	0.633024	+	0.774132i	$0.281814\pi$
$149$			18.8990i			1.54826i	−0.633024	+	0.774132i	$0.718186\pi$
$150$		0			0
$151$		−	22.6969i		−	1.84705i	−0.383537	−	0.923525i	$-0.625294\pi$
$151$		−	22.6969i		−	1.84705i	0.383537	−	0.923525i	$-0.374706\pi$
$152$		0			0
$153$	5.34847	+	5.34847i	0.432398	+	0.432398i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.24745	−	8.24745i	0.658218	−	0.658218i	−0.296740	−	0.954958i	$-0.595900\pi$
$157$	8.24745	−	8.24745i	0.658218	−	0.658218i	0.954958	+	0.296740i	$0.0958995\pi$
$158$		0			0
$159$	−28.6969			−2.27582
$160$		0			0
$161$	1.79796			0.141699
$162$		0			0
$163$	7.77526	−	7.77526i	0.609005	−	0.609005i	−0.333681	−	0.942686i	$-0.608291\pi$
$163$	7.77526	−	7.77526i	0.609005	−	0.609005i	0.942686	+	0.333681i	$0.108291\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.4495	+	10.4495i	0.808606	+	0.808606i	0.984423	−	0.175817i	$-0.0562567\pi$
$167$	10.4495	+	10.4495i	0.808606	+	0.808606i	−0.175817	+	0.984423i	$0.556257\pi$
$168$		0			0
$169$			26.5959i			2.04584i
$170$		0			0
$171$		−	17.5959i		−	1.34559i
$172$		0			0
$173$	−5.79796	−	5.79796i	−0.440811	−	0.440811i	0.451474	−	0.892284i	$-0.350898\pi$
$173$	−5.79796	−	5.79796i	−0.440811	−	0.440811i	−0.892284	+	0.451474i	$0.850898\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	17.3485	−	17.3485i	1.30399	−	1.30399i
$178$		0			0
$179$	15.2474			1.13965			0.569824	−	0.821767i	$-0.307011\pi$
$179$	15.2474			1.13965			0.569824	+	0.821767i	$0.307011\pi$
$180$		0			0
$181$	23.7980			1.76889			0.884444	−	0.466646i	$-0.154538\pi$
$181$	23.7980			1.76889			0.884444	+	0.466646i	$0.154538\pi$
$182$		0			0
$183$	−2.00000	+	2.00000i	−0.147844	+	0.147844i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.67423	−	2.67423i	−0.195559	−	0.195559i
$188$		0			0
$189$		−	34.6969i		−	2.52383i
$190$		0			0
$191$		−	13.1010i		−	0.947957i	−0.880537	−	0.473978i	$-0.842818\pi$
$191$		−	13.1010i		−	0.947957i	0.880537	−	0.473978i	$-0.157182\pi$
$192$		0			0
$193$	3.87628	+	3.87628i	0.279020	+	0.279020i	0.832718	−	0.553697i	$-0.186784\pi$
$193$	3.87628	+	3.87628i	0.279020	+	0.279020i	−0.553697	+	0.832718i	$0.686784\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.55051	−	1.55051i	0.110469	−	0.110469i	−0.649712	−	0.760181i	$-0.725110\pi$
$197$	1.55051	−	1.55051i	0.110469	−	0.110469i	0.760181	+	0.649712i	$0.225110\pi$
$198$		0			0
$199$	−9.79796			−0.694559			−0.347279	−	0.937762i	$-0.612894\pi$
$199$	−9.79796			−0.694559			−0.347279	+	0.937762i	$0.612894\pi$
$200$		0			0
$201$	18.7980			1.32591
$202$		0			0
$203$	5.79796	−	5.79796i	0.406937	−	0.406937i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.10102	+	3.10102i	0.215536	+	0.215536i
$208$		0			0
$209$			8.79796i			0.608568i
$210$		0			0
$211$			13.2474i			0.911992i	0.889982	+	0.455996i	$0.150717\pi$
$211$			13.2474i			0.911992i	−0.889982	+	0.455996i	$0.849283\pi$
$212$		0			0
$213$	4.44949	+	4.44949i	0.304874	+	0.304874i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5.79796	+	5.79796i	−0.393591	+	0.393591i
$218$		0			0
$219$	−25.2474			−1.70606
$220$		0			0
$221$	6.89898			0.464076
$222$		0			0
$223$	−16.8990	+	16.8990i	−1.13164	+	1.13164i	−0.141735	+	0.989905i	$0.545268\pi$
$223$	−16.8990	+	16.8990i	−1.13164	+	1.13164i	−0.989905	+	0.141735i	$0.954732\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0000	−	10.0000i	−0.663723	−	0.663723i	0.292532	−	0.956256i	$-0.405502\pi$
$227$	−10.0000	−	10.0000i	−0.663723	−	0.663723i	−0.956256	+	0.292532i	$0.905502\pi$
$228$		0			0
$229$		−	13.7980i		−	0.911795i	−0.890032	−	0.455897i	$-0.849318\pi$
$229$		−	13.7980i		−	0.911795i	0.890032	−	0.455897i	$-0.150682\pi$
$230$		0			0
$231$			30.6969i			2.01971i
$232$		0			0
$233$	−16.8990	−	16.8990i	−1.10709	−	1.10709i	−0.993531	−	0.113558i	$-0.963775\pi$
$233$	−16.8990	−	16.8990i	−1.10709	−	1.10709i	−0.113558	−	0.993531i	$-0.536225\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.00000	−	2.00000i	0.129914	−	0.129914i
$238$		0			0
$239$	25.5959			1.65566			0.827831	−	0.560977i	$-0.189575\pi$
$239$	25.5959			1.65566			0.827831	+	0.560977i	$0.189575\pi$
$240$		0			0
$241$	5.89898			0.379987			0.189993	−	0.981785i	$-0.439153\pi$
$241$	5.89898			0.379987			0.189993	+	0.981785i	$0.439153\pi$
$242$		0			0
$243$	13.7980	−	13.7980i	0.885139	−	0.885139i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−11.3485	−	11.3485i	−0.722086	−	0.722086i
$248$		0			0
$249$		−	20.7980i		−	1.31802i
$250$		0			0
$251$		−	17.4495i		−	1.10140i	−0.834703	−	0.550701i	$-0.814360\pi$
$251$		−	17.4495i		−	1.10140i	0.834703	−	0.550701i	$-0.185640\pi$
$252$		0			0
$253$	−1.55051	−	1.55051i	−0.0974797	−	0.0974797i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.6969	−	18.6969i	1.16628	−	1.16628i	0.183209	−	0.983074i	$-0.441351\pi$
