LMFDB - Embedded newform 2808.2.c.e.649.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(649,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 44x^{12} + 708x^{10} + 5026x^{8} + 15252x^{6} + 19688x^{4} + 8857x^{2} + 576 $ x^14 + 44x^12 + 708x^10 + 5026x^8 + 15252x^6 + 19688x^4 + 8857x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.14
Root			$3.85219i$ of defining polynomial
Character	$\chi$	$=$	2808.649
Dual form			2808.2.c.e.649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.85219i q^{5} +2.37854i q^{7} -0.733693i q^{11} +(-2.87661 - 2.17374i) q^{13} -4.14932 q^{17} +1.41046i q^{19} -5.23182 q^{23} -9.83934 q^{25} +1.20734 q^{29} -5.01326i q^{31} -9.16258 q^{35} -10.8085i q^{37} +4.46104i q^{41} +0.930997 q^{43} +6.57560i q^{47} +1.34255 q^{49} +4.81510 q^{53} +2.82632 q^{55} +0.634721i q^{59} -7.96040 q^{61} +(8.37364 - 11.0812i) q^{65} +5.70649i q^{67} -7.09577i q^{71} -7.60540i q^{73} +1.74512 q^{77} -13.5139 q^{79} +14.4942i q^{83} -15.9839i q^{85} -4.41066i q^{89} +(5.17032 - 6.84212i) q^{91} -5.43335 q^{95} +4.09129i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 16 q^{17} + 8 q^{23} - 18 q^{25} - 14 q^{29} - 16 q^{35} - 12 q^{43} - 10 q^{49} + 4 q^{53} - 4 q^{55} - 16 q^{61} - 8 q^{65} - 34 q^{79} - 32 q^{91} + 64 q^{95}+O(q^{100}) $ 14 * q - 16 * q^17 + 8 * q^23 - 18 * q^25 - 14 * q^29 - 16 * q^35 - 12 * q^43 - 10 * q^49 + 4 * q^53 - 4 * q^55 - 16 * q^61 - 8 * q^65 - 34 * q^79 - 32 * q^91 + 64 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$			3.85219i			1.72275i	0.507969	+	0.861375i	$0.330396\pi$
$5$			3.85219i			1.72275i	−0.507969	+	0.861375i	$0.669604\pi$
$6$		0			0
$7$			2.37854i			0.899003i	0.893280	+	0.449502i	$0.148398\pi$
$7$			2.37854i			0.899003i	−0.893280	+	0.449502i	$0.851602\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.733693i		−	0.221217i	−0.993864	−	0.110608i	$-0.964720\pi$
$11$		−	0.733693i		−	0.221217i	0.993864	−	0.110608i	$-0.0352799\pi$
$12$		0			0
$13$	−2.87661	−	2.17374i	−0.797827	−	0.602886i
$13$	−2.87661	−	2.17374i	−0.797827	−	0.602886i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.14932			−1.00636			−0.503178	−	0.864183i	$-0.667836\pi$
$17$	−4.14932			−1.00636			−0.503178	+	0.864183i	$0.667836\pi$
$18$		0			0
$19$			1.41046i			0.323581i	0.986825	+	0.161791i	$0.0517269\pi$
$19$			1.41046i			0.323581i	−0.986825	+	0.161791i	$0.948273\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.23182			−1.09091			−0.545455	−	0.838140i	$-0.683643\pi$
$23$	−5.23182			−1.09091			−0.545455	+	0.838140i	$0.683643\pi$
$24$		0			0
$25$	−9.83934			−1.96787
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.20734			0.224198			0.112099	−	0.993697i	$-0.464243\pi$
$29$	1.20734			0.224198			0.112099	+	0.993697i	$0.464243\pi$
$30$		0			0
$31$		−	5.01326i		−	0.900408i	−0.892926	−	0.450204i	$-0.851351\pi$
$31$		−	5.01326i		−	0.900408i	0.892926	−	0.450204i	$-0.148649\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.16258			−1.54876
$36$		0			0
$37$		−	10.8085i		−	1.77691i	−0.458964	−	0.888455i	$-0.651779\pi$
$37$		−	10.8085i		−	1.77691i	0.458964	−	0.888455i	$-0.348221\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.46104i			0.696698i	0.937365	+	0.348349i	$0.113258\pi$
$41$			4.46104i			0.696698i	−0.937365	+	0.348349i	$0.886742\pi$
$42$		0			0
$43$	0.930997			0.141976			0.0709879	−	0.997477i	$-0.477385\pi$
$43$	0.930997			0.141976			0.0709879	+	0.997477i	$0.477385\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.57560i			0.959150i	0.877501	+	0.479575i	$0.159209\pi$
$47$			6.57560i			0.959150i	−0.877501	+	0.479575i	$0.840791\pi$
$48$		0			0
$49$	1.34255			0.191793
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.81510			0.661405			0.330703	−	0.943735i	$-0.392714\pi$
$53$	4.81510			0.661405			0.330703	+	0.943735i	$0.392714\pi$
$54$		0			0
$55$	2.82632			0.381101
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.634721i			0.0826337i	0.999146	+	0.0413168i	$0.0131553\pi$
$59$			0.634721i			0.0826337i	−0.999146	+	0.0413168i	$0.986845\pi$
$60$		0			0
$61$	−7.96040			−1.01922			−0.509612	−	0.860404i	$-0.670211\pi$
$61$	−7.96040			−1.01922			−0.509612	+	0.860404i	$0.670211\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	8.37364	−	11.0812i	1.03862	−	1.37446i
$66$		0			0
$67$			5.70649i			0.697159i	0.937279	+	0.348579i	$0.113336\pi$
$67$			5.70649i			0.697159i	−0.937279	+	0.348579i	$0.886664\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.09577i		−	0.842113i	−0.907035	−	0.421056i	$-0.861659\pi$
$71$		−	7.09577i		−	0.842113i	0.907035	−	0.421056i	$-0.138341\pi$
$72$		0			0
$73$		−	7.60540i		−	0.890145i	−0.895495	−	0.445072i	$-0.853178\pi$
$73$		−	7.60540i		−	0.890145i	0.895495	−	0.445072i	$-0.146822\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.74512			0.198875
$78$		0			0
$79$	−13.5139			−1.52043			−0.760217	−	0.649669i	$-0.774907\pi$
