LMFDB - Embedded newform 4788.2.i.h.3457.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(3457,4788)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			3457.14
Root			$2.82448i$ of defining polynomial
Character	$\chi$	$=$	4788.3457
Dual form			4788.2.i.h.3457.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.32685i q^{5} +(-0.414214 - 2.61313i) q^{7} +2.51856 q^{11} +5.77180 q^{13} -5.61750i q^{17} +(4.08128 - 1.53073i) q^{19} -6.08034 q^{23} -0.414214 q^{25} -3.93358i q^{29} -2.39076 q^{31} +(6.08034 - 0.963811i) q^{35} -6.24735i q^{37} -10.2790 q^{41} +8.24264 q^{43} +2.89143i q^{47} +(-6.65685 + 2.16478i) q^{49} -9.49651i q^{53} +5.86030i q^{55} -14.5366 q^{59} +3.69552i q^{61} +13.4301i q^{65} -8.83509i q^{67} -9.49651i q^{71} +13.2513i q^{73} +(-1.04322 - 6.58132i) q^{77} -6.24735i q^{79} -6.98054i q^{83} +13.0711 q^{85} +10.2790 q^{89} +(-2.39076 - 15.0824i) q^{91} +(3.56178 + 9.49651i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			2.32685i			1.04060i	0.853984	+	0.520299i	$0.174179\pi$
$5$			2.32685i			1.04060i	−0.853984	+	0.520299i	$0.825821\pi$
$6$		0			0
$7$	−0.414214	−	2.61313i	−0.156558	−	0.987669i
$7$	−0.414214	−	2.61313i	−0.156558	−	0.987669i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.51856			0.759374			0.379687	−	0.925115i	$-0.376032\pi$
$11$	2.51856			0.759374			0.379687	+	0.925115i	$0.376032\pi$
$12$		0			0
$13$	5.77180			1.60081			0.800405	−	0.599460i	$-0.204618\pi$
$13$	5.77180			1.60081			0.800405	+	0.599460i	$0.204618\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.61750i		−	1.36244i	−0.732077	−	0.681222i	$-0.761449\pi$
$17$		−	5.61750i		−	1.36244i	0.732077	−	0.681222i	$-0.238551\pi$
$18$		0			0
$19$	4.08128	−	1.53073i	0.936310	−	0.351174i
$19$	4.08128	−	1.53073i	0.936310	−	0.351174i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.08034			−1.26784			−0.633920	−	0.773399i	$-0.718555\pi$
$23$	−6.08034			−1.26784			−0.633920	+	0.773399i	$0.718555\pi$
$24$		0			0
$25$	−0.414214			−0.0828427
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.93358i		−	0.730448i	−0.930920	−	0.365224i	$-0.880992\pi$
$29$		−	3.93358i		−	0.730448i	0.930920	−	0.365224i	$-0.119008\pi$
$30$		0			0
$31$	−2.39076			−0.429393			−0.214696	−	0.976681i	$-0.568876\pi$
$31$	−2.39076			−0.429393			−0.214696	+	0.976681i	$0.568876\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.08034	−	0.963811i	1.02777	−	0.162914i
$36$		0			0
$37$		−	6.24735i		−	1.02706i	−0.858072	−	0.513529i	$-0.828338\pi$
$37$		−	6.24735i		−	1.02706i	0.858072	−	0.513529i	$-0.171662\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.2790			−1.60530			−0.802651	−	0.596449i	$-0.796578\pi$
$41$	−10.2790			−1.60530			−0.802651	+	0.596449i	$0.796578\pi$
$42$		0			0
$43$	8.24264			1.25699			0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264			1.25699			0.628495	+	0.777813i	$0.283671\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.89143i			0.421759i	0.977512	+	0.210879i	$0.0676328\pi$
$47$			2.89143i			0.421759i	−0.977512	+	0.210879i	$0.932367\pi$
$48$		0			0
$49$	−6.65685	+	2.16478i	−0.950979	+	0.309255i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.49651i		−	1.30445i	−0.758027	−	0.652223i	$-0.773836\pi$
$53$		−	9.49651i		−	1.30445i	0.758027	−	0.652223i	$-0.226164\pi$
$54$		0			0
$55$			5.86030i			0.790203i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14.5366			−1.89251			−0.946254	−	0.323424i	$-0.895166\pi$
$59$	−14.5366			−1.89251			−0.946254	+	0.323424i	$0.895166\pi$
$60$		0			0
$61$			3.69552i			0.473163i	0.971612	+	0.236581i	$0.0760270\pi$
$61$			3.69552i			0.473163i	−0.971612	+	0.236581i	$0.923973\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			13.4301i			1.66580i
$66$		0			0
$67$		−	8.83509i		−	1.07938i	−0.841864	−	0.539689i	$-0.818542\pi$
$67$		−	8.83509i		−	1.07938i	0.841864	−	0.539689i	$-0.181458\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.49651i		−	1.12703i	−0.826106	−	0.563514i	$-0.809449\pi$
$71$		−	9.49651i		−	1.12703i	0.826106	−	0.563514i	$-0.190551\pi$
$72$		0			0
$73$			13.2513i			1.55095i	0.631377	+	0.775476i	$0.282490\pi$
$73$			13.2513i			1.55095i	−0.631377	+	0.775476i	$0.717510\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.04322	−	6.58132i	−0.118886	−	0.750010i
$78$		0			0
$79$		−	6.24735i		−	0.702882i	−0.936210	−	0.351441i	$-0.885692\pi$
$79$		−	6.24735i		−	0.702882i	0.936210	−	0.351441i	$-0.114308\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.98054i		−	0.766214i	−0.923704	−	0.383107i	$-0.874854\pi$
$83$		−	6.98054i		−	0.766214i	0.923704	−	0.383107i	$-0.125146\pi$
$84$		0			0
$85$	13.0711			1.41776
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.2790			1.08957			0.544783	−	0.838577i	$-0.316612\pi$
$89$	10.2790			1.08957			0.544783	+	0.838577i	$0.316612\pi$
$90$		0			0
