LMFDB - Embedded newform 4788.2.i.h.3457.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4788,2,Mod(3457,4788)] 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("4788.3457"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4788, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.i (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 860x^{12} + 7192x^{10} + 28448x^{8} + 45264x^{6} + 12088x^{4} + 784x^{2} + 4 $ x^16 + 48x^14 + 860x^12 + 7192x^10 + 28448x^8 + 45264x^6 + 12088x^4 + 784*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3457.5
Root			$2.34875i$ of defining polynomial
Character	$\chi$	$=$	4788.3457
Dual form			4788.2.i.h.3457.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60804i q^{5} +(2.41421 - 1.08239i) q^{7} +4.20201 q^{11} -3.26899 q^{13} -0.666071i q^{17} +(2.31152 - 3.69552i) q^{19} +1.74053 q^{23} +2.41421 q^{25} -8.97368i q^{29} -7.89204 q^{31} +(-1.74053 - 3.88215i) q^{35} -8.54228i q^{37} -9.71304 q^{41} -0.242641 q^{43} +11.6464i q^{47} +(4.65685 - 5.22625i) q^{49} +3.71702i q^{53} -6.75699i q^{55} +13.7363 q^{59} -1.53073i q^{61} +5.25666i q^{65} +12.0806i q^{67} +3.71702i q^{71} -9.81845i q^{73} +(10.1445 - 4.54822i) q^{77} -8.54228i q^{79} +4.82411i q^{83} -1.07107 q^{85} +9.71304 q^{89} +(-7.89204 + 3.53833i) q^{91} +(-5.94253 - 3.71702i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7} + 16 q^{25} + 64 q^{43} - 16 q^{49} + 96 q^{85}+O(q^{100}) $ 16 * q + 16 * q^7 + 16 * q^25 + 64 * q^43 - 16 * q^49 + 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$2395$	$4105$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$		−	1.60804i		−	0.719136i	−0.933119	−	0.359568i	$-0.882924\pi$
$5$		−	1.60804i		−	0.719136i	0.933119	−	0.359568i	$-0.117076\pi$
$6$		0			0
$7$	2.41421	−	1.08239i	0.912487	−	0.409106i
$7$	2.41421	−	1.08239i	0.912487	−	0.409106i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.20201			1.26695			0.633476	−	0.773762i	$-0.281627\pi$
$11$	4.20201			1.26695			0.633476	+	0.773762i	$0.281627\pi$
$12$		0			0
$13$	−3.26899			−0.906655			−0.453327	−	0.891344i	$-0.649763\pi$
$13$	−3.26899			−0.906655			−0.453327	+	0.891344i	$0.649763\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.666071i		−	0.161546i	−0.996733	−	0.0807730i	$-0.974261\pi$
$17$		−	0.666071i		−	0.161546i	0.996733	−	0.0807730i	$-0.0257389\pi$
$18$		0			0
$19$	2.31152	−	3.69552i	0.530300	−	0.847810i
$19$	2.31152	−	3.69552i	0.530300	−	0.847810i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.74053			0.362925			0.181463	−	0.983398i	$-0.441917\pi$
$23$	1.74053			0.362925			0.181463	+	0.983398i	$0.441917\pi$
$24$		0			0
$25$	2.41421			0.482843
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.97368i		−	1.66637i	−0.552994	−	0.833185i	$-0.686515\pi$
$29$		−	8.97368i		−	1.66637i	0.552994	−	0.833185i	$-0.313485\pi$
$30$		0			0
$31$	−7.89204			−1.41745			−0.708726	−	0.705484i	$-0.750730\pi$
$31$	−7.89204			−1.41745			−0.708726	+	0.705484i	$0.750730\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.74053	−	3.88215i	−0.294203	−	0.656203i
$36$		0			0
$37$		−	8.54228i		−	1.40434i	−0.712008	−	0.702171i	$-0.752214\pi$
$37$		−	8.54228i		−	1.40434i	0.712008	−	0.702171i	$-0.247786\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.71304			−1.51692			−0.758461	−	0.651718i	$-0.774048\pi$
$41$	−9.71304			−1.51692			−0.758461	+	0.651718i	$0.774048\pi$
$42$		0			0
$43$	−0.242641			−0.0370024			−0.0185012	−	0.999829i	$-0.505889\pi$
$43$	−0.242641			−0.0370024			−0.0185012	+	0.999829i	$0.505889\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			11.6464i			1.69881i	0.527743	+	0.849404i	$0.323038\pi$
$47$			11.6464i			1.69881i	−0.527743	+	0.849404i	$0.676962\pi$
$48$		0			0
$49$	4.65685	−	5.22625i	0.665265	−	0.746607i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.71702i			0.510572i	0.966866	+	0.255286i	$0.0821696\pi$
$53$			3.71702i			0.510572i	−0.966866	+	0.255286i	$0.917830\pi$
$54$		0			0
$55$		−	6.75699i		−	0.911112i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13.7363			1.78832			0.894158	−	0.447752i	$-0.147775\pi$
$59$	13.7363			1.78832			0.894158	+	0.447752i	$0.147775\pi$
$60$		0			0
$61$		−	1.53073i		−	0.195990i	−0.995187	−	0.0979952i	$-0.968757\pi$
$61$		−	1.53073i		−	0.195990i	0.995187	−	0.0979952i	$-0.0312430\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			5.25666i			0.652008i
$66$		0			0
$67$			12.0806i			1.47588i	0.674866	+	0.737941i	$0.264202\pi$
$67$			12.0806i			1.47588i	−0.674866	+	0.737941i	$0.735798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.71702i			0.441129i	0.975372	+	0.220565i	$0.0707900\pi$
$71$			3.71702i			0.441129i	−0.975372	+	0.220565i	$0.929210\pi$
$72$		0			0
$73$		−	9.81845i		−	1.14916i	−0.818447	−	0.574582i	$-0.805165\pi$
$73$		−	9.81845i		−	1.14916i	0.818447	−	0.574582i	$-0.194835\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.1445	−	4.54822i	1.15608	−	0.518318i
$78$		0			0