$79$	−13.5139			−1.52043			−0.760217	+	0.649669i	$0.774907\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			14.4942i			1.59095i	0.605988	+	0.795474i	$0.292778\pi$
$83$			14.4942i			1.59095i	−0.605988	+	0.795474i	$0.707222\pi$
$84$		0			0
$85$		−	15.9839i		−	1.73370i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.41066i		−	0.467529i	−0.972293	−	0.233765i	$-0.924895\pi$
$89$		−	4.41066i		−	0.467529i	0.972293	−	0.233765i	$-0.0751045\pi$
$90$		0			0
$91$	5.17032	−	6.84212i	0.541997	−	0.717249i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.43335			−0.557450
$96$		0			0
$97$			4.09129i			0.415408i	0.978192	+	0.207704i	$0.0665990\pi$
$97$			4.09129i			0.415408i	−0.978192	+	0.207704i	$0.933401\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.86976			0.186048			0.0930239	−	0.995664i	$-0.470347\pi$
$101$	1.86976			0.186048			0.0930239	+	0.995664i	$0.470347\pi$
$102$		0			0
$103$	−0.134968			−0.0132988			−0.00664941	−	0.999978i	$-0.502117\pi$
$103$	−0.134968			−0.0132988			−0.00664941	+	0.999978i	$0.502117\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.7307			1.42407			0.712034	−	0.702145i	$-0.247774\pi$
$107$	14.7307			1.42407			0.712034	+	0.702145i	$0.247774\pi$
$108$		0			0
$109$		−	13.4007i		−	1.28355i	−0.766893	−	0.641775i	$-0.778198\pi$
$109$		−	13.4007i		−	1.28355i	0.766893	−	0.641775i	$-0.221802\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.5834			1.56003			0.780016	−	0.625759i	$-0.215211\pi$
$113$	16.5834			1.56003			0.780016	+	0.625759i	$0.215211\pi$
$114$		0			0
$115$		−	20.1540i		−	1.87937i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.86931i		−	0.904718i
$120$		0			0
$121$	10.4617			0.951063
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	18.6420i		−	1.66740i
$126$		0			0
$127$	4.41855			0.392083			0.196041	−	0.980596i	$-0.437191\pi$
$127$	4.41855			0.392083			0.196041	+	0.980596i	$0.437191\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.0542482			−0.00473968			−0.00236984	−	0.999997i	$-0.500754\pi$
$131$	−0.0542482			−0.00473968			−0.00236984	+	0.999997i	$0.500754\pi$
$132$		0			0
$133$	−3.35483			−0.290901
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	16.3361i		−	1.39569i	−0.716249	−	0.697845i	$-0.754142\pi$
$137$		−	16.3361i		−	1.39569i	0.716249	−	0.697845i	$-0.245858\pi$
$138$		0			0
$139$	−15.1322			−1.28349			−0.641747	−	0.766917i	$-0.721790\pi$
$139$	−15.1322			−1.28349			−0.641747	+	0.766917i	$0.721790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.59486	+	2.11055i	−0.133369	+	0.176493i
$144$		0			0
$145$			4.65090i			0.386236i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			10.4869i			0.859121i	0.903038	+	0.429560i	$0.141332\pi$
$149$			10.4869i			0.859121i	−0.903038	+	0.429560i	$0.858668\pi$
$150$		0			0
$151$			5.52378i			0.449519i	0.974414	+	0.224759i	$0.0721596\pi$
$151$			5.52378i			0.449519i	−0.974414	+	0.224759i	$0.927840\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	19.3120			1.55118
$156$		0			0
$157$	−11.3891			−0.908947			−0.454474	−	0.890760i	$-0.650173\pi$
$157$	−11.3891			−0.908947			−0.454474	+	0.890760i	$0.650173\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	12.4441i		−	0.980732i
$162$		0			0
$163$		−	4.80958i		−	0.376715i	−0.982100	−	0.188358i	$-0.939684\pi$
$163$		−	4.80958i		−	0.376715i	0.982100	−	0.188358i	$-0.0603164\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.04538i			0.699953i	0.936759	+	0.349976i	$0.113810\pi$
$167$			9.04538i			0.699953i	−0.936759	+	0.349976i	$0.886190\pi$
$168$		0			0
$169$	3.54974	+	12.5060i	0.273057	+	0.961998i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−24.7847			−1.88435			−0.942174	−	0.335125i	$-0.891221\pi$
$173$	−24.7847			−1.88435			−0.942174	+	0.335125i	$0.891221\pi$
$174$		0			0
$175$		−	23.4033i		−	1.76912i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−25.6251			−1.91531			−0.957654	−	0.287920i	$-0.907036\pi$
$179$	−25.6251			−1.91531			−0.957654	+	0.287920i	$0.907036\pi$
$180$		0			0
$181$	−16.1523			−1.20059			−0.600296	−	0.799778i	$-0.704951\pi$
$181$	−16.1523			−1.20059			−0.600296	+	0.799778i	$0.704951\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	41.6364			3.06117
$186$		0			0
$187$			3.04433i			0.222623i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.0765			−1.38033			−0.690163	−	0.723654i	$-0.742461\pi$
$191$	−19.0765			−1.38033			−0.690163	+	0.723654i	$0.742461\pi$
$192$		0			0
$193$			19.7204i			1.41951i	0.704450	+	0.709754i	$0.251194\pi$
$193$			19.7204i			1.41951i	−0.704450	+	0.709754i	$0.748806\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.73609i		−	0.123691i	−0.998086	−	0.0618457i	$-0.980301\pi$
$197$		−	1.73609i		−	0.123691i	0.998086	−	0.0618457i	$-0.0196987\pi$
$198$		0			0
$199$	−20.7472			−1.47073			−0.735365	−	0.677672i	$-0.762989\pi$
$199$	−20.7472			−1.47073			−0.735365	+	0.677672i	$0.762989\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			2.87171i			0.201554i
