LMFDB - Embedded newform 544.2.cc.b.175.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [544,2,Mod(79,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 8, 7])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.79"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 544 = 2^{5} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	544.cc (of order $16$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$4.34386186996$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			175.1
Root			$-0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	544.175
Dual form			544.2.cc.b.143.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39635 - 2.08979i) q^{3} +(-1.26937 + 3.06453i) q^{9} +(-4.72705 - 3.15851i) q^{11} +(-3.76118 - 1.68925i) q^{17} +(2.83730 + 6.84984i) q^{19} +(-4.61940 - 1.91342i) q^{25} +(0.781491 - 0.155448i) q^{27} +14.2889i q^{33} +(0.252982 + 1.27183i) q^{41} +(-4.46447 + 10.7782i) q^{43} +(-6.46716 + 2.67878i) q^{49} +(1.72176 + 10.2189i) q^{51} +(10.3529 - 15.4942i) q^{57} +(-9.94975 - 4.12132i) q^{59} -16.1815i q^{67} +(2.96960 - 14.9292i) q^{73} +(2.45167 + 12.3254i) q^{75} +(5.62038 + 5.62038i) q^{81} +(14.3640 - 5.94975i) q^{83} +(8.56628 - 8.56628i) q^{89} +(-14.4112 - 2.86657i) q^{97} +(15.6797 - 10.4769i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 16 q^{9} + 32 q^{27} - 32 q^{41} - 64 q^{43} + 40 q^{51} - 40 q^{57} - 40 q^{59} + 48 q^{73} + 56 q^{81} + 64 q^{83} + 48 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + 16 * q^9 + 32 * q^27 - 32 * q^41 - 64 * q^43 + 40 * q^51 - 40 * q^57 - 40 * q^59 + 48 * q^73 + 56 * q^81 + 64 * q^83 + 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/544\mathbb{Z}\right)^\times$.

$n$	$69$	$511$	$513$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{16}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.39635	−	2.08979i	−0.806185	−	1.20654i	−0.975287	−	0.220942i	$-0.929087\pi$
$3$	−1.39635	−	2.08979i	−0.806185	−	1.20654i	0.169102	−	0.985599i	$-0.445913\pi$
$4$		0			0
$5$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$5$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$6$		0			0
$7$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$7$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$8$		0			0
$9$	−1.26937	+	3.06453i	−0.423124	+	1.02151i
$10$		0			0
$11$	−4.72705	−	3.15851i	−1.42526	−	0.952327i	−0.998857	−	0.0477934i	$-0.984781\pi$
$11$	−4.72705	−	3.15851i	−1.42526	−	0.952327i	−0.426401	−	0.904534i	$-0.640219\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.76118	−	1.68925i	−0.912219	−	0.409702i
$17$	−3.76118	−	1.68925i	−0.912219	−	0.409702i
$18$		0			0
$19$	2.83730	+	6.84984i	0.650921	+	1.57146i	0.811445	+	0.584429i	$0.198682\pi$
$19$	2.83730	+	6.84984i	0.650921	+	1.57146i	−0.160524	+	0.987032i	$0.551318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$23$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$24$		0			0
$25$	−4.61940	−	1.91342i	−0.923880	−	0.382683i
$26$		0			0
$27$	0.781491	−	0.155448i	0.150398	−	0.0299160i
$28$		0			0
$29$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$29$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$30$		0			0
$31$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$31$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$32$		0			0
$33$			14.2889i			2.48738i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$37$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.252982	+	1.27183i	0.0395091	+	0.198626i	0.995499	−	0.0947747i	$-0.0302131\pi$
$41$	0.252982	+	1.27183i	0.0395091	+	0.198626i	−0.955990	+	0.293400i	$0.905213\pi$
$42$		0			0
$43$	−4.46447	+	10.7782i	−0.680825	+	1.64366i	0.0816682	+	0.996660i	$0.473975\pi$
$43$	−4.46447	+	10.7782i	−0.680825	+	1.64366i	−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$	−6.46716	+	2.67878i	−0.923880	+	0.382683i
$50$		0			0
$51$	1.72176	+	10.2189i	0.241095	+	1.43093i
$52$		0			0
