LMFDB - Embedded newform 5328.2.h.b.2737.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5328,2,Mod(2737,5328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5328.2737"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-6,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0, 0,0,0,0,0,0,0,-12,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

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

Self dual:	no
Analytic conductor:	$42.5442941969$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 222)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5328.2737
Dual form			5328.2.h.b.2737.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} -3.00000 q^{7} -3.00000 q^{11} -1.00000i q^{13} +3.00000i q^{17} -1.00000i q^{19} -1.00000i q^{23} +1.00000 q^{25} -6.00000i q^{29} -10.0000i q^{31} -6.00000i q^{35} +(-6.00000 + 1.00000i) q^{37} -2.00000 q^{41} -4.00000i q^{43} +8.00000 q^{47} +2.00000 q^{49} +11.0000 q^{53} -6.00000i q^{55} +6.00000i q^{59} +10.0000i q^{61} +2.00000 q^{65} -8.00000 q^{67} +2.00000 q^{71} -1.00000 q^{73} +9.00000 q^{77} +14.0000i q^{79} -1.00000 q^{83} -6.00000 q^{85} +9.00000i q^{89} +3.00000i q^{91} +2.00000 q^{95} -8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7} - 6 q^{11} + 2 q^{25} - 12 q^{37} - 4 q^{41} + 16 q^{47} + 4 q^{49} + 22 q^{53} + 4 q^{65} - 16 q^{67} + 4 q^{71} - 2 q^{73} + 18 q^{77} - 2 q^{83} - 12 q^{85} + 4 q^{95}+O(q^{100}) $ 2 * q - 6 * q^7 - 6 * q^11 + 2 * q^25 - 12 * q^37 - 4 * q^41 + 16 * q^47 + 4 * q^49 + 22 * q^53 + 4 * q^65 - 16 * q^67 + 4 * q^71 - 2 * q^73 + 18 * q^77 - 2 * q^83 - 12 * q^85 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$		−	1.00000i		−	0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	6.00000i		−	1.01419i
$36$		0			0
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.0000			1.51097			0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000			1.51097			0.755483	+	0.655168i	$0.227402\pi$
$54$		0			0
$55$		−	6.00000i		−	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000			0.248069
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$	−1.00000			−0.117041			−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000			−0.117041			−0.0585206	+	0.998286i	$0.518638\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.00000			1.02565
$78$		0			0
$79$			14.0000i			1.57512i	0.616236	+	0.787562i	$0.288657\pi$
$79$			14.0000i			1.57512i	−0.616236	+	0.787562i	$0.711343\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.00000			−0.109764			−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000			−0.109764			−0.0548821	+	0.998493i	$0.517478\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$			3.00000i			0.314485i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000			0.205196
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.0000			1.25676			0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000			1.25676			0.628379	+	0.777908i	$0.283719\pi$
$108$		0			0
$109$		−	19.0000i		−	1.81987i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$		−	19.0000i		−	1.81987i	0.414751	−	0.909935i	$-0.363869\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$	2.00000			0.186501
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.00000i		−	0.825029i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$	7.00000			0.621150			0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000			0.621150			0.310575	+	0.950549i	$0.399478\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.0000i			0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$			10.0000i			0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$		0			0
$133$			3.00000i			0.260133i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.0000			1.02523			0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000			1.02523			0.512615	+	0.858619i	$0.328677\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3.00000i			0.250873i
$144$		0			0
$145$	12.0000			0.996546
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	13.0000			1.05792			0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000			1.05792			0.528962	+	0.848645i	$0.322581\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.0000			1.60644
$156$		0			0
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.00000i			0.236433i
$162$		0			0
$163$		−	9.00000i		−	0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$		−	9.00000i		−	0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.0000i			1.31550i	0.753237	+	0.657750i	$0.228492\pi$
$167$			17.0000i			1.31550i	−0.753237	+	0.657750i	$0.771508\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$	−3.00000			−0.226779
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	24.0000i		−	1.79384i	−0.442189	−	0.896922i	$-0.645798\pi$