$204$		0			0
$205$	−17.1848			−1.20024
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.03484			0.0715817
$210$		0			0
$211$	−6.82222			−0.469661			−0.234830	−	0.972036i	$-0.575453\pi$
$211$	−6.82222			−0.469661			−0.234830	+	0.972036i	$0.575453\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.58637i			0.244589i
$216$		0			0
$217$	11.9242			0.809470
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.9360	+	9.01952i	0.802899	+	0.606719i
$222$		0			0
$223$		−	14.2154i		−	0.951931i	−0.879464	−	0.475966i	$-0.842099\pi$
$223$		−	14.2154i		−	0.951931i	0.879464	−	0.475966i	$-0.157901\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			29.3012i			1.94479i	0.233342	+	0.972395i	$0.425034\pi$
$227$			29.3012i			1.94479i	−0.233342	+	0.972395i	$0.574966\pi$
$228$		0			0
$229$		−	11.4767i		−	0.758399i	−0.925315	−	0.379199i	$-0.876199\pi$
$229$		−	11.4767i		−	0.758399i	0.925315	−	0.379199i	$-0.123801\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.84128			0.579212			0.289606	−	0.957146i	$-0.406476\pi$
$233$	8.84128			0.579212			0.289606	+	0.957146i	$0.406476\pi$
$234$		0			0
$235$	−25.3304			−1.65238
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.12638i		−	0.266913i	−0.991055	−	0.133457i	$-0.957392\pi$
$239$		−	4.12638i		−	0.266913i	0.991055	−	0.133457i	$-0.0426077\pi$
$240$		0			0
$241$		−	20.0511i		−	1.29161i	−0.763503	−	0.645804i	$-0.776522\pi$
$241$		−	20.0511i		−	1.29161i	0.763503	−	0.645804i	$-0.223478\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			5.17176i			0.330412i
$246$		0			0
$247$	3.06597	−	4.05733i	0.195083	−	0.258162i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−29.0266			−1.83214			−0.916072	−	0.401013i	$-0.868658\pi$
$251$	−29.0266			−1.83214			−0.916072	+	0.401013i	$0.868658\pi$
$252$		0			0
$253$			3.83855i			0.241328i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.9551			−0.995249			−0.497625	−	0.867393i	$-0.665794\pi$
$257$	−15.9551			−0.995249			−0.497625	+	0.867393i	$0.665794\pi$
$258$		0			0
$259$	25.7085			1.59745
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.00100			0.0617243			0.0308622	−	0.999524i	$-0.490175\pi$
$263$	1.00100			0.0617243			0.0308622	+	0.999524i	$0.490175\pi$
$264$		0			0
$265$			18.5487i			1.13944i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−26.6615			−1.62558			−0.812791	−	0.582555i	$-0.802053\pi$
$269$	−26.6615			−1.62558			−0.812791	+	0.582555i	$0.802053\pi$
$270$		0			0
$271$		−	10.8679i		−	0.660176i	−0.943950	−	0.330088i	$-0.892922\pi$
$271$		−	10.8679i		−	0.660176i	0.943950	−	0.330088i	$-0.107078\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			7.21906i			0.435326i
$276$		0			0
$277$	17.8278			1.07117			0.535584	−	0.844482i	$-0.320092\pi$
$277$	17.8278			1.07117			0.535584	+	0.844482i	$0.320092\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.64894i			0.575607i	0.957689	+	0.287804i	$0.0929251\pi$
$281$			9.64894i			0.575607i	−0.957689	+	0.287804i	$0.907075\pi$
$282$		0			0
$283$	−24.6395			−1.46467			−0.732333	−	0.680947i	$-0.761568\pi$
$283$	−24.6395			−1.46467			−0.732333	+	0.680947i	$0.761568\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.6108			−0.626334
$288$		0			0
$289$	0.216825			0.0127544
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3.40432i			0.198883i	0.995043	+	0.0994413i	$0.0317055\pi$
$293$			3.40432i			0.198883i	−0.995043	+	0.0994413i	$0.968294\pi$
$294$		0			0
$295$	−2.44506			−0.142357
$296$		0			0
$297$		0			0
$298$		0			0
$299$	15.0499	+	11.3726i	0.870358	+	0.657695i
$300$		0			0
$301$			2.21441i			0.127637i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	30.6649i		−	1.75587i
$306$		0			0
$307$			32.2702i			1.84176i	0.389851	+	0.920878i	$0.372527\pi$
$307$			32.2702i			1.84176i	−0.389851	+	0.920878i	$0.627473\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.13005			−0.0640791			−0.0320395	−	0.999487i	$-0.510200\pi$
$311$	−1.13005			−0.0640791			−0.0320395	+	0.999487i	$0.510200\pi$
$312$		0			0
$313$	3.34519			0.189081			0.0945407	−	0.995521i	$-0.469862\pi$
$313$	3.34519			0.189081			0.0945407	+	0.995521i	$0.469862\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			9.04934i			0.508262i	0.967170	+	0.254131i	$0.0817894\pi$
$317$			9.04934i			0.508262i	−0.967170	+	0.254131i	$0.918211\pi$
$318$		0			0
$319$		−	0.885818i		−	0.0495963i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.85244i		−	0.325638i
$324$		0			0
$325$	28.3039	+	21.3881i	1.57002	+	1.18640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.6403			−0.862279
$330$		0			0
$331$		−	35.8544i		−	1.97074i	−0.170435	−	0.985369i	$-0.554517\pi$
$331$		−	35.8544i		−	1.97074i	0.170435	−	0.985369i	$-0.445483\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−21.9825			−1.20103
$336$		0			0
$337$	27.5377			1.50007			0.750037	−	0.661396i	$-0.230036\pi$
$337$	27.5377			1.50007			0.750037	+	0.661396i	$0.230036\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.67820			−0.199185
