LMFDB - Embedded newform 576.3.m.b.271.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,3,Mod(271,576)] 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("576.271"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$15.6948632272$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 $ x^16 - 6x^14 + 10x^12 + 88x^10 - 752x^8 + 1408x^6 + 2560x^4 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			271.2
Root			$1.66730 + 1.10459i$ of defining polynomial
Character	$\chi$	$=$	576.271
Dual form			576.3.m.b.559.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.23991 - 4.23991i) q^{5} +0.262225 q^{7} +(8.60531 - 8.60531i) q^{11} +(-15.9957 + 15.9957i) q^{13} -3.51534 q^{17} +(-10.7566 - 10.7566i) q^{19} +16.4968 q^{23} +10.9536i q^{25} +(-25.9522 + 25.9522i) q^{29} +46.2072i q^{31} +(-1.11181 - 1.11181i) q^{35} +(-2.99313 - 2.99313i) q^{37} +21.9026i q^{41} +(-48.7016 + 48.7016i) q^{43} +70.7760i q^{47} -48.9312 q^{49} +(-52.8193 - 52.8193i) q^{53} -72.9715 q^{55} +(61.7726 - 61.7726i) q^{59} +(-22.9004 + 22.9004i) q^{61} +135.640 q^{65} +(54.9939 + 54.9939i) q^{67} -84.2532 q^{71} +78.0341i q^{73} +(2.25653 - 2.25653i) q^{77} -59.2887i q^{79} +(-111.661 - 111.661i) q^{83} +(14.9047 + 14.9047i) q^{85} -34.5426i q^{89} +(-4.19447 + 4.19447i) q^{91} +91.2142i q^{95} -66.0805 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{19} + 96 q^{37} + 32 q^{43} + 112 q^{49} + 256 q^{55} - 32 q^{61} + 256 q^{67} + 160 q^{85} - 288 q^{91}+O(q^{100}) $ 16 * q + 32 * q^19 + 96 * q^37 + 32 * q^43 + 112 * q^49 + 256 * q^55 - 32 * q^61 + 256 * q^67 + 160 * q^85 - 288 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.23991	−	4.23991i	−0.847982	−	0.847982i	0.141900	−	0.989881i	$-0.454679\pi$
$5$	−4.23991	−	4.23991i	−0.847982	−	0.847982i	−0.989881	+	0.141900i	$0.954679\pi$
$6$		0			0
$7$	0.262225			0.0374608			0.0187304	−	0.999825i	$-0.494038\pi$
$7$	0.262225			0.0374608			0.0187304	+	0.999825i	$0.494038\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.60531	−	8.60531i	0.782301	−	0.782301i	−0.197917	−	0.980219i	$-0.563418\pi$
$11$	8.60531	−	8.60531i	0.782301	−	0.782301i	0.980219	+	0.197917i	$0.0634178\pi$
$12$		0			0
$13$	−15.9957	+	15.9957i	−1.23044	+	1.23044i	−0.266640	+	0.963796i	$0.585913\pi$
$13$	−15.9957	+	15.9957i	−1.23044	+	1.23044i	−0.963796	+	0.266640i	$0.914087\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.51534			−0.206785			−0.103392	−	0.994641i	$-0.532970\pi$
$17$	−3.51534			−0.206785			−0.103392	+	0.994641i	$0.532970\pi$
$18$		0			0
$19$	−10.7566	−	10.7566i	−0.566138	−	0.566138i	0.364906	−	0.931044i	$-0.381101\pi$
$19$	−10.7566	−	10.7566i	−0.566138	−	0.566138i	−0.931044	+	0.364906i	$0.881101\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	16.4968			0.717253			0.358626	−	0.933481i	$-0.383245\pi$
$23$	16.4968			0.717253			0.358626	+	0.933481i	$0.383245\pi$
$24$		0			0
$25$			10.9536i			0.438145i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−25.9522	+	25.9522i	−0.894903	+	0.894903i	−0.994980	−	0.100077i	$-0.968091\pi$
$29$	−25.9522	+	25.9522i	−0.894903	+	0.894903i	0.100077	+	0.994980i	$0.468091\pi$
$30$		0			0
$31$			46.2072i			1.49055i	0.666755	+	0.745277i	$0.267683\pi$
$31$			46.2072i			1.49055i	−0.666755	+	0.745277i	$0.732317\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.11181	−	1.11181i	−0.0317660	−	0.0317660i
$36$		0			0
$37$	−2.99313	−	2.99313i	−0.0808955	−	0.0808955i	0.665501	−	0.746397i	$-0.268218\pi$
$37$	−2.99313	−	2.99313i	−0.0808955	−	0.0808955i	−0.746397	+	0.665501i	$0.768218\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			21.9026i			0.534210i	0.963667	+	0.267105i	$0.0860670\pi$
$41$			21.9026i			0.534210i	−0.963667	+	0.267105i	$0.913933\pi$
$42$		0			0
$43$	−48.7016	+	48.7016i	−1.13260	+	1.13260i	−0.142851	+	0.989744i	$0.545627\pi$
$43$	−48.7016	+	48.7016i	−1.13260	+	1.13260i	−0.989744	+	0.142851i	$0.954373\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			70.7760i			1.50587i	0.658094	+	0.752936i	$0.271363\pi$
$47$			70.7760i			1.50587i	−0.658094	+	0.752936i	$0.728637\pi$
$48$		0			0
$49$	−48.9312			−0.998597
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−52.8193	−	52.8193i	−0.996590	−	0.996590i	0.00340427	−	0.999994i	$-0.498916\pi$
$53$	−52.8193	−	52.8193i	−0.996590	−	0.996590i	−0.999994	+	0.00340427i	$0.998916\pi$
$54$		0			0
$55$	−72.9715			−1.32675
$56$		0			0
$57$		0			0
$58$		0			0
$59$	61.7726	−	61.7726i	1.04699	−	1.04699i	0.0481541	−	0.998840i	$-0.484666\pi$
$59$	61.7726	−	61.7726i	1.04699	−	1.04699i	0.998840	−	0.0481541i	$-0.0153339\pi$
$60$		0			0
$61$	−22.9004	+	22.9004i	−0.375416	+	0.375416i	−0.869445	−	0.494029i	$-0.835524\pi$
$61$	−22.9004	+	22.9004i	−0.375416	+	0.375416i	0.494029	+	0.869445i	$0.335524\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	135.640			2.08677
