LMFDB - Embedded newform 336.2.b.d.223.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,2,Mod(223,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	336.b (of order $2$, degree $1$, 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:	$2.68297350792$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	336.223
Dual form			336.2.b.d.223.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +3.46410i q^{11} -6.92820i q^{17} +4.00000 q^{19} +(2.00000 + 1.73205i) q^{21} +3.46410i q^{23} +5.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} +3.46410i q^{33} -2.00000 q^{37} -6.92820i q^{41} -3.46410i q^{43} +(1.00000 + 6.92820i) q^{49} -6.92820i q^{51} -6.00000 q^{53} +4.00000 q^{57} -12.0000 q^{59} +13.8564i q^{61} +(2.00000 + 1.73205i) q^{63} -3.46410i q^{67} +3.46410i q^{69} -10.3923i q^{71} -13.8564i q^{73} +5.00000 q^{75} +(-6.00000 + 6.92820i) q^{77} -10.3923i q^{79} +1.00000 q^{81} -12.0000 q^{83} -6.00000 q^{87} +6.92820i q^{89} -4.00000 q^{93} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 8 q^{19} + 4 q^{21} + 10 q^{25} + 2 q^{27} - 12 q^{29} - 8 q^{31} - 4 q^{37} + 2 q^{49} - 12 q^{53} + 8 q^{57} - 24 q^{59} + 4 q^{63} + 10 q^{75} - 12 q^{77} + 2 q^{81} - 24 q^{83} - 12 q^{87} - 8 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 8 * q^19 + 4 * q^21 + 10 * q^25 + 2 * q^27 - 12 * q^29 - 8 * q^31 - 4 * q^37 + 2 * q^49 - 12 * q^53 + 8 * q^57 - 24 * q^59 + 4 * q^63 + 10 * q^75 - 12 * q^77 + 2 * q^81 - 24 * q^83 - 12 * q^87 - 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.92820i		−	1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$		−	6.92820i		−	1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	2.00000	+	1.73205i	0.436436	+	0.377964i
$22$		0			0
$23$			3.46410i			0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$			3.46410i			0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$			3.46410i			0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		−	6.92820i		−	0.970143i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$			13.8564i			1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$			13.8564i			1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$		0			0
$63$	2.00000	+	1.73205i	0.251976	+	0.218218i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$			3.46410i			0.417029i
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$		−	13.8564i		−	1.62177i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$		−	13.8564i		−	1.62177i	0.585206	−	0.810885i	$-0.301014\pi$
$74$		0			0
$75$	5.00000			0.577350
$76$		0			0
$77$	−6.00000	+	6.92820i	−0.683763	+	0.789542i
$78$		0			0
$79$		−	10.3923i		−	1.16923i	−0.811312	−	0.584613i	$-0.801246\pi$
$79$		−	10.3923i		−	1.16923i	0.811312	−	0.584613i	$-0.198754\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.00000			−0.643268
$88$		0			0
$89$			6.92820i			0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$			6.92820i			0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−4.00000			−0.414781
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$		−	13.8564i		−	1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$		−	13.8564i		−	1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			17.3205i			1.67444i	0.546869	+	0.837218i	$0.315820\pi$
$107$			17.3205i			1.67444i	−0.546869	+	0.837218i	$0.684180\pi$
$108$		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
$111$	−2.00000			−0.189832
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.0000	−	13.8564i	1.10004	−	1.27021i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	6.92820i		−	0.624695i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			17.3205i			1.53695i	0.639882	+	0.768473i	$0.278983\pi$