$342$		0			0
$343$			19.8431i			1.07143i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.9831			−0.965383			−0.482691	−	0.875791i	$-0.660341\pi$
$347$	−17.9831			−0.965383			−0.482691	+	0.875791i	$0.660341\pi$
$348$		0			0
$349$			5.61325i			0.300471i	0.988650	+	0.150235i	$0.0480031\pi$
$349$			5.61325i			0.300471i	−0.988650	+	0.150235i	$0.951997\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.64582i			0.140823i	0.997518	+	0.0704114i	$0.0224312\pi$
$353$			2.64582i			0.140823i	−0.997518	+	0.0704114i	$0.977569\pi$
$354$		0			0
$355$	27.3342			1.45075
$356$		0			0
$357$		0			0
$358$		0			0
$359$			21.3409i			1.12633i	0.826345	+	0.563164i	$0.190416\pi$
$359$			21.3409i			1.12633i	−0.826345	+	0.563164i	$0.809584\pi$
$360$		0			0
$361$	17.0106			0.895295
$362$		0			0
$363$		0			0
$364$		0			0
$365$	29.2974			1.53350
$366$		0			0
$367$	33.6037			1.75410			0.877048	−	0.480402i	$-0.159509\pi$
$367$	33.6037			1.75410			0.877048	+	0.480402i	$0.159509\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			11.4529i			0.594605i
$372$		0			0
$373$	−32.4930			−1.68242			−0.841211	−	0.540707i	$-0.818157\pi$
$373$	−32.4930			−1.68242			−0.841211	+	0.540707i	$0.818157\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.47305	−	2.62444i	−0.178871	−	0.135166i
$378$		0			0
$379$			11.8138i			0.606834i	0.952858	+	0.303417i	$0.0981276\pi$
$379$			11.8138i			0.606834i	−0.952858	+	0.303417i	$0.901872\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			16.5717i			0.846772i	0.905949	+	0.423386i	$0.139159\pi$
$383$			16.5717i			0.846772i	−0.905949	+	0.423386i	$0.860841\pi$
$384$		0			0
$385$			6.72252i			0.342611i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.7080			1.40485			0.702425	−	0.711758i	$-0.252101\pi$
$389$	27.7080			1.40485			0.702425	+	0.711758i	$0.252101\pi$
$390$		0			0
$391$	21.7085			1.09785
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	52.0581i		−	2.61933i
$396$		0			0
$397$		−	15.8526i		−	0.795618i	−0.917468	−	0.397809i	$-0.869771\pi$
$397$		−	15.8526i		−	0.795618i	0.917468	−	0.397809i	$-0.130229\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.9221i			0.845049i	0.906352	+	0.422524i	$0.138856\pi$
$401$			16.9221i			0.845049i	−0.906352	+	0.422524i	$0.861144\pi$
$402$		0			0
$403$	−10.8975	+	14.4212i	−0.542844	+	0.718370i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.93014			−0.393082
$408$		0			0
$409$		−	16.8358i		−	0.832479i	−0.909255	−	0.416240i	$-0.863348\pi$
$409$		−	16.8358i		−	0.832479i	0.909255	−	0.416240i	$-0.136652\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.50971			−0.0742879
$414$		0			0
$415$	−55.8345			−2.74081
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.6515			0.666920			0.333460	−	0.942764i	$-0.391784\pi$
$419$	13.6515			0.666920			0.333460	+	0.942764i	$0.391784\pi$
$420$		0			0
$421$		−	17.6913i		−	0.862222i	−0.902299	−	0.431111i	$-0.858122\pi$
$421$		−	17.6913i		−	0.862222i	0.902299	−	0.431111i	$-0.141878\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	40.8265			1.98038
$426$		0			0
$427$		−	18.9341i		−	0.916286i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	10.0763i		−	0.485356i	−0.970107	−	0.242678i	$-0.921974\pi$
$431$		−	10.0763i		−	0.485356i	0.970107	−	0.242678i	$-0.0780259\pi$
$432$		0			0
$433$	−6.92199			−0.332649			−0.166325	−	0.986071i	$-0.553190\pi$
$433$	−6.92199			−0.332649			−0.166325	+	0.986071i	$0.553190\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	7.37927i		−	0.352998i
$438$		0			0
$439$	37.2107			1.77597			0.887986	−	0.459871i	$-0.152104\pi$
$439$	37.2107			1.77597			0.887986	+	0.459871i	$0.152104\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.1257			−1.05122			−0.525612	−	0.850725i	$-0.676164\pi$
$443$	−22.1257			−1.05122			−0.525612	+	0.850725i	$0.676164\pi$
$444$		0			0
$445$	16.9907			0.805436
$446$		0			0
$447$		0			0
$448$		0			0
$449$			0.497997i			0.0235019i	0.999931	+	0.0117510i	$0.00374053\pi$
$449$			0.497997i			0.0235019i	−0.999931	+	0.0117510i	$0.996259\pi$
$450$		0			0
$451$	3.27304			0.154121
$452$		0			0
$453$		0			0
$454$		0			0
$455$	26.3571	+	19.9170i	1.23564	+	0.933725i
$456$		0			0
$457$			18.7599i			0.877549i	0.898597	+	0.438775i	$0.144587\pi$
$457$			18.7599i			0.877549i	−0.898597	+	0.438775i	$0.855413\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	34.2255i		−	1.59404i	−0.603952	−	0.797021i	$-0.706408\pi$
$461$		−	34.2255i		−	1.59404i	0.603952	−	0.797021i	$-0.293592\pi$
$462$		0			0
$463$			35.5281i			1.65113i	0.564308	+	0.825564i	$0.309143\pi$
$463$			35.5281i			1.65113i	−0.564308	+	0.825564i	$0.690857\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−30.9860			−1.43386			−0.716929	−	0.697146i	$-0.754453\pi$
$467$	−30.9860			−1.43386			−0.716929	+	0.697146i	$0.754453\pi$
$468$		0			0
$469$	−13.5731			−0.626748
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	0.683066i		−	0.0314074i