$91$	−2.39076	−	15.0824i	−0.250620	−	1.58107i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.56178	+	9.49651i	0.365431	+	0.974322i
$96$		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
$101$			7.54513i			0.750768i	0.926869	+	0.375384i	$0.122489\pi$
$101$			7.54513i			0.750768i	−0.926869	+	0.375384i	$0.877511\pi$
$102$		0			0
$103$	−8.16256			−0.804281			−0.402141	−	0.915578i	$-0.631734\pi$
$103$	−8.16256			−0.804281			−0.402141	+	0.915578i	$0.631734\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.0594i			1.45585i	0.685656	+	0.727926i	$0.259515\pi$
$107$			15.0594i			1.45585i	−0.685656	+	0.727926i	$0.740485\pi$
$108$		0			0
$109$			8.83509i			0.846248i	0.906072	+	0.423124i	$0.139067\pi$
$109$			8.83509i			0.846248i	−0.906072	+	0.423124i	$0.860933\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.93358i			0.370041i	0.982735	+	0.185020i	$0.0592351\pi$
$113$			3.93358i			0.370041i	−0.982735	+	0.185020i	$0.940765\pi$
$114$		0			0
$115$		−	14.1480i		−	1.31931i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.6792	+	2.32685i	−1.34564	+	0.213302i
$120$		0			0
$121$	−4.65685			−0.423350
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.6704i			0.954391i
$126$		0			0
$127$		−	12.4947i		−	1.10873i	−0.832275	−	0.554363i	$-0.812962\pi$
$127$		−	12.4947i		−	1.10873i	0.832275	−	0.554363i	$-0.187038\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	11.6342i		−	1.01649i	−0.861213	−	0.508244i	$-0.830295\pi$
$131$		−	11.6342i		−	1.01649i	0.861213	−	0.508244i	$-0.169705\pi$
$132$		0			0
$133$	−5.69052	−	10.0309i	−0.493431	−	0.869785i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.51246			−0.556397			−0.278198	−	0.960524i	$-0.589737\pi$
$137$	−6.51246			−0.556397			−0.278198	+	0.960524i	$0.589737\pi$
$138$		0			0
$139$		−	3.69552i		−	0.313450i	−0.987642	−	0.156725i	$-0.949906\pi$
$139$		−	3.69552i		−	0.313450i	0.987642	−	0.156725i	$-0.0500936\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.5366			1.21561
$144$		0			0
$145$	9.15285			0.760103
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.1607			0.996242			0.498121	−	0.867108i	$-0.334024\pi$
$149$	12.1607			0.996242			0.498121	+	0.867108i	$0.334024\pi$
$150$		0			0
$151$		−	23.9175i		−	1.94638i	−0.229999	−	0.973191i	$-0.573872\pi$
$151$		−	23.9175i		−	1.94638i	0.229999	−	0.973191i	$-0.426128\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.56293i		−	0.446825i
$156$		0			0
$157$		−	19.3743i		−	1.54624i	−0.634263	−	0.773118i	$-0.718696\pi$
$157$		−	19.3743i		−	1.54624i	0.634263	−	0.773118i	$-0.281304\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.51856	+	15.8887i	0.198490	+	1.25220i
$162$		0			0
$163$	19.0711			1.49376			0.746881	−	0.664958i	$-0.231551\pi$
$163$	19.0711			1.49376			0.746881	+	0.664958i	$0.231551\pi$
$164$		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
$169$	20.3137			1.56259
$170$		0			0
$171$		0			0
$172$		0			0
$173$	24.8156			1.88669			0.943347	−	0.331808i	$-0.107659\pi$
$173$	24.8156			1.88669			0.943347	+	0.331808i	$0.107659\pi$
$174$		0			0
$175$	0.171573	+	1.08239i	0.0129697	+	0.0818212i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	15.0594i		−	1.12560i	−0.826595	−	0.562798i	$-0.809725\pi$
$179$		−	15.0594i		−	1.12560i	0.826595	−	0.562798i	$-0.190275\pi$
$180$		0			0
$181$	−3.38104			−0.251311			−0.125656	−	0.992074i	$-0.540103\pi$
$181$	−3.38104			−0.251311			−0.125656	+	0.992074i	$0.540103\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.5366			1.06875
$186$		0			0
$187$		−	14.1480i		−	1.03461i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.16679			−0.590928			−0.295464	−	0.955354i	$-0.595474\pi$
$191$	−8.16679			−0.590928			−0.295464	+	0.955354i	$0.595474\pi$
$192$		0			0
$193$		−	15.0824i		−	1.08566i	−0.839843	−	0.542829i	$-0.817353\pi$
$193$		−	15.0824i		−	1.08566i	0.839843	−	0.542829i	$-0.182647\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.1607			0.866413			0.433206	−	0.901295i	$-0.357382\pi$
$197$	12.1607			0.866413			0.433206	+	0.901295i	$0.357382\pi$
$198$		0			0
$199$			8.65914i			0.613830i	0.951737	+	0.306915i	$0.0992967\pi$
$199$			8.65914i			0.613830i	−0.951737	+	0.306915i	$0.900703\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.2790	+	1.62934i	−0.721441	+	0.114358i
$204$		0			0
$205$		−	23.9175i		−	1.67047i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.2790	−	3.85525i	0.711010	−	0.266673i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			19.1794i			1.30802i
$216$		0			0
$217$	0.990285	+	6.24735i	0.0672249	+	0.424098i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	32.4231i		−	2.18102i
$222$		0			0
$223$	9.15285			0.612920			0.306460	−	0.951884i	$-0.400855\pi$
$223$	9.15285			0.612920			0.306460	+	0.951884i	$0.400855\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.5366			−0.964830			−0.482415	−	0.875943i	$-0.660240\pi$
$227$	−14.5366			−0.964830			−0.482415	+	0.875943i	$0.660240\pi$