$127$			17.3205i			1.53695i	−0.639882	+	0.768473i	$0.721017\pi$
$128$		0			0
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	8.00000	+	6.92820i	0.693688	+	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.00000	+	6.92820i	0.0824786	+	0.571429i
$148$		0			0
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$			3.46410i			0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$			3.46410i			0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$		0			0
$153$		−	6.92820i		−	0.560112i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	13.8564i		−	1.10586i	−0.833227	−	0.552931i	$-0.813509\pi$
$157$		−	13.8564i		−	1.10586i	0.833227	−	0.552931i	$-0.186491\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$	−6.00000	+	6.92820i	−0.472866	+	0.546019i
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	24.0000			1.85718			0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000			1.85718			0.928588	+	0.371113i	$0.121024\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$			13.8564i			1.05348i	0.850026	+	0.526742i	$0.176586\pi$
$173$			13.8564i			1.05348i	−0.850026	+	0.526742i	$0.823414\pi$
$174$		0			0
$175$	10.0000	+	8.66025i	0.755929	+	0.654654i
$176$		0			0
$177$	−12.0000			−0.901975
$178$		0			0
$179$			3.46410i			0.258919i	0.991585	+	0.129460i	$0.0413242\pi$
$179$			3.46410i			0.258919i	−0.991585	+	0.129460i	$0.958676\pi$
$180$		0			0
$181$		−	13.8564i		−	1.02994i	−0.857209	−	0.514969i	$-0.827803\pi$
$181$		−	13.8564i		−	1.02994i	0.857209	−	0.514969i	$-0.172197\pi$
$182$		0			0
$183$			13.8564i			1.02430i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	24.0000			1.75505
$188$		0			0
$189$	2.00000	+	1.73205i	0.145479	+	0.125988i
$190$		0			0
$191$			17.3205i			1.25327i	0.779314	+	0.626634i	$0.215568\pi$
$191$			17.3205i			1.25327i	−0.779314	+	0.626634i	$0.784432\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		−	3.46410i		−	0.244339i
$202$		0			0
$203$	−12.0000	−	10.3923i	−0.842235	−	0.729397i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			3.46410i			0.240772i
$208$		0			0
$209$			13.8564i			0.958468i
$210$		0			0
$211$			10.3923i			0.715436i	0.933830	+	0.357718i	$0.116445\pi$
$211$			10.3923i			0.715436i	−0.933830	+	0.357718i	$0.883555\pi$
$212$		0			0
$213$		−	10.3923i		−	0.712069i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8.00000	−	6.92820i	−0.543075	−	0.470317i
$218$		0			0
$219$		−	13.8564i		−	0.936329i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$		0			0
$225$	5.00000			0.333333
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$		−	13.8564i		−	0.915657i	−0.889041	−	0.457829i	$-0.848627\pi$
$229$		−	13.8564i		−	0.915657i	0.889041	−	0.457829i	$-0.151373\pi$
$230$		0			0
$231$	−6.00000	+	6.92820i	−0.394771	+	0.455842i
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	10.3923i		−	0.675053i
$238$		0			0
$239$		−	24.2487i		−	1.56852i	−0.620433	−	0.784259i	$-0.713043\pi$
$239$		−	24.2487i		−	1.56852i	0.620433	−	0.784259i	$-0.286957\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−12.0000			−0.760469
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.92820i			0.432169i	0.976375	+	0.216085i	$0.0693287\pi$
$257$			6.92820i			0.432169i	−0.976375	+	0.216085i	$0.930671\pi$
$258$		0			0
$259$	−4.00000	−	3.46410i	−0.248548	−	0.215249i
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		−	10.3923i		−	0.640817i	−0.947279	−	0.320408i	$-0.896180\pi$
$263$		−	10.3923i		−	0.640817i	0.947279	−	0.320408i	$-0.103820\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			6.92820i			0.423999i
$268$		0			0