$474$		0			0
$475$		−	13.8780i		−	0.636765i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	34.8803i		−	1.59372i	−0.604162	−	0.796861i	$-0.706492\pi$
$479$		−	34.8803i		−	1.59372i	0.604162	−	0.796861i	$-0.293508\pi$
$480$		0			0
$481$	−23.4949	+	31.0919i	−1.07127	+	1.41767i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.7604			−0.715644
$486$		0			0
$487$			3.61355i			0.163745i	0.996643	+	0.0818727i	$0.0260901\pi$
$487$			3.61355i			0.163745i	−0.996643	+	0.0818727i	$0.973910\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.7771			0.802267			0.401134	−	0.916020i	$-0.368616\pi$
$491$	17.7771			0.802267			0.401134	+	0.916020i	$0.368616\pi$
$492$		0			0
$493$	−5.00964			−0.225623
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.8776			0.757062
$498$		0			0
$499$			26.2876i			1.17680i	0.808571	+	0.588398i	$0.200241\pi$
$499$			26.2876i			1.17680i	−0.808571	+	0.588398i	$0.799759\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.52208			−0.246217			−0.123109	−	0.992393i	$-0.539286\pi$
$503$	−5.52208			−0.246217			−0.123109	+	0.992393i	$0.539286\pi$
$504$		0			0
$505$			7.20266i			0.320514i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			5.61785i			0.249007i	0.992219	+	0.124503i	$0.0397338\pi$
$509$			5.61785i			0.249007i	−0.992219	+	0.124503i	$0.960266\pi$
$510$		0			0
$511$	18.0897			0.800243
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	0.519923i		−	0.0229105i
$516$		0			0
$517$	4.82447			0.212180
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.2394			1.32481			0.662406	−	0.749145i	$-0.269535\pi$
$521$	30.2394			1.32481			0.662406	+	0.749145i	$0.269535\pi$
$522$		0			0
$523$	5.83073			0.254960			0.127480	−	0.991841i	$-0.459311\pi$
$523$	5.83073			0.254960			0.127480	+	0.991841i	$0.459311\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			20.8016i			0.906132i
$528$		0			0
$529$	4.37196			0.190085
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.69714	−	12.8327i	0.420030	−	0.555845i
$534$		0			0
$535$			56.7453i			2.45331i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	0.985021i		−	0.0424279i
$540$		0			0
$541$		−	19.1521i		−	0.823413i	−0.911317	−	0.411706i	$-0.864933\pi$
$541$		−	19.1521i		−	0.823413i	0.911317	−	0.411706i	$-0.135067\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	51.6218			2.21124
$546$		0			0
$547$	28.2196			1.20658			0.603292	−	0.797520i	$-0.293855\pi$
$547$	28.2196			1.20658			0.603292	+	0.797520i	$0.293855\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1.70290i			0.0725462i
$552$		0			0
$553$		−	32.1434i		−	1.36688i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	31.6364i		−	1.34048i	−0.742145	−	0.670239i	$-0.766192\pi$
$557$		−	31.6364i		−	1.34048i	0.742145	−	0.670239i	$-0.233808\pi$
$558$		0			0
$559$	−2.67811	−	2.02374i	−0.113272	−	0.0855952i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.42031			0.186294			0.0931469	−	0.995652i	$-0.470307\pi$
$563$	4.42031			0.186294			0.0931469	+	0.995652i	$0.470307\pi$
$564$		0			0
$565$			63.8822i			2.68755i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−35.5777			−1.49150			−0.745748	−	0.666228i	$-0.767908\pi$
$569$	−35.5777			−1.49150			−0.745748	+	0.666228i	$0.767908\pi$
$570$		0			0
$571$	27.0643			1.13260			0.566302	−	0.824198i	$-0.308373\pi$
$571$	27.0643			1.13260			0.566302	+	0.824198i	$0.308373\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	51.4777			2.14677
$576$		0			0
$577$			0.0886048i			0.00368866i	0.999998	+	0.00184433i	$0.000587069\pi$
$577$			0.0886048i			0.00368866i	−0.999998	+	0.00184433i	$0.999413\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−34.4751			−1.43027
$582$		0			0
$583$		−	3.53281i		−	0.146314i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	5.27073i		−	0.217546i	−0.994067	−	0.108773i	$-0.965308\pi$
$587$		−	5.27073i		−	0.217546i	0.994067	−	0.108773i	$-0.0346922\pi$
$588$		0			0
$589$	7.07100			0.291355
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	25.1204i		−	1.03157i	−0.856717	−	0.515786i	$-0.827500\pi$
$593$		−	25.1204i		−	1.03157i	0.856717	−	0.515786i	$-0.172500\pi$
$594$		0			0
$595$	38.0184			1.55860
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.14359			0.210162			0.105081	−	0.994464i	$-0.466490\pi$
$599$	5.14359			0.210162			0.105081	+	0.994464i	$0.466490\pi$
$600$		0			0
$601$	−20.8652			−0.851110			−0.425555	−	0.904933i	$-0.639921\pi$
$601$	−20.8652			−0.851110			−0.425555	+	0.904933i	$0.639921\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			40.3004i			1.63844i
$606$		0			0
$607$	−8.48225			−0.344284			−0.172142	−	0.985072i	$-0.555069\pi$
$607$	−8.48225			−0.344284			−0.172142	+	0.985072i	$0.555069\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	14.2936	−	18.9154i	0.578258	−	0.765236i
$612$		0			0
$613$			40.5600i			1.63820i	0.573649	+	0.819101i	$0.305527\pi$