$228$		0			0
$229$			5.22625i			0.345360i	0.984978	+	0.172680i	$0.0552427\pi$
$229$			5.22625i			0.345360i	−0.984978	+	0.172680i	$0.944757\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.1978			−1.12667			−0.563333	−	0.826230i	$-0.690481\pi$
$233$	−17.1978			−1.12667			−0.563333	+	0.826230i	$0.690481\pi$
$234$		0			0
$235$	−6.72792			−0.438881
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.6792			0.949521			0.474761	−	0.880115i	$-0.342535\pi$
$239$	14.6792			0.949521			0.474761	+	0.880115i	$0.342535\pi$
$240$		0			0
$241$	−12.9441			−0.833801			−0.416901	−	0.908952i	$-0.636884\pi$
$241$	−12.9441			−0.833801			−0.416901	+	0.908952i	$0.636884\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.03712	−	15.4895i	−0.321810	−	0.989586i
$246$		0			0
$247$	23.5563	−	8.83509i	1.49885	−	0.562164i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4.81906i		−	0.304176i	−0.988367	−	0.152088i	$-0.951400\pi$
$251$		−	4.81906i		−	0.304176i	0.988367	−	0.152088i	$-0.0485997\pi$
$252$		0			0
$253$	−15.3137			−0.962765
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.25768			−0.265587			−0.132793	−	0.991144i	$-0.542395\pi$
$257$	−4.25768			−0.265587			−0.132793	+	0.991144i	$0.542395\pi$
$258$		0			0
$259$	−16.3251	+	2.58774i	−1.01439	+	0.160794i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	28.9264			1.78368			0.891838	−	0.452354i	$-0.149416\pi$
$263$	28.9264			1.78368			0.891838	+	0.452354i	$0.149416\pi$
$264$		0			0
$265$	22.0969			1.35740
$266$		0			0
$267$		0			0
$268$		0			0
$269$	24.8156			1.51303			0.756516	−	0.653975i	$-0.226900\pi$
$269$	24.8156			1.51303			0.756516	+	0.653975i	$0.226900\pi$
$270$		0			0
$271$			19.7457i			1.19947i	0.800200	+	0.599733i	$0.204726\pi$
$271$			19.7457i			1.19947i	−0.800200	+	0.599733i	$0.795274\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.04322			−0.0629086
$276$		0			0
$277$	7.41421			0.445477			0.222738	−	0.974878i	$-0.428500\pi$
$277$	7.41421			0.445477			0.222738	+	0.974878i	$0.428500\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			28.4895i			1.69954i	0.527151	+	0.849772i	$0.323260\pi$
$281$			28.4895i			1.69954i	−0.527151	+	0.849772i	$0.676740\pi$
$282$		0			0
$283$			10.8239i			0.643415i	0.946839	+	0.321708i	$0.104257\pi$
$283$			10.8239i			0.643415i	−0.946839	+	0.321708i	$0.895743\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.25768	+	26.8602i	0.251323	+	1.58551i
$288$		0			0
$289$	−14.5563			−0.856256
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.2790			−0.600503			−0.300251	−	0.953860i	$-0.597071\pi$
$293$	−10.2790			−0.600503			−0.300251	+	0.953860i	$0.597071\pi$
$294$		0			0
$295$		−	33.8245i		−	1.96934i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−35.0945			−2.02957
$300$		0			0
$301$	−3.41421	−	21.5391i	−0.196792	−	1.24149i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.59890			−0.492372
$306$		0			0
$307$	25.4780			1.45410			0.727052	−	0.686582i	$-0.240890\pi$
$307$	25.4780			1.45410			0.727052	+	0.686582i	$0.240890\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	10.0373i		−	0.569165i	−0.958652	−	0.284583i	$-0.908145\pi$
$311$		−	10.0373i		−	0.569165i	0.958652	−	0.284583i	$-0.0918550\pi$
$312$		0			0
$313$			22.8072i			1.28914i	0.764547	+	0.644568i	$0.222963\pi$
$313$			22.8072i			1.28914i	−0.764547	+	0.644568i	$0.777037\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			11.8008i			0.662796i	0.943491	+	0.331398i	$0.107520\pi$
$317$			11.8008i			0.662796i	−0.943491	+	0.331398i	$0.892480\pi$
$318$		0			0
$319$		−	9.90697i		−	0.554684i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.59890	−	22.9266i	−0.478456	−	1.27567i
$324$		0			0
$325$	−2.39076			−0.132615
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.55568	−	1.19767i	0.416558	−	0.0660297i
$330$		0			0
$331$		−	6.24735i		−	0.343386i	−0.985151	−	0.171693i	$-0.945076\pi$
$331$		−	6.24735i		−	0.343386i	0.985151	−	0.171693i	$-0.0549237\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	20.5579			1.12320
$336$		0			0
$337$			21.3298i			1.16191i	0.813936	+	0.580954i	$0.197320\pi$
$337$			21.3298i			1.16191i	−0.813936	+	0.580954i	$0.802680\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.02127			−0.326070
$342$		0			0
$343$	8.41421	+	16.4985i	0.454325	+	0.890836i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−28.9264			−1.55285			−0.776425	−	0.630210i	$-0.782969\pi$
$347$	−28.9264			−1.55285			−0.776425	+	0.630210i	$0.782969\pi$
$348$		0			0
$349$			9.18440i			0.491630i	0.969317	+	0.245815i	$0.0790555\pi$
$349$			9.18440i			0.491630i	−0.969317	+	0.245815i	$0.920944\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			32.5074i			1.73019i	0.501607	+	0.865096i	$0.332743\pi$
$353$			32.5074i			1.73019i	−0.501607	+	0.865096i	$0.667257\pi$
$354$		0			0
$355$	22.0969			1.17278