$53$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$53$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	10.3529	−	15.4942i	1.37127	−	2.05225i
$58$		0			0
$59$	−9.94975	−	4.12132i	−1.29535	−	0.536550i	−0.374772	−	0.927117i	$-0.622279\pi$
$59$	−9.94975	−	4.12132i	−1.29535	−	0.536550i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$61$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	16.1815i		−	1.97688i	−0.151601	−	0.988442i	$-0.548443\pi$
$67$		−	16.1815i		−	1.97688i	0.151601	−	0.988442i	$-0.451557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$71$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$72$		0			0
$73$	2.96960	−	14.9292i	0.347565	−	1.74733i	−0.271921	−	0.962319i	$-0.587659\pi$
$73$	2.96960	−	14.9292i	0.347565	−	1.74733i	0.619486	−	0.785007i	$-0.287341\pi$
$74$		0			0
$75$	2.45167	+	12.3254i	0.283094	+	1.42321i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$79$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$80$		0			0
$81$	5.62038	+	5.62038i	0.624487	+	0.624487i
$82$		0			0
$83$	14.3640	−	5.94975i	1.57665	−	0.653070i	0.588771	−	0.808300i	$-0.299612\pi$
$83$	14.3640	−	5.94975i	1.57665	−	0.653070i	0.987878	+	0.155230i	$0.0496119\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.56628	−	8.56628i	0.908024	−	0.908024i	−0.0880885	−	0.996113i	$-0.528076\pi$
$89$	8.56628	−	8.56628i	0.908024	−	0.908024i	0.996113	+	0.0880885i	$0.0280758\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−14.4112	−	2.86657i	−1.46324	−	0.291057i	−0.601690	−	0.798730i	$-0.705506\pi$
$97$	−14.4112	−	2.86657i	−1.46324	−	0.291057i	−0.861550	+	0.507673i	$0.830506\pi$
$98$		0			0
$99$	15.6797	−	10.4769i	1.57587	−	1.05296i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.56001	−	17.8974i	0.344159	−	1.73021i	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	3.56001	−	17.8974i	0.344159	−	1.73021i	0.634180	−	0.773185i	$-0.281338\pi$
$108$		0			0
$109$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$109$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.30644	−	4.21383i	−0.593260	−	0.396404i	0.222383	−	0.974959i	$-0.428617\pi$
$113$	−6.30644	−	4.21383i	−0.593260	−	0.396404i	−0.815643	+	0.578556i	$0.803617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.15927	+	19.6982i	0.741751	+	1.79075i
$122$		0			0
$123$	2.30460	−	2.30460i	0.207798	−	0.207798i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$127$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$128$		0			0
$129$	28.7581	−	5.72034i	2.53201	−	0.503648i
$130$		0			0
$131$	−22.3849	−	4.45262i	−1.95577	−	0.389028i	−0.992874	−	0.119170i	$-0.961977\pi$
$131$	−22.3849	−	4.45262i	−1.95577	−	0.389028i	−0.962900	−	0.269858i	$-0.913023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.7574			−1.00450			−0.502249	−	0.864723i	$-0.667494\pi$
$137$	−11.7574			−1.00450			−0.502249	+	0.864723i	$0.667494\pi$
$138$		0			0
$139$	−2.23909	−	3.35103i	−0.189917	−	0.284231i	0.724276	−	0.689510i	$-0.242174\pi$
$139$	−2.23909	−	3.35103i	−0.189917	−	0.284231i	−0.914193	+	0.405279i	$0.867174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	14.6285	+	9.77447i	1.20654	+	0.806185i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		0			0
$153$	9.95108	−	9.38196i	0.804497	−	0.758487i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	24.8697	−	4.94689i	1.94794	−	0.387470i	0.951015	−	0.309144i	$-0.100042\pi$
$163$	24.8697	−	4.94689i	1.94794	−	0.387470i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$167$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	−24.5931			−1.88068
$172$		0			0
$173$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$173$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.28067	+	26.5477i	0.396919	+	1.99545i
$178$		0			0
$179$	−0.486803	+	1.17525i	−0.0363854	+	0.0878421i	−0.941028	−	0.338330i	$-0.890138\pi$
$179$	−0.486803	+	1.17525i	−0.0363854	+	0.0878421i	0.904642	+	0.426172i	$0.140138\pi$