$66$		0			0
$67$	54.9939	+	54.9939i	0.820804	+	0.820804i	0.986223	−	0.165419i	$-0.0528977\pi$
$67$	54.9939	+	54.9939i	0.820804	+	0.820804i	−0.165419	+	0.986223i	$0.552898\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−84.2532			−1.18666			−0.593332	−	0.804958i	$-0.702188\pi$
$71$	−84.2532			−1.18666			−0.593332	+	0.804958i	$0.702188\pi$
$72$		0			0
$73$			78.0341i			1.06896i	0.845181	+	0.534480i	$0.179493\pi$
$73$			78.0341i			1.06896i	−0.845181	+	0.534480i	$0.820507\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.25653	−	2.25653i	0.0293056	−	0.0293056i
$78$		0			0
$79$		−	59.2887i		−	0.750490i	−0.926926	−	0.375245i	$-0.877559\pi$
$79$		−	59.2887i		−	0.750490i	0.926926	−	0.375245i	$-0.122441\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−111.661	−	111.661i	−1.34531	−	1.34531i	−0.890676	−	0.454638i	$-0.849769\pi$
$83$	−111.661	−	111.661i	−1.34531	−	1.34531i	−0.454638	−	0.890676i	$-0.650231\pi$
$84$		0			0
$85$	14.9047	+	14.9047i	0.175350	+	0.175350i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	34.5426i		−	0.388119i	−0.980990	−	0.194060i	$-0.937834\pi$
$89$		−	34.5426i		−	0.388119i	0.980990	−	0.194060i	$-0.0621655\pi$
$90$		0			0
$91$	−4.19447	+	4.19447i	−0.0460931	+	0.0460931i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			91.2142i			0.960150i
$96$		0			0
$97$	−66.0805			−0.681242			−0.340621	−	0.940201i	$-0.610637\pi$
$97$	−66.0805			−0.681242			−0.340621	+	0.940201i	$0.610637\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−25.5254	−	25.5254i	−0.252727	−	0.252727i	0.569361	−	0.822088i	$-0.307191\pi$
$101$	−25.5254	−	25.5254i	−0.252727	−	0.252727i	−0.822088	+	0.569361i	$0.807191\pi$
$102$		0			0
$103$	14.2072			0.137934			0.0689670	−	0.997619i	$-0.478030\pi$
$103$	14.2072			0.137934			0.0689670	+	0.997619i	$0.478030\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−21.8489	+	21.8489i	−0.204195	+	0.204195i	−0.801795	−	0.597600i	$-0.796121\pi$
$107$	−21.8489	+	21.8489i	−0.204195	+	0.204195i	0.597600	+	0.801795i	$0.296121\pi$
$108$		0			0
$109$	17.8979	−	17.8979i	0.164201	−	0.164201i	−0.620224	−	0.784425i	$-0.712958\pi$
$109$	17.8979	−	17.8979i	0.164201	−	0.164201i	0.784425	+	0.620224i	$0.212958\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	174.546			1.54465			0.772325	−	0.635227i	$-0.219093\pi$
$113$	174.546			1.54465			0.772325	+	0.635227i	$0.219093\pi$
$114$		0			0
$115$	−69.9450	−	69.9450i	−0.608217	−	0.608217i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.921813			−0.00774632
$120$		0			0
$121$		−	27.1029i		−	0.223991i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−59.5553	+	59.5553i	−0.476442	+	0.476442i
$126$		0			0
$127$			45.3438i			0.357037i	0.983937	+	0.178519i	$0.0571305\pi$
$127$			45.3438i			0.357037i	−0.983937	+	0.178519i	$0.942869\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	32.6213	+	32.6213i	0.249017	+	0.249017i	0.820567	−	0.571550i	$-0.193658\pi$
$131$	32.6213	+	32.6213i	0.249017	+	0.249017i	−0.571550	+	0.820567i	$0.693658\pi$
$132$		0			0
$133$	−2.82066	−	2.82066i	−0.0212080	−	0.0212080i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			115.159i			0.840577i	0.907390	+	0.420289i	$0.138071\pi$
$137$			115.159i			0.840577i	−0.907390	+	0.420289i	$0.861929\pi$
$138$		0			0
$139$	122.841	−	122.841i	0.883748	−	0.883748i	−0.110165	−	0.993913i	$-0.535138\pi$
$139$	122.841	−	122.841i	0.883748	−	0.883748i	0.993913	+	0.110165i	$0.0351379\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			275.296i			1.92514i
$144$		0			0
$145$	220.070			1.51772
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−99.9710	−	99.9710i	−0.670946	−	0.670946i	0.286988	−	0.957934i	$-0.407346\pi$
$149$	−99.9710	−	99.9710i	−0.670946	−	0.670946i	−0.957934	+	0.286988i	$0.907346\pi$
$150$		0			0
$151$	222.762			1.47525			0.737623	−	0.675212i	$-0.235948\pi$
$151$	222.762			1.47525			0.737623	+	0.675212i	$0.235948\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	195.914	−	195.914i	1.26396	−	1.26396i
$156$		0			0
$157$	45.9080	−	45.9080i	0.292408	−	0.292408i	−0.545623	−	0.838031i	$-0.683707\pi$
$157$	45.9080	−	45.9080i	0.292408	−	0.292408i	0.838031	+	0.545623i	$0.183707\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.32588			0.0268689
$162$		0			0
$163$	−37.3289	−	37.3289i	−0.229012	−	0.229012i	0.583268	−	0.812280i	$-0.301774\pi$
$163$	−37.3289	−	37.3289i	−0.229012	−	0.229012i	−0.812280	+	0.583268i	$0.801774\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−71.6232			−0.428882			−0.214441	−	0.976737i	$-0.568793\pi$
$167$	−71.6232			−0.428882			−0.214441	+	0.976737i	$0.568793\pi$
$168$		0			0
$169$		−	342.723i		−	2.02795i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−100.398	+	100.398i	−0.580334	+	0.580334i	−0.934995	−	0.354661i	$-0.884596\pi$