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.99390			−0.210790			−0.105395	−	0.994430i	$-0.533611\pi$
$359$	−3.99390			−0.210790			−0.105395	+	0.994430i	$0.533611\pi$
$360$		0			0
$361$	14.3137	−	12.4947i	0.753353	−	0.657616i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−30.8338			−1.61392
$366$		0			0
$367$		−	9.18440i		−	0.479422i	−0.970844	−	0.239711i	$-0.922947\pi$
$367$		−	9.18440i		−	0.479422i	0.970844	−	0.239711i	$-0.0770527\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−24.8156	+	3.93358i	−1.28836	+	0.204222i
$372$		0			0
$373$			30.1649i			1.56188i	0.624607	+	0.780940i	$0.285259\pi$
$373$			30.1649i			1.56188i	−0.624607	+	0.780940i	$0.714741\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	22.7039i		−	1.16931i
$378$		0			0
$379$		−	36.4122i		−	1.87037i	−0.354157	−	0.935186i	$-0.615232\pi$
$379$		−	36.4122i		−	1.87037i	0.354157	−	0.935186i	$-0.384768\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.02127			0.307673			0.153836	−	0.988096i	$-0.450837\pi$
$383$	6.02127			0.307673			0.153836	+	0.988096i	$0.450837\pi$
$384$		0			0
$385$	15.3137	−	2.42742i	0.780459	−	0.123713i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	30.8338			1.56334			0.781669	−	0.623694i	$-0.214369\pi$
$389$	30.8338			1.56334			0.781669	+	0.623694i	$0.214369\pi$
$390$		0			0
$391$			34.1563i			1.72736i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.5366			0.731417
$396$		0			0
$397$		−	16.9469i		−	0.850538i	−0.905067	−	0.425269i	$-0.860179\pi$
$397$		−	16.9469i		−	0.850538i	0.905067	−	0.425269i	$-0.139821\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	25.2309i		−	1.25997i	−0.776608	−	0.629984i	$-0.783061\pi$
$401$		−	25.2309i		−	1.25997i	0.776608	−	0.629984i	$-0.216939\pi$
$402$		0			0
$403$	−13.7990			−0.687377
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	15.7343i		−	0.779922i
$408$		0			0
$409$	10.5533			0.521828			0.260914	−	0.965362i	$-0.415976\pi$
$409$	10.5533			0.521828			0.260914	+	0.965362i	$0.415976\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.02127	+	37.9861i	0.296287	+	1.86917i
$414$		0			0
$415$	16.2426			0.797320
$416$		0			0
$417$		0			0
$418$		0			0
$419$			17.9817i			0.878463i	0.898374	+	0.439231i	$0.144749\pi$
$419$			17.9817i			0.878463i	−0.898374	+	0.439231i	$0.855251\pi$
$420$		0			0
$421$		−	2.58774i		−	0.126119i	−0.998010	−	0.0630593i	$-0.979914\pi$
$421$		−	2.58774i		−	0.126119i	0.998010	−	0.0630593i	$-0.0200857\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			2.32685i			0.112869i
$426$		0			0
$427$	9.65685	−	1.53073i	0.467328	−	0.0740774i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	22.9266i		−	1.10434i	−0.833733	−	0.552168i	$-0.813801\pi$
$431$		−	22.9266i		−	1.10434i	0.833733	−	0.552168i	$-0.186199\pi$
$432$		0			0
$433$	29.2692			1.40659			0.703294	−	0.710899i	$-0.251712\pi$
$433$	29.2692			1.40659			0.703294	+	0.710899i	$0.251712\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.8156	+	9.30739i	−1.18709	+	0.445233i
$438$		0			0
$439$	9.15285			0.436842			0.218421	−	0.975855i	$-0.429909\pi$
$439$	9.15285			0.436842			0.218421	+	0.975855i	$0.429909\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.1917			1.00685			0.503424	−	0.864039i	$-0.332073\pi$
$443$	21.1917			1.00685			0.503424	+	0.864039i	$0.332073\pi$
$444$		0			0
$445$			23.9175i			1.13380i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			25.2309i			1.19072i	0.803460	+	0.595359i	$0.202990\pi$
$449$			25.2309i			1.19072i	−0.803460	+	0.595359i	$0.797010\pi$
$450$		0			0
$451$	−25.8882			−1.21903
$452$		0			0
$453$		0			0
$454$		0			0
$455$	35.0945	−	5.56293i	1.64526	−	0.260794i
$456$		0			0
$457$	−32.3848			−1.51490			−0.757448	−	0.652896i	$-0.773554\pi$
$457$	−32.3848			−1.51490			−0.757448	+	0.652896i	$0.773554\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	11.4004i		−	0.530968i	−0.964115	−	0.265484i	$-0.914468\pi$
$461$		−	11.4004i		−	0.530968i	0.964115	−	0.265484i	$-0.0855318\pi$
$462$		0			0
$463$	−32.2843			−1.50038			−0.750189	−	0.661224i	$-0.770037\pi$
$463$	−32.2843			−1.50038			−0.750189	+	0.661224i	$0.770037\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	24.7969i		−	1.14746i	−0.819044	−	0.573731i	$-0.805496\pi$
$467$		−	24.7969i		−	1.14746i	0.819044	−	0.573731i	$-0.194504\pi$
$468$		0			0
$469$	−23.0872	+	3.65962i	−1.06607	+	0.168985i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	20.7596			0.954527
$474$		0			0
$475$	−1.69052	+	0.634051i	−0.0775665	+	0.0290922i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	30.8136i		−	1.40791i	−0.710245	−	0.703954i	$-0.751416\pi$
$479$		−	30.8136i		−	1.40791i	0.710245	−	0.703954i	$-0.248584\pi$
$480$		0			0
$481$		−	36.0585i		−	1.64413i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	8.83509i		−	0.400356i	−0.979760	−	0.200178i	$-0.935848\pi$
$487$		−	8.83509i		−	0.400356i	0.979760	−	0.200178i	$-0.0641521\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	28.9264			1.30543			0.652714	−	0.757604i	$-0.273630\pi$