$180$		0			0
$181$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$181$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	12.4438	+	19.8649i	0.909978	+	1.45266i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	−10.1483	+	15.1880i	−0.730492	+	1.09326i	0.261283	+	0.965262i	$0.415854\pi$
$193$	−10.1483	+	15.1880i	−0.730492	+	1.09326i	−0.991775	+	0.127996i	$0.959146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$197$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$198$		0			0
$199$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$199$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$200$		0			0
$201$	−33.8159	+	22.5951i	−2.38519	+	1.59373i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.22327	−	41.3412i	0.568815	−	2.85963i
$210$		0			0
$211$	2.61183	+	13.1305i	0.179806	+	0.903943i	0.960340	+	0.278831i	$0.0899469\pi$
$211$	2.61183	+	13.1305i	0.179806	+	0.903943i	−0.780535	+	0.625112i	$0.785053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−35.3454	+	14.6406i	−2.38842	+	0.989317i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$223$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$224$		0			0
$225$	11.7275	−	11.7275i	0.781830	−	0.781830i
$226$		0			0
$227$	1.71040	−	2.55979i	0.113523	−	0.169899i	−0.770357	−	0.637613i	$-0.779922\pi$
$227$	1.71040	−	2.55979i	0.113523	−	0.169899i	0.883880	+	0.467714i	$0.154922\pi$
$228$		0			0
$229$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$229$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	27.7758	+	5.52495i	1.81965	+	0.361952i	0.982683	−	0.185296i	$-0.0593245\pi$
$233$	27.7758	+	5.52495i	1.81965	+	0.361952i	0.836971	+	0.547248i	$0.184325\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−6.77233	−	10.1355i	−0.436244	−	0.652885i	0.546585	−	0.837404i	$-0.315928\pi$
$241$	−6.77233	−	10.1355i	−0.436244	−	0.652885i	−0.982829	+	0.184518i	$0.940928\pi$
$242$		0			0
$243$	4.36373	−	21.9379i	0.279933	−	1.40732i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−32.4909	−	21.7097i	−2.05903	−	1.37580i
$250$		0			0
$251$	−12.3387	−	12.3387i	−0.778813	−	0.778813i	0.200816	−	0.979629i	$-0.435641\pi$
$251$	−12.3387	−	12.3387i	−0.778813	−	0.778813i	−0.979629	+	0.200816i	$0.935641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.2635	−	29.6066i	−0.764973	−	1.84681i	−0.412108	−	0.911135i	$-0.635208\pi$
$257$	−12.2635	−	29.6066i	−0.764973	−	1.84681i	−0.352865	−	0.935674i	$-0.614792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$263$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−29.8633	−	5.94018i	−1.82760	−	0.363533i
$268$		0			0
$269$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$269$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.7926	+	23.6352i	0.952327	+	1.42526i
$276$		0			0
$277$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$277$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.5244	+	30.2367i	−0.747145	+	1.80377i	−0.173176	+	0.984891i	$0.555403\pi$
$281$	−12.5244	+	30.2367i	−0.747145	+	1.80377i	−0.573969	+	0.818877i	$0.694597\pi$
$282$		0			0
$283$	3.45045	+	2.30552i	0.205108	+	0.137049i	0.653882	−	0.756596i	$-0.273139\pi$
$283$	3.45045	+	2.30552i	0.205108	+	0.137049i	−0.448774	+	0.893645i	$0.648139\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	11.2929	+	12.7071i	0.664288	+	0.747477i
$290$		0			0
$291$	14.1326	+	34.1192i	0.828470	+	2.00010i
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−4.18513	−	1.73354i	−0.242846	−	0.100590i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.48528			−0.484281			−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528			−0.484281			−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$311$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$312$		0			0
$313$	−0.621610	−	3.12504i	−0.0351355	−	0.176638i	0.959233	−	0.282617i	$-0.0912024\pi$