$269$			13.8564i			0.844840i	0.906400	+	0.422420i	$0.138819\pi$
$269$			13.8564i			0.844840i	−0.906400	+	0.422420i	$0.861181\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			17.3205i			1.04447i
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	−	13.8564i	0.708338	−	0.817918i
$288$		0			0
$289$	−31.0000			−1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			3.46410i			0.201008i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6.00000	−	6.92820i	0.345834	−	0.399335i
$302$		0			0
$303$		−	13.8564i		−	0.796030i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			13.8564i			0.783210i	0.920133	+	0.391605i	$0.128080\pi$
$313$			13.8564i			0.783210i	−0.920133	+	0.391605i	$0.871920\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$			17.3205i			0.966736i
$322$		0			0
$323$		−	27.7128i		−	1.54198i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−10.0000			−0.553001
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	31.1769i		−	1.71364i	−0.515617	−	0.856819i	$-0.672437\pi$
$331$		−	31.1769i		−	1.71364i	0.515617	−	0.856819i	$-0.327563\pi$
$332$		0			0
$333$	−2.00000			−0.109599
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$		−	13.8564i		−	0.750366i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			17.3205i			0.929814i	0.885360	+	0.464907i	$0.153912\pi$
$347$			17.3205i			0.929814i	−0.885360	+	0.464907i	$0.846088\pi$
$348$		0			0
$349$			27.7128i			1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$			27.7128i			1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.7846i			1.10625i	0.833097	+	0.553127i	$0.186565\pi$
$353$			20.7846i			1.10625i	−0.833097	+	0.553127i	$0.813435\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	12.0000	−	13.8564i	0.635107	−	0.733359i
$358$		0			0
$359$			3.46410i			0.182828i	0.995813	+	0.0914141i	$0.0291387\pi$
$359$			3.46410i			0.182828i	−0.995813	+	0.0914141i	$0.970861\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$		0			0
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$		0			0
$369$		−	6.92820i		−	0.360668i
$370$		0			0
$371$	−12.0000	−	10.3923i	−0.623009	−	0.539542i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	3.46410i		−	0.177939i	−0.996034	−	0.0889695i	$-0.971643\pi$
$379$		−	3.46410i		−	0.177939i	0.996034	−	0.0889695i	$-0.0283574\pi$
$380$		0			0
$381$			17.3205i			0.887357i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	3.46410i		−	0.176090i
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	27.7128i		−	1.39087i	−0.718591	−	0.695433i	$-0.755213\pi$
$397$		−	27.7128i		−	1.39087i	0.718591	−	0.695433i	$-0.244787\pi$
$398$		0			0
$399$	8.00000	+	6.92820i	0.400501	+	0.346844i
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	6.92820i		−	0.343418i
$408$		0			0
$409$			13.8564i			0.685155i	0.939490	+	0.342578i	$0.111300\pi$
$409$			13.8564i			0.685155i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$	18.0000			0.887875
$412$		0			0
$413$	−24.0000	−	20.7846i	−1.18096	−	1.02274i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	34.6410i		−	1.68034i
$426$		0			0
$427$	−24.0000	+	27.7128i	−1.16144	+	1.34112i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			31.1769i			1.50174i	0.660451	+	0.750870i	$0.270365\pi$
$431$			31.1769i			1.50174i	−0.660451	+	0.750870i	$0.729635\pi$
$432$		0			0
$433$		−	13.8564i		−	0.665896i	−0.942945	−	0.332948i	$-0.891957\pi$
$433$		−	13.8564i		−	0.665896i	0.942945	−	0.332948i	$-0.108043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			13.8564i			0.662842i
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$		0			0
$441$	1.00000	+	6.92820i	0.0476190	+	0.329914i
$442$		0			0