$491$	28.9264			1.30543			0.652714	+	0.757604i	$0.273630\pi$
$492$		0			0
$493$	−22.0969			−0.995196
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.8156	+	3.93358i	−1.11313	+	0.176445i
$498$		0			0
$499$	−2.34315			−0.104894			−0.0524468	−	0.998624i	$-0.516702\pi$
$499$	−2.34315			−0.104894			−0.0524468	+	0.998624i	$0.516702\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6.18209i			0.275646i	0.990457	+	0.137823i	$0.0440105\pi$
$503$			6.18209i			0.275646i	−0.990457	+	0.137823i	$0.955990\pi$
$504$		0			0
$505$	−17.5563			−0.781247
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−24.8156			−1.09993			−0.549966	−	0.835187i	$-0.685359\pi$
$509$	−24.8156			−1.09993			−0.549966	+	0.835187i	$0.685359\pi$
$510$		0			0
$511$	34.6274	−	5.48888i	1.53183	−	0.242814i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	18.9930i		−	0.836933i
$516$		0			0
$517$			7.28225i			0.320273i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.7943			−0.823394			−0.411697	−	0.911321i	$-0.635064\pi$
$521$	−18.7943			−0.823394			−0.411697	+	0.911321i	$0.635064\pi$
$522$		0			0
$523$	32.2401			1.40976			0.704880	−	0.709327i	$-0.251001\pi$
$523$	32.2401			1.40976			0.704880	+	0.709327i	$0.251001\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			13.4301i			0.585024i
$528$		0			0
$529$	13.9706			0.607416
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−59.3281			−2.56978
$534$		0			0
$535$	−35.0410			−1.51495
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.7657	+	5.45214i	−0.722149	+	0.234840i
$540$		0			0
$541$	26.6274			1.14480			0.572401	−	0.819974i	$-0.306012\pi$
$541$	26.6274			1.14480			0.572401	+	0.819974i	$0.306012\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.5579			−0.880604
$546$		0			0
$547$			9.90697i			0.423591i	0.977314	+	0.211796i	$0.0679311\pi$
$547$			9.90697i			0.423591i	−0.977314	+	0.211796i	$0.932069\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.02127	−	16.0541i	−0.256515	−	0.683926i
$552$		0			0
$553$	−16.3251	+	2.58774i	−0.694215	+	0.110042i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.8089			0.754588			0.377294	−	0.926094i	$-0.376855\pi$
$557$	17.8089			0.754588			0.377294	+	0.926094i	$0.376855\pi$
$558$		0			0
$559$	47.5749			2.01220
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.5579			−0.866412			−0.433206	−	0.901295i	$-0.642618\pi$
$563$	−20.5579			−0.866412			−0.433206	+	0.901295i	$0.642618\pi$
$564$		0			0
$565$	−9.15285			−0.385063
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.2309i		−	1.05773i	−0.848705	−	0.528866i	$-0.822617\pi$
$569$		−	25.2309i		−	1.05773i	0.848705	−	0.528866i	$-0.177383\pi$
$570$		0			0
$571$	−12.8284			−0.536853			−0.268426	−	0.963300i	$-0.586504\pi$
$571$	−12.8284			−0.536853			−0.268426	+	0.963300i	$0.586504\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.51856			0.105031
$576$		0			0
$577$		−	24.9719i		−	1.03960i	−0.854289	−	0.519798i	$-0.826007\pi$
$577$		−	24.9719i		−	1.03960i	0.854289	−	0.519798i	$-0.173993\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.2410	+	2.89143i	−0.756765	+	0.119957i
$582$		0			0
$583$		−	23.9175i		−	0.990563i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			42.3793i			1.74918i	0.484861	+	0.874591i	$0.338870\pi$
$587$			42.3793i			1.74918i	−0.484861	+	0.874591i	$0.661130\pi$
$588$		0			0
$589$	−9.75736	+	3.65962i	−0.402045	+	0.150792i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.6910i		−	0.603288i	−0.953421	−	0.301644i	$-0.902465\pi$
$593$		−	14.6910i		−	0.603288i	0.953421	−	0.301644i	$-0.0975354\pi$
$594$		0			0
$595$	−5.41421	−	34.1563i	−0.221961	−	1.40027i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			30.7938i			1.25820i	0.777324	+	0.629100i	$0.216576\pi$
$599$			30.7938i			1.25820i	−0.777324	+	0.629100i	$0.783424\pi$
$600$		0			0
$601$	9.56304			0.390084			0.195042	−	0.980795i	$-0.437516\pi$
$601$	9.56304			0.390084			0.195042	+	0.980795i	$0.437516\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	10.8358i		−	0.440537i
$606$		0			0
$607$	−13.5242			−0.548929			−0.274465	−	0.961597i	$-0.588501\pi$
$607$	−13.5242			−0.548929			−0.274465	+	0.961597i	$0.588501\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.6888i			0.675156i
$612$		0			0
$613$	22.5269			0.909853			0.454927	−	0.890529i	$-0.349665\pi$
$613$	22.5269			0.909853			0.454927	+	0.890529i	$0.349665\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.611105			−0.0246022			−0.0123011	−	0.999924i	$-0.503916\pi$
$617$	−0.611105			−0.0246022			−0.0123011	+	0.999924i	$0.503916\pi$
$618$		0			0
$619$			10.1899i			0.409566i	0.978807	+	0.204783i	$0.0656488\pi$
$619$			10.1899i			0.409566i	−0.978807	+	0.204783i	$0.934351\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.25768	−	26.8602i	−0.170580	−	1.07613i