$257$	18.6969	−	18.6969i	1.16628	−	1.16628i	0.983074	−	0.183209i	$-0.0586485\pi$
$258$		0			0
$259$	−24.0000			−1.49129
$260$		0			0
$261$	20.0000			1.23797
$262$		0			0
$263$	6.24745	−	6.24745i	0.385234	−	0.385234i	−0.487749	−	0.872984i	$-0.662182\pi$
$263$	6.24745	−	6.24745i	0.385234	−	0.385234i	0.872984	+	0.487749i	$0.162182\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−26.4722	−	26.4722i	−1.62007	−	1.62007i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$			26.8990i			1.63400i	0.576640	+	0.816998i	$0.304364\pi$
$271$			26.8990i			1.63400i	−0.576640	+	0.816998i	$0.695636\pi$
$272$		0			0
$273$	−39.5959	−	39.5959i	−2.39645	−	2.39645i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.55051	−	7.55051i	0.453666	−	0.453666i	−0.442903	−	0.896569i	$-0.646051\pi$
$277$	7.55051	−	7.55051i	0.453666	−	0.453666i	0.896569	+	0.442903i	$0.146051\pi$
$278$		0			0
$279$	−20.0000			−1.19737
$280$		0			0
$281$	31.7980			1.89691			0.948454	−	0.316916i	$-0.102647\pi$
$281$	31.7980			1.89691			0.948454	+	0.316916i	$0.102647\pi$
$282$		0			0
$283$	15.5732	−	15.5732i	0.925731	−	0.925731i	−0.0716951	−	0.997427i	$-0.522841\pi$
$283$	15.5732	−	15.5732i	0.925731	−	0.925731i	0.997427	+	0.0716951i	$0.0228409\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0000	−	10.0000i	−0.590281	−	0.590281i
$288$		0			0
$289$			15.7980i			0.929292i
$290$		0			0
$291$			21.7980i			1.27782i
$292$		0			0
$293$	6.89898	+	6.89898i	0.403043	+	0.403043i	0.879304	−	0.476261i	$-0.158008\pi$
$293$	6.89898	+	6.89898i	0.403043	+	0.403043i	−0.476261	+	0.879304i	$0.658008\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−29.9217	+	29.9217i	−1.73623	+	1.73623i
$298$		0			0
$299$	4.00000			0.231326
$300$		0			0
$301$	11.5959			0.668378
$302$		0			0
$303$	32.6969	−	32.6969i	1.87839	−	1.87839i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.426786	+	0.426786i	0.0243580	+	0.0243580i	0.719181	−	0.694823i	$-0.244517\pi$
$307$	0.426786	+	0.426786i	0.0243580	+	0.0243580i	−0.694823	+	0.719181i	$0.744517\pi$
$308$		0			0
$309$		−	10.8990i		−	0.620021i
$310$		0			0
$311$		−	15.1010i		−	0.856300i	−0.903708	−	0.428150i	$-0.859165\pi$
$311$		−	15.1010i		−	0.856300i	0.903708	−	0.428150i	$-0.140835\pi$
$312$		0			0
$313$	16.8990	+	16.8990i	0.955187	+	0.955187i	0.999038	−	0.0438513i	$-0.0139628\pi$
$313$	16.8990	+	16.8990i	0.955187	+	0.955187i	−0.0438513	+	0.999038i	$0.513963\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.69694	−	6.69694i	0.376138	−	0.376138i	−0.493569	−	0.869707i	$-0.664308\pi$
$317$	6.69694	−	6.69694i	0.376138	−	0.376138i	0.869707	+	0.493569i	$0.164308\pi$
$318$		0			0
$319$	−10.0000			−0.559893
$320$		0			0
$321$	−24.7980			−1.38409
$322$		0			0
$323$	−1.97730	+	1.97730i	−0.110020	+	0.110020i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.89898	−	6.89898i	−0.381514	−	0.381514i
$328$		0			0
$329$		−	29.3939i		−	1.62054i
$330$		0			0
$331$			10.3485i			0.568803i	0.958705	+	0.284402i	$0.0917949\pi$
$331$			10.3485i			0.568803i	−0.958705	+	0.284402i	$0.908205\pi$
$332$		0			0
$333$	−41.3939	−	41.3939i	−2.26837	−	2.26837i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.42679	+	7.42679i	−0.404563	+	0.404563i	−0.879837	−	0.475275i	$-0.842349\pi$
$337$	−7.42679	+	7.42679i	−0.404563	+	0.404563i	0.475275	+	0.879837i	$0.342349\pi$
$338$		0			0
$339$	−51.9444			−2.82123
$340$		0			0
$341$	10.0000			0.541530
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.6742	+	20.6742i	1.10985	+	1.10985i	0.993169	+	0.116682i	$0.0372257\pi$
$347$	20.6742	+	20.6742i	1.10985	+	1.10985i	0.116682	+	0.993169i	$0.462774\pi$
$348$		0			0
$349$			10.6969i			0.572594i	0.958141	+	0.286297i	$0.0924244\pi$
$349$			10.6969i			0.572594i	−0.958141	+	0.286297i	$0.907576\pi$
$350$		0			0
$351$		−	77.1918i		−	4.12020i
$352$		0			0
$353$	−7.10102	−	7.10102i	−0.377949	−	0.377949i	0.492413	−	0.870362i	$-0.336115\pi$
$353$	−7.10102	−	7.10102i	−0.377949	−	0.377949i	−0.870362	+	0.492413i	$0.836115\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−6.89898	+	6.89898i	−0.365133	+	0.365133i
$358$		0			0
$359$	−19.1010			−1.00811			−0.504057	−	0.863671i	$-0.668160\pi$
$359$	−19.1010			−1.00811			−0.504057	+	0.863671i	$0.668160\pi$
$360$		0			0
$361$	−12.4949			−0.657626
$362$		0			0
$363$	2.00000	−	2.00000i	0.104973	−	0.104973i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−18.6969	−	18.6969i	−0.975972	−	0.975972i	0.0237458	−	0.999718i	$-0.492441\pi$
$367$	−18.6969	−	18.6969i	−0.975972	−	0.975972i	−0.999718	+	0.0237458i	$0.992441\pi$
$368$		0			0
$369$		−	34.4949i		−	1.79573i
$370$		0			0
$371$		−	25.7980i		−	1.33936i
$372$		0			0