$443$			31.1769i			1.48126i	0.671913	+	0.740630i	$0.265473\pi$
$443$			31.1769i			1.48126i	−0.671913	+	0.740630i	$0.734527\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	18.0000			0.851371
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$			3.46410i			0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$		−	6.92820i		−	0.323381i
$460$		0			0
$461$		−	13.8564i		−	0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$		−	13.8564i		−	0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$		0			0
$463$		−	10.3923i		−	0.482971i	−0.970404	−	0.241486i	$-0.922365\pi$
$463$		−	10.3923i		−	0.482971i	0.970404	−	0.241486i	$-0.0776347\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	6.00000	−	6.92820i	0.277054	−	0.319915i
$470$		0			0
$471$		−	13.8564i		−	0.638470i
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−6.00000	+	6.92820i	−0.273009	+	0.315244i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			31.1769i			1.41276i	0.707832	+	0.706380i	$0.249673\pi$
$487$			31.1769i			1.41276i	−0.707832	+	0.706380i	$0.750327\pi$
$488$		0			0
$489$		−	3.46410i		−	0.156652i
$490$		0			0
$491$			3.46410i			0.156333i	0.996940	+	0.0781664i	$0.0249065\pi$
$491$			3.46410i			0.156333i	−0.996940	+	0.0781664i	$0.975093\pi$
$492$		0			0
$493$			41.5692i			1.87218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.0000	−	20.7846i	0.807410	−	0.932317i
$498$		0			0
$499$			10.3923i			0.465223i	0.972570	+	0.232612i	$0.0747271\pi$
$499$			10.3923i			0.465223i	−0.972570	+	0.232612i	$0.925273\pi$
$500$		0			0
$501$	24.0000			1.07224
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	13.0000			0.577350
$508$		0			0
$509$		−	27.7128i		−	1.22835i	−0.789170	−	0.614174i	$-0.789489\pi$
$509$		−	27.7128i		−	1.22835i	0.789170	−	0.614174i	$-0.210511\pi$
$510$		0			0
$511$	24.0000	−	27.7128i	1.06170	−	1.22594i
$512$		0			0
$513$	4.00000			0.176604
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			13.8564i			0.608229i
$520$		0			0
$521$		−	6.92820i		−	0.303530i	−0.988417	−	0.151765i	$-0.951504\pi$
$521$		−	6.92820i		−	0.303530i	0.988417	−	0.151765i	$-0.0484957\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$	10.0000	+	8.66025i	0.436436	+	0.377964i
$526$		0			0
$527$			27.7128i			1.20719i
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$			3.46410i			0.149487i
$538$		0			0
$539$	−24.0000	+	3.46410i	−1.03375	+	0.149209i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		−	13.8564i		−	0.594635i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			24.2487i			1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$			24.2487i			1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$			13.8564i			0.591377i
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$	18.0000	−	20.7846i	0.765438	−	0.883852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.00000			−0.254228			−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000			−0.254228			−0.127114	+	0.991888i	$0.540571\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.00000	+	1.73205i	0.0839921	+	0.0727393i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$		−	17.3205i		−	0.724841i	−0.932015	−	0.362420i	$-0.881950\pi$
$571$		−	17.3205i		−	0.724841i	0.932015	−	0.362420i	$-0.118050\pi$
$572$		0			0
$573$			17.3205i			0.723575i
$574$		0			0
$575$			17.3205i			0.722315i
$576$		0			0
$577$			13.8564i			0.576850i	0.957503	+	0.288425i	$0.0931316\pi$
$577$			13.8564i			0.576850i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	−14.0000			−0.581820
$580$		0			0
$581$	−24.0000	−	20.7846i	−0.995688	−	0.862291i
$582$		0			0
$583$		−	20.7846i		−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	18.0000			0.740421