$173$	−100.398	+	100.398i	−0.580334	+	0.580334i	0.354661	+	0.934995i	$0.384596\pi$
$174$		0			0
$175$			2.87232i			0.0164133i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−64.4199	−	64.4199i	−0.359888	−	0.359888i	0.503884	−	0.863771i	$-0.331904\pi$
$179$	−64.4199	−	64.4199i	−0.359888	−	0.359888i	−0.863771	+	0.503884i	$0.831904\pi$
$180$		0			0
$181$	−79.0033	−	79.0033i	−0.436482	−	0.436482i	0.454344	−	0.890826i	$-0.349874\pi$
$181$	−79.0033	−	79.0033i	−0.436482	−	0.436482i	−0.890826	+	0.454344i	$0.849874\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			25.3812i			0.137196i
$186$		0			0
$187$	−30.2506	+	30.2506i	−0.161768	+	0.161768i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	304.422i		−	1.59383i	−0.604089	−	0.796917i	$-0.706463\pi$
$191$		−	304.422i		−	1.59383i	0.604089	−	0.796917i	$-0.293537\pi$
$192$		0			0
$193$	253.689			1.31445			0.657225	−	0.753695i	$-0.271730\pi$
$193$	253.689			1.31445			0.657225	+	0.753695i	$0.271730\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−89.4218	−	89.4218i	−0.453918	−	0.453918i	0.442735	−	0.896653i	$-0.354008\pi$
$197$	−89.4218	−	89.4218i	−0.453918	−	0.453918i	−0.896653	+	0.442735i	$0.854008\pi$
$198$		0			0
$199$	−117.278			−0.589339			−0.294670	−	0.955599i	$-0.595210\pi$
$199$	−117.278			−0.589339			−0.294670	+	0.955599i	$0.595210\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.80533	+	6.80533i	−0.0335238	+	0.0335238i
$204$		0			0
$205$	92.8650	−	92.8650i	0.453000	−	0.453000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−185.128			−0.885781
$210$		0			0
$211$	80.8941	+	80.8941i	0.383384	+	0.383384i	0.872320	−	0.488936i	$-0.162615\pi$
$211$	80.8941	+	80.8941i	0.383384	+	0.383384i	−0.488936	+	0.872320i	$0.662615\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	412.981			1.92084
$216$		0			0
$217$			12.1167i			0.0558373i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	56.2303	−	56.2303i	0.254436	−	0.254436i
$222$		0			0
$223$		−	263.109i		−	1.17986i	−0.807453	−	0.589931i	$-0.799155\pi$
$223$		−	263.109i		−	1.17986i	0.807453	−	0.589931i	$-0.200845\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	112.085	+	112.085i	0.493765	+	0.493765i	0.909490	−	0.415725i	$-0.136472\pi$
$227$	112.085	+	112.085i	0.493765	+	0.493765i	−0.415725	+	0.909490i	$0.636472\pi$
$228$		0			0
$229$	100.869	+	100.869i	0.440477	+	0.440477i	0.892172	−	0.451695i	$-0.149180\pi$
$229$	100.869	+	100.869i	0.440477	+	0.440477i	−0.451695	+	0.892172i	$0.649180\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	325.669i		−	1.39772i	−0.715259	−	0.698860i	$-0.753691\pi$
$233$		−	325.669i		−	1.39772i	0.715259	−	0.698860i	$-0.246309\pi$
$234$		0			0
$235$	300.084	−	300.084i	1.27695	−	1.27695i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	25.2269i		−	0.105552i	−0.998606	−	0.0527760i	$-0.983193\pi$
$239$		−	25.2269i		−	0.105552i	0.998606	−	0.0527760i	$-0.0168069\pi$
$240$		0			0
$241$	−305.987			−1.26966			−0.634828	−	0.772653i	$-0.718929\pi$
$241$	−305.987			−1.26966			−0.634828	+	0.772653i	$0.718929\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	207.464	+	207.464i	0.846792	+	0.846792i
$246$		0			0
$247$	344.119			1.39319
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−99.5293	+	99.5293i	−0.396531	+	0.396531i	−0.877008	−	0.480476i	$-0.840464\pi$
$251$	−99.5293	+	99.5293i	−0.396531	+	0.396531i	0.480476	+	0.877008i	$0.340464\pi$
$252$		0			0
$253$	141.960	−	141.960i	0.561108	−	0.561108i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−265.635			−1.03360			−0.516799	−	0.856107i	$-0.672876\pi$
$257$	−265.635			−1.03360			−0.516799	+	0.856107i	$0.672876\pi$
$258$		0			0
$259$	−0.784876	−	0.784876i	−0.00303041	−	0.00303041i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−206.692			−0.785901			−0.392950	−	0.919560i	$-0.628546\pi$
$263$	−206.692			−0.785901			−0.392950	+	0.919560i	$0.628546\pi$
$264$		0			0
$265$			447.898i			1.69018i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.03470	−	8.03470i	0.0298688	−	0.0298688i	−0.692015	−	0.721883i	$-0.743277\pi$
$269$	8.03470	−	8.03470i	0.0298688	−	0.0298688i	0.721883	+	0.692015i	$0.243277\pi$
$270$		0			0
$271$		−	216.666i		−	0.799507i	−0.916623	−	0.399754i	$-0.869096\pi$
$271$		−	216.666i		−	0.799507i	0.916623	−	0.399754i	$-0.130904\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	94.2595	+	94.2595i	0.342762	+	0.342762i
$276$		0			0
$277$	65.0909	+	65.0909i	0.234985	+	0.234985i	0.814770	−	0.579785i	$-0.196863\pi$
$277$	65.0909	+	65.0909i	0.234985	+	0.234985i	−0.579785	+	0.814770i	$0.696863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			403.952i			1.43755i	0.695242	+	0.718776i	$0.255297\pi$
