LMFDB - Embedded newform 1680.2.k.a.209.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(209,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.3
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.a.209.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.41421i) q^{3} +(-1.73205 + 1.41421i) q^{5} +(1.00000 - 2.44949i) q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.41421i) q^{3} +(-1.73205 + 1.41421i) q^{5} +(1.00000 - 2.44949i) q^{7} +(-1.00000 - 2.82843i) q^{9} +2.82843i q^{11} +4.00000 q^{13} +(-0.267949 - 3.86370i) q^{15} +2.82843i q^{17} +(2.46410 + 3.86370i) q^{21} -3.46410 q^{23} +(1.00000 - 4.89898i) q^{25} +(5.00000 + 1.41421i) q^{27} +5.65685i q^{29} +9.79796i q^{31} +(-4.00000 - 2.82843i) q^{33} +(1.73205 + 5.65685i) q^{35} +(-4.00000 + 5.65685i) q^{39} -3.46410 q^{41} -4.89898i q^{43} +(5.73205 + 3.48477i) q^{45} -2.82843i q^{47} +(-5.00000 - 4.89898i) q^{49} +(-4.00000 - 2.82843i) q^{51} +(-4.00000 - 4.89898i) q^{55} -6.92820 q^{59} +9.79796i q^{61} +(-7.92820 - 0.378937i) q^{63} +(-6.92820 + 5.65685i) q^{65} +4.89898i q^{67} +(3.46410 - 4.89898i) q^{69} +2.82843i q^{71} -8.00000 q^{73} +(5.92820 + 6.31319i) q^{75} +(6.92820 + 2.82843i) q^{77} -8.00000 q^{79} +(-7.00000 + 5.65685i) q^{81} -2.82843i q^{83} +(-4.00000 - 4.89898i) q^{85} +(-8.00000 - 5.65685i) q^{87} +10.3923 q^{89} +(4.00000 - 9.79796i) q^{91} +(-13.8564 - 9.79796i) q^{93} -8.00000 q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^7 - 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{7} - 4 q^{9} + 16 q^{13} - 8 q^{15} - 4 q^{21} + 4 q^{25} + 20 q^{27} - 16 q^{33} - 16 q^{39} + 16 q^{45} - 20 q^{49} - 16 q^{51} - 16 q^{55} - 4 q^{63} - 32 q^{73} - 4 q^{75} - 32 q^{79} - 28 q^{81} - 16 q^{85} - 32 q^{87} + 16 q^{91} - 32 q^{97} + 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^7 - 4 * q^9 + 16 * q^13 - 8 * q^15 - 4 * q^21 + 4 * q^25 + 20 * q^27 - 16 * q^33 - 16 * q^39 + 16 * q^45 - 20 * q^49 - 16 * q^51 - 16 * q^55 - 4 * q^63 - 32 * q^73 - 4 * q^75 - 32 * q^79 - 28 * q^81 - 16 * q^85 - 32 * q^87 + 16 * q^91 - 32 * q^97 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.00000	+	1.41421i	−0.577350	+	0.816497i
$3$	−1.00000	+	1.41421i	−0.577350	+	0.816497i
$4$		0			0
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$6$		0			0
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$7$	1.00000	−	2.44949i	0.377964	−	0.925820i
$8$		0			0
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$	−0.267949	−	3.86370i	−0.0691842	−	0.997604i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	2.46410	+	3.86370i	0.537711	+	0.843129i
$22$		0			0
$23$	−3.46410			−0.722315			−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410			−0.722315			−0.361158	+	0.932505i	$0.617618\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$	5.00000	+	1.41421i	0.962250	+	0.272166i
$28$		0			0
$29$			5.65685i			1.05045i	0.850963	+	0.525226i	$0.176019\pi$
$29$			5.65685i			1.05045i	−0.850963	+	0.525226i	$0.823981\pi$
$30$		0			0
$31$			9.79796i			1.75977i	0.475191	+	0.879883i	$0.342379\pi$
$31$			9.79796i			1.75977i	−0.475191	+	0.879883i	$0.657621\pi$
$32$		0			0
$33$	−4.00000	−	2.82843i	−0.696311	−	0.492366i
$34$		0			0
$35$	1.73205	+	5.65685i	0.292770	+	0.956183i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−4.00000	+	5.65685i	−0.640513	+	0.905822i
$40$		0			0
$41$	−3.46410			−0.541002			−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410			−0.541002			−0.270501	+	0.962720i	$0.587189\pi$
$42$		0			0
$43$		−	4.89898i		−	0.747087i	−0.927613	−	0.373544i	$-0.878143\pi$
$43$		−	4.89898i		−	0.747087i	0.927613	−	0.373544i	$-0.121857\pi$
$44$		0			0
$45$	5.73205	+	3.48477i	0.854484	+	0.519478i
$46$		0			0
$47$		−	2.82843i		−	0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$		−	2.82843i		−	0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$		0			0
$51$	−4.00000	−	2.82843i	−0.560112	−	0.396059i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−4.00000	−	4.89898i	−0.539360	−	0.660578i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.92820			−0.901975			−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820			−0.901975			−0.450988	+	0.892530i	$0.648928\pi$
$60$		0			0
$61$			9.79796i			1.25450i	0.778818	+	0.627250i	$0.215820\pi$
$61$			9.79796i			1.25450i	−0.778818	+	0.627250i	$0.784180\pi$
$62$		0			0
$63$	−7.92820	−	0.378937i	−0.998860	−	0.0477416i
$64$		0			0
$65$	−6.92820	+	5.65685i	−0.859338	+	0.701646i
$66$		0			0
$67$			4.89898i			0.598506i	0.954174	+	0.299253i	$0.0967374\pi$
$67$			4.89898i			0.598506i	−0.954174	+	0.299253i	$0.903263\pi$
$68$		0			0
$69$	3.46410	−	4.89898i	0.417029	−	0.589768i
$70$		0			0
$71$			2.82843i			0.335673i	0.985815	+	0.167836i	$0.0536780\pi$
$71$			2.82843i			0.335673i	−0.985815	+	0.167836i	$0.946322\pi$
$72$		0			0
$73$	−8.00000			−0.936329			−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000			−0.936329			−0.468165	+	0.883641i	$0.655085\pi$
$74$		0			0
$75$	5.92820	+	6.31319i	0.684530	+	0.728985i
$76$		0			0
$77$	6.92820	+	2.82843i	0.789542	+	0.322329i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$		−	2.82843i		−	0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$		−	2.82843i		−	0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$		0			0
$85$	−4.00000	−	4.89898i	−0.433861	−	0.531369i
$86$		0			0
$87$	−8.00000	−	5.65685i	−0.857690	−	0.606478i
$88$		0			0
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$	4.00000	−	9.79796i	0.419314	−	1.02711i
$92$		0			0
$93$	−13.8564	−	9.79796i	−1.43684	−	1.01600i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$	8.00000	−	2.82843i	0.804030	−	0.284268i