$313$	−0.621610	−	3.12504i	−0.0351355	−	0.176638i	−0.994368	+	0.105979i	$0.966202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$317$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−42.3728	+	17.5514i	−2.36502	+	0.979623i
$322$		0			0
$323$	0.899495	−	30.5563i	0.0500492	−	1.70020i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.1924	−	5.46447i	−0.725119	−	0.300354i	−0.0105746	−	0.999944i	$-0.503366\pi$
$331$	−13.1924	−	5.46447i	−0.725119	−	0.300354i	−0.714545	+	0.699590i	$0.753366\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	29.9497	−	20.0118i	1.63147	−	1.09011i	0.708154	−	0.706058i	$-0.249528\pi$
$337$	29.9497	−	20.0118i	1.63147	−	1.09011i	0.923312	−	0.384052i	$-0.125472\pi$
$338$		0			0
$339$			19.0631i			1.03537i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.95859	+	24.9285i	0.266191	+	1.33823i	0.850189	+	0.526477i	$0.176488\pi$
$347$	4.95859	+	24.9285i	0.266191	+	1.33823i	−0.583998	+	0.811755i	$0.698512\pi$
$348$		0			0
$349$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$349$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.66413	−	6.66413i	−0.354696	−	0.354696i	0.507157	−	0.861853i	$-0.330696\pi$
$353$	−6.66413	−	6.66413i	−0.354696	−	0.354696i	−0.861853	+	0.507157i	$0.830696\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$359$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$360$		0			0
$361$	−25.4350	+	25.4350i	−1.33869	+	1.33869i
$362$		0			0
$363$	29.7719	−	44.5568i	1.56262	−	2.33863i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$367$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$368$		0			0
$369$	−4.21868	−	0.839147i	−0.219616	−	0.0436843i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−7.08373	+	35.6123i	−0.363867	+	1.82928i	0.172145	+	0.985072i	$0.444930\pi$
$379$	−7.08373	+	35.6123i	−0.363867	+	1.82928i	−0.536011	+	0.844211i	$0.680070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$383$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−27.3630	−	27.3630i	−1.39094	−	1.39094i
$388$		0			0
$389$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$389$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	21.9521	+	52.9971i	1.10734	+	2.67335i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$397$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.80511	−	0.359058i	0.0901427	−	0.0179305i	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	1.80511	−	0.359058i	0.0901427	−	0.0179305i	0.239956	+	0.970784i	$0.422867\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	33.3141			1.64727			0.823637	−	0.567117i	$-0.191941\pi$
$409$	33.3141			1.64727			0.823637	+	0.567117i	$0.191941\pi$
$410$		0			0
$411$	16.4174	+	24.5704i	0.809812	+	1.21197i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.87640	+	9.35845i	−0.189828	+	0.458285i
$418$		0			0
$419$	−32.9051	−	21.9865i	−1.60752	−	1.07411i	−0.945991	−	0.324192i	$-0.894908\pi$
$419$	−32.9051	−	21.9865i	−1.60752	−	1.07411i	−0.661529	−	0.749919i	$-0.730092\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.1421	+	15.0000i	0.685994	+	0.727607i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$431$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$432$		0			0
$433$	35.9113	+	14.8749i	1.72578	+	0.714843i	0.999627	+	0.0273152i	$0.00869578\pi$
$433$	35.9113	+	14.8749i	1.72578	+	0.714843i	0.726158	+	0.687528i	$0.241304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$439$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$440$		0			0
$441$		−	23.2192i		−	1.10568i
$442$		0			0
$443$	−13.4596			−0.639484			−0.319742	−	0.947505i	$-0.603596\pi$
$443$	−13.4596			−0.639484			−0.319742	+	0.947505i	$0.603596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.68092	+	13.4779i	0.126521	+	0.636062i	0.991051	+	0.133482i	$0.0426157\pi$
$449$	2.68092	+	13.4779i	0.126521	+	0.636062i	−0.864531	+	0.502580i	$0.832384\pi$
$450$		0			0
$451$	2.82122	−	6.81103i	0.132846	−	0.320719i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.18779	+	2.14885i	−0.242675	+	0.100519i	−0.500706	−	0.865617i	$-0.666926\pi$