$373$	−2.44949	−	2.44949i	−0.126830	−	0.126830i	0.640843	−	0.767672i	$-0.278585\pi$
$373$	−2.44949	−	2.44949i	−0.126830	−	0.126830i	−0.767672	+	0.640843i	$0.778585\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.8990	−	12.8990i	0.664331	−	0.664331i
$378$		0			0
$379$	33.0454			1.69743			0.848714	−	0.528852i	$-0.177377\pi$
$379$	33.0454			1.69743			0.848714	+	0.528852i	$0.177377\pi$
$380$		0			0
$381$	−37.5959			−1.92610
$382$		0			0
$383$	11.7980	−	11.7980i	0.602848	−	0.602848i	−0.338220	−	0.941067i	$-0.609825\pi$
$383$	11.7980	−	11.7980i	0.602848	−	0.602848i	0.941067	+	0.338220i	$0.109825\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20.0000	+	20.0000i	1.01666	+	1.01666i
$388$		0			0
$389$			11.7980i			0.598180i	0.954225	+	0.299090i	$0.0966831\pi$
$389$			11.7980i			0.598180i	−0.954225	+	0.299090i	$0.903317\pi$
$390$		0			0
$391$		−	0.696938i		−	0.0352457i
$392$		0			0
$393$	35.1464	+	35.1464i	1.77290	+	1.77290i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.79796	+	5.79796i	−0.290991	+	0.290991i	−0.837472	−	0.546481i	$-0.815967\pi$
$397$	−5.79796	+	5.79796i	−0.290991	+	0.290991i	0.546481	+	0.837472i	$0.315967\pi$
$398$		0			0
$399$	22.6969			1.13627
$400$		0			0
$401$	−4.10102			−0.204795			−0.102398	−	0.994744i	$-0.532651\pi$
$401$	−4.10102			−0.204795			−0.102398	+	0.994744i	$0.532651\pi$
$402$		0			0
$403$	−12.8990	+	12.8990i	−0.642544	+	0.642544i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	20.6969	+	20.6969i	1.02591	+	1.02591i
$408$		0			0
$409$		−	10.7980i		−	0.533925i	−0.963707	−	0.266962i	$-0.913980\pi$
$409$		−	10.7980i		−	0.533925i	0.963707	−	0.266962i	$-0.0860199\pi$
$410$		0			0
$411$		−	9.44949i		−	0.466109i
$412$		0			0
$413$	15.5959	+	15.5959i	0.767425	+	0.767425i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−20.5732	+	20.5732i	−1.00747	+	1.00747i
$418$		0			0
$419$	−22.1464			−1.08192			−0.540962	−	0.841047i	$-0.681940\pi$
$419$	−22.1464			−1.08192			−0.540962	+	0.841047i	$0.681940\pi$
$420$		0			0
$421$	27.5959			1.34494			0.672471	−	0.740123i	$-0.265233\pi$
$421$	27.5959			1.34494			0.672471	+	0.740123i	$0.265233\pi$
$422$		0			0
$423$	50.6969	−	50.6969i	2.46497	−	2.46497i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.79796	−	1.79796i	−0.0870093	−	0.0870093i
$428$		0			0
$429$			68.2929i			3.29721i
$430$		0			0
$431$			12.8990i			0.621322i	0.950521	+	0.310661i	$0.100550\pi$
$431$			12.8990i			0.621322i	−0.950521	+	0.310661i	$0.899450\pi$
$432$		0			0
$433$	6.32577	+	6.32577i	0.303997	+	0.303997i	0.842575	−	0.538578i	$-0.181038\pi$
$433$	6.32577	+	6.32577i	0.303997	+	0.303997i	−0.538578	+	0.842575i	$0.681038\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.14643	+	1.14643i	−0.0548411	+	0.0548411i
$438$		0			0
$439$	−26.0000			−1.24091			−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000			−1.24091			−0.620456	+	0.784241i	$0.713053\pi$
$440$		0			0
$441$	6.89898			0.328523
$442$		0			0
$443$	16.9217	−	16.9217i	0.803973	−	0.803973i	−0.179741	−	0.983714i	$-0.557526\pi$
$443$	16.9217	−	16.9217i	0.803973	−	0.803973i	0.983714	+	0.179741i	$0.0575258\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−42.0454	−	42.0454i	−1.98868	−	1.98868i
$448$		0			0
$449$		−	5.89898i		−	0.278390i	−0.990265	−	0.139195i	$-0.955549\pi$
$449$		−	5.89898i		−	0.278390i	0.990265	−	0.139195i	$-0.0444515\pi$
$450$		0			0
$451$			17.2474i			0.812151i
$452$		0			0
$453$	50.4949	+	50.4949i	2.37246	+	2.37246i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.0227	−	23.0227i	1.07696	−	1.07696i	0.0801759	−	0.996781i	$-0.474452\pi$
$457$	23.0227	−	23.0227i	1.07696	−	1.07696i	0.996781	−	0.0801759i	$-0.0255482\pi$
$458$		0			0
$459$	−13.4495			−0.627768
$460$		0			0
$461$	−2.20204			−0.102559			−0.0512796	−	0.998684i	$-0.516330\pi$
$461$	−2.20204			−0.102559			−0.0512796	+	0.998684i	$0.516330\pi$
$462$		0			0
$463$	2.69694	−	2.69694i	0.125337	−	0.125337i	−0.641656	−	0.766993i	$-0.721752\pi$
$463$	2.69694	−	2.69694i	0.125337	−	0.125337i	0.766993	+	0.641656i	$0.221752\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	26.4949	+	26.4949i	1.22604	+	1.22604i	0.965450	+	0.260587i	$0.0839162\pi$
$467$	26.4949	+	26.4949i	1.22604	+	1.22604i	0.260587	+	0.965450i	$0.416084\pi$
$468$		0			0
$469$			16.8990i			0.780322i
$470$		0			0
$471$			36.6969i			1.69091i
$472$		0			0
$473$	−10.0000	−	10.0000i	−0.459800	−	0.459800i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	44.4949	−	44.4949i	2.03728	−	2.03728i
$478$		0			0
$479$	2.89898			0.132458			0.0662289	−	0.997804i	$-0.478903\pi$
$479$	2.89898			0.132458			0.0662289	+	0.997804i	$0.478903\pi$
$480$		0			0
$481$	−53.3939			−2.43455
$482$		0			0