$79$		−	8.54228i		−	0.961082i	−0.876972	−	0.480541i	$-0.840440\pi$
$79$		−	8.54228i		−	0.961082i	0.876972	−	0.480541i	$-0.159560\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.82411i			0.529515i	0.964315	+	0.264758i	$0.0852920\pi$
$83$			4.82411i			0.529515i	−0.964315	+	0.264758i	$0.914708\pi$
$84$		0			0
$85$	−1.07107			−0.116174
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.71304			1.02958			0.514790	−	0.857316i	$-0.327870\pi$
$89$	9.71304			1.02958			0.514790	+	0.857316i	$0.327870\pi$
$90$		0			0
$91$	−7.89204	+	3.53833i	−0.827310	+	0.370918i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.94253	−	3.71702i	−0.609691	−	0.381358i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.43037i			0.838853i	0.907789	+	0.419426i	$0.137769\pi$
$101$			8.43037i			0.838853i	−0.907789	+	0.419426i	$0.862231\pi$
$102$		0			0
$103$	−4.62305			−0.455523			−0.227761	−	0.973717i	$-0.573141\pi$
$103$	−4.62305			−0.455523			−0.227761	+	0.973717i	$0.573141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	16.4077i		−	1.58619i	−0.609095	−	0.793097i	$-0.708467\pi$
$107$		−	16.4077i		−	1.58619i	0.609095	−	0.793097i	$-0.291533\pi$
$108$		0			0
$109$		−	12.0806i		−	1.15711i	−0.815642	−	0.578556i	$-0.803616\pi$
$109$		−	12.0806i		−	1.15711i	0.815642	−	0.578556i	$-0.196384\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.97368i			0.844173i	0.906556	+	0.422086i	$0.138702\pi$
$113$			8.97368i			0.844173i	−0.906556	+	0.422086i	$0.861298\pi$
$114$		0			0
$115$		−	2.79884i		−	0.260993i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.720950	−	1.60804i	−0.0660894	−	0.147409i
$120$		0			0
$121$	6.65685			0.605169
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.9223i		−	1.06637i
$126$		0			0
$127$		−	17.0846i		−	1.51601i	−0.652249	−	0.758005i	$-0.726174\pi$
$127$		−	17.0846i		−	1.51601i	0.652249	−	0.758005i	$-0.273826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.04019i			0.702475i	0.936286	+	0.351237i	$0.114239\pi$
$131$			8.04019i			0.702475i	−0.936286	+	0.351237i	$0.885761\pi$
$132$		0			0
$133$	1.58051	−	11.4237i	0.137048	−	0.990564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−22.7506			−1.94371			−0.971856	−	0.235577i	$-0.924302\pi$
$137$	−22.7506			−1.94371			−0.971856	+	0.235577i	$0.924302\pi$
$138$		0			0
$139$			1.53073i			0.129835i	0.997891	+	0.0649176i	$0.0206784\pi$
$139$			1.53073i			0.129835i	−0.997891	+	0.0649176i	$0.979322\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−13.7363			−1.14869
$144$		0			0
$145$	−14.4300			−1.19835
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.48106			−0.285179			−0.142590	−	0.989782i	$-0.545543\pi$
$149$	−3.48106			−0.285179			−0.142590	+	0.989782i	$0.545543\pi$
$150$		0			0
$151$			15.6189i			1.27105i	0.772080	+	0.635526i	$0.219217\pi$
$151$			15.6189i			1.27105i	−0.772080	+	0.635526i	$0.780783\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			12.6907i			1.01934i
$156$		0			0
$157$		−	4.96362i		−	0.396140i	−0.980188	−	0.198070i	$-0.936533\pi$
$157$		−	4.96362i		−	0.396140i	0.980188	−	0.198070i	$-0.0634673\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.20201	−	1.88393i	0.331164	−	0.148475i
$162$		0			0
$163$	4.92893			0.386064			0.193032	−	0.981192i	$-0.438168\pi$
$163$	4.92893			0.386064			0.193032	+	0.981192i	$0.438168\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−2.31371			−0.177978
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.02327			−0.305884			−0.152942	−	0.988235i	$-0.548875\pi$
$173$	−4.02327			−0.305884			−0.152942	+	0.988235i	$0.548875\pi$
$174$		0			0
$175$	5.82843	−	2.61313i	0.440588	−	0.197534i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.4077i			1.22637i	0.789939	+	0.613185i	$0.210112\pi$
$179$			16.4077i			1.22637i	−0.789939	+	0.613185i	$0.789888\pi$
$180$		0			0
$181$	11.1610			0.829593			0.414796	−	0.909914i	$-0.363853\pi$
$181$	11.1610			0.829593			0.414796	+	0.909914i	$0.363853\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.7363			−1.00991
$186$		0			0
$187$		−	2.79884i		−	0.204671i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.0296			1.59401			0.797003	−	0.603975i	$-0.206417\pi$
$191$	22.0296			1.59401			0.797003	+	0.603975i	$0.206417\pi$
$192$		0			0
$193$			3.53833i			0.254694i	0.991858	+	0.127347i	$0.0406463\pi$
$193$			3.53833i			0.254694i	−0.991858	+	0.127347i	$0.959354\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.48106			−0.248015			−0.124007	−	0.992281i	$-0.539575\pi$
$197$	−3.48106			−0.248015			−0.124007	+	0.992281i	$0.539575\pi$
$198$		0			0
$199$		−	20.9050i		−	1.48192i	−0.671551	−	0.740958i	$-0.734372\pi$
$199$		−	20.9050i		−	1.48192i	0.671551	−	0.740958i	$-0.265628\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.71304	−	21.6644i	−0.681722	−	1.52054i
$204$		0			0
$205$			15.6189i			1.09087i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.71304	−	15.5286i	0.671865	−	1.07413i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.390175i			0.0266097i