$592$		0			0
$593$		−	6.92820i		−	0.284507i	−0.989830	−	0.142254i	$-0.954565\pi$
$593$		−	6.92820i		−	0.284507i	0.989830	−	0.142254i	$-0.0454349\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−20.0000			−0.818546
$598$		0			0
$599$		−	10.3923i		−	0.424618i	−0.977203	−	0.212309i	$-0.931902\pi$
$599$		−	10.3923i		−	0.424618i	0.977203	−	0.212309i	$-0.0680983\pi$
$600$		0			0
$601$			27.7128i			1.13043i	0.824944	+	0.565215i	$0.191207\pi$
$601$			27.7128i			1.13043i	−0.824944	+	0.565215i	$0.808793\pi$
$602$		0			0
$603$		−	3.46410i		−	0.141069i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	4.00000			0.162355			0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000			0.162355			0.0811775	+	0.996700i	$0.474132\pi$
$608$		0			0
$609$	−12.0000	−	10.3923i	−0.486265	−	0.421117i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$			3.46410i			0.139010i
$622$		0			0
$623$	−12.0000	+	13.8564i	−0.480770	+	0.555145i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$			13.8564i			0.553372i
$628$		0			0
$629$			13.8564i			0.552491i
$630$		0			0
$631$			3.46410i			0.137904i	0.997620	+	0.0689519i	$0.0219655\pi$
$631$			3.46410i			0.137904i	−0.997620	+	0.0689519i	$0.978035\pi$
$632$		0			0
$633$			10.3923i			0.413057i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	10.3923i		−	0.411113i
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$	−44.0000			−1.73519			−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000			−1.73519			−0.867595	+	0.497271i	$0.834335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		−	41.5692i		−	1.63173i
$650$		0			0
$651$	−8.00000	−	6.92820i	−0.313545	−	0.271538i
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	13.8564i		−	0.540590i
$658$		0			0
$659$		−	24.2487i		−	0.944596i	−0.881439	−	0.472298i	$-0.843425\pi$
$659$		−	24.2487i		−	0.944596i	0.881439	−	0.472298i	$-0.156575\pi$
$660$		0			0
$661$			27.7128i			1.07790i	0.842337	+	0.538952i	$0.181179\pi$
$661$			27.7128i			1.07790i	−0.842337	+	0.538952i	$0.818821\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	20.7846i		−	0.804783i
$668$		0			0
$669$	4.00000			0.154649
$670$		0			0
$671$	−48.0000			−1.85302
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$	5.00000			0.192450
$676$		0			0
$677$			13.8564i			0.532545i	0.963898	+	0.266272i	$0.0857921\pi$
$677$			13.8564i			0.532545i	−0.963898	+	0.266272i	$0.914208\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$		−	24.2487i		−	0.927851i	−0.885874	−	0.463926i	$-0.846441\pi$
$683$		−	24.2487i		−	0.927851i	0.885874	−	0.463926i	$-0.153559\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	13.8564i		−	0.528655i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−4.00000			−0.152167			−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000			−0.152167			−0.0760836	+	0.997101i	$0.524242\pi$
$692$		0			0
$693$	−6.00000	+	6.92820i	−0.227921	+	0.263181i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−48.0000			−1.81813
$698$		0			0
$699$	18.0000			0.680823
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.0000	−	27.7128i	0.902613	−	1.04225i
$708$		0			0
$709$	−34.0000			−1.27690			−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000			−1.27690			−0.638448	+	0.769665i	$0.720423\pi$
$710$		0			0
$711$		−	10.3923i		−	0.389742i
$712$		0			0
$713$		−	13.8564i		−	0.518927i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	24.2487i		−	0.905585i
$718$		0			0
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	−8.00000	−	6.92820i	−0.297936	−	0.258020i
$722$		0			0
$723$			13.8564i			0.515325i
$724$		0			0
$725$	−30.0000			−1.11417