$483$	−4.00000	+	4.00000i	−0.182006	+	0.182006i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−12.4495	−	12.4495i	−0.564140	−	0.564140i	0.366341	−	0.930481i	$-0.380611\pi$
$487$	−12.4495	−	12.4495i	−0.564140	−	0.564140i	−0.930481	+	0.366341i	$0.880611\pi$
$488$		0			0
$489$			34.5959i			1.56448i
$490$		0			0
$491$			1.59592i			0.0720228i	0.999351	+	0.0360114i	$0.0114653\pi$
$491$			1.59592i			0.0720228i	−0.999351	+	0.0360114i	$0.988535\pi$
$492$		0			0
$493$	−2.24745	−	2.24745i	−0.101220	−	0.101220i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.00000	+	4.00000i	−0.179425	+	0.179425i
$498$		0			0
$499$	−7.79796			−0.349085			−0.174542	−	0.984650i	$-0.555845\pi$
$499$	−7.79796			−0.349085			−0.174542	+	0.984650i	$0.555845\pi$
$500$		0			0
$501$	−46.4949			−2.07724
$502$		0			0
$503$	−13.7980	+	13.7980i	−0.615221	+	0.615221i	−0.944302	−	0.329081i	$-0.893261\pi$
$503$	−13.7980	+	13.7980i	−0.615221	+	0.615221i	0.329081	+	0.944302i	$0.393261\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−59.1691	−	59.1691i	−2.62779	−	2.62779i
$508$		0			0
$509$		−	20.2020i		−	0.895440i	−0.894174	−	0.447720i	$-0.852236\pi$
$509$		−	20.2020i		−	0.895440i	0.894174	−	0.447720i	$-0.147764\pi$
$510$		0			0
$511$		−	22.6969i		−	1.00405i
$512$		0			0
$513$	22.1237	+	22.1237i	0.976786	+	0.976786i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−25.3485	+	25.3485i	−1.11482	+	1.11482i
$518$		0			0
$519$	25.7980			1.13240
$520$		0			0
$521$	−19.0000			−0.832405			−0.416203	−	0.909272i	$-0.636639\pi$
$521$	−19.0000			−0.832405			−0.416203	+	0.909272i	$0.636639\pi$
$522$		0			0
$523$	8.22474	−	8.22474i	0.359643	−	0.359643i	−0.504038	−	0.863681i	$-0.668153\pi$
$523$	8.22474	−	8.22474i	0.359643	−	0.359643i	0.863681	+	0.504038i	$0.168153\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.24745	+	2.24745i	0.0979004	+	0.0979004i
$528$		0			0
$529$			22.5959i			0.982431i
$530$		0			0
$531$			53.7980i			2.33463i
$532$		0			0
$533$	−22.2474	−	22.2474i	−0.963644	−	0.963644i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−33.9217	+	33.9217i	−1.46383	+	1.46383i
$538$		0			0
$539$	−3.44949			−0.148580
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	−52.9444	+	52.9444i	−2.27206	+	2.27206i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.32577	+	1.32577i	0.0566856	+	0.0566856i	0.734881	−	0.678196i	$-0.237238\pi$
$547$	1.32577	+	1.32577i	0.0566856	+	0.0566856i	−0.678196	+	0.734881i	$0.737238\pi$
$548$		0			0
$549$		−	6.20204i		−	0.264697i
$550$		0			0
$551$			7.39388i			0.314990i
$552$		0			0
$553$	1.79796	+	1.79796i	0.0764570	+	0.0764570i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.44949	+	8.44949i	−0.358016	+	0.358016i	−0.863081	−	0.505065i	$-0.831469\pi$
$557$	−8.44949	+	8.44949i	−0.358016	+	0.358016i	0.505065	+	0.863081i	$0.331469\pi$
$558$		0			0
$559$	25.7980			1.09114
$560$		0			0
$561$	11.8990			0.502375
$562$		0			0
$563$	10.8990	−	10.8990i	0.459337	−	0.459337i	−0.439101	−	0.898438i	$-0.644703\pi$
$563$	10.8990	−	10.8990i	0.459337	−	0.459337i	0.898438	+	0.439101i	$0.144703\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	35.7980	+	35.7980i	1.50337	+	1.50337i
$568$		0			0
$569$			44.5959i			1.86956i	0.355230	+	0.934779i	$0.384403\pi$
$569$			44.5959i			1.86956i	−0.355230	+	0.934779i	$0.615597\pi$
$570$		0			0
$571$			14.0000i			0.585882i	0.956131	+	0.292941i	$0.0946339\pi$
$571$			14.0000i			0.585882i	−0.956131	+	0.292941i	$0.905366\pi$
$572$		0			0
$573$	29.1464	+	29.1464i	1.21761	+	1.21761i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	9.22474	−	9.22474i	0.384031	−	0.384031i	−0.488521	−	0.872552i	$-0.662463\pi$
$577$	9.22474	−	9.22474i	0.384031	−	0.384031i	0.872552	+	0.488521i	$0.162463\pi$
$578$		0			0
$579$	−17.2474			−0.716780
$580$		0			0
$581$	18.6969			0.775680
$582$		0			0
$583$	−22.2474	+	22.2474i	−0.921395	+	0.921395i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.5732	+	15.5732i	0.642775	+	0.642775i	0.951237	−	0.308461i	$-0.0998141\pi$
$587$	15.5732	+	15.5732i	0.642775	+	0.642775i	−0.308461	+	0.951237i	$0.599814\pi$
$588$		0			0
$589$		−	7.39388i		−	0.304659i
$590$		0			0
$591$			6.89898i			0.283786i
$592$		0			0
$593$	−5.22474	−	5.22474i	−0.214555	−	0.214555i	0.591644	−	0.806199i	$-0.298479\pi$
$593$	−5.22474	−	5.22474i	−0.214555	−	0.214555i	−0.806199	+	0.591644i	$0.798479\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	21.7980	−	21.7980i	0.892131	−	0.892131i
$598$		0			0
$599$	1.30306			0.0532417			0.0266208	−	0.999646i	$-0.491525\pi$
$599$	1.30306			0.0532417			0.0266208	+	0.999646i	$0.491525\pi$
$600$		0			0
$601$	−13.6969			−0.558710			−0.279355	−	0.960188i	$-0.590121\pi$
$601$	−13.6969			−0.558710			−0.279355	+	0.960188i	$0.590121\pi$
$602$		0			0