$613$			40.5600i			1.63820i	−0.573649	+	0.819101i	$0.694473\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	4.68479i		−	0.188602i	−0.995544	−	0.0943012i	$-0.969938\pi$
$617$		−	4.68479i		−	0.188602i	0.995544	−	0.0943012i	$-0.0300617\pi$
$618$		0			0
$619$			42.6310i			1.71349i	0.515744	+	0.856743i	$0.327515\pi$
$619$			42.6310i			1.71349i	−0.515744	+	0.856743i	$0.672485\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.4909			0.420310
$624$		0			0
$625$	22.6159			0.904637
$626$		0			0
$627$		0			0
$628$		0			0
$629$			44.8480i			1.78821i
$630$		0			0
$631$			20.5307i			0.817315i	0.912688	+	0.408657i	$0.134003\pi$
$631$			20.5307i			0.817315i	−0.912688	+	0.408657i	$0.865997\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			17.0211i			0.675460i
$636$		0			0
$637$	−3.86199	−	2.91835i	−0.153018	−	0.115629i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.93699			0.313492			0.156746	−	0.987639i	$-0.449900\pi$
$641$	7.93699			0.313492			0.156746	+	0.987639i	$0.449900\pi$
$642$		0			0
$643$			15.5249i			0.612243i	0.951993	+	0.306121i	$0.0990313\pi$
$643$			15.5249i			0.612243i	−0.951993	+	0.306121i	$0.900969\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.0481431			−0.00189270			−0.000946350	−	1.00000i	$-0.500301\pi$
$647$	−0.0481431			−0.00189270			−0.000946350		1.00000i	$0.500301\pi$
$648$		0			0
$649$	0.465691			0.0182800
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−42.2517			−1.65344			−0.826719	−	0.562615i	$-0.809795\pi$
$653$	−42.2517			−1.65344			−0.826719	+	0.562615i	$0.809795\pi$
$654$		0			0
$655$		−	0.208974i		−	0.00816529i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.3704			0.754563			0.377281	−	0.926099i	$-0.376859\pi$
$659$	19.3704			0.754563			0.377281	+	0.926099i	$0.376859\pi$
$660$		0			0
$661$		−	30.8136i		−	1.19851i	−0.800558	−	0.599255i	$-0.795464\pi$
$661$		−	30.8136i		−	1.19851i	0.800558	−	0.599255i	$-0.204536\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	12.9234i		−	0.501149i
$666$		0			0
$667$	−6.31659			−0.244579
$668$		0			0
$669$		0			0
$670$		0			0
$671$			5.84049i			0.225470i
$672$		0			0
$673$	−35.4926			−1.36814			−0.684069	−	0.729417i	$-0.739791\pi$
$673$	−35.4926			−1.36814			−0.684069	+	0.729417i	$0.739791\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.1009			−0.541942			−0.270971	−	0.962588i	$-0.587345\pi$
$677$	−14.1009			−0.541942			−0.270971	+	0.962588i	$0.587345\pi$
$678$		0			0
$679$	−9.73130			−0.373453
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.5553i		−	1.20743i	−0.797200	−	0.603715i	$-0.793686\pi$
$683$		−	31.5553i		−	1.20743i	0.797200	−	0.603715i	$-0.206314\pi$
$684$		0			0
$685$	62.9298			2.40442
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.8512	−	10.4668i	−0.527687	−	0.398752i
$690$		0			0
$691$			23.3407i			0.887923i	0.896046	+	0.443962i	$0.146427\pi$
$691$			23.3407i			0.887923i	−0.896046	+	0.443962i	$0.853573\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	58.2919i		−	2.21114i
$696$		0			0
$697$		−	18.5103i		−	0.701127i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−23.4201			−0.884565			−0.442283	−	0.896876i	$-0.645831\pi$
$701$	−23.4201			−0.884565			−0.442283	+	0.896876i	$0.645831\pi$
$702$		0			0
$703$	15.2450			0.574975
$704$		0			0
$705$		0			0
$706$		0			0
$707$			4.44729i			0.167258i
$708$		0			0
$709$		−	7.71832i		−	0.289868i	−0.989441	−	0.144934i	$-0.953703\pi$
$709$		−	7.71832i		−	0.289868i	0.989441	−	0.144934i	$-0.0462969\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			26.2285i			0.982264i
$714$		0			0
$715$	−8.13022	−	6.14368i	−0.304053	−	0.229761i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.1346			−0.900068			−0.450034	−	0.893012i	$-0.648588\pi$
$719$	−24.1346			−0.900068			−0.450034	+	0.893012i	$0.648588\pi$
$720$		0			0
$721$		−	0.321027i		−	0.0119557i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11.8794			−0.441191
$726$		0			0
$727$	2.92222			0.108379			0.0541896	−	0.998531i	$-0.482742\pi$
$727$	2.92222			0.108379			0.0541896	+	0.998531i	$0.482742\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.86300			−0.142878
$732$		0			0
$733$		−	2.28352i		−	0.0843438i	−0.999110	−	0.0421719i	$-0.986572\pi$
$733$		−	2.28352i		−	0.0843438i	0.999110	−	0.0421719i	$-0.0134277\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.18682			0.154223
$738$		0			0
$739$		−	19.6635i		−	0.723333i	−0.932307	−	0.361667i	$-0.882208\pi$
$739$		−	19.6635i		−	0.723333i	0.932307	−	0.361667i	$-0.117792\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16.4662i			0.604086i	0.953294	+	0.302043i	$0.0976686\pi$
$743$			16.4662i			0.604086i	−0.953294	+	0.302043i	$0.902331\pi$
$744$		0			0
$745$	−40.3975			−1.48005
$746$		0			0
$747$		0			0
$748$		0			0
$749$			35.0375i			1.28024i
$750$		0			0
$751$	−33.7213			−1.23051			−0.615254	−	0.788329i	$-0.710947\pi$
$751$	−33.7213			−1.23051			−0.615254	+	0.788329i	$0.710947\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−21.2786			−0.774408