$100$		0			0
$101$	−17.3205			−1.72345			−0.861727	−	0.507371i	$-0.830617\pi$
$101$	−17.3205			−1.72345			−0.861727	+	0.507371i	$0.830617\pi$
$102$		0			0
$103$	−10.0000			−0.985329			−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000			−0.985329			−0.492665	+	0.870219i	$0.663977\pi$
$104$		0			0
$105$	−9.73205	−	3.20736i	−0.949751	−	0.313007i
$106$		0			0
$107$	10.3923			1.00466			0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923			1.00466			0.502331	+	0.864675i	$0.332476\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.92820			−0.651751			−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820			−0.651751			−0.325875	+	0.945413i	$0.605659\pi$
$114$		0			0
$115$	6.00000	−	4.89898i	0.559503	−	0.456832i
$116$		0			0
$117$	−4.00000	−	11.3137i	−0.369800	−	1.04595i
$118$		0			0
$119$	6.92820	+	2.82843i	0.635107	+	0.259281i
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	3.46410	−	4.89898i	0.312348	−	0.441726i
$124$		0			0
$125$	5.19615	+	9.89949i	0.464758	+	0.885438i
$126$		0			0
$127$		−	14.6969i		−	1.30414i	−0.758158	−	0.652071i	$-0.773900\pi$
$127$		−	14.6969i		−	1.30414i	0.758158	−	0.652071i	$-0.226100\pi$
$128$		0			0
$129$	6.92820	+	4.89898i	0.609994	+	0.431331i
$130$		0			0
$131$	−6.92820			−0.605320			−0.302660	−	0.953099i	$-0.597875\pi$
$131$	−6.92820			−0.605320			−0.302660	+	0.953099i	$0.597875\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−10.6603	+	4.62158i	−0.917489	+	0.397762i
$136$		0			0
$137$	6.92820			0.591916			0.295958	−	0.955201i	$-0.404361\pi$
$137$	6.92820			0.591916			0.295958	+	0.955201i	$0.404361\pi$
$138$		0			0
$139$			9.79796i			0.831052i	0.909581	+	0.415526i	$0.136402\pi$
$139$			9.79796i			0.831052i	−0.909581	+	0.415526i	$0.863598\pi$
$140$		0			0
$141$	4.00000	+	2.82843i	0.336861	+	0.238197i
$142$		0			0
$143$			11.3137i			0.946100i
$144$		0			0
$145$	−8.00000	−	9.79796i	−0.664364	−	0.813676i
$146$		0			0
$147$	11.9282	−	2.17209i	0.983822	−	0.179151i
$148$		0			0
$149$		−	11.3137i		−	0.926855i	−0.886135	−	0.463428i	$-0.846619\pi$
$149$		−	11.3137i		−	0.926855i	0.886135	−	0.463428i	$-0.153381\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$	8.00000	−	2.82843i	0.646762	−	0.228665i
$154$		0			0
$155$	−13.8564	−	16.9706i	−1.11297	−	1.36311i
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.46410	+	8.48528i	−0.273009	+	0.668734i
$162$		0			0
$163$			14.6969i			1.15115i	0.817748	+	0.575577i	$0.195222\pi$
$163$			14.6969i			1.15115i	−0.817748	+	0.575577i	$0.804778\pi$
$164$		0			0
$165$	10.9282	−	0.757875i	0.850759	−	0.0590005i
$166$		0			0
$167$			14.1421i			1.09435i	0.837018	+	0.547176i	$0.184297\pi$
$167$			14.1421i			1.09435i	−0.837018	+	0.547176i	$0.815703\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$			19.7990i			1.50529i	0.658427	+	0.752645i	$0.271222\pi$
$173$			19.7990i			1.50529i	−0.658427	+	0.752645i	$0.728778\pi$
$174$		0			0
$175$	−11.0000	−	7.34847i	−0.831522	−	0.555492i
$176$		0			0
$177$	6.92820	−	9.79796i	0.520756	−	0.736460i
$178$		0			0
$179$			2.82843i			0.211407i	0.994398	+	0.105703i	$0.0337094\pi$
$179$			2.82843i			0.211407i	−0.994398	+	0.105703i	$0.966291\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−13.8564	−	9.79796i	−1.02430	−	0.724286i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−8.00000			−0.585018
$188$		0			0
$189$	8.46410	−	10.8332i	0.615673	−	0.788002i
$190$		0			0
$191$			19.7990i			1.43260i	0.697790	+	0.716302i	$0.254167\pi$
$191$			19.7990i			1.43260i	−0.697790	+	0.716302i	$0.745833\pi$
$192$		0			0
$193$			9.79796i			0.705273i	0.935760	+	0.352636i	$0.114715\pi$
$193$			9.79796i			0.705273i	−0.935760	+	0.352636i	$0.885285\pi$
$194$		0			0
$195$	−1.07180	−	15.4548i	−0.0767530	−	1.10674i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−6.92820	−	4.89898i	−0.488678	−	0.345547i
$202$		0			0
$203$	13.8564	+	5.65685i	0.972529	+	0.397033i
$204$		0			0
$205$	6.00000	−	4.89898i	0.419058	−	0.342160i
$206$		0			0
$207$	3.46410	+	9.79796i	0.240772	+	0.681005i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	−4.00000	−	2.82843i	−0.274075	−	0.193801i
$214$		0			0
$215$	6.92820	+	8.48528i	0.472500	+	0.578691i
$216$		0			0
$217$	24.0000	+	9.79796i	1.62923	+	0.665129i
$218$		0			0
$219$	8.00000	−	11.3137i	0.540590	−	0.764510i
$220$		0			0
$221$			11.3137i			0.761042i
$222$		0			0
$223$	26.0000			1.74109			0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000			1.74109			0.870544	+	0.492090i	$0.163767\pi$
$224$		0			0
$225$	−14.8564	+	2.07055i	−0.990427	+	0.138037i
$226$		0			0
$227$		−	2.82843i		−	0.187729i	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$		−	2.82843i		−	0.187729i	0.995585	−	0.0938647i	$-0.0299221\pi$
$228$		0			0
$229$			19.5959i			1.29493i	0.762093	+	0.647467i	$0.224172\pi$
$229$			19.5959i			1.29493i	−0.762093	+	0.647467i	$0.775828\pi$
$230$		0			0
$231$	−10.9282	+	6.96953i	−0.719023	+	0.458562i
$232$		0			0
$233$	20.7846			1.36165			0.680823	−	0.732448i	$-0.261622\pi$
$233$	20.7846			1.36165			0.680823	+	0.732448i	$0.261622\pi$
$234$		0			0
$235$	4.00000	+	4.89898i	0.260931	+	0.319574i
$236$		0			0
$237$	8.00000	−	11.3137i	0.519656	−	0.734904i
$238$		0			0
$239$			2.82843i			0.182956i	0.995807	+	0.0914779i	$0.0291591\pi$
$239$			2.82843i			0.182956i	−0.995807	+	0.0914779i	$0.970841\pi$
$240$		0			0
$241$			9.79796i			0.631142i	0.948902	+	0.315571i	$0.102196\pi$
$241$			9.79796i			0.631142i	−0.948902	+	0.315571i	$0.897804\pi$
$242$		0			0
$243$	−1.00000	−	15.5563i	−0.0641500	−	0.997940i