$179$		−	24.0000i		−	1.79384i	0.442189	−	0.896922i	$-0.354202\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.00000	−	12.0000i	−0.147043	−	0.882258i
$186$		0			0
$187$		−	9.00000i		−	0.658145i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.0000i			1.08536i	0.839939	+	0.542681i	$0.182591\pi$
$191$			15.0000i			1.08536i	−0.839939	+	0.542681i	$0.817409\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.0000			1.21120			0.605600	−	0.795769i	$-0.292933\pi$
$197$	17.0000			1.21120			0.605600	+	0.795769i	$0.292933\pi$
$198$		0			0
$199$			14.0000i			0.992434i	0.868199	+	0.496217i	$0.165278\pi$
$199$			14.0000i			0.992434i	−0.868199	+	0.496217i	$0.834722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			18.0000i			1.26335i
$204$		0			0
$205$		−	4.00000i		−	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.00000i			0.207514i
$210$		0			0
$211$	−22.0000			−1.51454			−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000			−1.51454			−0.757271	+	0.653101i	$0.773468\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$			30.0000i			2.03653i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−24.0000			−1.57229			−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000			−1.57229			−0.786146	+	0.618041i	$0.787927\pi$
$234$		0			0
$235$			16.0000i			1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			4.00000i			0.255551i
$246$		0			0
$247$	−1.00000			−0.0636285
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$			3.00000i			0.188608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.00000i			0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$			3.00000i			0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$		0			0
$259$	18.0000	−	3.00000i	1.11847	−	0.186411i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$			22.0000i			1.35145i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.00000			0.304855			0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000			0.304855			0.152428	+	0.988315i	$0.451291\pi$
$270$		0			0
$271$	28.0000			1.70088			0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000			1.70088			0.850439	+	0.526073i	$0.176336\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000			−0.180907
$276$		0			0
$277$			7.00000i			0.420589i	0.977638	+	0.210295i	$0.0674423\pi$
$277$			7.00000i			0.420589i	−0.977638	+	0.210295i	$0.932558\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			15.0000i			0.894825i	0.894328	+	0.447412i	$0.147654\pi$
$281$			15.0000i			0.894825i	−0.894328	+	0.447412i	$0.852346\pi$
$282$		0			0
$283$		−	29.0000i		−	1.72387i	−0.507018	−	0.861936i	$-0.669252\pi$
$283$		−	29.0000i		−	1.72387i	0.507018	−	0.861936i	$-0.330748\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.00000			−0.525786			−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000			−0.525786			−0.262893	+	0.964825i	$0.584677\pi$
$294$		0			0
$295$	−12.0000			−0.698667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.00000			−0.0578315
$300$		0			0
$301$			12.0000i			0.691669i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−20.0000			−1.14520
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	20.0000i		−	1.13410i	−0.823685	−	0.567048i	$-0.808085\pi$
$311$		−	20.0000i		−	1.13410i	0.823685	−	0.567048i	$-0.191915\pi$
$312$		0			0
$313$		−	6.00000i		−	0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$		−	6.00000i		−	0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$			18.0000i			1.00781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.00000			0.166924
$324$		0			0
$325$		−	1.00000i		−	0.0554700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.0000			−1.32316
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	16.0000i		−	0.874173i
$336$		0			0
$337$	3.00000			0.163420			0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000			0.163420			0.0817102	+	0.996656i	$0.473962\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			30.0000i			1.62459i
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	34.0000i		−	1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$		−	34.0000i		−	1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		0			0
$355$			4.00000i			0.212298i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.0000			0.527780			0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000			0.527780			0.263890	+	0.964553i	$0.414994\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.00000i		−	0.104685i
$366$		0			0
$367$	17.0000			0.887393			0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000			0.887393			0.443696	+	0.896177i	$0.353667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−33.0000			−1.71327
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.00000			−0.309016
$378$		0			0