$281$			403.952i			1.43755i	−0.695242	+	0.718776i	$0.744703\pi$
$282$		0			0
$283$	−245.705	+	245.705i	−0.868214	+	0.868214i	−0.992275	−	0.124061i	$-0.960408\pi$
$283$	−245.705	+	245.705i	−0.868214	+	0.868214i	0.124061	+	0.992275i	$0.460408\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.74342i			0.0200119i
$288$		0			0
$289$	−276.642			−0.957240
$290$		0			0
$291$		0			0
$292$		0			0
$293$	67.6849	+	67.6849i	0.231007	+	0.231007i	0.813113	−	0.582106i	$-0.197771\pi$
$293$	67.6849	+	67.6849i	0.231007	+	0.231007i	−0.582106	+	0.813113i	$0.697771\pi$
$294$		0			0
$295$	−523.821			−1.77566
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−263.878	+	263.878i	−0.882534	+	0.882534i
$300$		0			0
$301$	−12.7708	+	12.7708i	−0.0424279	+	0.0424279i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	194.191			0.636692
$306$		0			0
$307$	4.98162	+	4.98162i	0.0162268	+	0.0162268i	0.715174	−	0.698947i	$-0.246348\pi$
$307$	4.98162	+	4.98162i	0.0162268	+	0.0162268i	−0.698947	+	0.715174i	$0.746348\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−380.390			−1.22312			−0.611560	−	0.791198i	$-0.709458\pi$
$311$	−380.390			−1.22312			−0.611560	+	0.791198i	$0.709458\pi$
$312$		0			0
$313$		−	145.761i		−	0.465691i	−0.972514	−	0.232845i	$-0.925196\pi$
$313$		−	145.761i		−	0.465691i	0.972514	−	0.232845i	$-0.0748036\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	283.175	−	283.175i	0.893295	−	0.893295i	−0.101537	−	0.994832i	$-0.532376\pi$
$317$	283.175	−	283.175i	0.893295	−	0.893295i	0.994832	+	0.101537i	$0.0323759\pi$
$318$		0			0
$319$			446.654i			1.40017i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	37.8132	+	37.8132i	0.117069	+	0.117069i
$324$		0			0
$325$	−175.211	−	175.211i	−0.539110	−	0.539110i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			18.5593i			0.0564111i
$330$		0			0
$331$	−105.963	+	105.963i	−0.320131	+	0.320131i	−0.848817	−	0.528687i	$-0.822685\pi$
$331$	−105.963	+	105.963i	−0.320131	+	0.320131i	0.528687	+	0.848817i	$0.322685\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	466.338i		−	1.39205i
$336$		0			0
$337$	249.581			0.740595			0.370298	−	0.928913i	$-0.379256\pi$
$337$	249.581			0.740595			0.370298	+	0.928913i	$0.379256\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	397.627	+	397.627i	1.16606	+	1.16606i
$342$		0			0
$343$	−25.6801			−0.0748690
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−297.586	+	297.586i	−0.857597	+	0.857597i	−0.991054	−	0.133458i	$-0.957392\pi$
$347$	−297.586	+	297.586i	−0.857597	+	0.857597i	0.133458	+	0.991054i	$0.457392\pi$
$348$		0			0
$349$	−288.797	+	288.797i	−0.827498	+	0.827498i	−0.987170	−	0.159672i	$-0.948956\pi$
$349$	−288.797	+	288.797i	−0.827498	+	0.827498i	0.159672	+	0.987170i	$0.448956\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−255.362			−0.723404			−0.361702	−	0.932294i	$-0.617804\pi$
$353$	−255.362			−0.723404			−0.361702	+	0.932294i	$0.617804\pi$
$354$		0			0
$355$	357.226	+	357.226i	1.00627	+	1.00627i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	381.798			1.06350			0.531752	−	0.846900i	$-0.321534\pi$
$359$	381.798			1.06350			0.531752	+	0.846900i	$0.321534\pi$
$360$		0			0
$361$		−	129.590i		−	0.358975i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	330.857	−	330.857i	0.906459	−	0.906459i
$366$		0			0
$367$		−	18.6026i		−	0.0506884i	−0.999679	−	0.0253442i	$-0.991932\pi$
$367$		−	18.6026i		−	0.0506884i	0.999679	−	0.0253442i	$-0.00806817\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.8506	−	13.8506i	−0.0373330	−	0.0373330i
$372$		0			0
$373$	150.079	+	150.079i	0.402357	+	0.402357i	0.879063	−	0.476706i	$-0.158169\pi$
$373$	150.079	+	150.079i	0.402357	+	0.402357i	−0.476706	+	0.879063i	$0.658169\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	830.245i		−	2.20224i
$378$		0			0
$379$	−379.359	+	379.359i	−1.00095	+	1.00095i	−0.000948851		1.00000i	$0.500302\pi$
$379$	−379.359	+	379.359i	−1.00095	+	1.00095i	−1.00000		0.000948851i	$0.999698\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	427.421i		−	1.11598i	−0.829847	−	0.557991i	$-0.811572\pi$
$383$		−	427.421i		−	1.11598i	0.829847	−	0.557991i	$-0.188428\pi$
$384$		0			0
$385$	−19.1350			−0.0497012
$386$		0			0
$387$		0			0
$388$		0			0
$389$	236.247	+	236.247i	0.607318	+	0.607318i	0.942244	−	0.334926i	$-0.108711\pi$
$389$	236.247	+	236.247i	0.607318	+	0.607318i	−0.334926	+	0.942244i	$0.608711\pi$
$390$		0			0
$391$	−57.9920			−0.148317
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−251.379	+	251.379i	−0.636402	+	0.636402i
$396$		0			0
$397$	−427.593	+	427.593i	−1.07706	+	1.07706i	−0.0802895	+	0.996772i	$0.525584\pi$
$397$	−427.593	+	427.593i	−1.07706	+	1.07706i	−0.996772	+	0.0802895i	$0.974416\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−638.396			−1.59201			−0.796005	−	0.605290i	$-0.793057\pi$