$756$		0			0
$757$	16.8620			0.612859			0.306430	−	0.951893i	$-0.400866\pi$
$757$	16.8620			0.612859			0.306430	+	0.951893i	$0.400866\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	26.2971i		−	0.953269i	−0.879102	−	0.476634i	$-0.841857\pi$
$761$		−	26.2971i		−	0.953269i	0.879102	−	0.476634i	$-0.158143\pi$
$762$		0			0
$763$	31.8740			1.15392
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.37972	−	1.82584i	0.0498187	−	0.0659274i
$768$		0			0
$769$			4.84344i			0.174659i	0.996179	+	0.0873295i	$0.0278333\pi$
$769$			4.84344i			0.174659i	−0.996179	+	0.0873295i	$0.972167\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			29.5662i			1.06342i	0.846926	+	0.531711i	$0.178451\pi$
$773$			29.5662i			1.06342i	−0.846926	+	0.531711i	$0.821549\pi$
$774$		0			0
$775$			49.3272i			1.77188i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.29212			−0.225439
$780$		0			0
$781$	−5.20612			−0.186290
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	43.8729i		−	1.56589i
$786$		0			0
$787$			44.1573i			1.57404i	0.616930	+	0.787018i	$0.288376\pi$
$787$			44.1573i			1.57404i	−0.616930	+	0.787018i	$0.711624\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			39.4442i			1.40247i
$792$		0			0
$793$	22.8989	+	17.3038i	0.813165	+	0.614476i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.8236			−0.773033			−0.386516	−	0.922283i	$-0.626322\pi$
$797$	−21.8236			−0.773033			−0.386516	+	0.922283i	$0.626322\pi$
$798$		0			0
$799$		−	27.2842i		−	0.965247i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.58003			−0.196915
$804$		0			0
$805$	47.9370			1.68956
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.7001			0.446511			0.223256	−	0.974760i	$-0.428331\pi$
$809$	12.7001			0.446511			0.223256	+	0.974760i	$0.428331\pi$
$810$		0			0
$811$		−	7.33711i		−	0.257641i	−0.991668	−	0.128820i	$-0.958881\pi$
$811$		−	7.33711i		−	0.257641i	0.991668	−	0.128820i	$-0.0411191\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.5274			0.648987
$816$		0			0
$817$			1.31313i			0.0459407i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			56.8052i			1.98251i	0.131947	+	0.991257i	$0.457877\pi$
$821$			56.8052i			1.98251i	−0.131947	+	0.991257i	$0.542123\pi$
$822$		0			0
$823$	48.4651			1.68939			0.844694	−	0.535249i	$-0.179782\pi$
$823$	48.4651			1.68939			0.844694	+	0.535249i	$0.179782\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	15.7732i		−	0.548489i	−0.961660	−	0.274245i	$-0.911572\pi$
$827$		−	15.7732i		−	0.548489i	0.961660	−	0.274245i	$-0.0884278\pi$
$828$		0			0
$829$	48.4962			1.68434			0.842171	−	0.539211i	$-0.181277\pi$
$829$	48.4962			1.68434			0.842171	+	0.539211i	$0.181277\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.57067			−0.193012
$834$		0			0
$835$	−34.8445			−1.20584
$836$		0			0
$837$		0			0
$838$		0			0
$839$			35.6874i			1.23207i	0.787721	+	0.616033i	$0.211261\pi$
$839$			35.6874i			1.23207i	−0.787721	+	0.616033i	$0.788739\pi$
$840$		0			0
$841$	−27.5423			−0.949735
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−48.1753	+	13.6743i	−1.65728	+	0.470409i
$846$		0			0
$847$			24.8835i			0.855009i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			56.5482i			1.93845i
$852$		0			0
$853$			20.4790i			0.701189i	0.936527	+	0.350595i	$0.114020\pi$
$853$			20.4790i			0.701189i	−0.936527	+	0.350595i	$0.885980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.6485			0.807817			0.403908	−	0.914799i	$-0.367651\pi$
$857$	23.6485			0.807817			0.403908	+	0.914799i	$0.367651\pi$
$858$		0			0
$859$	−11.8778			−0.405266			−0.202633	−	0.979255i	$-0.564950\pi$
$859$	−11.8778			−0.405266			−0.202633	+	0.979255i	$0.564950\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			32.3103i			1.09985i	0.835213	+	0.549927i	$0.185345\pi$
$863$			32.3103i			1.09985i	−0.835213	+	0.549927i	$0.814655\pi$
$864$		0			0
$865$		−	95.4754i		−	3.24626i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.91507i			0.336346i
$870$		0			0
$871$	12.4044	−	16.4153i	0.420307	−	0.556212i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	44.3408			1.49899
$876$		0			0
$877$			56.5079i			1.90814i	0.299591	+	0.954068i	$0.403150\pi$
$877$			56.5079i			1.90814i	−0.299591	+	0.954068i	$0.596850\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.67108			−0.123682			−0.0618409	−	0.998086i	$-0.519697\pi$
$881$	−3.67108			−0.123682			−0.0618409	+	0.998086i	$0.519697\pi$
$882$		0			0
$883$	−34.6595			−1.16639			−0.583193	−	0.812334i	$-0.698197\pi$
$883$	−34.6595			−1.16639			−0.583193	+	0.812334i	$0.698197\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.25039			−0.209868			−0.104934	−	0.994479i	$-0.533463\pi$
$887$	−6.25039			−0.209868			−0.104934	+	0.994479i	$0.533463\pi$
$888$		0			0
$889$			10.5097i			0.352483i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.27461			−0.310363
$894$		0			0
$895$		−	98.7126i		−	3.29960i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.05271i		−	0.201869i