$379$	30.0000			1.54100			0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000			1.54100			0.770498	+	0.637442i	$0.220007\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.0000i		−	0.562074i	−0.959697	−	0.281037i	$-0.909322\pi$
$383$		−	11.0000i		−	0.562074i	0.959697	−	0.281037i	$-0.0906783\pi$
$384$		0			0
$385$			18.0000i			0.917365i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			4.00000i			0.202808i	0.994845	+	0.101404i	$0.0323335\pi$
$389$			4.00000i			0.202808i	−0.994845	+	0.101404i	$0.967667\pi$
$390$		0			0
$391$	3.00000			0.151717
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−28.0000			−1.40883
$396$		0			0
$397$	−32.0000			−1.60603			−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000			−1.60603			−0.803017	+	0.595956i	$0.796773\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	15.0000i		−	0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$		−	15.0000i		−	0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$		0			0
$403$	−10.0000			−0.498135
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.0000	−	3.00000i	0.892227	−	0.148704i
$408$		0			0
$409$		−	24.0000i		−	1.18672i	−0.804936	−	0.593362i	$-0.797800\pi$
$409$		−	24.0000i		−	1.18672i	0.804936	−	0.593362i	$-0.202200\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	18.0000i		−	0.885722i
$414$		0			0
$415$		−	2.00000i		−	0.0981761i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−35.0000			−1.70986			−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000			−1.70986			−0.854931	+	0.518742i	$0.826401\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.00000i			0.145521i
$426$		0			0
$427$		−	30.0000i		−	1.45180i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			15.0000i			0.722525i	0.932464	+	0.361262i	$0.117654\pi$
$431$			15.0000i			0.722525i	−0.932464	+	0.361262i	$0.882346\pi$
$432$		0			0
$433$	−1.00000			−0.0480569			−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000			−0.0480569			−0.0240285	+	0.999711i	$0.507649\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.00000			−0.0478365
$438$		0			0
$439$		−	26.0000i		−	1.24091i	−0.784241	−	0.620456i	$-0.786947\pi$
$439$		−	26.0000i		−	1.24091i	0.784241	−	0.620456i	$-0.213053\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$	−18.0000			−0.853282
$446$		0			0
$447$		0			0
$448$		0			0
$449$			34.0000i			1.60456i	0.596948	+	0.802280i	$0.296380\pi$
$449$			34.0000i			1.60456i	−0.596948	+	0.802280i	$0.703620\pi$
$450$		0			0
$451$	6.00000			0.282529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000			−0.281284
$456$		0			0
$457$			2.00000i			0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$			2.00000i			0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	40.0000i		−	1.86299i	−0.363760	−	0.931493i	$-0.618507\pi$
$461$		−	40.0000i		−	1.86299i	0.363760	−	0.931493i	$-0.381493\pi$
$462$		0			0
$463$		−	34.0000i		−	1.58011i	−0.613033	−	0.790057i	$-0.710051\pi$
$463$		−	34.0000i		−	1.58011i	0.613033	−	0.790057i	$-0.289949\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	8.00000i		−	0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$		−	8.00000i		−	0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$		0			0
$469$	24.0000			1.10822
$470$		0			0
$471$		0			0
$472$		0			0
$473$			12.0000i			0.551761i
$474$		0			0
$475$		−	1.00000i		−	0.0458831i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			21.0000i			0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$			21.0000i			0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$		0			0
$481$	1.00000	+	6.00000i	0.0455961	+	0.273576i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000			0.726523
$486$		0			0
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	27.0000			1.21849			0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000			1.21849			0.609246	+	0.792981i	$0.291472\pi$
$492$		0			0
$493$	18.0000			0.810679
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000			−0.269137
$498$		0			0
$499$			39.0000i			1.74588i	0.487828	+	0.872940i	$0.337789\pi$
$499$			39.0000i			1.74588i	−0.487828	+	0.872940i	$0.662211\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$			36.0000i			1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.00000			0.221621			0.110811	−	0.993842i	$-0.464655\pi$
$509$	5.00000			0.221621			0.110811	+	0.993842i	$0.464655\pi$
$510$		0			0
$511$	3.00000			0.132712
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−32.0000			−1.41009
$516$		0			0
$517$	−24.0000			−1.05552
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	30.0000			1.30682
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.00000i			0.0866296i
$534$		0			0
$535$			26.0000i			1.12408i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$		−	45.0000i		−	1.93470i	−0.253442	−	0.967351i	$-0.581563\pi$