$624$		0			0
$625$	−26.8995			−1.07598
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−35.0945			−1.39931
$630$		0			0
$631$	−13.2132			−0.526009			−0.263005	−	0.964795i	$-0.584713\pi$
$631$	−13.2132			−0.526009			−0.263005	+	0.964795i	$0.584713\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	29.0733			1.15374
$636$		0			0
$637$	−38.4221	+	12.4947i	−1.52234	+	0.495058i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	34.0525i		−	1.34499i	−0.740101	−	0.672496i	$-0.765222\pi$
$641$		−	34.0525i		−	1.34499i	0.740101	−	0.672496i	$-0.234778\pi$
$642$		0			0
$643$		−	5.59767i		−	0.220751i	−0.993890	−	0.110375i	$-0.964795\pi$
$643$		−	5.59767i		−	0.220751i	0.993890	−	0.110375i	$-0.0352053\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.2657i			1.42575i	0.701290	+	0.712876i	$0.252608\pi$
$647$			36.2657i			1.42575i	−0.701290	+	0.712876i	$0.747392\pi$
$648$		0			0
$649$	−36.6114			−1.43712
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.1114			0.591353			0.295677	−	0.955288i	$-0.404455\pi$
$653$	15.1114			0.591353			0.295677	+	0.955288i	$0.404455\pi$
$654$		0			0
$655$	27.0711			1.05775
$656$		0			0
$657$		0			0
$658$		0			0
$659$			9.49651i			0.369932i	0.982745	+	0.184966i	$0.0592174\pi$
$659$			9.49651i			0.369932i	−0.982745	+	0.184966i	$0.940783\pi$
$660$		0			0
$661$	−7.75237			−0.301532			−0.150766	−	0.988569i	$-0.548174\pi$
$661$	−7.75237			−0.301532			−0.150766	+	0.988569i	$0.548174\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.3402	−	13.2410i	0.905096	−	0.513463i
$666$		0			0
$667$			23.9175i			0.926091i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			9.30739i			0.359308i
$672$		0			0
$673$			11.4228i			0.440318i	0.975464	+	0.220159i	$0.0706576\pi$
$673$			11.4228i			0.440318i	−0.975464	+	0.220159i	$0.929342\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−30.8369			−1.18516			−0.592578	−	0.805513i	$-0.701890\pi$
$677$	−30.8369			−1.18516			−0.592578	+	0.805513i	$0.701890\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			7.19227i			0.275205i	0.990488	+	0.137602i	$0.0439396\pi$
$683$			7.19227i			0.275205i	−0.990488	+	0.137602i	$0.956060\pi$
$684$		0			0
$685$		−	15.1535i		−	0.578985i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	54.8120i		−	2.08817i
$690$		0			0
$691$			25.4972i			0.969960i	0.874525	+	0.484980i	$0.161173\pi$
$691$			25.4972i			0.969960i	−0.874525	+	0.484980i	$0.838827\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.59890			0.326175
$696$		0			0
$697$			57.7421i			2.18714i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	47.1674			1.78149			0.890744	−	0.454505i	$-0.150184\pi$
$701$	47.1674			1.78149			0.890744	+	0.454505i	$0.150184\pi$
$702$		0			0
$703$	−9.56304	−	25.4972i	−0.360677	−	0.961645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.7164	−	3.12529i	0.741510	−	0.117539i
$708$		0			0
$709$	−33.0711			−1.24201			−0.621005	−	0.783807i	$-0.713275\pi$
$709$	−33.0711			−1.24201			−0.621005	+	0.783807i	$0.713275\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.5366			0.544401
$714$		0			0
$715$			33.8245i			1.26496i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			10.8358i			0.404107i	0.979375	+	0.202053i	$0.0647614\pi$
$719$			10.8358i			0.404107i	−0.979375	+	0.202053i	$0.935239\pi$
$720$		0			0
$721$	3.38104	+	21.3298i	0.125917	+	0.794363i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1.62934i			0.0605123i
$726$		0			0
$727$		−	19.7457i		−	0.732327i	−0.930550	−	0.366164i	$-0.880671\pi$
$727$		−	19.7457i		−	0.732327i	0.930550	−	0.366164i	$-0.119329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	46.3031i		−	1.71258i
$732$		0			0
$733$			22.0643i			0.814964i	0.913213	+	0.407482i	$0.133593\pi$
$733$			22.0643i			0.814964i	−0.913213	+	0.407482i	$0.866407\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22.2517i		−	0.819652i
$738$		0			0
$739$	19.5563			0.719392			0.359696	−	0.933070i	$-0.382880\pi$
$739$	19.5563			0.719392			0.359696	+	0.933070i	$0.382880\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.19227i		−	0.263859i	−0.991259	−	0.131929i	$-0.957883\pi$
$743$		−	7.19227i		−	0.263859i	0.991259	−	0.131929i	$-0.0421172\pi$
$744$		0			0
$745$			28.2960i			1.03669i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	39.3522	−	6.23783i	1.43790	−	0.227925i
$750$		0			0
$751$		−	27.5772i		−	1.00630i	−0.864198	−	0.503152i	$-0.832173\pi$
$751$		−	27.5772i		−	1.00630i	0.864198	−	0.503152i	$-0.167827\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	55.6524			2.02540
$756$		0			0
$757$	−37.6569			−1.36866			−0.684331	−	0.729172i	$-0.739906\pi$
$757$	−37.6569			−1.36866			−0.684331	+	0.729172i	$0.739906\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.52840i		−	0.0554044i	−0.999616	−	0.0277022i	$-0.991181\pi$
$761$		−	1.52840i		−	0.0554044i	0.999616	−	0.0277022i	$-0.00881901\pi$
$762$		0			0