$603$	−29.1464	+	29.1464i	−1.18693	+	1.18693i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.79796	−	5.79796i	−0.235332	−	0.235332i	0.579582	−	0.814914i	$-0.303216\pi$
$607$	−5.79796	−	5.79796i	−0.235332	−	0.235332i	−0.814914	+	0.579582i	$0.803216\pi$
$608$		0			0
$609$			25.7980i			1.04539i
$610$		0			0
$611$		−	65.3939i		−	2.64555i
$612$		0			0
$613$	5.14643	+	5.14643i	0.207862	+	0.207862i	0.803358	−	0.595496i	$-0.203044\pi$
$613$	5.14643	+	5.14643i	0.207862	+	0.207862i	−0.595496	+	0.803358i	$0.703044\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.5959	+	23.5959i	−0.949936	+	0.949936i	−0.998805	−	0.0488693i	$-0.984438\pi$
$617$	−23.5959	+	23.5959i	−0.949936	+	0.949936i	0.0488693	+	0.998805i	$0.484438\pi$
$618$		0			0
$619$	18.0000			0.723481			0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000			0.723481			0.361741	+	0.932279i	$0.382183\pi$
$620$		0			0
$621$	−7.79796			−0.312921
$622$		0			0
$623$	23.7980	−	23.7980i	0.953445	−	0.953445i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−19.5732	−	19.5732i	−0.781679	−	0.781679i
$628$		0			0
$629$			9.30306i			0.370937i
$630$		0			0
$631$		−	4.69694i		−	0.186982i	−0.995620	−	0.0934911i	$-0.970197\pi$
$631$		−	4.69694i		−	0.186982i	0.995620	−	0.0934911i	$-0.0298027\pi$
$632$		0			0
$633$	−29.4722	−	29.4722i	−1.17141	−	1.17141i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.44949	−	4.44949i	0.176295	−	0.176295i
$638$		0			0
$639$	−13.7980			−0.545839
$640$		0			0
$641$	15.7980			0.623982			0.311991	−	0.950085i	$-0.399004\pi$
$641$	15.7980			0.623982			0.311991	+	0.950085i	$0.399004\pi$
$642$		0			0
$643$	−0.696938	+	0.696938i	−0.0274846	+	0.0274846i	−0.720716	−	0.693231i	$-0.756187\pi$
$643$	−0.696938	+	0.696938i	−0.0274846	+	0.0274846i	0.693231	+	0.720716i	$0.256187\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	27.5959	+	27.5959i	1.08491	+	1.08491i	0.996044	+	0.0888637i	$0.0283236\pi$
$647$	27.5959	+	27.5959i	1.08491	+	1.08491i	0.0888637	+	0.996044i	$0.471676\pi$
$648$		0			0
$649$		−	26.8990i		−	1.05588i
$650$		0			0
$651$		−	25.7980i		−	1.01110i
$652$		0			0
$653$	26.0000	+	26.0000i	1.01746	+	1.01746i	0.999845	+	0.0176138i	$0.00560692\pi$
$653$	26.0000	+	26.0000i	1.01746	+	1.01746i	0.0176138	+	0.999845i	$0.494393\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	39.1464	−	39.1464i	1.52725	−	1.52725i
$658$		0			0
$659$	4.14643			0.161522			0.0807610	−	0.996733i	$-0.474265\pi$
$659$	4.14643			0.161522			0.0807610	+	0.996733i	$0.474265\pi$
$660$		0			0
$661$	−40.2929			−1.56721			−0.783605	−	0.621259i	$-0.786621\pi$
$661$	−40.2929			−1.56721			−0.783605	+	0.621259i	$0.786621\pi$
$662$		0			0
$663$	−15.3485	+	15.3485i	−0.596085	+	0.596085i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.30306	−	1.30306i	−0.0504547	−	0.0504547i
$668$		0			0
$669$		−	75.1918i		−	2.90708i
$670$		0			0
$671$			3.10102i			0.119714i
$672$		0			0
$673$	−13.7980	−	13.7980i	−0.531872	−	0.531872i	0.389257	−	0.921129i	$-0.372732\pi$
$673$	−13.7980	−	13.7980i	−0.531872	−	0.531872i	−0.921129	+	0.389257i	$0.872732\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.0000	+	28.0000i	−1.07613	+	1.07613i	−0.0792746	+	0.996853i	$0.525260\pi$
$677$	−28.0000	+	28.0000i	−1.07613	+	1.07613i	−0.996853	+	0.0792746i	$0.974740\pi$
$678$		0			0
$679$	−19.5959			−0.752022
$680$		0			0
$681$	44.4949			1.70505
$682$		0			0
$683$	12.2247	−	12.2247i	0.467767	−	0.467767i	−0.433424	−	0.901190i	$-0.642695\pi$
$683$	12.2247	−	12.2247i	0.467767	−	0.467767i	0.901190	+	0.433424i	$0.142695\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	30.6969	+	30.6969i	1.17116	+	1.17116i
$688$		0			0
$689$		−	57.3939i		−	2.18653i
$690$		0			0
$691$		−	39.9444i		−	1.51956i	−0.650183	−	0.759778i	$-0.725308\pi$
$691$		−	39.9444i		−	1.51956i	0.650183	−	0.759778i	$-0.274692\pi$
$692$		0			0
$693$	−47.5959	−	47.5959i	−1.80802	−	1.80802i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.87628	+	3.87628i	−0.146824	+	0.146824i
$698$		0			0
$699$	75.1918			2.84402
$700$		0			0
$701$	13.1010			0.494819			0.247409	−	0.968911i	$-0.420421\pi$
$701$	13.1010			0.494819			0.247409	+	0.968911i	$0.420421\pi$
$702$		0			0
$703$	15.3031	−	15.3031i	0.577166	−	0.577166i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	29.3939	+	29.3939i	1.10547	+	1.10547i
$708$		0			0
$709$		−	12.0000i		−	0.450669i	−0.974281	−	0.225335i	$-0.927652\pi$
$709$		−	12.0000i		−	0.450669i	0.974281	−	0.225335i	$-0.0723476\pi$
$710$		0			0
$711$			6.20204i			0.232595i
$712$		0			0
$713$	1.30306	+	1.30306i	0.0488000	+	0.0488000i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−56.9444	+	56.9444i	−2.12663	+	2.12663i
$718$		0			0
$719$	31.1010			1.15987			0.579936	−	0.814662i	$-0.303077\pi$