$541$		−	45.0000i		−	1.93470i	0.253442	−	0.967351i	$-0.418437\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	38.0000			1.62774
$546$		0			0
$547$		−	17.0000i		−	0.726868i	−0.931620	−	0.363434i	$-0.881604\pi$
$547$		−	17.0000i		−	0.726868i	0.931620	−	0.363434i	$-0.118396\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.00000			−0.255609
$552$		0			0
$553$		−	42.0000i		−	1.78602i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			38.0000i			1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$			38.0000i			1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$			14.0000i			0.590030i	0.955493	+	0.295015i	$0.0953246\pi$
$563$			14.0000i			0.590030i	−0.955493	+	0.295015i	$0.904675\pi$
$564$		0			0
$565$	28.0000			1.17797
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	21.0000i		−	0.880366i	−0.897908	−	0.440183i	$-0.854914\pi$
$569$		−	21.0000i		−	0.880366i	0.897908	−	0.440183i	$-0.145086\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1.00000i		−	0.0417029i
$576$		0			0
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.00000			0.124461
$582$		0			0
$583$	−33.0000			−1.36672
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		0			0
$589$	−10.0000			−0.412043
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	18.0000			0.737928
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.0000			0.408589			0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000			0.408589			0.204294	+	0.978909i	$0.434510\pi$
$600$		0			0
$601$	7.00000			0.285536			0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000			0.285536			0.142768	+	0.989756i	$0.454400\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	4.00000i		−	0.162623i
$606$		0			0
$607$		−	12.0000i		−	0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$		−	12.0000i		−	0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	8.00000i		−	0.323645i
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	−40.0000			−1.60774			−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000			−1.60774			−0.803868	+	0.594808i	$0.797228\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	27.0000i		−	1.08173i
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.00000	−	18.0000i	−0.119618	−	0.717707i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			14.0000i			0.555573i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$			21.0000i			0.828159i	0.910241	+	0.414080i	$0.135896\pi$
$643$			21.0000i			0.828159i	−0.910241	+	0.414080i	$0.864104\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	3.00000i		−	0.117942i	−0.998260	−	0.0589711i	$-0.981218\pi$
$647$		−	3.00000i		−	0.117942i	0.998260	−	0.0589711i	$-0.0187820\pi$
$648$		0			0
$649$		−	18.0000i		−	0.706562i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$	−20.0000			−0.781465
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			25.0000i			0.972387i	0.873851	+	0.486194i	$0.161615\pi$
$661$			25.0000i			0.972387i	−0.873851	+	0.486194i	$0.838385\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000			−0.232670
$666$		0			0
$667$	−6.00000			−0.232321
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	30.0000i		−	1.15814i
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.00000			0.269032			0.134516	−	0.990911i	$-0.457052\pi$
$677$	7.00000			0.269032			0.134516	+	0.990911i	$0.457052\pi$
$678$		0			0
$679$			24.0000i			0.921035i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	16.0000i		−	0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$		−	16.0000i		−	0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$		0			0
$685$			24.0000i			0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	11.0000i		−	0.419067i
$690$		0			0
$691$	−2.00000			−0.0760836			−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000			−0.0760836			−0.0380418	+	0.999276i	$0.512112\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	6.00000i		−	0.227266i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	1.00000	+	6.00000i	0.0377157	+	0.226294i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−54.0000			−2.03088
$708$		0			0
$709$		−	49.0000i		−	1.84023i	−0.391644	−	0.920117i	$-0.628094\pi$
$709$		−	49.0000i		−	1.84023i	0.391644	−	0.920117i	$-0.371906\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.0000			−0.374503
$714$		0			0
$715$	−6.00000			−0.224387
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	48.0000i		−	1.78761i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	6.00000i		−	0.222834i
$726$		0			0
$727$		−	42.0000i		−	1.55769i	−0.627214	−	0.778847i	$-0.715805\pi$
$727$		−	42.0000i		−	1.55769i	0.627214	−	0.778847i	$-0.284195\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000			0.443836