$763$	23.0872	−	3.65962i	0.835813	−	0.132487i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−83.9026			−3.02955
$768$		0			0
$769$			29.4554i			1.06219i	0.847313	+	0.531094i	$0.178219\pi$
$769$			29.4554i			1.06219i	−0.847313	+	0.531094i	$0.821781\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.25768			−0.153138			−0.0765691	−	0.997064i	$-0.524397\pi$
$773$	−4.25768			−0.153138			−0.0765691	+	0.997064i	$0.524397\pi$
$774$		0			0
$775$	0.990285			0.0355721
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−41.9513	+	15.7343i	−1.50306	+	0.563741i
$780$		0			0
$781$		−	23.9175i		−	0.855837i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	45.0810			1.60901
$786$		0			0
$787$	−27.4585			−0.978791			−0.489396	−	0.872062i	$-0.662783\pi$
$787$	−27.4585			−0.978791			−0.489396	+	0.872062i	$0.662783\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	10.2790	−	1.62934i	0.365477	−	0.0579328i
$792$		0			0
$793$			21.3298i			0.757443i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.3002			0.577384			0.288692	−	0.957422i	$-0.406780\pi$
$797$	16.3002			0.577384			0.288692	+	0.957422i	$0.406780\pi$
$798$		0			0
$799$	16.2426			0.574623
$800$		0			0
$801$		0			0
$802$		0			0
$803$			33.3743i			1.17775i
$804$		0			0
$805$	−36.9706	+	5.86030i	−1.30304	+	0.206549i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.7225			−0.552772			−0.276386	−	0.961047i	$-0.589137\pi$
$809$	−15.7225			−0.552772			−0.276386	+	0.961047i	$0.589137\pi$
$810$		0			0
$811$	−5.36161			−0.188272			−0.0941359	−	0.995559i	$-0.530009\pi$
$811$	−5.36161			−0.188272			−0.0941359	+	0.995559i	$0.530009\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			44.3754i			1.55440i
$816$		0			0
$817$	33.6405	−	12.6173i	1.17693	−	0.441423i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.59890			0.300104			0.150052	−	0.988678i	$-0.452056\pi$
$821$	8.59890			0.300104			0.150052	+	0.988678i	$0.452056\pi$
$822$		0			0
$823$	−31.0711			−1.08307			−0.541535	−	0.840678i	$-0.682157\pi$
$823$	−31.0711			−1.08307			−0.541535	+	0.840678i	$0.682157\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	28.4895i		−	0.990678i	−0.868700	−	0.495339i	$-0.835044\pi$
$827$		−	28.4895i		−	0.990678i	0.868700	−	0.495339i	$-0.164956\pi$
$828$		0			0
$829$	−6.76209			−0.234857			−0.117429	−	0.993081i	$-0.537465\pi$
$829$	−6.76209			−0.234857			−0.117429	+	0.993081i	$0.537465\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.1607	+	37.3949i	0.421343	+	1.29566i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.02127			−0.207877			−0.103939	−	0.994584i	$-0.533145\pi$
$839$	−6.02127			−0.207877			−0.103939	+	0.994584i	$0.533145\pi$
$840$		0			0
$841$	13.5269			0.466445
$842$		0			0
$843$		0			0
$844$		0			0
$845$			47.2669i			1.62603i
$846$		0			0
$847$	1.92893	+	12.1689i	0.0662789	+	0.418130i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			37.9861i			1.30215i
$852$		0			0
$853$			54.4273i			1.86356i	0.363029	+	0.931778i	$0.381742\pi$
$853$			54.4273i			1.86356i	−0.363029	+	0.931778i	$0.618258\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.2790			0.351122			0.175561	−	0.984469i	$-0.443826\pi$
$857$	10.2790			0.351122			0.175561	+	0.984469i	$0.443826\pi$
$858$		0			0
$859$			30.9861i			1.05723i	0.848861	+	0.528616i	$0.177289\pi$
$859$			30.9861i			1.05723i	−0.848861	+	0.528616i	$0.822711\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			38.6609i			1.31603i	0.753003	+	0.658017i	$0.228605\pi$
$863$			38.6609i			1.31603i	−0.753003	+	0.658017i	$0.771395\pi$
$864$		0			0
$865$			57.7421i			1.96329i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	15.7343i		−	0.533751i
$870$		0			0
$871$		−	50.9944i		−	1.72788i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	27.8832	−	4.41983i	0.942623	−	0.149418i
$876$		0			0
$877$			33.8245i			1.14217i	0.820890	+	0.571086i	$0.193478\pi$
$877$			33.8245i			1.14217i	−0.820890	+	0.571086i	$0.806522\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5.38364i			0.181380i	0.995879	+	0.0906898i	$0.0289072\pi$
$881$			5.38364i			0.181380i	−0.995879	+	0.0906898i	$0.971093\pi$
$882$		0			0
$883$	−22.6274			−0.761473			−0.380737	−	0.924684i	$-0.624330\pi$
$883$	−22.6274			−0.761473			−0.380737	+	0.924684i	$0.624330\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	55.6524			1.86863			0.934313	−	0.356454i	$-0.116014\pi$
$887$	55.6524			1.86863			0.934313	+	0.356454i	$0.116014\pi$
$888$		0			0
$889$	−32.6502	+	5.17548i	−1.09505	+	0.173580i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.42602	+	11.8008i	0.148111	+	0.394897i
$894$		0			0
$895$	35.0410			1.17129
$896$		0			0
$897$		0			0
$898$		0			0
$899$			9.40425i			0.313649i
$900$		0			0
$901$	−53.3467			−1.77724
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7.86717i		−	0.261514i
$906$		0			0
$907$		−	8.83509i		−	0.293364i	−0.989184	−	0.146682i	$-0.953141\pi$