$719$	31.1010			1.15987			0.579936	+	0.814662i	$0.303077\pi$
$720$		0			0
$721$	9.79796			0.364895
$722$		0			0
$723$	−13.1237	+	13.1237i	−0.488077	+	0.488077i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−17.1464	−	17.1464i	−0.635926	−	0.635926i	0.313622	−	0.949548i	$-0.398458\pi$
$727$	−17.1464	−	17.1464i	−0.635926	−	0.635926i	−0.949548	+	0.313622i	$0.898458\pi$
$728$		0			0
$729$			7.69694i			0.285072i
$730$		0			0
$731$		−	4.49490i		−	0.166250i
$732$		0			0
$733$	−15.5505	−	15.5505i	−0.574371	−	0.574371i	0.358976	−	0.933347i	$-0.383126\pi$
$733$	−15.5505	−	15.5505i	−0.574371	−	0.574371i	−0.933347	+	0.358976i	$0.883126\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.5732	−	14.5732i	0.536811	−	0.536811i
$738$		0			0
$739$	33.5959			1.23585			0.617923	−	0.786239i	$-0.287974\pi$
$739$	33.5959			1.23585			0.617923	+	0.786239i	$0.287974\pi$
$740$		0			0
$741$	50.4949			1.85498
$742$		0			0
$743$	5.34847	−	5.34847i	0.196216	−	0.196216i	−0.602160	−	0.798376i	$-0.705693\pi$
$743$	5.34847	−	5.34847i	0.196216	−	0.196216i	0.798376	+	0.602160i	$0.205693\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	32.2474	+	32.2474i	1.17987	+	1.17987i
$748$		0			0
$749$		−	22.2929i		−	0.814563i
$750$		0			0
$751$			41.1918i			1.50311i	0.659670	+	0.751556i	$0.270696\pi$
$751$			41.1918i			1.50311i	−0.659670	+	0.751556i	$0.729304\pi$
$752$		0			0
$753$	38.8207	+	38.8207i	1.41470	+	1.41470i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	14.6969	−	14.6969i	0.534169	−	0.534169i	−0.387641	−	0.921810i	$-0.626710\pi$
$757$	14.6969	−	14.6969i	0.534169	−	0.534169i	0.921810	+	0.387641i	$0.126710\pi$
$758$		0			0
$759$	6.89898			0.250417
$760$		0			0
$761$	−17.0000			−0.616250			−0.308125	−	0.951346i	$-0.599701\pi$
$761$	−17.0000			−0.616250			−0.308125	+	0.951346i	$0.599701\pi$
$762$		0			0
$763$	6.20204	−	6.20204i	0.224529	−	0.224529i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	34.6969	+	34.6969i	1.25283	+	1.25283i
$768$		0			0
$769$		−	25.6969i		−	0.926655i	−0.886187	−	0.463328i	$-0.846655\pi$
$769$		−	25.6969i		−	0.926655i	0.886187	−	0.463328i	$-0.153345\pi$
$770$		0			0
$771$			83.1918i			2.99608i
$772$		0			0
$773$	−28.9444	−	28.9444i	−1.04106	−	1.04106i	−0.999120	−	0.0419370i	$-0.986647\pi$
$773$	−28.9444	−	28.9444i	−1.04106	−	1.04106i	−0.0419370	−	0.999120i	$-0.513353\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	53.3939	−	53.3939i	1.91549	−	1.91549i
$778$		0			0
$779$	12.7526			0.456908
$780$		0			0
$781$	6.89898			0.246865
$782$		0			0
$783$	−25.1464	+	25.1464i	−0.898660	+	0.898660i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−6.89898	−	6.89898i	−0.245922	−	0.245922i	0.573373	−	0.819295i	$-0.305635\pi$
$787$	−6.89898	−	6.89898i	−0.245922	−	0.245922i	−0.819295	+	0.573373i	$0.805635\pi$
$788$		0			0
$789$			27.7980i			0.989634i
$790$		0			0
$791$		−	46.6969i		−	1.66035i
$792$		0			0
$793$	−4.00000	−	4.00000i	−0.142044	−	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.3485	−	11.3485i	0.401983	−	0.401983i	−0.476948	−	0.878931i	$-0.658257\pi$
$797$	11.3485	−	11.3485i	0.401983	−	0.401983i	0.878931	+	0.476948i	$0.158257\pi$
$798$		0			0
$799$	−11.3939			−0.403086
$800$		0			0
$801$	82.0908			2.90054
$802$		0			0
$803$	−19.5732	+	19.5732i	−0.690724	+	0.690724i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	0.202041i		−	0.00710338i	−0.999994	−	0.00355169i	$-0.998869\pi$
$809$		−	0.202041i		−	0.00710338i	0.999994	−	0.00355169i	$-0.00113054\pi$
$810$		0			0
$811$			26.0000i			0.912983i	0.889728	+	0.456492i	$0.150894\pi$
$811$			26.0000i			0.912983i	−0.889728	+	0.456492i	$0.849106\pi$
$812$		0			0
$813$	−59.8434	−	59.8434i	−2.09880	−	2.09880i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−7.39388	+	7.39388i	−0.258679	+	0.258679i
$818$		0			0
$819$	122.788			4.29055
$820$		0			0
$821$	−15.7980			−0.551353			−0.275676	−	0.961251i	$-0.588902\pi$
$821$	−15.7980			−0.551353			−0.275676	+	0.961251i	$0.588902\pi$
$822$		0			0
$823$	−10.6969	+	10.6969i	−0.372872	+	0.372872i	−0.868522	−	0.495650i	$-0.834930\pi$
$823$	−10.6969	+	10.6969i	−0.372872	+	0.372872i	0.495650	+	0.868522i	$0.334930\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.07832	−	3.07832i	−0.107044	−	0.107044i	0.651557	−	0.758600i	$-0.274116\pi$
$827$	−3.07832	−	3.07832i	−0.107044	−	0.107044i	−0.758600	+	0.651557i	$0.774116\pi$
$828$		0			0
$829$		−	37.1010i		−	1.28857i	−0.764785	−	0.644286i	$-0.777155\pi$
$829$		−	37.1010i		−	1.28857i	0.764785	−	0.644286i	$-0.222845\pi$
$830$		0			0
$831$			33.5959i			1.16543i
$832$		0			0
$833$	−0.775255	−	0.775255i	−0.0268610	−	0.0268610i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	25.1464	−	25.1464i	0.869188	−	0.869188i
$838$		0			0