$726$		0			0
$727$	20.0000			0.741759			0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000			0.741759			0.370879	+	0.928681i	$0.379056\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$			27.7128i			1.02360i	0.859106	+	0.511798i	$0.171020\pi$
$733$			27.7128i			1.02360i	−0.859106	+	0.511798i	$0.828980\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000			0.442026
$738$		0			0
$739$		−	3.46410i		−	0.127429i	−0.997968	−	0.0637145i	$-0.979705\pi$
$739$		−	3.46410i		−	0.127429i	0.997968	−	0.0637145i	$-0.0202947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.46410i			0.127086i	0.997979	+	0.0635428i	$0.0202399\pi$
$743$			3.46410i			0.127086i	−0.997979	+	0.0635428i	$0.979760\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−12.0000			−0.439057
$748$		0			0
$749$	−30.0000	+	34.6410i	−1.09618	+	1.26576i
$750$		0			0
$751$			17.3205i			0.632034i	0.948753	+	0.316017i	$0.102346\pi$
$751$			17.3205i			0.632034i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$	−12.0000			−0.437304
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$			6.92820i			0.251147i	0.992084	+	0.125574i	$0.0400771\pi$
$761$			6.92820i			0.251147i	−0.992084	+	0.125574i	$0.959923\pi$
$762$		0			0
$763$	−20.0000	−	17.3205i	−0.724049	−	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$			6.92820i			0.249513i
$772$		0			0
$773$			27.7128i			0.996761i	0.866959	+	0.498380i	$0.166072\pi$
$773$			27.7128i			0.996761i	−0.866959	+	0.498380i	$0.833928\pi$
$774$		0			0
$775$	−20.0000			−0.718421
$776$		0			0
$777$	−4.00000	−	3.46410i	−0.143499	−	0.124274i
$778$		0			0
$779$		−	27.7128i		−	0.992915i
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−4.00000			−0.142585			−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000			−0.142585			−0.0712923	+	0.997455i	$0.522712\pi$
$788$		0			0
$789$		−	10.3923i		−	0.369976i
$790$		0			0
$791$	−12.0000	−	10.3923i	−0.426671	−	0.369508i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			27.7128i			0.981638i	0.871262	+	0.490819i	$0.163302\pi$
$797$			27.7128i			0.981638i	−0.871262	+	0.490819i	$0.836698\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			6.92820i			0.244796i
$802$		0			0
$803$	48.0000			1.69388
$804$		0			0
$805$		0			0
$806$		0			0
$807$			13.8564i			0.487769i
$808$		0			0
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$	−28.0000			−0.982003
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.00000			−0.209401			−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000			−0.209401			−0.104701	+	0.994504i	$0.533388\pi$
$822$		0			0
$823$		−	51.9615i		−	1.81126i	−0.424064	−	0.905632i	$-0.639397\pi$
$823$		−	51.9615i		−	1.81126i	0.424064	−	0.905632i	$-0.360603\pi$
$824$		0			0
$825$			17.3205i			0.603023i
$826$		0			0
$827$			17.3205i			0.602293i	0.953578	+	0.301147i	$0.0973693\pi$
$827$			17.3205i			0.602293i	−0.953578	+	0.301147i	$0.902631\pi$
$828$		0			0
$829$		−	41.5692i		−	1.44376i	−0.692019	−	0.721879i	$-0.743279\pi$
$829$		−	41.5692i		−	1.44376i	0.692019	−	0.721879i	$-0.256721\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	48.0000	−	6.92820i	1.66310	−	0.240048i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.00000			−0.138260
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−30.0000			−1.03325
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2.00000	−	1.73205i	−0.0687208	−	0.0595140i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$		−	6.92820i		−	0.237496i
$852$		0			0
$853$		−	13.8564i		−	0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$		−	13.8564i		−	0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			34.6410i			1.18331i	0.806190	+	0.591657i	$0.201526\pi$