$216$		0			0
$217$	−19.0531	+	8.54228i	−1.29341	+	0.579888i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.17738i			0.146466i
$222$		0			0
$223$	−14.4300			−0.966305			−0.483153	−	0.875536i	$-0.660508\pi$
$223$	−14.4300			−0.966305			−0.483153	+	0.875536i	$0.660508\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.7363			0.911711			0.455856	−	0.890054i	$-0.349333\pi$
$227$	13.7363			0.911711			0.455856	+	0.890054i	$0.349333\pi$
$228$		0			0
$229$			2.16478i			0.143053i	0.997439	+	0.0715265i	$0.0227870\pi$
$229$			2.16478i			0.143053i	−0.997439	+	0.0715265i	$0.977213\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.92296			−0.322514			−0.161257	−	0.986912i	$-0.551555\pi$
$233$	−4.92296			−0.322514			−0.161257	+	0.986912i	$0.551555\pi$
$234$		0			0
$235$	18.7279			1.22167
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.720950			0.0466344			0.0233172	−	0.999728i	$-0.492577\pi$
$239$	0.720950			0.0466344			0.0233172	+	0.999728i	$0.492577\pi$
$240$		0			0
$241$	−20.4071			−1.31454			−0.657269	−	0.753656i	$-0.728289\pi$
$241$	−20.4071			−1.31454			−0.657269	+	0.753656i	$0.728289\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.40401	−	7.48840i	−0.536913	−	0.478416i
$246$		0			0
$247$	−7.55635	+	12.0806i	−0.480799	+	0.768671i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	19.4107i		−	1.22519i	−0.790395	−	0.612597i	$-0.790125\pi$
$251$		−	19.4107i		−	1.22519i	0.790395	−	0.612597i	$-0.209875\pi$
$252$		0			0
$253$	7.31371			0.459809
$254$		0			0
$255$		0			0
$256$		0			0
$257$	23.4494			1.46273			0.731365	−	0.681986i	$-0.238884\pi$
$257$	23.4494			1.46273			0.731365	+	0.681986i	$0.238884\pi$
$258$		0			0
$259$	−9.24610	−	20.6229i	−0.574525	−	1.28144i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−23.0492			−1.42127			−0.710637	−	0.703559i	$-0.751593\pi$
$263$	−23.0492			−1.42127			−0.710637	+	0.703559i	$0.751593\pi$
$264$		0			0
$265$	5.97711			0.367171
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.02327			−0.245303			−0.122652	−	0.992450i	$-0.539140\pi$
$269$	−4.02327			−0.245303			−0.122652	+	0.992450i	$0.539140\pi$
$270$		0			0
$271$		−	25.4972i		−	1.54885i	−0.632669	−	0.774423i	$-0.718041\pi$
$271$		−	25.4972i		−	1.54885i	0.632669	−	0.774423i	$-0.281959\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.1445			0.611739
$276$		0			0
$277$	4.58579			0.275533			0.137767	−	0.990465i	$-0.456008\pi$
$277$	4.58579			0.275533			0.137767	+	0.990465i	$0.456008\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.1511i		−	0.665216i	−0.943065	−	0.332608i	$-0.892071\pi$
$281$		−	11.1511i		−	0.665216i	0.943065	−	0.332608i	$-0.107929\pi$
$282$		0			0
$283$		−	26.1313i		−	1.55334i	−0.629906	−	0.776671i	$-0.716907\pi$
$283$		−	26.1313i		−	1.55334i	0.629906	−	0.776671i	$-0.283093\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−23.4494	+	10.5133i	−1.38417	+	0.620582i
$288$		0			0
$289$	16.5563			0.973903
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.71304			−0.567442			−0.283721	−	0.958907i	$-0.591569\pi$
$293$	−9.71304			−0.567442			−0.283721	+	0.958907i	$0.591569\pi$
$294$		0			0
$295$		−	22.0885i		−	1.28604i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.68977			−0.329048
$300$		0			0
$301$	−0.585786	+	0.262632i	−0.0337642	+	0.0151379i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.46148			−0.140944
$306$		0			0
$307$	−5.18392			−0.295862			−0.147931	−	0.988998i	$-0.547261\pi$
$307$	−5.18392			−0.295862			−0.147931	+	0.988998i	$0.547261\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	29.4491i		−	1.66991i	−0.550320	−	0.834954i	$-0.685494\pi$
$311$		−	29.4491i		−	1.66991i	0.550320	−	0.834954i	$-0.314506\pi$
$312$		0			0
$313$		−	18.1062i		−	1.02342i	−0.859158	−	0.511711i	$-0.829012\pi$
$313$		−	18.1062i		−	1.02342i	0.859158	−	0.511711i	$-0.170988\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			26.9210i			1.51204i	0.654551	+	0.756018i	$0.272858\pi$
$317$			26.9210i			1.51204i	−0.654551	+	0.756018i	$0.727142\pi$
$318$		0			0
$319$		−	37.7075i		−	2.11121i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.46148	−	1.53964i	−0.136960	−	0.0856679i
$324$		0			0
$325$	−7.89204			−0.437772
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.6060	+	28.1170i	0.694992	+	1.55014i
$330$		0			0
$331$		−	8.54228i		−	0.469526i	−0.972053	−	0.234763i	$-0.924569\pi$
$331$		−	8.54228i		−	0.469526i	0.972053	−	0.234763i	$-0.0754314\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	19.4261			1.06136
$336$		0			0
$337$			5.00395i			0.272583i	0.990669	+	0.136291i	$0.0435183\pi$
$337$			5.00395i			0.272583i	−0.990669	+	0.136291i	$0.956482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−33.1624			−1.79584
$342$		0			0
$343$	5.58579	−	17.6578i	0.301604	−	0.953433i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.0492			1.23735			0.618673	−	0.785649i	$-0.287671\pi$
$347$	23.0492			1.23735			0.618673	+	0.785649i	$0.287671\pi$