$907$		−	8.83509i		−	0.293364i	0.989184	−	0.146682i	$-0.0468595\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			44.2239i			1.46520i	0.680658	+	0.732601i	$0.261694\pi$
$911$			44.2239i			1.46520i	−0.680658	+	0.732601i	$0.738306\pi$
$912$		0			0
$913$		−	17.5809i		−	0.581843i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−30.4017	+	4.81906i	−1.00395	+	0.159139i
$918$		0			0
$919$	7.17157			0.236568			0.118284	−	0.992980i	$-0.462261\pi$
$919$	7.17157			0.236568			0.118284	+	0.992980i	$0.462261\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	54.8120i		−	1.80416i
$924$		0			0
$925$			2.58774i			0.0850843i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			5.61750i			0.184304i	0.995745	+	0.0921522i	$0.0293746\pi$
$929$			5.61750i			0.184304i	−0.995745	+	0.0921522i	$0.970625\pi$
$930$		0			0
$931$	−23.8548	+	19.0250i	−0.781809	+	0.623518i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	32.9203			1.07661
$936$		0			0
$937$		−	9.03056i		−	0.295015i	−0.989061	−	0.147508i	$-0.952875\pi$
$937$		−	9.03056i		−	0.295015i	0.989061	−	0.147508i	$-0.0471251\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.25768			0.138796			0.0693982	−	0.997589i	$-0.477892\pi$
$941$	4.25768			0.138796			0.0693982	+	0.997589i	$0.477892\pi$
$942$		0			0
$943$	62.4995			2.03527
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.7657			0.544812			0.272406	−	0.962182i	$-0.412181\pi$
$947$	16.7657			0.544812			0.272406	+	0.962182i	$0.412181\pi$
$948$		0			0
$949$			76.4841i			2.48278i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			9.49651i			0.307622i	0.988100	+	0.153811i	$0.0491547\pi$
$953$			9.49651i			0.307622i	−0.988100	+	0.153811i	$0.950845\pi$
$954$		0			0
$955$		−	19.0029i		−	0.614918i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.69755	+	17.0179i	0.0871084	+	0.549536i
$960$		0			0
$961$	−25.2843			−0.815622
$962$		0			0
$963$		0			0
$964$		0			0
$965$	35.0945			1.12973
$966$		0			0
$967$	16.0416			0.515864			0.257932	−	0.966163i	$-0.416959\pi$
$967$	16.0416			0.515864			0.257932	+	0.966163i	$0.416959\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.02127			0.193232			0.0966159	−	0.995322i	$-0.469198\pi$
$971$	6.02127			0.193232			0.0966159	+	0.995322i	$0.469198\pi$
$972$		0			0
$973$	−9.65685	+	1.53073i	−0.309585	+	0.0490731i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			28.4895i			0.911461i	0.890118	+	0.455731i	$0.150622\pi$
$977$			28.4895i			0.911461i	−0.890118	+	0.455731i	$0.849378\pi$
$978$		0			0
$979$	25.8882			0.827389
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.02127			−0.192049			−0.0960244	−	0.995379i	$-0.530613\pi$
$983$	−6.02127			−0.192049			−0.0960244	+	0.995379i	$0.530613\pi$
$984$		0			0
$985$			28.2960i			0.901587i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−50.1181			−1.59366
$990$		0			0
$991$			8.83509i			0.280656i	0.990105	+	0.140328i	$0.0448157\pi$
$991$			8.83509i			0.280656i	−0.990105	+	0.140328i	$0.955184\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.1485			−0.638750
$996$		0			0
$997$			38.4859i			1.21886i	0.792839	+	0.609431i	$0.208602\pi$
$997$			38.4859i			1.21886i	−0.792839	+	0.609431i	$0.791398\pi$
$998$		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.3457.14	yes	16
3.2	odd	2	inner	4788.2.i.h.3457.2	yes	16
7.6	odd	2	inner	4788.2.i.h.3457.3	yes	16
19.18	odd	2	inner	4788.2.i.h.3457.13	yes	16
21.20	even	2	inner	4788.2.i.h.3457.15	yes	16
57.56	even	2	inner	4788.2.i.h.3457.1	✓	16
133.132	even	2	inner	4788.2.i.h.3457.4	yes	16
399.398	odd	2	inner	4788.2.i.h.3457.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.i.h.3457.1	✓	16	57.56	even	2	inner
4788.2.i.h.3457.2	yes	16	3.2	odd	2	inner
4788.2.i.h.3457.3	yes	16	7.6	odd	2	inner
4788.2.i.h.3457.4	yes	16	133.132	even	2	inner
4788.2.i.h.3457.13	yes	16	19.18	odd	2	inner
4788.2.i.h.3457.14	yes	16	1.1	even	1	trivial
4788.2.i.h.3457.15	yes	16	21.20	even	2	inner
4788.2.i.h.3457.16	yes	16	399.398	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+2.32685i q^{5} +(-0.414214 - 2.61313i) q^{7} +2.51856 q^{11} +5.77180 q^{13} -5.61750i q^{17} +(4.08128 - 1.53073i) q^{19} -6.08034 q^{23} -0.414214 q^{25} -3.93358i q^{29} -2.39076 q^{31} +(6.08034 - 0.963811i) q^{35} -6.24735i q^{37} -10.2790 q^{41} +8.24264 q^{43} +2.89143i q^{47} +(-6.65685 + 2.16478i) q^{49} -9.49651i q^{53} +5.86030i q^{55} -14.5366 q^{59} +3.69552i q^{61} +13.4301i q^{65} -8.83509i q^{67} -9.49651i q^{71} +13.2513i q^{73} +(-1.04322 - 6.58132i) q^{77} -6.24735i q^{79} -6.98054i q^{83} +13.0711 q^{85} +10.2790 q^{89} +(-2.39076 - 15.0824i) q^{91} +(3.56178 + 9.49651i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7} + 16 q^{25} + 64 q^{43} - 16 q^{49} + 96 q^{85}+O(q^{100}) \) 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

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