$401$	−638.396			−1.59201			−0.796005	+	0.605290i	$0.793057\pi$
$402$		0			0
$403$	−739.115	−	739.115i	−1.83403	−	1.83403i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−51.5137			−0.126569
$408$		0			0
$409$			273.582i			0.668905i	0.942413	+	0.334453i	$0.108551\pi$
$409$			273.582i			0.668905i	−0.942413	+	0.334453i	$0.891449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.1984	−	16.1984i	0.0392212	−	0.0392212i
$414$		0			0
$415$			946.865i			2.28160i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	150.314	+	150.314i	0.358745	+	0.358745i	0.863350	−	0.504606i	$-0.168362\pi$
$419$	150.314	+	150.314i	0.358745	+	0.358745i	−0.504606	+	0.863350i	$0.668362\pi$
$420$		0			0
$421$	−220.826	−	220.826i	−0.524527	−	0.524527i	0.394408	−	0.918935i	$-0.370950\pi$
$421$	−220.826	−	220.826i	−0.524527	−	0.524527i	−0.918935	+	0.394408i	$0.870950\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	38.5058i		−	0.0906018i
$426$		0			0
$427$	−6.00507	+	6.00507i	−0.0140634	+	0.0140634i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	463.722i		−	1.07592i	−0.842970	−	0.537960i	$-0.819195\pi$
$431$		−	463.722i		−	1.07592i	0.842970	−	0.537960i	$-0.180805\pi$
$432$		0			0
$433$	−310.114			−0.716199			−0.358099	−	0.933683i	$-0.616575\pi$
$433$	−310.114			−0.716199			−0.358099	+	0.933683i	$0.616575\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−177.450	−	177.450i	−0.406064	−	0.406064i
$438$		0			0
$439$	−89.9103			−0.204807			−0.102404	−	0.994743i	$-0.532653\pi$
$439$	−89.9103			−0.204807			−0.102404	+	0.994743i	$0.532653\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−78.9267	+	78.9267i	−0.178164	+	0.178164i	−0.790555	−	0.612391i	$-0.790208\pi$
$443$	−78.9267	+	78.9267i	−0.178164	+	0.178164i	0.612391	+	0.790555i	$0.290208\pi$
$444$		0			0
$445$	−146.458	+	146.458i	−0.329118	+	0.329118i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−187.048			−0.416587			−0.208294	−	0.978066i	$-0.566791\pi$
$449$	−187.048			−0.416587			−0.208294	+	0.978066i	$0.566791\pi$
$450$		0			0
$451$	188.479	+	188.479i	0.417913	+	0.417913i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	35.5683			0.0781722
$456$		0			0
$457$			153.318i			0.335489i	0.985831	+	0.167744i	$0.0536483\pi$
$457$			153.318i			0.335489i	−0.985831	+	0.167744i	$0.946352\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−99.6025	+	99.6025i	−0.216057	+	0.216057i	−0.806835	−	0.590777i	$-0.798821\pi$
$461$	−99.6025	+	99.6025i	−0.216057	+	0.216057i	0.590777	+	0.806835i	$0.298821\pi$
$462$		0			0
$463$		−	255.682i		−	0.552229i	−0.961125	−	0.276114i	$-0.910953\pi$
$463$		−	255.682i		−	0.552229i	0.961125	−	0.276114i	$-0.0890469\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	285.533	+	285.533i	0.611420	+	0.611420i	0.943316	−	0.331896i	$-0.107688\pi$
$467$	285.533	+	285.533i	0.611420	+	0.611420i	−0.331896	+	0.943316i	$0.607688\pi$
$468$		0			0
$469$	14.4208	+	14.4208i	0.0307480	+	0.0307480i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			838.185i			1.77206i
$474$		0			0
$475$	117.824	−	117.824i	0.248051	−	0.248051i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			418.575i			0.873852i	0.899498	+	0.436926i	$0.143933\pi$
$479$			418.575i			0.873852i	−0.899498	+	0.436926i	$0.856067\pi$
$480$		0			0
$481$	95.7543			0.199073
$482$		0			0
$483$		0			0
$484$		0			0
$485$	280.175	+	280.175i	0.577681	+	0.577681i
$486$		0			0
$487$	162.560			0.333800			0.166900	−	0.985974i	$-0.446624\pi$
$487$	162.560			0.333800			0.166900	+	0.985974i	$0.446624\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	559.393	−	559.393i	1.13929	−	1.13929i	0.150715	−	0.988577i	$-0.451842\pi$
$491$	559.393	−	559.393i	1.13929	−	1.13929i	0.988577	−	0.150715i	$-0.0481577\pi$
$492$		0			0
$493$	91.2309	−	91.2309i	0.185052	−	0.185052i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−22.0933			−0.0444534
$498$		0			0
$499$	−247.957	−	247.957i	−0.496908	−	0.496908i	0.413566	−	0.910474i	$-0.364283\pi$
$499$	−247.957	−	247.957i	−0.496908	−	0.496908i	−0.910474	+	0.413566i	$0.864283\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	260.971			0.518829			0.259415	−	0.965766i	$-0.416470\pi$
$503$	260.971			0.518829			0.259415	+	0.965766i	$0.416470\pi$
$504$		0			0
$505$			216.451i			0.428616i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−149.941	+	149.941i	−0.294580	+	0.294580i	−0.838886	−	0.544306i	$-0.816793\pi$
$509$	−149.941	+	149.941i	−0.294580	+	0.294580i	0.544306	+	0.838886i	$0.316793\pi$
$510$		0			0
$511$			20.4625i			0.0400441i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−60.2372	−	60.2372i	−0.116965	−	0.116965i
$516$		0			0
$517$	609.049	+	609.049i	1.17805	+	1.17805i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			391.686i			0.751797i	0.926661	+	0.375899i	$0.122666\pi$