$348$		0			0
$349$			22.1731i			1.18690i	0.804871	+	0.593450i	$0.202234\pi$
$349$			22.1731i			1.18690i	−0.804871	+	0.593450i	$0.797766\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			32.1134i			1.70923i	0.519266	+	0.854613i	$0.326205\pi$
$353$			32.1134i			1.70923i	−0.519266	+	0.854613i	$0.673795\pi$
$354$		0			0
$355$	5.97711			0.317232
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.5486			−0.978955			−0.489478	−	0.872016i	$-0.662812\pi$
$359$	−18.5486			−0.978955			−0.489478	+	0.872016i	$0.662812\pi$
$360$		0			0
$361$	−8.31371	−	17.0846i	−0.437564	−	0.899187i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.7884			−0.826405
$366$		0			0
$367$		−	22.1731i		−	1.15743i	−0.815531	−	0.578713i	$-0.803555\pi$
$367$		−	22.1731i		−	1.15743i	0.815531	−	0.578713i	$-0.196445\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.02327	+	8.97368i	0.208878	+	0.465890i
$372$		0			0
$373$		−	7.07666i		−	0.366415i	−0.983074	−	0.183208i	$-0.941352\pi$
$373$		−	7.07666i		−	0.366415i	0.983074	−	0.183208i	$-0.0586481\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			29.3349i			1.51082i
$378$		0			0
$379$		−	1.46562i		−	0.0752840i	−0.999291	−	0.0376420i	$-0.988015\pi$
$379$		−	1.46562i		−	0.0752840i	0.999291	−	0.0376420i	$-0.0119847\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	33.1624			1.69452			0.847260	−	0.531179i	$-0.178251\pi$
$383$	33.1624			1.69452			0.847260	+	0.531179i	$0.178251\pi$
$384$		0			0
$385$	−7.31371	−	16.3128i	−0.372741	−	0.831378i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.7884			0.800506			0.400253	−	0.916405i	$-0.368922\pi$
$389$	15.7884			0.800506			0.400253	+	0.916405i	$0.368922\pi$
$390$		0			0
$391$		−	1.15932i		−	0.0586291i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.7363			−0.691149
$396$		0			0
$397$			11.3492i			0.569599i	0.958587	+	0.284800i	$0.0919271\pi$
$397$			11.3492i			0.569599i	−0.958587	+	0.284800i	$0.908073\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	32.1777i		−	1.60688i	−0.595387	−	0.803439i	$-0.703001\pi$
$401$		−	32.1777i		−	1.60688i	0.595387	−	0.803439i	$-0.296999\pi$
$402$		0			0
$403$	25.7990			1.28514
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	35.8947i		−	1.77923i
$408$		0			0
$409$	12.5151			0.618831			0.309416	−	0.950927i	$-0.399867\pi$
$409$	12.5151			0.618831			0.309416	+	0.950927i	$0.399867\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	33.1624	−	14.8681i	1.63181	−	0.731610i
$414$		0			0
$415$	7.75736			0.380794
$416$		0			0
$417$		0			0
$418$		0			0
$419$			28.5072i			1.39267i	0.717719	+	0.696333i	$0.245186\pi$
$419$			28.5072i			1.39267i	−0.717719	+	0.696333i	$0.754814\pi$
$420$		0			0
$421$			20.6229i			1.00510i	0.864549	+	0.502549i	$0.167604\pi$
$421$			20.6229i			1.00510i	−0.864549	+	0.502549i	$0.832396\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.60804i		−	0.0780013i
$426$		0			0
$427$	−1.65685	−	3.69552i	−0.0801808	−	0.178839i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.53964i		−	0.0741618i	−0.999312	−	0.0370809i	$-0.988194\pi$
$431$		−	1.53964i		−	0.0741618i	0.999312	−	0.0370809i	$-0.0118059\pi$
$432$		0			0
$433$	29.6532			1.42504			0.712522	−	0.701650i	$-0.247553\pi$
$433$	29.6532			1.42504			0.712522	+	0.701650i	$0.247553\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.02327	−	6.43215i	0.192459	−	0.307692i
$438$		0			0
$439$	−14.4300			−0.688707			−0.344354	−	0.938840i	$-0.611902\pi$
$439$	−14.4300			−0.688707			−0.344354	+	0.938840i	$0.611902\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	23.4715			1.11516			0.557582	−	0.830122i	$-0.311729\pi$
$443$	23.4715			1.11516			0.557582	+	0.830122i	$0.311729\pi$
$444$		0			0
$445$		−	15.6189i		−	0.740409i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			32.1777i			1.51856i	0.650764	+	0.759280i	$0.274449\pi$
$449$			32.1777i			1.51856i	−0.650764	+	0.759280i	$0.725551\pi$
$450$		0			0
$451$	−40.8143			−1.92187
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.68977	+	12.6907i	0.266740	+	0.594949i
$456$		0			0
$457$	4.38478			0.205111			0.102556	−	0.994727i	$-0.467298\pi$
$457$	4.38478			0.205111			0.102556	+	0.994727i	$0.467298\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	23.9590i		−	1.11588i	−0.829881	−	0.557940i	$-0.811592\pi$
$461$		−	23.9590i		−	1.11588i	0.829881	−	0.557940i	$-0.188408\pi$
$462$		0			0
$463$	24.2843			1.12859			0.564293	−	0.825575i	$-0.309149\pi$
$463$	24.2843			1.12859			0.564293	+	0.825575i	$0.309149\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.05625i		−	0.0488773i	−0.999701	−	0.0244386i	$-0.992220\pi$
$467$		−	1.05625i		−	0.0488773i	0.999701	−	0.0244386i	$-0.00777983\pi$
$468$		0			0
$469$	13.0760	+	29.1652i	0.603792	+	1.34672i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.01958			−0.0468802
$474$		0			0
$475$	5.58051	−	8.92177i	0.256052	−	0.409359i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			7.65001i			0.349538i	0.984609	+	0.174769i	$0.0559179\pi$