$244$		0			0
$245$	15.5885	+	1.41421i	0.995910	+	0.0903508i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	4.00000	+	2.82843i	0.253490	+	0.179244i
$250$		0			0
$251$	−20.7846			−1.31191			−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846			−1.31191			−0.655956	+	0.754799i	$0.727735\pi$
$252$		0			0
$253$		−	9.79796i		−	0.615992i
$254$		0			0
$255$	10.9282	−	0.757875i	0.684351	−	0.0474600i
$256$		0			0
$257$		−	14.1421i		−	0.882162i	−0.897467	−	0.441081i	$-0.854595\pi$
$257$		−	14.1421i		−	0.882162i	0.897467	−	0.441081i	$-0.145405\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	16.0000	−	5.65685i	0.990375	−	0.350150i
$262$		0			0
$263$	3.46410			0.213606			0.106803	−	0.994280i	$-0.465939\pi$
$263$	3.46410			0.213606			0.106803	+	0.994280i	$0.465939\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−10.3923	+	14.6969i	−0.635999	+	0.899438i
$268$		0			0
$269$	−10.3923			−0.633630			−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923			−0.633630			−0.316815	+	0.948487i	$0.602613\pi$
$270$		0			0
$271$			29.3939i			1.78555i	0.450502	+	0.892775i	$0.351245\pi$
$271$			29.3939i			1.78555i	−0.450502	+	0.892775i	$0.648755\pi$
$272$		0			0
$273$	9.85641	+	15.4548i	0.596537	+	0.935368i
$274$		0			0
$275$	13.8564	+	2.82843i	0.835573	+	0.170561i
$276$		0			0
$277$			19.5959i			1.17740i	0.808350	+	0.588702i	$0.200361\pi$
$277$			19.5959i			1.17740i	−0.808350	+	0.588702i	$0.799639\pi$
$278$		0			0
$279$	27.7128	−	9.79796i	1.65912	−	0.586588i
$280$		0			0
$281$		−	28.2843i		−	1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$		−	28.2843i		−	1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.46410	+	8.48528i	−0.204479	+	0.500870i
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	8.00000	−	11.3137i	0.468968	−	0.663221i
$292$		0			0
$293$			2.82843i			0.165238i	0.996581	+	0.0826192i	$0.0263285\pi$
$293$			2.82843i			0.165238i	−0.996581	+	0.0826192i	$0.973671\pi$
$294$		0			0
$295$	12.0000	−	9.79796i	0.698667	−	0.570459i
$296$		0			0
$297$	−4.00000	+	14.1421i	−0.232104	+	0.820610i
$298$		0			0
$299$	−13.8564			−0.801337
$300$		0			0
$301$	−12.0000	−	4.89898i	−0.691669	−	0.282372i
$302$		0			0
$303$	17.3205	−	24.4949i	0.995037	−	1.40720i
$304$		0			0
$305$	−13.8564	−	16.9706i	−0.793416	−	0.971732i
$306$		0			0
$307$	−10.0000			−0.570730			−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000			−0.570730			−0.285365	+	0.958419i	$0.592115\pi$
$308$		0			0
$309$	10.0000	−	14.1421i	0.568880	−	0.804518i
$310$		0			0
$311$	−27.7128			−1.57145			−0.785725	−	0.618576i	$-0.787710\pi$
$311$	−27.7128			−1.57145			−0.785725	+	0.618576i	$0.787710\pi$
$312$		0			0
$313$	16.0000			0.904373			0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000			0.904373			0.452187	+	0.891923i	$0.350644\pi$
$314$		0			0
$315$	14.2679	−	10.5558i	0.803908	−	0.594754i
$316$		0			0
$317$	−13.8564			−0.778253			−0.389127	−	0.921184i	$-0.627223\pi$
$317$	−13.8564			−0.778253			−0.389127	+	0.921184i	$0.627223\pi$
$318$		0			0
$319$	−16.0000			−0.895828
$320$		0			0
$321$	−10.3923	+	14.6969i	−0.580042	+	0.820303i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.00000	−	19.5959i	0.221880	−	1.08699i
$326$		0			0
$327$	10.0000	−	14.1421i	0.553001	−	0.782062i
$328$		0			0
$329$	−6.92820	−	2.82843i	−0.381964	−	0.155936i
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.92820	−	8.48528i	−0.378528	−	0.463600i
$336$		0			0
$337$		−	19.5959i		−	1.06746i	−0.845656	−	0.533729i	$-0.820790\pi$
$337$		−	19.5959i		−	1.06746i	0.845656	−	0.533729i	$-0.179210\pi$
$338$		0			0
$339$	6.92820	−	9.79796i	0.376288	−	0.532152i
$340$		0			0
$341$	−27.7128			−1.50073
$342$		0			0
$343$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$344$		0			0
$345$	0.928203	+	13.3843i	0.0499728	+	0.720584i
$346$		0			0
$347$	17.3205			0.929814			0.464907	−	0.885360i	$-0.346088\pi$
$347$	17.3205			0.929814			0.464907	+	0.885360i	$0.346088\pi$
$348$		0			0
$349$		−	19.5959i		−	1.04895i	−0.851427	−	0.524473i	$-0.824262\pi$
$349$		−	19.5959i		−	1.04895i	0.851427	−	0.524473i	$-0.175738\pi$
$350$		0			0
$351$	20.0000	+	5.65685i	1.06752	+	0.301941i
$352$		0			0
$353$		−	31.1127i		−	1.65596i	−0.560756	−	0.827981i	$-0.689490\pi$
$353$		−	31.1127i		−	1.65596i	0.560756	−	0.827981i	$-0.310510\pi$
$354$		0			0
$355$	−4.00000	−	4.89898i	−0.212298	−	0.260011i
$356$		0			0
$357$	−10.9282	+	6.96953i	−0.578382	+	0.368867i
$358$		0			0
$359$		−	31.1127i		−	1.64207i	−0.570881	−	0.821033i	$-0.693398\pi$
$359$		−	31.1127i		−	1.64207i	0.570881	−	0.821033i	$-0.306602\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	−3.00000	+	4.24264i	−0.157459	+	0.222681i
$364$		0			0
$365$	13.8564	−	11.3137i	0.725277	−	0.592187i
$366$		0			0
$367$	−10.0000			−0.521996			−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000			−0.521996			−0.260998	+	0.965339i	$0.584052\pi$
$368$		0			0
$369$	3.46410	+	9.79796i	0.180334	+	0.510061i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			9.79796i			0.507319i	0.967294	+	0.253660i	$0.0816343\pi$
$373$			9.79796i			0.507319i	−0.967294	+	0.253660i	$0.918366\pi$
$374$		0			0
$375$	−19.1962	−	2.55103i	−0.991285	−	0.131734i
$376$		0			0
$377$			22.6274i			1.16537i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$	20.7846	+	14.6969i	1.06483	+	0.752947i
$382$		0			0
$383$			14.1421i			0.722629i	0.932444	+	0.361315i	$0.117672\pi$
$383$			14.1421i			0.722629i	−0.932444	+	0.361315i	$0.882328\pi$
$384$		0			0
$385$	−16.0000	+	4.89898i	−0.815436	+	0.249675i
$386$		0			0
$387$	−13.8564	+	4.89898i	−0.704361	+	0.249029i