$732$		0			0
$733$	−26.0000			−0.960332			−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000			−0.960332			−0.480166	+	0.877178i	$0.659424\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	34.0000			1.24734			0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000			1.24734			0.623670	+	0.781688i	$0.285641\pi$
$744$		0			0
$745$			20.0000i			0.732743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−39.0000			−1.42503
$750$		0			0
$751$	48.0000			1.75154			0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000			1.75154			0.875772	+	0.482724i	$0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			26.0000i			0.946237i
$756$		0			0
$757$			17.0000i			0.617876i	0.951082	+	0.308938i	$0.0999735\pi$
$757$			17.0000i			0.617876i	−0.951082	+	0.308938i	$0.900027\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$			57.0000i			2.06354i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.00000			0.216647
$768$		0			0
$769$			26.0000i			0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$			26.0000i			0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−39.0000			−1.40273			−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000			−1.40273			−0.701366	+	0.712801i	$0.747426\pi$
$774$		0			0
$775$		−	10.0000i		−	0.359211i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2.00000i			0.0716574i
$780$		0			0
$781$	−6.00000			−0.214697
$782$		0			0
$783$		0			0
$784$		0			0
$785$			16.0000i			0.571064i
$786$		0			0
$787$	−38.0000			−1.35455			−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000			−1.35455			−0.677277	+	0.735728i	$0.736840\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			42.0000i			1.49335i
$792$		0			0
$793$	10.0000			0.355110
$794$		0			0
$795$		0			0
$796$		0			0
$797$			8.00000i			0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$			8.00000i			0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000			0.105868
$804$		0			0
$805$	−6.00000			−0.211472
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	31.0000i		−	1.08990i	−0.838468	−	0.544951i	$-0.816548\pi$
$809$		−	31.0000i		−	1.08990i	0.838468	−	0.544951i	$-0.183452\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.0000			0.630512
$816$		0			0
$817$	−4.00000			−0.139942
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000			0.104701			0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000			0.104701			0.0523504	+	0.998629i	$0.483329\pi$
$822$		0			0
$823$	−29.0000			−1.01088			−0.505438	−	0.862863i	$-0.668669\pi$
$823$	−29.0000			−1.01088			−0.505438	+	0.862863i	$0.668669\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.00000i			0.0695468i	0.999395	+	0.0347734i	$0.0110710\pi$
$827$			2.00000i			0.0695468i	−0.999395	+	0.0347734i	$0.988929\pi$
$828$		0			0
$829$			11.0000i			0.382046i	0.981586	+	0.191023i	$0.0611805\pi$
$829$			11.0000i			0.382046i	−0.981586	+	0.191023i	$0.938820\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			6.00000i			0.207888i
$834$		0			0
$835$	−34.0000			−1.17662
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.0000			−0.690477			−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000			−0.690477			−0.345238	+	0.938515i	$0.612202\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$			24.0000i			0.825625i
$846$		0			0
$847$	6.00000			0.206162
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.00000	+	6.00000i	0.0342796	+	0.205677i
$852$		0			0
$853$			49.0000i			1.67773i	0.544341	+	0.838864i	$0.316780\pi$
$853$			49.0000i			1.67773i	−0.544341	+	0.838864i	$0.683220\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			13.0000i			0.444072i	0.975039	+	0.222036i	$0.0712702\pi$
$857$			13.0000i			0.444072i	−0.975039	+	0.222036i	$0.928730\pi$
$858$		0			0
$859$		−	31.0000i		−	1.05771i	−0.848713	−	0.528853i	$-0.822622\pi$
$859$		−	31.0000i		−	1.05771i	0.848713	−	0.528853i	$-0.177378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$		−	18.0000i		−	0.612018i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	42.0000i		−	1.42475i
$870$		0			0
$871$			8.00000i			0.271070i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	36.0000i		−	1.21702i
$876$		0			0
$877$	−32.0000			−1.08056			−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000			−1.08056			−0.540282	+	0.841484i	$0.681682\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$			11.0000i			0.370179i	0.982722	+	0.185090i	$0.0592576\pi$
$883$			11.0000i			0.370179i	−0.982722	+	0.185090i	$0.940742\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.0000			0.940148			0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000			0.940148			0.470074	+	0.882627i	$0.344227\pi$
$888$		0			0
$889$	−21.0000			−0.704317
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	48.0000			1.60446