$479$			7.65001i			0.349538i	−0.984609	+	0.174769i	$0.944082\pi$
$480$		0			0
$481$			27.9246i			1.27325i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			12.0806i			0.547425i	0.961812	+	0.273712i	$0.0882516\pi$
$487$			12.0806i			0.547425i	−0.961812	+	0.273712i	$0.911748\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−23.0492			−1.04020			−0.520098	−	0.854107i	$-0.674104\pi$
$491$	−23.0492			−1.04020			−0.520098	+	0.854107i	$0.674104\pi$
$492$		0			0
$493$	−5.97711			−0.269195
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.02327	+	8.97368i	0.180468	+	0.402525i
$498$		0			0
$499$	−13.6569			−0.611365			−0.305682	−	0.952134i	$-0.598885\pi$
$499$	−13.6569			−0.611365			−0.305682	+	0.952134i	$0.598885\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			13.9206i			0.620687i	0.950625	+	0.310343i	$0.100444\pi$
$503$			13.9206i			0.620687i	−0.950625	+	0.310343i	$0.899556\pi$
$504$		0			0
$505$	13.5563			0.603250
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.02327			0.178328			0.0891642	−	0.996017i	$-0.471580\pi$
$509$	4.02327			0.178328			0.0891642	+	0.996017i	$0.471580\pi$
$510$		0			0
$511$	−10.6274	−	23.7038i	−0.470129	−	1.04860i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			7.43404i			0.327583i
$516$		0			0
$517$			48.9384i			2.15231i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	37.1857			1.62913			0.814567	−	0.580070i	$-0.196975\pi$
$521$	37.1857			1.62913			0.814567	+	0.580070i	$0.196975\pi$
$522$		0			0
$523$	−27.5060			−1.20275			−0.601376	−	0.798966i	$-0.705381\pi$
$523$	−27.5060			−1.20275			−0.601376	+	0.798966i	$0.705381\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.25666i			0.228984i
$528$		0			0
$529$	−19.9706			−0.868285
$530$		0			0
$531$		0			0
$532$		0			0
$533$	31.7518			1.37532
$534$		0			0
$535$	−26.3842			−1.14069
$536$		0			0
$537$		0			0
$538$		0			0
$539$	19.5681	−	21.9607i	0.842859	−	0.945916i
$540$		0			0
$541$	−18.6274			−0.800855			−0.400428	−	0.916328i	$-0.631138\pi$
$541$	−18.6274			−0.800855			−0.400428	+	0.916328i	$0.631138\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.4261			−0.832122
$546$		0			0
$547$			37.7075i			1.61225i	0.591742	+	0.806127i	$0.298440\pi$
$547$			37.7075i			1.61225i	−0.591742	+	0.806127i	$0.701560\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−33.1624	−	20.7429i	−1.41277	−	0.883676i
$552$		0			0
$553$	−9.24610	−	20.6229i	−0.393184	−	0.876975i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−29.7127			−1.25897			−0.629483	−	0.777014i	$-0.716733\pi$
$557$	−29.7127			−1.25897			−0.629483	+	0.777014i	$0.716733\pi$
$558$		0			0
$559$	0.793190			0.0335484
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.4261			−0.818712			−0.409356	−	0.912375i	$-0.634247\pi$
$563$	−19.4261			−0.818712			−0.409356	+	0.912375i	$0.634247\pi$
$564$		0			0
$565$	14.4300			0.607075
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	32.1777i		−	1.34896i	−0.738293	−	0.674480i	$-0.764368\pi$
$569$		−	32.1777i		−	1.34896i	0.738293	−	0.674480i	$-0.235632\pi$
$570$		0			0
$571$	−7.17157			−0.300121			−0.150060	−	0.988677i	$-0.547947\pi$
$571$	−7.17157			−0.300121			−0.150060	+	0.988677i	$0.547947\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.20201			0.175236
$576$		0			0
$577$			23.3324i			0.971342i	0.874142	+	0.485671i	$0.161425\pi$
$577$			23.3324i			0.971342i	−0.874142	+	0.485671i	$0.838575\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.22158	+	11.6464i	0.216628	+	0.483176i
$582$		0			0
$583$			15.6189i			0.646870i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			38.9358i			1.60705i	0.595271	+	0.803525i	$0.297045\pi$
$587$			38.9358i			1.60705i	−0.595271	+	0.803525i	$0.702955\pi$
$588$		0			0
$589$	−18.2426	+	29.1652i	−0.751675	+	1.20173i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	26.2331i		−	1.07726i	−0.842542	−	0.538631i	$-0.818942\pi$
$593$		−	26.2331i		−	1.07726i	0.842542	−	0.538631i	$-0.181058\pi$
$594$		0			0
$595$	−2.58579	+	1.15932i	−0.106007	+	0.0475273i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			19.4870i			0.796217i	0.917338	+	0.398109i	$0.130333\pi$
$599$			19.4870i			0.796217i	−0.917338	+	0.398109i	$0.869667\pi$
$600$		0			0
$601$	31.5682			1.28769			0.643846	−	0.765155i	$-0.277338\pi$
$601$	31.5682			1.28769			0.643846	+	0.765155i	$0.277338\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	10.7045i		−	0.435199i
$606$		0			0
$607$	44.6441			1.81205			0.906024	−	0.423226i	$-0.139102\pi$
$607$	44.6441			1.81205			0.906024	+	0.423226i	$0.139102\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	38.0721i		−	1.54023i
$612$		0			0
$613$	−42.5269			−1.71765			−0.858823	−	0.512273i	$-0.828804\pi$
$613$	−42.5269			−1.71765			−0.858823	+	0.512273i	$0.828804\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.6356			1.39438			0.697189	−	0.716887i	$-0.254434\pi$
$617$	34.6356			1.39438			0.697189	+	0.716887i	$0.254434\pi$