$388$		0			0
$389$			22.6274i			1.14726i	0.819116	+	0.573628i	$0.194464\pi$
$389$			22.6274i			1.14726i	−0.819116	+	0.573628i	$0.805536\pi$
$390$		0			0
$391$		−	9.79796i		−	0.495504i
$392$		0			0
$393$	6.92820	−	9.79796i	0.349482	−	0.494242i
$394$		0			0
$395$	13.8564	−	11.3137i	0.697191	−	0.569254i
$396$		0			0
$397$	−20.0000			−1.00377			−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000			−1.00377			−0.501886	+	0.864934i	$0.667360\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.6274i			1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$			22.6274i			1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$		0			0
$403$			39.1918i			1.95228i
$404$		0			0
$405$	4.12436	−	19.6975i	0.204941	−	0.978774i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	9.79796i		−	0.484478i	−0.970217	−	0.242239i	$-0.922118\pi$
$409$		−	9.79796i		−	0.484478i	0.970217	−	0.242239i	$-0.0778818\pi$
$410$		0			0
$411$	−6.92820	+	9.79796i	−0.341743	+	0.483298i
$412$		0			0
$413$	−6.92820	+	16.9706i	−0.340915	+	0.835067i
$414$		0			0
$415$	4.00000	+	4.89898i	0.196352	+	0.240481i
$416$		0			0
$417$	−13.8564	−	9.79796i	−0.678551	−	0.479808i
$418$		0			0
$419$	−6.92820			−0.338465			−0.169232	−	0.985576i	$-0.554129\pi$
$419$	−6.92820			−0.338465			−0.169232	+	0.985576i	$0.554129\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$		0			0
$423$	−8.00000	+	2.82843i	−0.388973	+	0.137523i
$424$		0			0
$425$	13.8564	+	2.82843i	0.672134	+	0.137199i
$426$		0			0
$427$	24.0000	+	9.79796i	1.16144	+	0.474156i
$428$		0			0
$429$	−16.0000	−	11.3137i	−0.772487	−	0.546231i
$430$		0			0
$431$			2.82843i			0.136241i	0.997677	+	0.0681203i	$0.0217002\pi$
$431$			2.82843i			0.136241i	−0.997677	+	0.0681203i	$0.978300\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$	21.8564	−	1.51575i	1.04793	−	0.0726746i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	39.1918i		−	1.87052i	−0.353956	−	0.935262i	$-0.615164\pi$
$439$		−	39.1918i		−	1.87052i	0.353956	−	0.935262i	$-0.384836\pi$
$440$		0			0
$441$	−8.85641	+	19.0411i	−0.421734	+	0.906720i
$442$		0			0
$443$	−17.3205			−0.822922			−0.411461	−	0.911427i	$-0.634981\pi$
$443$	−17.3205			−0.822922			−0.411461	+	0.911427i	$0.634981\pi$
$444$		0			0
$445$	−18.0000	+	14.6969i	−0.853282	+	0.696702i
$446$		0			0
$447$	16.0000	+	11.3137i	0.756774	+	0.535120i
$448$		0			0
$449$			5.65685i			0.266963i	0.991051	+	0.133482i	$0.0426157\pi$
$449$			5.65685i			0.266963i	−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$		−	9.79796i		−	0.461368i
$452$		0			0
$453$	8.00000	−	11.3137i	0.375873	−	0.531564i
$454$		0			0
$455$	6.92820	+	22.6274i	0.324799	+	1.06079i
$456$		0			0
$457$			19.5959i			0.916658i	0.888783	+	0.458329i	$0.151552\pi$
$457$			19.5959i			0.916658i	−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$	−4.00000	+	14.1421i	−0.186704	+	0.660098i
$460$		0			0
$461$	3.46410			0.161339			0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410			0.161339			0.0806696	+	0.996741i	$0.474294\pi$
$462$		0			0
$463$			4.89898i			0.227675i	0.993499	+	0.113837i	$0.0363143\pi$
$463$			4.89898i			0.227675i	−0.993499	+	0.113837i	$0.963686\pi$
$464$		0			0
$465$	37.8564	−	2.62536i	1.75555	−	0.121748i
$466$		0			0
$467$		−	2.82843i		−	0.130884i	−0.997856	−	0.0654420i	$-0.979154\pi$
$467$		−	2.82843i		−	0.130884i	0.997856	−	0.0654420i	$-0.0208457\pi$
$468$		0			0
$469$	12.0000	+	4.89898i	0.554109	+	0.226214i
$470$		0			0
$471$	−4.00000	+	5.65685i	−0.184310	+	0.260654i
$472$		0			0
$473$	13.8564			0.637118
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	27.7128			1.26623			0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128			1.26623			0.633115	+	0.774057i	$0.281776\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−8.53590	−	13.3843i	−0.388397	−	0.609005i
$484$		0			0
$485$	13.8564	−	11.3137i	0.629187	−	0.513729i
$486$		0			0
$487$		−	14.6969i		−	0.665982i	−0.942930	−	0.332991i	$-0.891942\pi$
$487$		−	14.6969i		−	0.665982i	0.942930	−	0.332991i	$-0.108058\pi$
$488$		0			0
$489$	−20.7846	−	14.6969i	−0.939913	−	0.664619i
$490$		0			0
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$	−16.0000			−0.720604
$494$		0			0
$495$	−9.85641	+	16.2127i	−0.443013	+	0.728706i
$496$		0			0
$497$	6.92820	+	2.82843i	0.310772	+	0.126872i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	−20.0000	−	14.1421i	−0.893534	−	0.631824i
$502$		0			0
$503$		−	19.7990i		−	0.882793i	−0.897312	−	0.441397i	$-0.854483\pi$
$503$		−	19.7990i		−	0.882793i	0.897312	−	0.441397i	$-0.145517\pi$
$504$		0			0
$505$	30.0000	−	24.4949i	1.33498	−	1.09001i
$506$		0			0
$507$	−3.00000	+	4.24264i	−0.133235	+	0.188422i
$508$		0			0
$509$	−3.46410			−0.153544			−0.0767718	−	0.997049i	$-0.524461\pi$
$509$	−3.46410			−0.153544			−0.0767718	+	0.997049i	$0.524461\pi$
$510$		0			0
$511$	−8.00000	+	19.5959i	−0.353899	+	0.866872i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.3205	−	14.1421i	0.763233	−	0.623177i
$516$		0			0
$517$	8.00000			0.351840
$518$		0			0
$519$	−28.0000	−	19.7990i	−1.22906	−	0.869079i
$520$		0			0
$521$	−10.3923			−0.455295			−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923			−0.455295			−0.227648	+	0.973744i	$0.573103\pi$
$522$		0			0
$523$	26.0000			1.13690			0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000			1.13690			0.568450	+	0.822718i	$0.307543\pi$
$524$		0			0
$525$	21.3923	−	8.20788i	0.933637	−	0.358221i
$526$		0			0
$527$	−27.7128			−1.20719
$528$		0			0
$529$	−11.0000			−0.478261
$530$		0			0
$531$	6.92820	+	19.5959i	0.300658	+	0.850390i
$532$		0			0