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−60.0000			−2.00111
$900$		0			0
$901$			33.0000i			1.09939i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			4.00000i			0.132964i
$906$		0			0
$907$		−	7.00000i		−	0.232431i	−0.993224	−	0.116216i	$-0.962924\pi$
$907$		−	7.00000i		−	0.232431i	0.993224	−	0.116216i	$-0.0370764\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	3.00000			0.0992855
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	30.0000i		−	0.990687i
$918$		0			0
$919$		−	36.0000i		−	1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$919$		−	36.0000i		−	1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2.00000i		−	0.0658308i
$924$		0			0
$925$	−6.00000	+	1.00000i	−0.197279	+	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	60.0000			1.96854			0.984268	−	0.176682i	$-0.0565363\pi$
$929$	60.0000			1.96854			0.984268	+	0.176682i	$0.0565363\pi$
$930$		0			0
$931$		−	2.00000i		−	0.0655474i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18.0000			0.588663
$936$		0			0
$937$	18.0000			0.588034			0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000			0.588034			0.294017	+	0.955800i	$0.405008\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	38.0000			1.23876			0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000			1.23876			0.619382	+	0.785090i	$0.287383\pi$
$942$		0			0
$943$			2.00000i			0.0651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$			1.00000i			0.0324614i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−44.0000			−1.42530			−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000			−1.42530			−0.712650	+	0.701520i	$0.752505\pi$
$954$		0			0
$955$	−30.0000			−0.970777
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.0000			−1.16250
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$		0			0
$964$		0			0
$965$	32.0000			1.03012
$966$		0			0
$967$		−	52.0000i		−	1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$		−	52.0000i		−	1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			53.0000i			1.69562i	0.530300	+	0.847810i	$0.322079\pi$
$977$			53.0000i			1.69562i	−0.530300	+	0.847810i	$0.677921\pi$
$978$		0			0
$979$		−	27.0000i		−	0.862924i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.0000			−0.510321			−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000			−0.510321			−0.255160	+	0.966899i	$0.582128\pi$
$984$		0			0
$985$			34.0000i			1.08333i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.00000			−0.127193
$990$		0			0
$991$		−	10.0000i		−	0.317660i	−0.987306	−	0.158830i	$-0.949228\pi$
$991$		−	10.0000i		−	0.317660i	0.987306	−	0.158830i	$-0.0507723\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28.0000			−0.887660
$996$		0			0
$997$			17.0000i			0.538395i	0.963085	+	0.269198i	$0.0867585\pi$
$997$			17.0000i			0.538395i	−0.963085	+	0.269198i	$0.913241\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.h.b.2737.2		2
3.2	odd	2		1776.2.h.a.961.1		2
4.3	odd	2		666.2.c.a.73.2		2
12.11	even	2		222.2.c.a.73.1	✓	2
37.36	even	2	inner	5328.2.h.b.2737.1		2
111.110	odd	2		1776.2.h.a.961.2		2
148.147	odd	2		666.2.c.a.73.1		2
444.179	odd	4		8214.2.a.g.1.1		1
444.191	odd	4		8214.2.a.c.1.1		1
444.443	even	2		222.2.c.a.73.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.c.a.73.1	✓	2	12.11	even	2
222.2.c.a.73.2	yes	2	444.443	even	2
666.2.c.a.73.1		2	148.147	odd	2
666.2.c.a.73.2		2	4.3	odd	2
1776.2.h.a.961.1		2	3.2	odd	2
1776.2.h.a.961.2		2	111.110	odd	2
5328.2.h.b.2737.1		2	37.36	even	2	inner
5328.2.h.b.2737.2		2	1.1	even	1	trivial
8214.2.a.c.1.1		1	444.191	odd	4
8214.2.a.g.1.1		1	444.179	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} -3.00000 q^{7} -3.00000 q^{11} -1.00000i q^{13} +3.00000i q^{17} -1.00000i q^{19} -1.00000i q^{23} +1.00000 q^{25} -6.00000i q^{29} -10.0000i q^{31} -6.00000i q^{35} +(-6.00000 + 1.00000i) q^{37} -2.00000 q^{41} -4.00000i q^{43} +8.00000 q^{47} +2.00000 q^{49} +11.0000 q^{53} -6.00000i q^{55} +6.00000i q^{59} +10.0000i q^{61} +2.00000 q^{65} -8.00000 q^{67} +2.00000 q^{71} -1.00000 q^{73} +9.00000 q^{77} +14.0000i q^{79} -1.00000 q^{83} -6.00000 q^{85} +9.00000i q^{89} +3.00000i q^{91} +2.00000 q^{95} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7} - 6 q^{11} + 2 q^{25} - 12 q^{37} - 4 q^{41} + 16 q^{47} + 4 q^{49} + 22 q^{53} + 4 q^{65} - 16 q^{67} + 4 q^{71} - 2 q^{73} + 18 q^{77} - 2 q^{83} - 12 q^{85} + 4 q^{95}+O(q^{100}) \) 2 * q - 6 * q^7 - 6 * q^11 + 2 * q^25 - 12 * q^37 - 4 * q^41 + 16 * q^47 + 4 * q^49 + 22 * q^53 + 4 * q^65 - 16 * q^67 + 4 * q^71 - 2 * q^73 + 18 * q^77 - 2 * q^83 - 12 * q^85 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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