$618$		0			0
$619$		−	17.2095i		−	0.691708i	−0.938288	−	0.345854i	$-0.887589\pi$
$619$		−	17.2095i		−	0.691708i	0.938288	−	0.345854i	$-0.112411\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	23.4494	−	10.5133i	0.939478	−	0.421207i
$624$		0			0
$625$	−7.10051			−0.284020
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.68977			−0.226866
$630$		0			0
$631$	29.2132			1.16296			0.581480	−	0.813561i	$-0.302474\pi$
$631$	29.2132			1.16296			0.581480	+	0.813561i	$0.302474\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.4726			−1.09022
$636$		0			0
$637$	−15.2232	+	17.0846i	−0.603165	+	0.676915i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			23.8418i			0.941693i	0.882215	+	0.470846i	$0.156051\pi$
$641$			23.8418i			0.941693i	−0.882215	+	0.470846i	$0.843949\pi$
$642$		0			0
$643$			28.2960i			1.11589i	0.829879	+	0.557944i	$0.188409\pi$
$643$			28.2960i			1.11589i	−0.829879	+	0.557944i	$0.811591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	29.6108i		−	1.16412i	−0.813146	−	0.582059i	$-0.802247\pi$
$647$		−	29.6108i		−	1.16412i	0.813146	−	0.582059i	$-0.197753\pi$
$648$		0			0
$649$	57.7201			2.26571
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.2120			0.986623			0.493312	−	0.869853i	$-0.335786\pi$
$653$	25.2120			0.986623			0.493312	+	0.869853i	$0.335786\pi$
$654$		0			0
$655$	12.9289			0.505175
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3.71702i		−	0.144795i	−0.997376	−	0.0723973i	$-0.976935\pi$
$659$		−	3.71702i		−	0.144795i	0.997376	−	0.0723973i	$-0.0230649\pi$
$660$		0			0
$661$	41.3751			1.60931			0.804653	−	0.593746i	$-0.202351\pi$
$661$	41.3751			1.60931			0.804653	+	0.593746i	$0.202351\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−18.3698	−	2.54153i	−0.712351	−	0.0985562i
$666$		0			0
$667$		−	15.6189i		−	0.604768i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	6.43215i		−	0.248310i
$672$		0			0
$673$		−	32.7035i		−	1.26063i	−0.776341	−	0.630314i	$-0.782926\pi$
$673$		−	32.7035i		−	1.26063i	0.776341	−	0.630314i	$-0.217074\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−29.1391			−1.11991			−0.559954	−	0.828524i	$-0.689181\pi$
$677$	−29.1391			−1.11991			−0.559954	+	0.828524i	$0.689181\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	34.3551i		−	1.31456i	−0.753646	−	0.657280i	$-0.771707\pi$
$683$		−	34.3551i		−	1.31456i	0.753646	−	0.657280i	$-0.228293\pi$
$684$		0			0
$685$			36.5838i			1.39779i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	12.1509i		−	0.462912i
$690$		0			0
$691$			19.7457i			0.751162i	0.926790	+	0.375581i	$0.122557\pi$
$691$			19.7457i			0.751162i	−0.926790	+	0.375581i	$0.877443\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.46148			0.0933692
$696$		0			0
$697$			6.46958i			0.245053i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28.2708			−1.06777			−0.533886	−	0.845556i	$-0.679269\pi$
$701$	−28.2708			−1.06777			−0.533886	+	0.845556i	$0.679269\pi$
$702$		0			0
$703$	−31.5682	−	19.7457i	−1.19062	−	0.744723i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.12496	+	20.3527i	0.343180	+	0.765442i
$708$		0			0
$709$	−18.9289			−0.710891			−0.355445	−	0.934697i	$-0.615671\pi$
$709$	−18.9289			−0.710891			−0.355445	+	0.934697i	$0.615671\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.7363			−0.514429
$714$		0			0
$715$			22.0885i			0.826064i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			10.7045i			0.399210i	0.979876	+	0.199605i	$0.0639658\pi$
$719$			10.7045i			0.399210i	−0.979876	+	0.199605i	$0.936034\pi$
$720$		0			0
$721$	−11.1610	+	5.00395i	−0.415658	+	0.186357i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	21.6644i		−	0.804595i
$726$		0			0
$727$			25.4972i			0.945639i	0.881159	+	0.472820i	$0.156764\pi$
$727$			25.4972i			0.945639i	−0.881159	+	0.472820i	$0.843236\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			0.161616i			0.00597758i
$732$		0			0
$733$			42.8155i			1.58143i	0.612187	+	0.790713i	$0.290290\pi$
$733$			42.8155i			1.58143i	−0.612187	+	0.790713i	$0.709710\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			50.7628i			1.86987i
$738$		0			0
$739$	−11.5563			−0.425107			−0.212554	−	0.977149i	$-0.568178\pi$
$739$	−11.5563			−0.425107			−0.212554	+	0.977149i	$0.568178\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			34.3551i			1.26037i	0.776447	+	0.630183i	$0.217020\pi$
$743$			34.3551i			1.26037i	−0.776447	+	0.630183i	$0.782980\pi$
$744$		0			0
$745$			5.59767i			0.205083i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17.7596	−	39.6117i	−0.648921	−	1.44738i
$750$		0			0
$751$		−	13.5462i		−	0.494309i	−0.968976	−	0.247155i	$-0.920504\pi$
$751$		−	13.5462i		−	0.494309i	0.968976	−	0.247155i	$-0.0794955\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25.1158			0.914059
$756$		0			0
$757$	−26.3431			−0.957458			−0.478729	−	0.877963i	$-0.658902\pi$