$533$	−13.8564			−0.600188
$534$		0			0
$535$	−18.0000	+	14.6969i	−0.778208	+	0.635404i
$536$		0			0
$537$	−4.00000	−	2.82843i	−0.172613	−	0.122056i
$538$		0			0
$539$	13.8564	−	14.1421i	0.596838	−	0.609145i
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	17.3205	−	14.1421i	0.741929	−	0.605783i
$546$		0			0
$547$		−	34.2929i		−	1.46626i	−0.680090	−	0.733128i	$-0.738059\pi$
$547$		−	34.2929i		−	1.46626i	0.680090	−	0.733128i	$-0.261941\pi$
$548$		0			0
$549$	27.7128	−	9.79796i	1.18275	−	0.418167i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8.00000	+	19.5959i	−0.340195	+	0.833303i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	41.5692			1.76134			0.880672	−	0.473726i	$-0.157091\pi$
$557$	41.5692			1.76134			0.880672	+	0.473726i	$0.157091\pi$
$558$		0			0
$559$		−	19.5959i		−	0.828819i
$560$		0			0
$561$	8.00000	−	11.3137i	0.337760	−	0.477665i
$562$		0			0
$563$			14.1421i			0.596020i	0.954563	+	0.298010i	$0.0963229\pi$
$563$			14.1421i			0.596020i	−0.954563	+	0.298010i	$0.903677\pi$
$564$		0			0
$565$	12.0000	−	9.79796i	0.504844	−	0.412203i
$566$		0			0
$567$	6.85641	+	22.8033i	0.287942	+	0.957648i
$568$		0			0
$569$		−	28.2843i		−	1.18574i	−0.805299	−	0.592869i	$-0.797995\pi$
$569$		−	28.2843i		−	1.18574i	0.805299	−	0.592869i	$-0.202005\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	−28.0000	−	19.7990i	−1.16972	−	0.827115i
$574$		0			0
$575$	−3.46410	+	16.9706i	−0.144463	+	0.707721i
$576$		0			0
$577$	−8.00000			−0.333044			−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000			−0.333044			−0.166522	+	0.986038i	$0.553254\pi$
$578$		0			0
$579$	−13.8564	−	9.79796i	−0.575853	−	0.407189i
$580$		0			0
$581$	−6.92820	−	2.82843i	−0.287430	−	0.117343i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	22.9282	+	13.9391i	0.947965	+	0.576309i
$586$		0			0
$587$		−	19.7990i		−	0.817192i	−0.912715	−	0.408596i	$-0.866019\pi$
$587$		−	19.7990i		−	0.817192i	0.912715	−	0.408596i	$-0.133981\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			19.7990i			0.813047i	0.913640	+	0.406524i	$0.133259\pi$
$593$			19.7990i			0.813047i	−0.913640	+	0.406524i	$0.866741\pi$
$594$		0			0
$595$	−16.0000	+	4.89898i	−0.655936	+	0.200839i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			36.7696i			1.50236i	0.660096	+	0.751182i	$0.270516\pi$
$599$			36.7696i			1.50236i	−0.660096	+	0.751182i	$0.729484\pi$
$600$		0			0
$601$			9.79796i			0.399667i	0.979830	+	0.199834i	$0.0640401\pi$
$601$			9.79796i			0.399667i	−0.979830	+	0.199834i	$0.935960\pi$
$602$		0			0
$603$	13.8564	−	4.89898i	0.564276	−	0.199502i
$604$		0			0
$605$	−5.19615	+	4.24264i	−0.211254	+	0.172488i
$606$		0			0
$607$	−10.0000			−0.405887			−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000			−0.405887			−0.202944	+	0.979190i	$0.565051\pi$
$608$		0			0
$609$	−21.8564	+	13.9391i	−0.885666	+	0.564839i
$610$		0			0
$611$		−	11.3137i		−	0.457704i
$612$		0			0
$613$			29.3939i			1.18721i	0.804757	+	0.593604i	$0.202295\pi$
$613$			29.3939i			1.18721i	−0.804757	+	0.593604i	$0.797705\pi$
$614$		0			0
$615$	0.928203	+	13.3843i	0.0374288	+	0.539705i
$616$		0			0
$617$	−48.4974			−1.95243			−0.976216	−	0.216799i	$-0.930439\pi$
$617$	−48.4974			−1.95243			−0.976216	+	0.216799i	$0.930439\pi$
$618$		0			0
$619$		−	9.79796i		−	0.393813i	−0.980422	−	0.196907i	$-0.936910\pi$
$619$		−	9.79796i		−	0.393813i	0.980422	−	0.196907i	$-0.0630896\pi$
$620$		0			0
$621$	−17.3205	−	4.89898i	−0.695048	−	0.196589i
$622$		0			0
$623$	10.3923	−	25.4558i	0.416359	−	1.01987i
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$	−4.00000	+	5.65685i	−0.158986	+	0.224840i
$634$		0			0
$635$	20.7846	+	25.4558i	0.824812	+	1.01018i
$636$		0			0
$637$	−20.0000	−	19.5959i	−0.792429	−	0.776419i
$638$		0			0
$639$	8.00000	−	2.82843i	0.316475	−	0.111891i
$640$		0			0
$641$			5.65685i			0.223432i	0.993740	+	0.111716i	$0.0356347\pi$
$641$			5.65685i			0.223432i	−0.993740	+	0.111716i	$0.964365\pi$
$642$		0			0
$643$	−22.0000			−0.867595			−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000			−0.867595			−0.433798	+	0.901010i	$0.642827\pi$
$644$		0			0
$645$	−18.9282	+	1.31268i	−0.745297	+	0.0516866i
$646$		0			0
$647$		−	19.7990i		−	0.778379i	−0.921158	−	0.389189i	$-0.872755\pi$
$647$		−	19.7990i		−	0.778379i	0.921158	−	0.389189i	$-0.127245\pi$
$648$		0			0
$649$		−	19.5959i		−	0.769207i
$650$		0			0
$651$	−37.8564	+	24.1432i	−1.48371	+	0.946245i
$652$		0			0
$653$	−27.7128			−1.08449			−0.542243	−	0.840222i	$-0.682425\pi$
$653$	−27.7128			−1.08449			−0.542243	+	0.840222i	$0.682425\pi$
$654$		0			0
$655$	12.0000	−	9.79796i	0.468879	−	0.382838i
$656$		0			0
$657$	8.00000	+	22.6274i	0.312110	+	0.882780i
$658$		0			0
$659$			2.82843i			0.110180i	0.998481	+	0.0550899i	$0.0175446\pi$
$659$			2.82843i			0.110180i	−0.998481	+	0.0550899i	$0.982455\pi$
$660$		0			0
$661$		−	9.79796i		−	0.381096i	−0.981678	−	0.190548i	$-0.938973\pi$
$661$		−	9.79796i		−	0.381096i	0.981678	−	0.190548i	$-0.0610266\pi$
$662$		0			0
$663$	−16.0000	−	11.3137i	−0.621389	−	0.439388i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	19.5959i		−	0.758757i
$668$		0			0
$669$	−26.0000	+	36.7696i	−1.00522	+	1.42159i
$670$		0			0
$671$	−27.7128			−1.06984
$672$		0			0
$673$		−	9.79796i		−	0.377684i	−0.982008	−	0.188842i	$-0.939527\pi$
$673$		−	9.79796i		−	0.377684i	0.982008	−	0.188842i	$-0.0604733\pi$
$674$		0			0
$675$	11.9282	−	23.0807i	0.459117	−	0.888376i
$676$		0			0
$677$			2.82843i			0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$			2.82843i			0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$		0			0