$857$			34.6410i			1.18331i	−0.806190	+	0.591657i	$0.798474\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	12.0000	−	13.8564i	0.408959	−	0.472225i
$862$		0			0
$863$		−	10.3923i		−	0.353758i	−0.984233	−	0.176879i	$-0.943400\pi$
$863$		−	10.3923i		−	0.353758i	0.984233	−	0.176879i	$-0.0566002\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−31.0000			−1.05282
$868$		0			0
$869$	36.0000			1.22122
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	20.7846i		−	0.700251i	−0.936703	−	0.350126i	$-0.886139\pi$
$881$		−	20.7846i		−	0.700251i	0.936703	−	0.350126i	$-0.113861\pi$
$882$		0			0
$883$			10.3923i			0.349729i	0.984593	+	0.174864i	$0.0559487\pi$
$883$			10.3923i			0.349729i	−0.984593	+	0.174864i	$0.944051\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0000			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$888$		0			0
$889$	−30.0000	+	34.6410i	−1.00617	+	1.16182i
$890$		0			0
$891$			3.46410i			0.116052i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.0000			0.800445
$900$		0			0
$901$			41.5692i			1.38487i
$902$		0			0
$903$	6.00000	−	6.92820i	0.199667	−	0.230556i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			51.9615i			1.72535i	0.505755	+	0.862677i	$0.331214\pi$
$907$			51.9615i			1.72535i	−0.505755	+	0.862677i	$0.668786\pi$
$908$		0			0
$909$		−	13.8564i		−	0.459588i
$910$		0			0
$911$			31.1769i			1.03294i	0.856306	+	0.516469i	$0.172754\pi$
$911$			31.1769i			1.03294i	−0.856306	+	0.516469i	$0.827246\pi$
$912$		0			0
$913$		−	41.5692i		−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.0000	+	20.7846i	0.792550	+	0.686368i
$918$		0			0
$919$			31.1769i			1.02843i	0.857661	+	0.514216i	$0.171917\pi$
$919$			31.1769i			1.02843i	−0.857661	+	0.514216i	$0.828083\pi$
$920$		0			0
$921$	−20.0000			−0.659022
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−10.0000			−0.328798
$926$		0			0
$927$	−4.00000			−0.131377
$928$		0			0
$929$			6.92820i			0.227307i	0.993520	+	0.113653i	$0.0362554\pi$
$929$			6.92820i			0.227307i	−0.993520	+	0.113653i	$0.963745\pi$
$930$		0			0
$931$	4.00000	+	27.7128i	0.131095	+	0.908251i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	13.8564i		−	0.452669i	−0.974050	−	0.226335i	$-0.927326\pi$
$937$		−	13.8564i		−	0.452669i	0.974050	−	0.226335i	$-0.0726743\pi$
$938$		0			0
$939$			13.8564i			0.452187i
$940$		0			0
$941$		−	27.7128i		−	0.903412i	−0.892167	−	0.451706i	$-0.850816\pi$
$941$		−	27.7128i		−	0.903412i	0.892167	−	0.451706i	$-0.149184\pi$
$942$		0			0
$943$	24.0000			0.781548
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	51.9615i		−	1.68852i	−0.535932	−	0.844261i	$-0.680040\pi$
$947$		−	51.9615i		−	1.68852i	0.535932	−	0.844261i	$-0.319960\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	18.0000			0.583690
$952$		0			0
$953$	−30.0000			−0.971795			−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000			−0.971795			−0.485898	+	0.874016i	$0.661507\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	20.7846i		−	0.671871i
$958$		0			0
$959$	36.0000	+	31.1769i	1.16250	+	1.00676i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$			17.3205i			0.558146i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	24.2487i		−	0.779786i	−0.920860	−	0.389893i	$-0.872512\pi$
$967$		−	24.2487i		−	0.779786i	0.920860	−	0.389893i	$-0.127488\pi$
$968$		0			0
$969$		−	27.7128i		−	0.890264i
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$	−8.00000	−	6.92820i	−0.256468	−	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.0000			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$	−10.0000			−0.319275