$521$			391.686i			0.751797i	−0.926661	+	0.375899i	$0.877334\pi$
$522$		0			0
$523$	128.962	−	128.962i	0.246582	−	0.246582i	−0.572985	−	0.819566i	$-0.694214\pi$
$523$	128.962	−	128.962i	0.246582	−	0.246582i	0.819566	+	0.572985i	$0.194214\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	162.434i		−	0.308224i
$528$		0			0
$529$	−256.855			−0.485548
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−350.347	−	350.347i	−0.657311	−	0.657311i
$534$		0			0
$535$	185.274			0.346307
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−421.069	+	421.069i	−0.781204	+	0.781204i
$540$		0			0
$541$	71.0166	−	71.0166i	0.131269	−	0.131269i	−0.638420	−	0.769689i	$-0.720411\pi$
$541$	71.0166	−	71.0166i	0.131269	−	0.131269i	0.769689	+	0.638420i	$0.220411\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−151.771			−0.278478
$546$		0			0
$547$	672.846	+	672.846i	1.23007	+	1.23007i	0.963938	+	0.266128i	$0.0857445\pi$
$547$	672.846	+	672.846i	1.23007	+	1.23007i	0.266128	+	0.963938i	$0.414255\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	558.316			1.01328
$552$		0			0
$553$		−	15.5470i		−	0.0281140i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−294.546	+	294.546i	−0.528809	+	0.528809i	−0.920217	−	0.391408i	$-0.871988\pi$
$557$	−294.546	+	294.546i	−0.528809	+	0.528809i	0.391408	+	0.920217i	$0.371988\pi$
$558$		0			0
$559$		−	1558.03i		−	2.78717i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	613.943	+	613.943i	1.09048	+	1.09048i	0.995477	+	0.0950080i	$0.0302877\pi$
$563$	613.943	+	613.943i	1.09048	+	1.09048i	0.0950080	+	0.995477i	$0.469712\pi$
$564$		0			0
$565$	−740.057	−	740.057i	−1.30984	−	1.30984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1044.74i		−	1.83610i	−0.396460	−	0.918052i	$-0.629761\pi$
$569$		−	1044.74i		−	1.83610i	0.396460	−	0.918052i	$-0.370239\pi$
$570$		0			0
$571$	−131.781	+	131.781i	−0.230789	+	0.230789i	−0.813022	−	0.582233i	$-0.802179\pi$
$571$	−131.781	+	131.781i	−0.230789	+	0.230789i	0.582233	+	0.813022i	$0.302179\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			180.700i			0.314261i
$576$		0			0
$577$	−114.453			−0.198359			−0.0991794	−	0.995070i	$-0.531622\pi$
$577$	−114.453			−0.198359			−0.0991794	+	0.995070i	$0.531622\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−29.2804	−	29.2804i	−0.0503965	−	0.0503965i
$582$		0			0
$583$	−909.053			−1.55927
$584$		0			0
$585$		0			0
$586$		0			0
$587$	308.249	−	308.249i	0.525125	−	0.525125i	−0.393990	−	0.919115i	$-0.628905\pi$
$587$	308.249	−	308.249i	0.525125	−	0.525125i	0.919115	+	0.393990i	$0.128905\pi$
$588$		0			0
$589$	497.034	−	497.034i	0.843860	−	0.843860i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−646.173			−1.08967			−0.544834	−	0.838544i	$-0.683407\pi$
$593$	−646.173			−1.08967			−0.544834	+	0.838544i	$0.683407\pi$
$594$		0			0
$595$	3.90840	+	3.90840i	0.00656874	+	0.00656874i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−528.055			−0.881561			−0.440780	−	0.897615i	$-0.645298\pi$
$599$	−528.055			−0.881561			−0.440780	+	0.897615i	$0.645298\pi$
$600$		0			0
$601$		−	280.764i		−	0.467161i	−0.972338	−	0.233580i	$-0.924956\pi$
$601$		−	280.764i		−	0.467161i	0.972338	−	0.233580i	$-0.0750442\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−114.914	+	114.914i	−0.189940	+	0.189940i
$606$		0			0
$607$			662.871i			1.09204i	0.837771	+	0.546022i	$0.183859\pi$
$607$			662.871i			1.09204i	−0.837771	+	0.546022i	$0.816141\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1132.11	−	1132.11i	−1.85288	−	1.85288i
$612$		0			0
$613$	128.389	+	128.389i	0.209443	+	0.209443i	0.804031	−	0.594588i	$-0.202685\pi$
$613$	128.389	+	128.389i	0.209443	+	0.209443i	−0.594588	+	0.804031i	$0.702685\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20.8646i		−	0.0338162i	−0.999857	−	0.0169081i	$-0.994618\pi$
$617$		−	20.8646i		−	0.0338162i	0.999857	−	0.0169081i	$-0.00538228\pi$
$618$		0			0
$619$	472.367	−	472.367i	0.763113	−	0.763113i	−0.213771	−	0.976884i	$-0.568575\pi$
$619$	472.367	−	472.367i	0.763113	−	0.763113i	0.976884	+	0.213771i	$0.0685746\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	9.05795i		−	0.0145393i
$624$		0			0
$625$	778.859			1.24617
$626$		0			0
$627$		0			0
$628$		0			0
$629$	10.5219	+	10.5219i	0.0167280	+	0.0167280i
$630$		0			0
$631$	906.653			1.43685			0.718426	−	0.695604i	$-0.244863\pi$
$631$	906.653			1.43685			0.718426	+	0.695604i	$0.244863\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	192.253	−	192.253i	0.302761	−	0.302761i
$636$		0			0
$637$	782.688	−	782.688i	1.22871	−	1.22871i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1125.11			1.75524			0.877622	−	0.479353i	$-0.159129\pi$