$679$	−8.00000	+	19.5959i	−0.307012	+	0.752022i
$680$		0			0
$681$	4.00000	+	2.82843i	0.153280	+	0.108386i
$682$		0			0
$683$	10.3923			0.397650			0.198825	−	0.980035i	$-0.436287\pi$
$683$	10.3923			0.397650			0.198825	+	0.980035i	$0.436287\pi$
$684$		0			0
$685$	−12.0000	+	9.79796i	−0.458496	+	0.374361i
$686$		0			0
$687$	−27.7128	−	19.5959i	−1.05731	−	0.747631i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	9.79796i		−	0.372732i	−0.982480	−	0.186366i	$-0.940329\pi$
$691$		−	9.79796i		−	0.372732i	0.982480	−	0.186366i	$-0.0596710\pi$
$692$		0			0
$693$	1.07180	−	22.4243i	0.0407142	−	0.851830i
$694$		0			0
$695$	−13.8564	−	16.9706i	−0.525603	−	0.643730i
$696$		0			0
$697$		−	9.79796i		−	0.371124i
$698$		0			0
$699$	−20.7846	+	29.3939i	−0.786146	+	1.11178i
$700$		0			0
$701$			22.6274i			0.854626i	0.904104	+	0.427313i	$0.140540\pi$
$701$			22.6274i			0.854626i	−0.904104	+	0.427313i	$0.859460\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−10.9282	+	0.757875i	−0.411580	+	0.0285432i
$706$		0			0
$707$	−17.3205	+	42.4264i	−0.651405	+	1.59561i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$		0			0
$711$	8.00000	+	22.6274i	0.300023	+	0.848594i
$712$		0			0
$713$		−	33.9411i		−	1.27111i
$714$		0			0
$715$	−16.0000	−	19.5959i	−0.598366	−	0.732846i
$716$		0			0
$717$	−4.00000	−	2.82843i	−0.149383	−	0.105630i
$718$		0			0
$719$	41.5692			1.55027			0.775135	−	0.631795i	$-0.217682\pi$
$719$	41.5692			1.55027			0.775135	+	0.631795i	$0.217682\pi$
$720$		0			0
$721$	−10.0000	+	24.4949i	−0.372419	+	0.912238i
$722$		0			0
$723$	−13.8564	−	9.79796i	−0.515325	−	0.364390i
$724$		0			0
$725$	27.7128	+	5.65685i	1.02923	+	0.210090i
$726$		0			0
$727$	−10.0000			−0.370879			−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000			−0.370879			−0.185440	+	0.982656i	$0.559371\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	13.8564			0.512498
$732$		0			0
$733$	28.0000			1.03420			0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000			1.03420			0.517102	+	0.855924i	$0.327011\pi$
$734$		0			0
$735$	−17.5885	+	20.6312i	−0.648760	+	0.760993i
$736$		0			0
$737$	−13.8564			−0.510407
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.2487			−0.889599			−0.444799	−	0.895630i	$-0.646725\pi$
$743$	−24.2487			−0.889599			−0.444799	+	0.895630i	$0.646725\pi$
$744$		0			0
$745$	16.0000	+	19.5959i	0.586195	+	0.717939i
$746$		0			0
$747$	−8.00000	+	2.82843i	−0.292705	+	0.103487i
$748$		0			0
$749$	10.3923	−	25.4558i	0.379727	−	0.930136i
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		0			0
$753$	20.7846	−	29.3939i	0.757433	−	1.07117i
$754$		0			0
$755$	13.8564	−	11.3137i	0.504286	−	0.411748i
$756$		0			0
$757$		−	29.3939i		−	1.06834i	−0.845378	−	0.534169i	$-0.820624\pi$
$757$		−	29.3939i		−	1.06834i	0.845378	−	0.534169i	$-0.179376\pi$
$758$		0			0
$759$	13.8564	+	9.79796i	0.502956	+	0.355643i
$760$		0			0
$761$	38.1051			1.38131			0.690655	−	0.723185i	$-0.257322\pi$
$761$	38.1051			1.38131			0.690655	+	0.723185i	$0.257322\pi$
$762$		0			0
$763$	−10.0000	+	24.4949i	−0.362024	+	0.886775i
$764$		0			0
$765$	−9.85641	+	16.2127i	−0.356359	+	0.586171i
$766$		0			0
$767$	−27.7128			−1.00065
$768$		0			0
$769$			19.5959i			0.706647i	0.935501	+	0.353323i	$0.114948\pi$
$769$			19.5959i			0.706647i	−0.935501	+	0.353323i	$0.885052\pi$
$770$		0			0
$771$	20.0000	+	14.1421i	0.720282	+	0.509317i
$772$		0			0
$773$			2.82843i			0.101731i	0.998706	+	0.0508657i	$0.0161981\pi$
$773$			2.82843i			0.101731i	−0.998706	+	0.0508657i	$0.983802\pi$
$774$		0			0
$775$	48.0000	+	9.79796i	1.72421	+	0.351953i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$	−8.00000	+	28.2843i	−0.285897	+	1.01080i
$784$		0			0
$785$	−6.92820	+	5.65685i	−0.247278	+	0.201902i
$786$		0			0
$787$	14.0000			0.499046			0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000			0.499046			0.249523	+	0.968369i	$0.419726\pi$
$788$		0			0
$789$	−3.46410	+	4.89898i	−0.123325	+	0.174408i
$790$		0			0
$791$	−6.92820	+	16.9706i	−0.246339	+	0.603404i
$792$		0			0
$793$			39.1918i			1.39174i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			36.7696i			1.30244i	0.758887	+	0.651222i	$0.225743\pi$
$797$			36.7696i			1.30244i	−0.758887	+	0.651222i	$0.774257\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$	−10.3923	−	29.3939i	−0.367194	−	1.03858i
$802$		0			0
$803$		−	22.6274i		−	0.798504i
$804$		0			0
$805$	−6.00000	−	19.5959i	−0.211472	−	0.690665i
$806$		0			0
$807$	10.3923	−	14.6969i	0.365826	−	0.517357i
$808$		0			0
$809$			22.6274i			0.795538i	0.917486	+	0.397769i	$0.130215\pi$
$809$			22.6274i			0.795538i	−0.917486	+	0.397769i	$0.869785\pi$
$810$		0			0
$811$			29.3939i			1.03216i	0.856541	+	0.516079i	$0.172609\pi$
$811$			29.3939i			1.03216i	−0.856541	+	0.516079i	$0.827391\pi$
$812$		0			0
$813$	−41.5692	−	29.3939i	−1.45790	−	1.03089i
$814$		0			0
$815$	−20.7846	−	25.4558i	−0.728053	−	0.891679i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−31.7128	−	1.51575i	−1.10814	−	0.0529646i
$820$		0			0
$821$			22.6274i			0.789702i	0.918745	+	0.394851i	$0.129204\pi$
$821$			22.6274i			0.789702i	−0.918745	+	0.394851i	$0.870796\pi$
$822$		0			0
$823$			4.89898i			0.170768i	0.996348	+	0.0853838i	$0.0272117\pi$
$823$			4.89898i			0.170768i	−0.996348	+	0.0853838i	$0.972788\pi$
$824$		0			0
$825$	−17.8564	+	16.7675i	−0.621680	+	0.583769i
$826$		0			0
$827$	10.3923			0.361376			0.180688	−	0.983540i	$-0.442168\pi$
$827$	10.3923			0.361376			0.180688	+	0.983540i	$0.442168\pi$
$828$		0			0
$829$			29.3939i			1.02089i	0.859910	+	0.510446i	$0.170520\pi$