$900$		0			0
$901$	−19.9794			−0.665610
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	62.2218i		−	2.06832i
$906$		0			0
$907$	−28.7579			−0.954890			−0.477445	−	0.878662i	$-0.658437\pi$
$907$	−28.7579			−0.954890			−0.477445	+	0.878662i	$0.658437\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.2451			−1.33338			−0.666690	−	0.745335i	$-0.732290\pi$
$911$	−40.2451			−1.33338			−0.666690	+	0.745335i	$0.732290\pi$
$912$		0			0
$913$	10.6343			0.351945
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	0.129031i		−	0.00426099i
$918$		0			0
$919$	54.7098			1.80471			0.902355	−	0.430993i	$-0.141837\pi$
$919$	54.7098			1.80471			0.902355	+	0.430993i	$0.141837\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−15.4243	+	20.4117i	−0.507698	+	0.671860i
$924$		0			0
$925$			106.349i			3.49672i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			5.77324i			0.189414i	0.995505	+	0.0947069i	$0.0301914\pi$
$929$			5.77324i			0.189414i	−0.995505	+	0.0947069i	$0.969809\pi$
$930$		0			0
$931$			1.89361i			0.0620607i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−11.7273			−0.383524
$936$		0			0
$937$	4.80561			0.156992			0.0784962	−	0.996914i	$-0.474988\pi$
$937$	4.80561			0.156992			0.0784962	+	0.996914i	$0.474988\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			27.1858i			0.886233i	0.896464	+	0.443117i	$0.146127\pi$
$941$			27.1858i			0.886233i	−0.896464	+	0.443117i	$0.853873\pi$
$942$		0			0
$943$		−	23.3394i		−	0.760035i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	23.2238i		−	0.754672i	−0.926076	−	0.377336i	$-0.876840\pi$
$947$		−	23.2238i		−	0.754672i	0.926076	−	0.377336i	$-0.123160\pi$
$948$		0			0
$949$	−16.5321	+	21.8778i	−0.536656	+	0.710182i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.1555			0.393757			0.196878	−	0.980428i	$-0.436920\pi$
$953$	12.1555			0.393757			0.196878	+	0.980428i	$0.436920\pi$
$954$		0			0
$955$		−	73.4862i		−	2.37796i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38.8561			1.25473
$960$		0			0
$961$	5.86722			0.189265
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−75.9668			−2.44546
$966$		0			0
$967$		−	12.3286i		−	0.396460i	−0.980156	−	0.198230i	$-0.936481\pi$
$967$		−	12.3286i		−	0.396460i	0.980156	−	0.198230i	$-0.0635193\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.6845			−0.407065			−0.203532	−	0.979068i	$-0.565242\pi$
$971$	−12.6845			−0.407065			−0.203532	+	0.979068i	$0.565242\pi$
$972$		0			0
$973$		−	35.9924i		−	1.15386i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			6.80239i			0.217628i	0.994062	+	0.108814i	$0.0347052\pi$
$977$			6.80239i			0.217628i	−0.994062	+	0.108814i	$0.965295\pi$
$978$		0			0
$979$	−3.23607			−0.103425
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	20.9510i		−	0.668233i	−0.942532	−	0.334116i	$-0.891562\pi$
$983$		−	20.9510i		−	0.668233i	0.942532	−	0.334116i	$-0.108438\pi$
$984$		0			0
$985$	6.68774			0.213089
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.87081			−0.154883
$990$		0			0
$991$	−30.2796			−0.961863			−0.480931	−	0.876758i	$-0.659701\pi$
$991$	−30.2796			−0.961863			−0.480931	+	0.876758i	$0.659701\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	79.9221i		−	2.53370i
$996$		0			0
$997$	−34.6912			−1.09868			−0.549341	−	0.835598i	$-0.685121\pi$
$997$	−34.6912			−1.09868			−0.549341	+	0.835598i	$0.685121\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.c.e.649.14	yes	14
3.2	odd	2		2808.2.c.f.649.1	yes	14
13.12	even	2	inner	2808.2.c.e.649.1	✓	14
39.38	odd	2		2808.2.c.f.649.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2808.2.c.e.649.1	✓	14	13.12	even	2	inner
2808.2.c.e.649.14	yes	14	1.1	even	1	trivial
2808.2.c.f.649.1	yes	14	3.2	odd	2
2808.2.c.f.649.14	yes	14	39.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.85219i q^{5} +2.37854i q^{7} -0.733693i q^{11} +(-2.87661 - 2.17374i) q^{13} -4.14932 q^{17} +1.41046i q^{19} -5.23182 q^{23} -9.83934 q^{25} +1.20734 q^{29} -5.01326i q^{31} -9.16258 q^{35} -10.8085i q^{37} +4.46104i q^{41} +0.930997 q^{43} +6.57560i q^{47} +1.34255 q^{49} +4.81510 q^{53} +2.82632 q^{55} +0.634721i q^{59} -7.96040 q^{61} +(8.37364 - 11.0812i) q^{65} +5.70649i q^{67} -7.09577i q^{71} -7.60540i q^{73} +1.74512 q^{77} -13.5139 q^{79} +14.4942i q^{83} -15.9839i q^{85} -4.41066i q^{89} +(5.17032 - 6.84212i) q^{91} -5.43335 q^{95} +4.09129i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 16 q^{17} + 8 q^{23} - 18 q^{25} - 14 q^{29} - 16 q^{35} - 12 q^{43} - 10 q^{49} + 4 q^{53} - 4 q^{55} - 16 q^{61} - 8 q^{65} - 34 q^{79} - 32 q^{91} + 64 q^{95}+O(q^{100}) \) 14 * q - 16 * q^17 + 8 * q^23 - 18 * q^25 - 14 * q^29 - 16 * q^35 - 12 * q^43 - 10 * q^49 + 4 * q^53 - 4 * q^55 - 16 * q^61 - 8 * q^65 - 34 * q^79 - 32 * q^91 + 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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