$757$	−26.3431			−0.957458			−0.478729	+	0.877963i	$0.658902\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	17.1366i		−	0.621202i	−0.950540	−	0.310601i	$-0.899470\pi$
$761$		−	17.1366i		−	0.621202i	0.950540	−	0.310601i	$-0.100530\pi$
$762$		0			0
$763$	−13.0760	−	29.1652i	−0.473381	−	1.05585i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−44.9039			−1.62138
$768$		0			0
$769$			39.7540i			1.43357i	0.697296	+	0.716783i	$0.254386\pi$
$769$			39.7540i			1.43357i	−0.697296	+	0.716783i	$0.745614\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23.4494			0.843415			0.421707	−	0.906732i	$-0.361431\pi$
$773$	23.4494			0.843415			0.421707	+	0.906732i	$0.361431\pi$
$774$		0			0
$775$	−19.0531			−0.684406
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.4519	+	35.8947i	−0.804424	+	1.28606i
$780$		0			0
$781$			15.6189i			0.558890i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.98169			−0.284879
$786$		0			0
$787$	43.2901			1.54312			0.771562	−	0.636154i	$-0.219476\pi$
$787$	43.2901			1.54312			0.771562	+	0.636154i	$0.219476\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.71304	+	21.6644i	0.345356	+	0.770297i
$792$		0			0
$793$			5.00395i			0.177696i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	42.8754			1.51873			0.759363	−	0.650667i	$-0.225511\pi$
$797$	42.8754			1.51873			0.759363	+	0.650667i	$0.225511\pi$
$798$		0			0
$799$	7.75736			0.274436
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	41.2572i		−	1.45593i
$804$		0			0
$805$	−3.02944	−	6.75699i	−0.106774	−	0.238152i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.42359			0.331316			0.165658	−	0.986183i	$-0.447025\pi$
$809$	9.42359			0.331316			0.165658	+	0.986183i	$0.447025\pi$
$810$		0			0
$811$	49.2672			1.73000			0.865002	−	0.501768i	$-0.167317\pi$
$811$	49.2672			1.73000			0.865002	+	0.501768i	$0.167317\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	7.92591i		−	0.277633i
$816$		0			0
$817$	−0.560870	+	0.896683i	−0.0196224	+	0.0313710i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.46148			0.0859062			0.0429531	−	0.999077i	$-0.486323\pi$
$821$	2.46148			0.0859062			0.0429531	+	0.999077i	$0.486323\pi$
$822$		0			0
$823$	−16.9289			−0.590105			−0.295053	−	0.955481i	$-0.595337\pi$
$823$	−16.9289			−0.590105			−0.295053	+	0.955481i	$0.595337\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			11.1511i			0.387760i	0.981025	+	0.193880i	$0.0621073\pi$
$827$			11.1511i			0.387760i	−0.981025	+	0.193880i	$0.937893\pi$
$828$		0			0
$829$	22.3221			0.775277			0.387639	−	0.921811i	$-0.373291\pi$
$829$	22.3221			0.775277			0.387639	+	0.921811i	$0.373291\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.48106	−	3.10180i	−0.120611	−	0.107471i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−33.1624			−1.14489			−0.572446	−	0.819942i	$-0.694006\pi$
$839$	−33.1624			−1.14489			−0.572446	+	0.819942i	$0.694006\pi$
$840$		0			0
$841$	−51.5269			−1.77679
$842$		0			0
$843$		0			0
$844$		0			0
$845$			3.72053i			0.127990i
$846$		0			0
$847$	16.0711	−	7.20533i	0.552208	−	0.247578i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	14.8681i		−	0.509671i
$852$		0			0
$853$			16.4216i			0.562265i	0.959669	+	0.281132i	$0.0907100\pi$
$853$			16.4216i			0.562265i	−0.959669	+	0.281132i	$0.909290\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.71304			0.331791			0.165896	−	0.986143i	$-0.446949\pi$
$857$	9.71304			0.331791			0.165896	+	0.986143i	$0.446949\pi$
$858$		0			0
$859$			43.4495i			1.48248i	0.671241	+	0.741239i	$0.265762\pi$
$859$			43.4495i			1.48248i	−0.671241	+	0.741239i	$0.734238\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			37.4344i			1.27428i	0.770748	+	0.637140i	$0.219883\pi$
$863$			37.4344i			1.27428i	−0.770748	+	0.637140i	$0.780117\pi$
$864$		0			0
$865$			6.46958i			0.219972i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	35.8947i		−	1.21764i
$870$		0			0
$871$		−	39.4914i		−	1.33811i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.9046	−	28.7831i	−0.436257	−	0.973045i
$876$		0			0
$877$			22.0885i			0.745876i	0.927856	+	0.372938i	$0.121650\pi$
$877$			22.0885i			0.745876i	−0.927856	+	0.372938i	$0.878350\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			32.6652i			1.10052i	0.834994	+	0.550260i	$0.185471\pi$
$881$			32.6652i			1.10052i	−0.834994	+	0.550260i	$0.814529\pi$
$882$		0			0
$883$	22.6274			0.761473			0.380737	−	0.924684i	$-0.375670\pi$
$883$	22.6274			0.761473			0.380737	+	0.924684i	$0.375670\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.1158			0.843308			0.421654	−	0.906757i	$-0.361450\pi$
$887$	25.1158			0.843308			0.421654	+	0.906757i	$0.361450\pi$
$888$		0			0
$889$	−18.4922	−	41.2458i	−0.620208	−	1.38334i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	43.0396	+	26.9210i	1.44027	+	0.900878i
$894$		0			0
$895$	26.3842			0.881928
$896$		0			0
$897$		0			0
$898$		0			0
$899$			70.8206i			2.36200i
$900$		0			0
$901$	2.47580			0.0824808
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	17.9474i		−	0.596590i