$839$	−39.1918			−1.35305			−0.676526	−	0.736419i	$-0.736515\pi$
$839$	−39.1918			−1.35305			−0.676526	+	0.736419i	$0.736515\pi$
$840$		0			0
$841$	20.5959			0.710204
$842$		0			0
$843$	−70.7423	+	70.7423i	−2.43650	+	2.43650i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.79796	+	1.79796i	0.0617786	+	0.0617786i
$848$		0			0
$849$			69.2929i			2.37812i
$850$		0			0
$851$			5.39388i			0.184900i
$852$		0			0
$853$	4.65153	+	4.65153i	0.159265	+	0.159265i	0.782241	−	0.622976i	$-0.214076\pi$
$853$	4.65153	+	4.65153i	0.159265	+	0.159265i	−0.622976	+	0.782241i	$0.714076\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.42679	−	7.42679i	0.253694	−	0.253694i	−0.568789	−	0.822483i	$-0.692588\pi$
$857$	7.42679	−	7.42679i	0.253694	−	0.253694i	0.822483	+	0.568789i	$0.192588\pi$
$858$		0			0
$859$	−42.3485			−1.44491			−0.722456	−	0.691417i	$-0.756987\pi$
$859$	−42.3485			−1.44491			−0.722456	+	0.691417i	$0.756987\pi$
$860$		0			0
$861$	44.4949			1.51638
$862$		0			0
$863$	−5.55051	+	5.55051i	−0.188942	+	0.188942i	−0.795238	−	0.606297i	$-0.792654\pi$
$863$	−5.55051	+	5.55051i	−0.188942	+	0.188942i	0.606297	+	0.795238i	$0.292654\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−35.1464	−	35.1464i	−1.19364	−	1.19364i
$868$		0			0
$869$		−	3.10102i		−	0.105195i
$870$		0			0
$871$			37.5959i			1.27389i
$872$		0			0
$873$	−33.7980	−	33.7980i	−1.14389	−	1.14389i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10.6969	+	10.6969i	−0.361210	+	0.361210i	−0.864258	−	0.503048i	$-0.832212\pi$
$877$	−10.6969	+	10.6969i	−0.361210	+	0.361210i	0.503048	+	0.864258i	$0.332212\pi$
$878$		0			0
$879$	−30.6969			−1.03538
$880$		0			0
$881$	16.2020			0.545861			0.272930	−	0.962034i	$-0.412007\pi$
$881$	16.2020			0.545861			0.272930	+	0.962034i	$0.412007\pi$
$882$		0			0
$883$	−21.8207	+	21.8207i	−0.734324	+	0.734324i	−0.971473	−	0.237149i	$-0.923787\pi$
$883$	−21.8207	+	21.8207i	−0.734324	+	0.734324i	0.237149	+	0.971473i	$0.423787\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−40.0000	−	40.0000i	−1.34307	−	1.34307i	−0.892988	−	0.450081i	$-0.851395\pi$
$887$	−40.0000	−	40.0000i	−1.34307	−	1.34307i	−0.450081	−	0.892988i	$-0.648605\pi$
$888$		0			0
$889$		−	33.7980i		−	1.13355i
$890$		0			0
$891$		−	61.7423i		−	2.06845i
$892$		0			0
$893$	18.7423	+	18.7423i	0.627189	+	0.627189i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−8.89898	+	8.89898i	−0.297128	+	0.297128i
$898$		0			0
$899$	8.40408			0.280292
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$	−25.7980	+	25.7980i	−0.858502	+	0.858502i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−10.8990	−	10.8990i	−0.361895	−	0.361895i	0.502615	−	0.864510i	$-0.332371\pi$
$907$	−10.8990	−	10.8990i	−0.361895	−	0.361895i	−0.864510	+	0.502615i	$0.832371\pi$
$908$		0			0
$909$			101.394i			3.36302i
$910$		0			0
$911$		−	29.7980i		−	0.987250i	−0.869675	−	0.493625i	$-0.835671\pi$
$911$		−	29.7980i		−	0.987250i	0.869675	−	0.493625i	$-0.164329\pi$
$912$		0			0
$913$	−16.1237	−	16.1237i	−0.533617	−	0.533617i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−31.5959	+	31.5959i	−1.04339	+	1.04339i
$918$		0			0
$919$	56.6969			1.87026			0.935130	−	0.354306i	$-0.115283\pi$
$919$	56.6969			1.87026			0.935130	+	0.354306i	$0.115283\pi$
$920$		0			0
$921$	−1.89898			−0.0625735
$922$		0			0
$923$	−8.89898	+	8.89898i	−0.292913	+	0.292913i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	16.8990	+	16.8990i	0.555035	+	0.555035i
$928$		0			0
$929$		−	6.40408i		−	0.210111i	−0.994466	−	0.105056i	$-0.966498\pi$
$929$		−	6.40408i		−	0.210111i	0.994466	−	0.105056i	$-0.0335020\pi$
$930$		0			0
$931$			2.55051i			0.0835896i
$932$		0			0
$933$	33.5959	+	33.5959i	1.09988	+	1.09988i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−14.3258	+	14.3258i	−0.468002	+	0.468002i	−0.901267	−	0.433264i	$-0.857362\pi$
$937$	−14.3258	+	14.3258i	−0.468002	+	0.468002i	0.433264	+	0.901267i	$0.357362\pi$
$938$		0			0
$939$	−75.1918			−2.45379
$940$		0			0
$941$	−30.4949			−0.994105			−0.497053	−	0.867720i	$-0.665584\pi$
$941$	−30.4949			−0.994105			−0.497053	+	0.867720i	$0.665584\pi$
$942$		0			0
$943$	−2.24745	+	2.24745i	−0.0731870	+	0.0731870i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.8990	−	26.8990i	−0.874099	−	0.874099i	0.118817	−	0.992916i	$-0.462090\pi$
$947$	−26.8990	−	26.8990i	−0.874099	−	0.874099i	−0.992916	+	0.118817i	$0.962090\pi$
$948$		0			0
$949$		−	50.4949i		−	1.63913i
$950$		0			0
$951$			29.7980i			0.966265i
$952$		0			0
$953$	17.8763	+	17.8763i	0.579069	+	0.579069i	0.934647	−	0.355577i	$-0.115716\pi$
$953$	17.8763	+	17.8763i	0.579069	+	0.579069i	−0.355577	+	0.934647i	$0.615716\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	22.2474	−	22.2474i	0.719158	−	0.719158i