$982$		0			0
$983$	−24.0000			−0.765481			−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000			−0.765481			−0.382741	+	0.923856i	$0.625020\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$			17.3205i			0.550204i	0.961415	+	0.275102i	$0.0887116\pi$
$991$			17.3205i			0.550204i	−0.961415	+	0.275102i	$0.911288\pi$
$992$		0			0
$993$		−	31.1769i		−	0.989369i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	41.5692i		−	1.31651i	−0.752795	−	0.658255i	$-0.771295\pi$
$997$		−	41.5692i		−	1.31651i	0.752795	−	0.658255i	$-0.228705\pi$
$998$		0			0
$999$	−2.00000			−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.b.d.223.2	yes	2
3.2	odd	2		1008.2.b.h.559.2		2
4.3	odd	2		336.2.b.a.223.1	✓	2
7.2	even	3		2352.2.bl.c.31.1		2
7.3	odd	6		2352.2.bl.j.607.1		2
7.4	even	3		2352.2.bl.d.607.1		2
7.5	odd	6		2352.2.bl.i.31.1		2
7.6	odd	2		336.2.b.a.223.2	yes	2
8.3	odd	2		1344.2.b.c.895.1		2
8.5	even	2		1344.2.b.b.895.2		2
12.11	even	2		1008.2.b.a.559.1		2
21.20	even	2		1008.2.b.a.559.2		2
24.5	odd	2		4032.2.b.g.3583.2		2
24.11	even	2		4032.2.b.c.3583.1		2
28.3	even	6		2352.2.bl.c.607.1		2
28.11	odd	6		2352.2.bl.i.607.1		2
28.19	even	6		2352.2.bl.d.31.1		2
28.23	odd	6		2352.2.bl.j.31.1		2
28.27	even	2	inner	336.2.b.d.223.1	yes	2
56.13	odd	2		1344.2.b.c.895.2		2
56.27	even	2		1344.2.b.b.895.1		2
84.83	odd	2		1008.2.b.h.559.1		2
168.83	odd	2		4032.2.b.g.3583.1		2
168.125	even	2		4032.2.b.c.3583.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.b.a.223.1	✓	2	4.3	odd	2
336.2.b.a.223.2	yes	2	7.6	odd	2
336.2.b.d.223.1	yes	2	28.27	even	2	inner
336.2.b.d.223.2	yes	2	1.1	even	1	trivial
1008.2.b.a.559.1		2	12.11	even	2
1008.2.b.a.559.2		2	21.20	even	2
1008.2.b.h.559.1		2	84.83	odd	2
1008.2.b.h.559.2		2	3.2	odd	2
1344.2.b.b.895.1		2	56.27	even	2
1344.2.b.b.895.2		2	8.5	even	2
1344.2.b.c.895.1		2	8.3	odd	2
1344.2.b.c.895.2		2	56.13	odd	2
2352.2.bl.c.31.1		2	7.2	even	3
2352.2.bl.c.607.1		2	28.3	even	6
2352.2.bl.d.31.1		2	28.19	even	6
2352.2.bl.d.607.1		2	7.4	even	3
2352.2.bl.i.31.1		2	7.5	odd	6
2352.2.bl.i.607.1		2	28.11	odd	6
2352.2.bl.j.31.1		2	28.23	odd	6
2352.2.bl.j.607.1		2	7.3	odd	6
4032.2.b.c.3583.1		2	24.11	even	2
4032.2.b.c.3583.2		2	168.125	even	2
4032.2.b.g.3583.1		2	168.83	odd	2
4032.2.b.g.3583.2		2	24.5	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +(2.00000 + 1.73205i) q^{7} +1.00000 q^{9} +3.46410i q^{11} -6.92820i q^{17} +4.00000 q^{19} +(2.00000 + 1.73205i) q^{21} +3.46410i q^{23} +5.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} +3.46410i q^{33} -2.00000 q^{37} -6.92820i q^{41} -3.46410i q^{43} +(1.00000 + 6.92820i) q^{49} -6.92820i q^{51} -6.00000 q^{53} +4.00000 q^{57} -12.0000 q^{59} +13.8564i q^{61} +(2.00000 + 1.73205i) q^{63} -3.46410i q^{67} +3.46410i q^{69} -10.3923i q^{71} -13.8564i q^{73} +5.00000 q^{75} +(-6.00000 + 6.92820i) q^{77} -10.3923i q^{79} +1.00000 q^{81} -12.0000 q^{83} -6.00000 q^{87} +6.92820i q^{89} -4.00000 q^{93} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 8 q^{19} + 4 q^{21} + 10 q^{25} + 2 q^{27} - 12 q^{29} - 8 q^{31} - 4 q^{37} + 2 q^{49} - 12 q^{53} + 8 q^{57} - 24 q^{59} + 4 q^{63} + 10 q^{75} - 12 q^{77} + 2 q^{81} - 24 q^{83} - 12 q^{87} - 8 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 8 * q^19 + 4 * q^21 + 10 * q^25 + 2 * q^27 - 12 * q^29 - 8 * q^31 - 4 * q^37 + 2 * q^49 - 12 * q^53 + 8 * q^57 - 24 * q^59 + 4 * q^63 + 10 * q^75 - 12 * q^77 + 2 * q^81 - 24 * q^83 - 12 * q^87 - 8 * q^93

Introduction

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Representations

Groups

Database

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