$906$		0			0
$907$			12.0806i			0.401130i	0.979680	+	0.200565i	$0.0642778\pi$
$907$			12.0806i			0.401130i	−0.979680	+	0.200565i	$0.935722\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			24.7437i			0.819794i	0.912132	+	0.409897i	$0.134435\pi$
$911$			24.7437i			0.819794i	−0.912132	+	0.409897i	$0.865565\pi$
$912$		0			0
$913$			20.2710i			0.670871i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.70264	+	19.4107i	0.287386	+	0.640999i
$918$		0			0
$919$	12.8284			0.423171			0.211585	−	0.977360i	$-0.432137\pi$
$919$	12.8284			0.423171			0.211585	+	0.977360i	$0.432137\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	12.1509i		−	0.399952i
$924$		0			0
$925$		−	20.6229i		−	0.678076i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			0.666071i			0.0218531i	0.999940	+	0.0109265i	$0.00347810\pi$
$929$			0.666071i			0.0218531i	−0.999940	+	0.0109265i	$0.996522\pi$
$930$		0			0
$931$	−8.54928	−	29.2901i	−0.280191	−	0.959944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.50063			−0.147186
$936$		0			0
$937$			51.3658i			1.67805i	0.544094	+	0.839024i	$0.316874\pi$
$937$			51.3658i			1.67805i	−0.544094	+	0.839024i	$0.683126\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.4494			−0.764427			−0.382214	−	0.924074i	$-0.624838\pi$
$941$	−23.4494			−0.764427			−0.382214	+	0.924074i	$0.624838\pi$
$942$		0			0
$943$	−16.9058			−0.550529
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.5681			−0.635879			−0.317939	−	0.948111i	$-0.602991\pi$
$947$	−19.5681			−0.635879			−0.317939	+	0.948111i	$0.602991\pi$
$948$		0			0
$949$			32.0964i			1.04189i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	3.71702i		−	0.120406i	−0.998186	−	0.0602030i	$-0.980825\pi$
$953$		−	3.71702i		−	0.120406i	0.998186	−	0.0602030i	$-0.0191748\pi$
$954$		0			0
$955$		−	35.4244i		−	1.14631i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−54.9247	+	24.6250i	−1.77361	+	0.795184i
$960$		0			0
$961$	31.2843			1.00917
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.68977			0.183160
$966$		0			0
$967$	−32.0416			−1.03039			−0.515195	−	0.857073i	$-0.672280\pi$
$967$	−32.0416			−1.03039			−0.515195	+	0.857073i	$0.672280\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.1624			1.06423			0.532116	−	0.846672i	$-0.321397\pi$
$971$	33.1624			1.06423			0.532116	+	0.846672i	$0.321397\pi$
$972$		0			0
$973$	1.65685	+	3.69552i	0.0531163	+	0.118473i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	11.1511i		−	0.356754i	−0.983962	−	0.178377i	$-0.942915\pi$
$977$		−	11.1511i		−	0.356754i	0.983962	−	0.178377i	$-0.0570847\pi$
$978$		0			0
$979$	40.8143			1.30443
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−33.1624			−1.05772			−0.528858	−	0.848710i	$-0.677380\pi$
$983$	−33.1624			−1.05772			−0.528858	+	0.848710i	$0.677380\pi$
$984$		0			0
$985$			5.59767i			0.178357i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.422323			−0.0134291
$990$		0			0
$991$		−	12.0806i		−	0.383753i	−0.981419	−	0.191877i	$-0.938543\pi$
$991$		−	12.0806i		−	0.383753i	0.981419	−	0.191877i	$-0.0614573\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.6160			−1.06570
$996$		0			0
$997$		−	11.6118i		−	0.367750i	−0.982950	−	0.183875i	$-0.941136\pi$
$997$		−	11.6118i		−	0.367750i	0.982950	−	0.183875i	$-0.0588642\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.i.h.3457.5	✓	16
3.2	odd	2	inner	4788.2.i.h.3457.9	yes	16
7.6	odd	2	inner	4788.2.i.h.3457.12	yes	16
19.18	odd	2	inner	4788.2.i.h.3457.6	yes	16
21.20	even	2	inner	4788.2.i.h.3457.8	yes	16
57.56	even	2	inner	4788.2.i.h.3457.10	yes	16
133.132	even	2	inner	4788.2.i.h.3457.11	yes	16
399.398	odd	2	inner	4788.2.i.h.3457.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.i.h.3457.5	✓	16	1.1	even	1	trivial
4788.2.i.h.3457.6	yes	16	19.18	odd	2	inner
4788.2.i.h.3457.7	yes	16	399.398	odd	2	inner
4788.2.i.h.3457.8	yes	16	21.20	even	2	inner
4788.2.i.h.3457.9	yes	16	3.2	odd	2	inner
4788.2.i.h.3457.10	yes	16	57.56	even	2	inner
4788.2.i.h.3457.11	yes	16	133.132	even	2	inner
4788.2.i.h.3457.12	yes	16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.60804i q^{5} +(2.41421 - 1.08239i) q^{7} +4.20201 q^{11} -3.26899 q^{13} -0.666071i q^{17} +(2.31152 - 3.69552i) q^{19} +1.74053 q^{23} +2.41421 q^{25} -8.97368i q^{29} -7.89204 q^{31} +(-1.74053 - 3.88215i) q^{35} -8.54228i q^{37} -9.71304 q^{41} -0.242641 q^{43} +11.6464i q^{47} +(4.65685 - 5.22625i) q^{49} +3.71702i q^{53} -6.75699i q^{55} +13.7363 q^{59} -1.53073i q^{61} +5.25666i q^{65} +12.0806i q^{67} +3.71702i q^{71} -9.81845i q^{73} +(10.1445 - 4.54822i) q^{77} -8.54228i q^{79} +4.82411i q^{83} -1.07107 q^{85} +9.71304 q^{89} +(-7.89204 + 3.53833i) q^{91} +(-5.94253 - 3.71702i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7} + 16 q^{25} + 64 q^{43} - 16 q^{49} + 96 q^{85}+O(q^{100}) \) 16 * q + 16 * q^7 + 16 * q^25 + 64 * q^43 - 16 * q^49 + 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more