$457$	−5.18779	+	2.14885i	−0.242675	+	0.100519i	0.258031	+	0.966137i	$0.416926\pi$
$458$		0			0
$459$	−3.20192	−	0.735463i	−0.149453	−	0.0343285i
$460$		0			0
$461$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$461$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	39.9238	+	16.5370i	1.84745	+	0.765240i	0.930693	+	0.365801i	$0.119205\pi$
$467$	39.9238	+	16.5370i	1.84745	+	0.765240i	0.916760	+	0.399439i	$0.130795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	55.1467	−	36.8479i	2.53565	−	1.69427i
$474$		0			0
$475$		−	37.0711i		−	1.70094i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$479$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$487$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$488$		0			0
$489$	−45.0648	−	45.0648i	−2.03790	−	2.03790i
$490$		0			0
$491$	18.1990	−	7.53828i	0.821311	−	0.340198i	0.0678537	−	0.997695i	$-0.478385\pi$
$491$	18.1990	−	7.53828i	0.821311	−	0.340198i	0.753457	+	0.657497i	$0.228385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.62811	−	9.91966i	0.296715	−	0.444065i	−0.652919	−	0.757428i	$-0.726456\pi$
$499$	6.62811	−	9.91966i	0.296715	−	0.444065i	0.949633	+	0.313363i	$0.101456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$503$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	27.1673	−	18.1526i	1.20654	−	0.806185i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	3.28212	+	4.91204i	0.144909	+	0.216872i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.5149	−	19.0531i	−1.24926	−	0.834730i	−0.257936	−	0.966162i	$-0.583042\pi$
$521$	−28.5149	−	19.0531i	−1.24926	−	0.834730i	−0.991325	+	0.131432i	$0.958042\pi$
$522$		0			0
$523$	−6.34711	−	6.34711i	−0.277540	−	0.277540i	0.554587	−	0.832126i	$-0.312876\pi$
$523$	−6.34711	−	6.34711i	−0.277540	−	0.277540i	−0.832126	+	0.554587i	$0.812876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−8.80172	−	21.2492i	−0.382683	−	0.923880i
$530$		0			0
$531$	25.2598	−	25.2598i	1.09618	−	1.09618i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.13577	−	0.623743i	0.135318	−	0.0269165i
$538$		0			0
$539$	39.0315	+	7.76386i	1.68121	+	0.334413i
$540$		0			0
$541$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$541$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−9.60936	−	14.3814i	−0.410867	−	0.614905i	0.567104	−	0.823646i	$-0.308064\pi$
$547$	−9.60936	−	14.3814i	−0.410867	−	0.614905i	−0.977971	+	0.208741i	$0.933064\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	24.1375	−	53.7432i	1.01909	−	2.26904i
$562$		0			0
$563$	5.22183	+	12.6066i	0.220074	+	0.531305i	0.994900	−	0.100870i	$-0.0321625\pi$
$563$	5.22183	+	12.6066i	0.220074	+	0.531305i	−0.774826	+	0.632175i	$0.782163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.8492	−	11.5355i	−1.16750	−	0.483595i	−0.287135	−	0.957890i	$-0.592703\pi$
$569$	−27.8492	−	11.5355i	−1.16750	−	0.483595i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	−46.5861	+	9.26656i	−1.94957	+	0.387793i	−0.953237	+	0.302224i	$0.902271\pi$
$571$	−46.5861	+	9.26656i	−1.94957	+	0.387793i	−0.996332	+	0.0855694i	$0.972729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	33.9411i		−	1.41299i	−0.707719	−	0.706494i	$-0.750276\pi$
$577$		−	33.9411i		−	1.41299i	0.707719	−	0.706494i	$-0.249724\pi$
$578$		0			0
$579$	45.9104			1.90797
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.29610	−	5.54328i	0.0947702	−	0.228796i	−0.869385	−	0.494136i	$-0.835484\pi$
$587$	2.29610	−	5.54328i	0.0947702	−	0.228796i	0.964155	+	0.265341i	$0.0854844\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.3640	+	12.9914i	−1.28796	+	0.533492i	−0.918378	−	0.395705i	$-0.870500\pi$
$593$	−31.3640	+	12.9914i	−1.28796	+	0.533492i	−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$		0			0
$601$	26.9046	−	40.2655i	1.09746	−	1.64246i	0.418655	−	0.908145i	$-0.362502\pi$