$958$		0			0
$959$	8.49490			0.274315
$960$		0			0
$961$	22.5959			0.728901
$962$		0			0
$963$	38.4495	−	38.4495i	1.23902	−	1.23902i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	16.8990	+	16.8990i	0.543435	+	0.543435i	0.924534	−	0.381099i	$-0.124454\pi$
$967$	16.8990	+	16.8990i	0.543435	+	0.543435i	−0.381099	+	0.924534i	$0.624454\pi$
$968$		0			0
$969$		−	8.79796i		−	0.282631i
$970$		0			0
$971$			32.1464i			1.03163i	0.856701	+	0.515814i	$0.172510\pi$
$971$			32.1464i			1.03163i	−0.856701	+	0.515814i	$0.827490\pi$
$972$		0			0
$973$	−18.4949	−	18.4949i	−0.592919	−	0.592919i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.6742	+	33.6742i	−1.07733	+	1.07733i	−0.0805866	+	0.996748i	$0.525679\pi$
$977$	−33.6742	+	33.6742i	−1.07733	+	1.07733i	−0.996748	+	0.0805866i	$0.974321\pi$
$978$		0			0
$979$	−41.0454			−1.31182
$980$		0			0
$981$	21.3939			0.683054
$982$		0			0
$983$	−2.89898	+	2.89898i	−0.0924631	+	0.0924631i	−0.751825	−	0.659362i	$-0.770826\pi$
$983$	−2.89898	+	2.89898i	−0.0924631	+	0.0924631i	0.659362	+	0.751825i	$0.270826\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	65.3939	+	65.3939i	2.08151	+	2.08151i
$988$		0			0
$989$		−	2.60612i		−	0.0828699i
$990$		0			0
$991$			32.8990i			1.04507i	0.852618	+	0.522535i	$0.175014\pi$
$991$			32.8990i			1.04507i	−0.852618	+	0.522535i	$0.824986\pi$
$992$		0			0
$993$	−23.0227	−	23.0227i	−0.730603	−	0.730603i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	17.7980	−	17.7980i	0.563667	−	0.563667i	−0.366680	−	0.930347i	$-0.619506\pi$
$997$	17.7980	−	17.7980i	0.563667	−	0.563667i	0.930347	+	0.366680i	$0.119506\pi$
$998$		0			0
$999$	104.091			3.29329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.n.o.1343.1		4
4.3	odd	2		1600.2.n.s.1343.2		4
5.2	odd	4		1600.2.n.s.1407.2		4
5.3	odd	4		1600.2.n.p.1407.1		4
5.4	even	2		1600.2.n.t.1343.2		4
8.3	odd	2		800.2.n.l.543.1	yes	4
8.5	even	2		800.2.n.n.543.2	yes	4
20.3	even	4		1600.2.n.t.1407.2		4
20.7	even	4	inner	1600.2.n.o.1407.1		4
20.19	odd	2		1600.2.n.p.1343.1		4
40.3	even	4		800.2.n.k.607.1	yes	4
40.13	odd	4		800.2.n.m.607.2	yes	4
40.19	odd	2		800.2.n.m.543.2	yes	4
40.27	even	4		800.2.n.n.607.2	yes	4
40.29	even	2		800.2.n.k.543.1	✓	4
40.37	odd	4		800.2.n.l.607.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.2.n.k.543.1	✓	4	40.29	even	2
800.2.n.k.607.1	yes	4	40.3	even	4
800.2.n.l.543.1	yes	4	8.3	odd	2
800.2.n.l.607.1	yes	4	40.37	odd	4
800.2.n.m.543.2	yes	4	40.19	odd	2
800.2.n.m.607.2	yes	4	40.13	odd	4
800.2.n.n.543.2	yes	4	8.5	even	2
800.2.n.n.607.2	yes	4	40.27	even	4
1600.2.n.o.1343.1		4	1.1	even	1	trivial
1600.2.n.o.1407.1		4	20.7	even	4	inner
1600.2.n.p.1343.1		4	20.19	odd	2
1600.2.n.p.1407.1		4	5.3	odd	4
1600.2.n.s.1343.2		4	4.3	odd	2
1600.2.n.s.1407.2		4	5.2	odd	4
1600.2.n.t.1343.2		4	5.4	even	2
1600.2.n.t.1407.2		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.22474 + 2.22474i) q^{3} +(-2.00000 - 2.00000i) q^{7} -6.89898i q^{9} +O(q^{10})\)	\(q+(-2.22474 + 2.22474i) q^{3} +(-2.00000 - 2.00000i) q^{7} -6.89898i q^{9} +3.44949i q^{11} +(-4.44949 - 4.44949i) q^{13} +(-0.775255 + 0.775255i) q^{17} +2.55051 q^{19} +8.89898 q^{21} +(-0.449490 + 0.449490i) q^{23} +(8.67423 + 8.67423i) q^{27} +2.89898i q^{29} -2.89898i q^{31} +(-7.67423 - 7.67423i) q^{33} +(6.00000 - 6.00000i) q^{37} +19.7980 q^{39} +5.00000 q^{41} +(-2.89898 + 2.89898i) q^{43} +(7.34847 + 7.34847i) q^{47} +1.00000i q^{49} -3.44949i q^{51} +(6.44949 + 6.44949i) q^{53} +(-5.67423 + 5.67423i) q^{57} -7.79796 q^{59} +0.898979 q^{61} +(-13.7980 + 13.7980i) q^{63} +(-4.22474 - 4.22474i) q^{67} -2.00000i q^{69} -2.00000i q^{71} +(5.67423 + 5.67423i) q^{73} +(6.89898 - 6.89898i) q^{77} -0.898979 q^{79} -17.8990 q^{81} +(-4.67423 + 4.67423i) q^{83} +(-6.44949 - 6.44949i) q^{87} +11.8990i q^{89} +17.7980i q^{91} +(6.44949 + 6.44949i) q^{93} +(4.89898 - 4.89898i) q^{97} +23.7980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \) 4 * q - 4 * q^3 - 8 * q^7	\( 4 q - 4 q^{3} - 8 q^{7} - 8 q^{13} - 8 q^{17} + 20 q^{19} + 16 q^{21} + 8 q^{23} + 20 q^{27} - 16 q^{33} + 24 q^{37} + 40 q^{39} + 20 q^{41} + 8 q^{43} + 16 q^{53} - 8 q^{57} + 8 q^{59} - 16 q^{61} - 16 q^{63} - 12 q^{67} + 8 q^{73} + 8 q^{77} + 16 q^{79} - 52 q^{81} - 4 q^{83} - 16 q^{87} + 16 q^{93} + 56 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 - 8 * q^13 - 8 * q^17 + 20 * q^19 + 16 * q^21 + 8 * q^23 + 20 * q^27 - 16 * q^33 + 24 * q^37 + 40 * q^39 + 20 * q^41 + 8 * q^43 + 16 * q^53 - 8 * q^57 + 8 * q^59 - 16 * q^61 - 16 * q^63 - 12 * q^67 + 8 * q^73 + 8 * q^77 + 16 * q^79 - 52 * q^81 - 4 * q^83 - 16 * q^87 + 16 * q^93 + 56 * q^99

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