$829$			29.3939i			1.02089i	−0.859910	+	0.510446i	$0.829480\pi$
$830$		0			0
$831$	−27.7128	−	19.5959i	−0.961347	−	0.679775i
$832$		0			0
$833$	13.8564	−	14.1421i	0.480096	−	0.489996i
$834$		0			0
$835$	−20.0000	−	24.4949i	−0.692129	−	0.847681i
$836$		0			0
$837$	−13.8564	+	48.9898i	−0.478947	+	1.69334i
$838$		0			0
$839$	27.7128			0.956753			0.478376	−	0.878155i	$-0.341226\pi$
$839$	27.7128			0.956753			0.478376	+	0.878155i	$0.341226\pi$
$840$		0			0
$841$	−3.00000			−0.103448
$842$		0			0
$843$	40.0000	+	28.2843i	1.37767	+	0.974162i
$844$		0			0
$845$	−5.19615	+	4.24264i	−0.178753	+	0.145951i
$846$		0			0
$847$	3.00000	−	7.34847i	0.103081	−	0.252496i
$848$		0			0
$849$	−14.0000	+	19.7990i	−0.480479	+	0.679500i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−20.0000			−0.684787			−0.342393	−	0.939557i	$-0.611238\pi$
$853$	−20.0000			−0.684787			−0.342393	+	0.939557i	$0.611238\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	48.0833i		−	1.64249i	−0.570574	−	0.821246i	$-0.693279\pi$
$857$		−	48.0833i		−	1.64249i	0.570574	−	0.821246i	$-0.306721\pi$
$858$		0			0
$859$			39.1918i			1.33721i	0.743619	+	0.668604i	$0.233108\pi$
$859$			39.1918i			1.33721i	−0.743619	+	0.668604i	$0.766892\pi$
$860$		0			0
$861$	−8.53590	−	13.3843i	−0.290903	−	0.456134i
$862$		0			0
$863$	10.3923			0.353758			0.176879	−	0.984233i	$-0.443400\pi$
$863$	10.3923			0.353758			0.176879	+	0.984233i	$0.443400\pi$
$864$		0			0
$865$	−28.0000	−	34.2929i	−0.952029	−	1.16599i
$866$		0			0
$867$	−9.00000	+	12.7279i	−0.305656	+	0.432263i
$868$		0			0
$869$		−	22.6274i		−	0.767583i
$870$		0			0
$871$			19.5959i			0.663982i
$872$		0			0
$873$	8.00000	+	22.6274i	0.270759	+	0.765822i
$874$		0			0
$875$	29.4449	−	2.82843i	0.995418	−	0.0956183i
$876$		0			0
$877$		−	48.9898i		−	1.65427i	−0.562005	−	0.827134i	$-0.689970\pi$
$877$		−	48.9898i		−	1.65427i	0.562005	−	0.827134i	$-0.310030\pi$
$878$		0			0
$879$	−4.00000	−	2.82843i	−0.134917	−	0.0954005i
$880$		0			0
$881$	−10.3923			−0.350126			−0.175063	−	0.984557i	$-0.556013\pi$
$881$	−10.3923			−0.350126			−0.175063	+	0.984557i	$0.556013\pi$
$882$		0			0
$883$		−	14.6969i		−	0.494591i	−0.968940	−	0.247296i	$-0.920458\pi$
$883$		−	14.6969i		−	0.494591i	0.968940	−	0.247296i	$-0.0795419\pi$
$884$		0			0
$885$	1.85641	+	26.7685i	0.0624024	+	0.899814i
$886$		0			0
$887$		−	2.82843i		−	0.0949693i	−0.998872	−	0.0474846i	$-0.984879\pi$
$887$		−	2.82843i		−	0.0949693i	0.998872	−	0.0474846i	$-0.0151205\pi$
$888$		0			0
$889$	−36.0000	−	14.6969i	−1.20740	−	0.492919i
$890$		0			0
$891$	−16.0000	−	19.7990i	−0.536020	−	0.663291i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−4.00000	−	4.89898i	−0.133705	−	0.163755i
$896$		0			0
$897$	13.8564	−	19.5959i	0.462652	−	0.654289i
$898$		0			0
$899$	−55.4256			−1.84855
$900$		0			0
$901$		0			0
$902$		0			0
$903$	18.9282	−	12.0716i	0.629891	−	0.401717i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			24.4949i			0.813340i	0.913575	+	0.406670i	$0.133310\pi$
$907$			24.4949i			0.813340i	−0.913575	+	0.406670i	$0.866690\pi$
$908$		0			0
$909$	17.3205	+	48.9898i	0.574485	+	1.62489i
$910$		0			0
$911$		−	48.0833i		−	1.59307i	−0.604593	−	0.796535i	$-0.706664\pi$
$911$		−	48.0833i		−	1.59307i	0.604593	−	0.796535i	$-0.293336\pi$
$912$		0			0
$913$	8.00000			0.264761
$914$		0			0
$915$	37.8564	−	2.62536i	1.25149	−	0.0867916i
$916$		0			0
$917$	−6.92820	+	16.9706i	−0.228789	+	0.560417i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$	10.0000	−	14.1421i	0.329511	−	0.465999i
$922$		0			0
$923$			11.3137i			0.372395i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	10.0000	+	28.2843i	0.328443	+	0.928977i
$928$		0			0
$929$	45.0333			1.47750			0.738748	−	0.673982i	$-0.235418\pi$
$929$	45.0333			1.47750			0.738748	+	0.673982i	$0.235418\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	27.7128	−	39.1918i	0.907277	−	1.28308i
$934$		0			0
$935$	13.8564	−	11.3137i	0.453153	−	0.369998i
$936$		0			0
$937$	−8.00000			−0.261349			−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000			−0.261349			−0.130674	+	0.991425i	$0.541714\pi$
$938$		0			0
$939$	−16.0000	+	22.6274i	−0.522140	+	0.738418i
$940$		0			0
$941$	−24.2487			−0.790485			−0.395243	−	0.918577i	$-0.629340\pi$
$941$	−24.2487			−0.790485			−0.395243	+	0.918577i	$0.629340\pi$
$942$		0			0
$943$	12.0000			0.390774
$944$		0			0
$945$	0.660254	+	30.7338i	0.0214781	+	0.999769i
$946$		0			0
$947$	3.46410			0.112568			0.0562841	−	0.998415i	$-0.482075\pi$
$947$	3.46410			0.112568			0.0562841	+	0.998415i	$0.482075\pi$
$948$		0			0
$949$	−32.0000			−1.03876
$950$		0			0
$951$	13.8564	−	19.5959i	0.449325	−	0.635441i
$952$		0			0
$953$	20.7846			0.673280			0.336640	−	0.941634i	$-0.390710\pi$
$953$	20.7846			0.673280			0.336640	+	0.941634i	$0.390710\pi$
$954$		0			0
$955$	−28.0000	−	34.2929i	−0.906059	−	1.10969i
$956$		0			0
$957$	16.0000	−	22.6274i	0.517207	−	0.731441i
$958$		0			0
$959$	6.92820	−	16.9706i	0.223723	−	0.548008i
$960$		0			0
$961$	−65.0000			−2.09677
$962$		0			0
$963$	−10.3923	−	29.3939i	−0.334887	−	0.947204i
$964$		0			0
$965$	−13.8564	−	16.9706i	−0.446054	−	0.546302i
$966$		0			0
$967$			34.2929i			1.10278i	0.834246	+	0.551392i	$0.185903\pi$
$967$			34.2929i			1.10278i	−0.834246	+	0.551392i	$0.814097\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.7846			0.667010			0.333505	−	0.942748i	$-0.391769\pi$
$971$	20.7846			0.667010			0.333505	+	0.942748i	$0.391769\pi$
$972$		0			0
$973$	24.0000	+	9.79796i	0.769405	+	0.314108i
$974$		0			0
$975$	23.7128	+	25.2528i	0.759418	+	0.808736i