$601$	26.9046	−	40.2655i	1.09746	−	1.64246i	0.678804	−	0.734319i	$-0.262498\pi$
$602$		0			0
$603$	49.5887	+	20.5403i	2.01941	+	0.836466i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$607$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.15823	+	46.0415i	−0.368697	+	1.85356i	0.136613	+	0.990624i	$0.456378\pi$
$617$	−9.15823	+	46.0415i	−0.368697	+	1.85356i	−0.505310	+	0.862938i	$0.668622\pi$
$618$		0			0
$619$	9.10751	+	45.7865i	0.366062	+	1.84032i	0.522514	+	0.852631i	$0.324994\pi$
$619$	9.10751	+	45.7865i	0.366062	+	1.84032i	−0.156452	+	0.987685i	$0.550006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.6777	+	17.6777i	0.707107	+	0.707107i
$626$		0			0
$627$	−97.8769	+	40.5420i	−3.90883	+	1.61909i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$631$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$632$		0			0
$633$	23.7930	−	23.7930i	0.945688	−	0.945688i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.180035	+	0.0358112i	0.00711096	+	0.00141446i	0.198645	−	0.980072i	$-0.436346\pi$
$641$	0.180035	+	0.0358112i	0.00711096	+	0.00141446i	−0.191534	+	0.981486i	$0.561346\pi$
$642$		0			0
$643$	−17.2308	+	11.5132i	−0.679515	+	0.454037i	−0.846828	−	0.531866i	$-0.821491\pi$
$643$	−17.2308	+	11.5132i	−0.679515	+	0.454037i	0.167313	+	0.985904i	$0.446491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	34.0157	+	50.9081i	1.33523	+	1.99832i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$653$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	41.9814	+	28.0511i	1.63785	+	1.09438i
$658$		0			0
$659$	12.7279	+	12.7279i	0.495809	+	0.495809i	0.910131	−	0.414321i	$-0.135981\pi$
$659$	12.7279	+	12.7279i	0.495809	+	0.495809i	−0.414321	+	0.910131i	$0.635981\pi$
$660$		0			0
$661$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$661$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−19.5866	+	3.89602i	−0.755008	+	0.150180i	−0.557564	−	0.830134i	$-0.688264\pi$
$673$	−19.5866	+	3.89602i	−0.755008	+	0.150180i	−0.197444	+	0.980314i	$0.563264\pi$
$674$		0			0
$675$	−3.90746	−	0.777242i	−0.150398	−	0.0299160i
$676$		0			0
$677$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$677$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−7.73773			−0.296511
$682$		0			0
$683$	−9.79828	−	14.6642i	−0.374921	−	0.561108i	0.595247	−	0.803543i	$-0.297054\pi$
$683$	−9.79828	−	14.6642i	−0.374921	−	0.561108i	−0.970168	+	0.242434i	$0.922054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−43.5608	−	29.1064i	−1.65713	−	1.10726i	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−43.5608	−	29.1064i	−1.65713	−	1.10726i	−0.782169	−	0.623066i	$-0.785887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.19692	−	5.21091i	0.0453365	−	0.197377i
$698$		0			0
$699$	−27.2388	−	65.7604i	−1.03027	−	2.48729i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$709$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$719$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−11.7245	+	28.3055i	−0.436040	+	1.05269i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−29.9088	+	12.3886i	−1.10773	+	0.458839i
$730$		0			0
$731$	34.9986	−	32.9970i	1.29447	−	1.22044i
$732$		0			0
$733$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−51.1094	+	76.4907i	−1.88264	+	2.81757i
$738$		0			0
$739$	39.1969	+	16.2359i	1.44188	+	0.597247i	0.960253	−	0.279129i	$-0.0900459\pi$
$739$	39.1969	+	16.2359i	1.44188	+	0.597247i	0.481627	+	0.876376i	$0.340046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$743$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			51.5713i			1.88689i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$751$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$752$		0			0
$753$	−8.55613	+	43.0146i	−0.311803	+	1.56754i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$757$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.3431	−	21.3431i	−0.773688	−	0.773688i	0.205061	−	0.978749i	$-0.434261\pi$