$641$	1125.11			1.75524			0.877622	+	0.479353i	$0.159129\pi$
$642$		0			0
$643$	607.794	+	607.794i	0.945247	+	0.945247i	0.998577	−	0.0533302i	$-0.0169836\pi$
$643$	607.794	+	607.794i	0.945247	+	0.945247i	−0.0533302	+	0.998577i	$0.516984\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	383.377			0.592545			0.296273	−	0.955103i	$-0.404256\pi$
$647$	383.377			0.592545			0.296273	+	0.955103i	$0.404256\pi$
$648$		0			0
$649$		−	1063.15i		−	1.63813i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	755.121	−	755.121i	1.15639	−	1.15639i	0.171141	−	0.985246i	$-0.445254\pi$
$653$	755.121	−	755.121i	1.15639	−	1.15639i	0.985246	−	0.171141i	$-0.0547455\pi$
$654$		0			0
$655$		−	276.622i		−	0.422324i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.8233	−	31.8233i	−0.0482902	−	0.0482902i	0.682549	−	0.730840i	$-0.260871\pi$
$659$	−31.8233	−	31.8233i	−0.0482902	−	0.0482902i	−0.730840	+	0.682549i	$0.760871\pi$
$660$		0			0
$661$	579.890	+	579.890i	0.877292	+	0.877292i	0.993254	−	0.115961i	$-0.0369949\pi$
$661$	579.890	+	579.890i	0.877292	+	0.877292i	−0.115961	+	0.993254i	$0.536995\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			23.9187i			0.0359680i
$666$		0			0
$667$	−428.129	+	428.129i	−0.641872	+	0.641872i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			394.130i			0.587377i
$672$		0			0
$673$	262.375			0.389859			0.194930	−	0.980817i	$-0.437552\pi$
$673$	262.375			0.389859			0.194930	+	0.980817i	$0.437552\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−574.436	−	574.436i	−0.848502	−	0.848502i	0.141444	−	0.989946i	$-0.454825\pi$
$677$	−574.436	−	574.436i	−0.848502	−	0.848502i	−0.989946	+	0.141444i	$0.954825\pi$
$678$		0			0
$679$	−17.3280			−0.0255199
$680$		0			0
$681$		0			0
$682$		0			0
$683$	253.790	−	253.790i	0.371582	−	0.371582i	−0.496471	−	0.868053i	$-0.665371\pi$
$683$	253.790	−	253.790i	0.371582	−	0.371582i	0.868053	+	0.496471i	$0.165371\pi$
$684$		0			0
$685$	488.264	−	488.264i	0.712794	−	0.712794i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1689.76			2.45248
$690$		0			0
$691$	−734.512	−	734.512i	−1.06297	−	1.06297i	−0.997879	−	0.0650902i	$-0.979266\pi$
$691$	−734.512	−	734.512i	−1.06297	−	1.06297i	−0.0650902	−	0.997879i	$-0.520734\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1041.67			−1.49880
$696$		0			0
$697$		−	76.9952i		−	0.110467i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−745.410	+	745.410i	−1.06335	+	1.06335i	−0.0655005	+	0.997853i	$0.520864\pi$
$701$	−745.410	+	745.410i	−1.06335	+	1.06335i	−0.997853	+	0.0655005i	$0.979136\pi$
$702$		0			0
$703$			64.3920i			0.0915961i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.69342	−	6.69342i	−0.00946735	−	0.00946735i
$708$		0			0
$709$	−809.881	−	809.881i	−1.14229	−	1.14229i	−0.988031	−	0.154255i	$-0.950702\pi$
$709$	−809.881	−	809.881i	−1.14229	−	1.14229i	−0.154255	−	0.988031i	$-0.549298\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			762.272i			1.06910i
$714$		0			0
$715$	1167.23	−	1167.23i	1.63249	−	1.63249i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			662.400i			0.921280i	0.887587	+	0.460640i	$0.152380\pi$
$719$			662.400i			0.921280i	−0.887587	+	0.460640i	$0.847620\pi$
$720$		0			0
$721$	3.72549			0.00516711
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−284.271	−	284.271i	−0.392098	−	0.392098i
$726$		0			0
$727$	1253.93			1.72481			0.862403	−	0.506222i	$-0.168958\pi$
$727$	1253.93			1.72481			0.862403	+	0.506222i	$0.168958\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	171.203	−	171.203i	0.234204	−	0.234204i
$732$		0			0
$733$	−509.300	+	509.300i	−0.694816	+	0.694816i	−0.963288	−	0.268472i	$-0.913481\pi$
$733$	−509.300	+	509.300i	−0.694816	+	0.694816i	0.268472	+	0.963288i	$0.413481\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	946.479			1.28423
$738$		0			0
$739$	−818.847	−	818.847i	−1.10805	−	1.10805i	−0.993407	−	0.114640i	$-0.963428\pi$
$739$	−818.847	−	818.847i	−1.10805	−	1.10805i	−0.114640	−	0.993407i	$-0.536572\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1110.03			−1.49398			−0.746991	−	0.664834i	$-0.768502\pi$
$743$	−1110.03			−1.49398			−0.746991	+	0.664834i	$0.768502\pi$
$744$		0			0
$745$			847.736i			1.13790i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.72933	+	5.72933i	−0.00764931	+	0.00764931i
$750$		0			0
$751$		−	34.0597i		−	0.0453525i	−0.999743	−	0.0226762i	$-0.992781\pi$
$751$		−	34.0597i		−	0.0453525i	0.999743	−	0.0226762i	$-0.00721869\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−944.491	−	944.491i	−1.25098	−	1.25098i
$756$		0			0
$757$	398.543	+	398.543i	0.526477	+	0.526477i	0.919520	−	0.393043i	$-0.128578\pi$
$757$	398.543	+	398.543i	0.526477	+	0.526477i	−0.393043	+	0.919520i	$0.628578\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			903.338i			1.18704i	0.804819	+	0.593520i	$0.202262\pi$