$976$		0			0
$977$	−48.4974			−1.55157			−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974			−1.55157			−0.775785	+	0.630997i	$0.782646\pi$
$978$		0			0
$979$			29.3939i			0.939432i
$980$		0			0
$981$	10.0000	+	28.2843i	0.319275	+	0.903047i
$982$		0			0
$983$		−	2.82843i		−	0.0902128i	−0.998982	−	0.0451064i	$-0.985637\pi$
$983$		−	2.82843i		−	0.0902128i	0.998982	−	0.0451064i	$-0.0143627\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	10.9282	−	6.96953i	0.347849	−	0.221843i
$988$		0			0
$989$			16.9706i			0.539633i
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$	−28.0000	+	39.5980i	−0.888553	+	1.25660i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	52.0000			1.64686			0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000			1.64686			0.823428	+	0.567420i	$0.192059\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.k.a.209.3		4
3.2	odd	2	inner	1680.2.k.a.209.2		4
4.3	odd	2		105.2.g.c.104.3	yes	4
5.4	even	2		1680.2.k.c.209.1		4
7.6	odd	2		1680.2.k.c.209.2		4
12.11	even	2		105.2.g.c.104.2	yes	4
15.14	odd	2		1680.2.k.c.209.4		4
20.3	even	4		525.2.b.j.251.4		8
20.7	even	4		525.2.b.j.251.5		8
20.19	odd	2		105.2.g.a.104.2	yes	4
21.20	even	2		1680.2.k.c.209.3		4
28.3	even	6		735.2.p.c.509.2		8
28.11	odd	6		735.2.p.a.509.1		8
28.19	even	6		735.2.p.c.374.1		8
28.23	odd	6		735.2.p.a.374.2		8
28.27	even	2		105.2.g.a.104.4	yes	4
35.34	odd	2	inner	1680.2.k.a.209.4		4
60.23	odd	4		525.2.b.j.251.6		8
60.47	odd	4		525.2.b.j.251.3		8
60.59	even	2		105.2.g.a.104.3	yes	4
84.11	even	6		735.2.p.a.509.4		8
84.23	even	6		735.2.p.a.374.3		8
84.47	odd	6		735.2.p.c.374.4		8
84.59	odd	6		735.2.p.c.509.3		8
84.83	odd	2		105.2.g.a.104.1	✓	4
105.104	even	2	inner	1680.2.k.a.209.1		4
140.19	even	6		735.2.p.a.374.4		8
140.27	odd	4		525.2.b.j.251.8		8
140.39	odd	6		735.2.p.c.509.4		8
140.59	even	6		735.2.p.a.509.3		8
140.79	odd	6		735.2.p.c.374.3		8
140.83	odd	4		525.2.b.j.251.1		8
140.139	even	2		105.2.g.c.104.1	yes	4
420.59	odd	6		735.2.p.a.509.2		8
420.83	even	4		525.2.b.j.251.7		8
420.167	even	4		525.2.b.j.251.2		8
420.179	even	6		735.2.p.c.509.1		8
420.299	odd	6		735.2.p.a.374.1		8
420.359	even	6		735.2.p.c.374.2		8
420.419	odd	2		105.2.g.c.104.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.g.a.104.1	✓	4	84.83	odd	2
105.2.g.a.104.2	yes	4	20.19	odd	2
105.2.g.a.104.3	yes	4	60.59	even	2
105.2.g.a.104.4	yes	4	28.27	even	2
105.2.g.c.104.1	yes	4	140.139	even	2
105.2.g.c.104.2	yes	4	12.11	even	2
105.2.g.c.104.3	yes	4	4.3	odd	2
105.2.g.c.104.4	yes	4	420.419	odd	2
525.2.b.j.251.1		8	140.83	odd	4
525.2.b.j.251.2		8	420.167	even	4
525.2.b.j.251.3		8	60.47	odd	4
525.2.b.j.251.4		8	20.3	even	4
525.2.b.j.251.5		8	20.7	even	4
525.2.b.j.251.6		8	60.23	odd	4
525.2.b.j.251.7		8	420.83	even	4
525.2.b.j.251.8		8	140.27	odd	4
735.2.p.a.374.1		8	420.299	odd	6
735.2.p.a.374.2		8	28.23	odd	6
735.2.p.a.374.3		8	84.23	even	6
735.2.p.a.374.4		8	140.19	even	6
735.2.p.a.509.1		8	28.11	odd	6
735.2.p.a.509.2		8	420.59	odd	6
735.2.p.a.509.3		8	140.59	even	6
735.2.p.a.509.4		8	84.11	even	6
735.2.p.c.374.1		8	28.19	even	6
735.2.p.c.374.2		8	420.359	even	6
735.2.p.c.374.3		8	140.79	odd	6
735.2.p.c.374.4		8	84.47	odd	6
735.2.p.c.509.1		8	420.179	even	6
735.2.p.c.509.2		8	28.3	even	6
735.2.p.c.509.3		8	84.59	odd	6
735.2.p.c.509.4		8	140.39	odd	6
1680.2.k.a.209.1		4	105.104	even	2	inner
1680.2.k.a.209.2		4	3.2	odd	2	inner
1680.2.k.a.209.3		4	1.1	even	1	trivial
1680.2.k.a.209.4		4	35.34	odd	2	inner
1680.2.k.c.209.1		4	5.4	even	2
1680.2.k.c.209.2		4	7.6	odd	2
1680.2.k.c.209.3		4	21.20	even	2
1680.2.k.c.209.4		4	15.14	odd	2

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.41421i) q^{3} +(-1.73205 + 1.41421i) q^{5} +(1.00000 - 2.44949i) q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.41421i) q^{3} +(-1.73205 + 1.41421i) q^{5} +(1.00000 - 2.44949i) q^{7} +(-1.00000 - 2.82843i) q^{9} +2.82843i q^{11} +4.00000 q^{13} +(-0.267949 - 3.86370i) q^{15} +2.82843i q^{17} +(2.46410 + 3.86370i) q^{21} -3.46410 q^{23} +(1.00000 - 4.89898i) q^{25} +(5.00000 + 1.41421i) q^{27} +5.65685i q^{29} +9.79796i q^{31} +(-4.00000 - 2.82843i) q^{33} +(1.73205 + 5.65685i) q^{35} +(-4.00000 + 5.65685i) q^{39} -3.46410 q^{41} -4.89898i q^{43} +(5.73205 + 3.48477i) q^{45} -2.82843i q^{47} +(-5.00000 - 4.89898i) q^{49} +(-4.00000 - 2.82843i) q^{51} +(-4.00000 - 4.89898i) q^{55} -6.92820 q^{59} +9.79796i q^{61} +(-7.92820 - 0.378937i) q^{63} +(-6.92820 + 5.65685i) q^{65} +4.89898i q^{67} +(3.46410 - 4.89898i) q^{69} +2.82843i q^{71} -8.00000 q^{73} +(5.92820 + 6.31319i) q^{75} +(6.92820 + 2.82843i) q^{77} -8.00000 q^{79} +(-7.00000 + 5.65685i) q^{81} -2.82843i q^{83} +(-4.00000 - 4.89898i) q^{85} +(-8.00000 - 5.65685i) q^{87} +10.3923 q^{89} +(4.00000 - 9.79796i) q^{91} +(-13.8564 - 9.79796i) q^{93} -8.00000 q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^7 - 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{7} - 4 q^{9} + 16 q^{13} - 8 q^{15} - 4 q^{21} + 4 q^{25} + 20 q^{27} - 16 q^{33} - 16 q^{39} + 16 q^{45} - 20 q^{49} - 16 q^{51} - 16 q^{55} - 4 q^{63} - 32 q^{73} - 4 q^{75} - 32 q^{79} - 28 q^{81} - 16 q^{85} - 32 q^{87} + 16 q^{91} - 32 q^{97} + 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^7 - 4 * q^9 + 16 * q^13 - 8 * q^15 - 4 * q^21 + 4 * q^25 + 20 * q^27 - 16 * q^33 - 16 * q^39 + 16 * q^45 - 20 * q^49 - 16 * q^51 - 16 * q^55 - 4 * q^63 - 32 * q^73 - 4 * q^75 - 32 * q^79 - 28 * q^81 - 16 * q^85 - 32 * q^87 + 16 * q^91 - 32 * q^97 + 32 * q^99

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