$761$	−21.3431	−	21.3431i	−0.773688	−	0.773688i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.4558	+	14.4558i	−0.521291	+	0.521291i	−0.917961	−	0.396670i	$-0.870166\pi$
$769$	−14.4558	+	14.4558i	−0.521291	+	0.521291i	0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	−44.7475	+	66.9693i	−1.61154	+	2.41184i
$772$		0			0
$773$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.99402	+	5.34143i	−0.286415	+	0.191377i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	8.59823	−	43.2262i	0.306494	−	1.54085i	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	8.59823	−	43.2262i	0.306494	−	1.54085i	0.760195	−	0.649695i	$-0.225103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$797$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	15.3779	+	37.1254i	0.543350	+	1.31176i
$802$		0			0
$803$	−61.1914	+	61.1914i	−2.15940	+	2.15940i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.5956	+	3.30106i	−0.583469	+	0.116059i	−0.477994	−	0.878363i	$-0.658636\pi$
$809$	−16.5956	+	3.30106i	−0.583469	+	0.116059i	−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	28.4358	+	5.65623i	0.998515	+	0.198617i	0.667180	−	0.744896i	$-0.267501\pi$
$811$	28.4358	+	5.65623i	0.998515	+	0.198617i	0.331335	+	0.943513i	$0.392501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−86.4958			−3.02610
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$821$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$822$		0			0
$823$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$823$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$824$		0			0
$825$	27.3407	−	66.0063i	0.951881	−	2.29804i
$826$		0			0
$827$	−40.8250	−	27.2784i	−1.41962	−	0.948563i	−0.999148	−	0.0412591i	$-0.986863\pi$
$827$	−40.8250	−	27.2784i	−1.41962	−	0.948563i	−0.420476	−	0.907304i	$-0.638137\pi$
$828$		0			0
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	28.8492	+	0.849242i	0.999567	+	0.0294245i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$839$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$840$		0			0
$841$	26.7925	+	11.0978i	0.923880	+	0.382683i
$842$		0			0
$843$	80.6768	−	16.0476i	2.77866	−	0.552709i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	10.4300i		−	0.357958i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$853$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.18243	−	10.9718i	−0.0745505	−	0.374791i	0.925441	−	0.378892i	$-0.123695\pi$
$857$	−2.18243	−	10.9718i	−0.0745505	−	0.374791i	−0.999992	+	0.00410087i	$0.998695\pi$
$858$		0			0
$859$	5.49390	−	13.2635i	0.187450	−	0.452543i	−0.802018	−	0.597300i	$-0.796240\pi$
$859$	5.49390	−	13.2635i	0.187450	−	0.452543i	0.989467	+	0.144757i	$0.0462401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	10.7863	−	41.3434i	0.366323	−	1.40410i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	27.0779	−	40.5250i	0.916449	−	1.37156i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$877$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−47.4231	+	31.6871i	−1.59773	+	1.06757i	−0.644804	+	0.764348i	$0.723061\pi$
$881$	−47.4231	+	31.6871i	−1.59773	+	1.06757i	−0.952921	+	0.303218i	$0.901939\pi$
$882$		0			0
$883$		−	55.6410i		−	1.87247i	−0.351376	−	0.936235i	$-0.614286\pi$
$883$		−	55.6410i		−	1.87247i	0.351376	−	0.936235i	$-0.385714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$887$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−8.81577	−	44.3199i	−0.295339	−	1.48477i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−24.3512	+	36.4442i	−0.808570	+	1.21011i	0.166022	+	0.986122i	$0.446908\pi$
$907$	−24.3512	+	36.4442i	−0.808570	+	1.21011i	−0.974592	+	0.223988i	$0.928092\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$911$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$912$		0			0
$913$	−86.6915	−	17.2440i	−2.86907	−	0.570693i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	11.8484	+	17.7325i	0.390420	+	0.584304i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48.6247	+	32.4900i	1.59532	+	1.06596i	0.954474	+	0.298295i	$0.0964179\pi$
$929$	48.6247	+	32.4900i	1.59532	+	1.06596i	0.640850	+	0.767666i	$0.278582\pi$
$930$		0			0
$931$	−36.6985	−	36.6985i	−1.20274	−	1.20274i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−19.4831	−	47.0363i	−0.636484	−	1.53661i	−0.831333	−	0.555775i	$-0.812422\pi$
$937$	−19.4831	−	47.0363i	−0.636484	−	1.53661i	0.194849	−	0.980833i	$-0.437578\pi$
$938$		0			0
$939$	−5.66270	+	5.66270i	−0.184795	+	0.184795i
$940$		0			0
$941$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$941$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.5755	−	3.69489i	−0.603622	−	0.120068i	−0.116187	−	0.993227i	$-0.537067\pi$
$947$	−18.5755	−	3.69489i	−0.603622	−	0.120068i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	57.8827			1.87501			0.937503	−	0.347978i	$-0.113132\pi$
$953$	57.8827			1.87501			0.937503	+	0.347978i	$0.113132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.8632	−	28.6403i	0.382683	−	0.923880i
$962$		0			0
$963$	50.3281	+	33.6282i	1.62180	+	1.08365i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$967$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$968$		0			0
$969$	−65.1124	+	40.7877i	−2.09171	+	1.31029i
$970$		0			0
$971$	3.09188	+	7.46447i	0.0992233	+	0.239546i	0.965694	−	0.259681i	$-0.0836174\pi$
$971$	3.09188	+	7.46447i	0.0992233	+	0.239546i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	50.9914	+	21.1213i	1.63136	+	0.675731i	0.995383	−	0.0959785i	$-0.0305980\pi$
$977$	50.9914	+	21.1213i	1.63136	+	0.675731i	0.635975	+	0.771709i	$0.280598\pi$
$978$		0			0
$979$	−67.5499	+	13.4365i	−2.15891	+	0.429433i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$983$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$991$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$992$		0			0
$993$	7.00165	+	35.1996i	0.222190	+	1.11703i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$997$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	544.2.cc.b.175.1		8
4.3	odd	2		136.2.s.a.107.1	yes	8
8.3	odd	2	CM	544.2.cc.b.175.1		8
8.5	even	2		136.2.s.a.107.1	yes	8
17.7	odd	16	inner	544.2.cc.b.143.1		8
68.7	even	16		136.2.s.a.75.1	✓	8
136.75	even	16	inner	544.2.cc.b.143.1		8
136.109	odd	16		136.2.s.a.75.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.s.a.75.1	✓	8	68.7	even	16
136.2.s.a.75.1	✓	8	136.109	odd	16
136.2.s.a.107.1	yes	8	4.3	odd	2
136.2.s.a.107.1	yes	8	8.5	even	2
544.2.cc.b.143.1		8	17.7	odd	16	inner
544.2.cc.b.143.1		8	136.75	even	16	inner
544.2.cc.b.175.1		8	1.1	even	1	trivial
544.2.cc.b.175.1		8	8.3	odd	2	CM

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 544 = 2^{5} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	544.cc (of order \(16\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.39635 - 2.08979i) q^{3} +(-1.26937 + 3.06453i) q^{9} +(-4.72705 - 3.15851i) q^{11} +(-3.76118 - 1.68925i) q^{17} +(2.83730 + 6.84984i) q^{19} +(-4.61940 - 1.91342i) q^{25} +(0.781491 - 0.155448i) q^{27} +14.2889i q^{33} +(0.252982 + 1.27183i) q^{41} +(-4.46447 + 10.7782i) q^{43} +(-6.46716 + 2.67878i) q^{49} +(1.72176 + 10.2189i) q^{51} +(10.3529 - 15.4942i) q^{57} +(-9.94975 - 4.12132i) q^{59} -16.1815i q^{67} +(2.96960 - 14.9292i) q^{73} +(2.45167 + 12.3254i) q^{75} +(5.62038 + 5.62038i) q^{81} +(14.3640 - 5.94975i) q^{83} +(8.56628 - 8.56628i) q^{89} +(-14.4112 - 2.86657i) q^{97} +(15.6797 - 10.4769i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 16 q^{9} + 32 q^{27} - 32 q^{41} - 64 q^{43} + 40 q^{51} - 40 q^{57} - 40 q^{59} + 48 q^{73} + 56 q^{81} + 64 q^{83} + 48 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + 16 * q^9 + 32 * q^27 - 32 * q^41 - 64 * q^43 + 40 * q^51 - 40 * q^57 - 40 * q^59 + 48 * q^73 + 56 * q^81 + 64 * q^83 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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