$761$			903.338i			1.18704i	−0.804819	+	0.593520i	$0.797738\pi$
$762$		0			0
$763$	4.69328	−	4.69328i	0.00615108	−	0.00615108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1976.19i			2.57652i
$768$		0			0
$769$	−98.9621			−0.128689			−0.0643447	−	0.997928i	$-0.520496\pi$
$769$	−98.9621			−0.128689			−0.0643447	+	0.997928i	$0.520496\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	111.945	+	111.945i	0.144818	+	0.144818i	0.775799	−	0.630980i	$-0.217347\pi$
$773$	111.945	+	111.945i	0.144818	+	0.144818i	−0.630980	+	0.775799i	$0.717347\pi$
$774$		0			0
$775$	−506.137			−0.653080
$776$		0			0
$777$		0			0
$778$		0			0
$779$	235.598	−	235.598i	0.302437	−	0.302437i
$780$		0			0
$781$	−725.025	+	725.025i	−0.928329	+	0.928329i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−389.291			−0.495913
$786$		0			0
$787$	163.931	+	163.931i	0.208299	+	0.208299i	0.803544	−	0.595245i	$-0.202945\pi$
$787$	163.931	+	163.931i	0.208299	+	0.208299i	−0.595245	+	0.803544i	$0.702945\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	45.7703			0.0578638
$792$		0			0
$793$		−	732.614i		−	0.923852i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	271.277	−	271.277i	0.340373	−	0.340373i	−0.516135	−	0.856507i	$-0.672630\pi$
$797$	271.277	−	271.277i	0.340373	−	0.340373i	0.856507	+	0.516135i	$0.172630\pi$
$798$		0			0
$799$		−	248.802i		−	0.311392i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	671.508	+	671.508i	0.836249	+	0.836249i
$804$		0			0
$805$	−18.3414	−	18.3414i	−0.0227843	−	0.0227843i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1505.90i		−	1.86144i	−0.365736	−	0.930718i	$-0.619183\pi$
$809$		−	1505.90i		−	1.86144i	0.365736	−	0.930718i	$-0.380817\pi$
$810$		0			0
$811$	492.677	−	492.677i	0.607493	−	0.607493i	−0.334797	−	0.942290i	$-0.608668\pi$
$811$	492.677	−	492.677i	0.607493	−	0.607493i	0.942290	+	0.334797i	$0.108668\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			316.542i			0.388396i
$816$		0			0
$817$	1047.73			1.28241
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−139.500	−	139.500i	−0.169914	−	0.169914i	0.617027	−	0.786942i	$-0.288337\pi$
$821$	−139.500	−	139.500i	−0.169914	−	0.169914i	−0.786942	+	0.617027i	$0.788337\pi$
$822$		0			0
$823$	1262.09			1.53353			0.766763	−	0.641931i	$-0.221866\pi$
$823$	1262.09			1.53353			0.766763	+	0.641931i	$0.221866\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−179.936	+	179.936i	−0.217577	+	0.217577i	−0.807477	−	0.589900i	$-0.799167\pi$
$827$	−179.936	+	179.936i	−0.217577	+	0.217577i	0.589900	+	0.807477i	$0.299167\pi$
$828$		0			0
$829$	−144.901	+	144.901i	−0.174790	+	0.174790i	−0.789080	−	0.614290i	$-0.789443\pi$
$829$	−144.901	+	144.901i	−0.174790	+	0.174790i	0.614290	+	0.789080i	$0.289443\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	172.010			0.206495
$834$		0			0
$835$	303.676	+	303.676i	0.363684	+	0.363684i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1006.03			1.19908			0.599540	−	0.800345i	$-0.295350\pi$
$839$	1006.03			1.19908			0.599540	+	0.800345i	$0.295350\pi$
$840$		0			0
$841$		−	506.033i		−	0.601703i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1453.11	+	1453.11i	−1.71966	+	1.71966i
$846$		0			0
$847$		−	7.10706i		−	0.00839087i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−49.3772	−	49.3772i	−0.0580225	−	0.0580225i
$852$		0			0
$853$	77.4150	+	77.4150i	0.0907561	+	0.0907561i	0.751027	−	0.660271i	$-0.229559\pi$
$853$	77.4150	+	77.4150i	0.0907561	+	0.0907561i	−0.660271	+	0.751027i	$0.729559\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			175.567i			0.204863i	0.994740	+	0.102431i	$0.0326622\pi$
$857$			175.567i			0.204863i	−0.994740	+	0.102431i	$0.967338\pi$
$858$		0			0
$859$	−1058.18	+	1058.18i	−1.23188	+	1.23188i	−0.268634	+	0.963242i	$0.586572\pi$
$859$	−1058.18	+	1058.18i	−1.23188	+	1.23188i	−0.963242	+	0.268634i	$0.913428\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	569.219i		−	0.659581i	−0.944054	−	0.329791i	$-0.893022\pi$
$863$		−	569.219i		−	0.659581i	0.944054	−	0.329791i	$-0.106978\pi$
$864$		0			0
$865$	851.355			0.984225
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−510.198	−	510.198i	−0.587110	−	0.587110i
$870$		0			0
$871$	−1759.33			−2.01989
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−15.6169	+	15.6169i	−0.0178479	+	0.0178479i
$876$		0			0
$877$	−166.695	+	166.695i	−0.190074	+	0.190074i	−0.795728	−	0.605654i	$-0.792912\pi$
$877$	−166.695	+	166.695i	−0.190074	+	0.190074i	0.605654	+	0.795728i	$0.292912\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1495.53			1.69754			0.848771	−	0.528761i	$-0.177343\pi$
$881$	1495.53			1.69754			0.848771	+	0.528761i	$0.177343\pi$
$882$		0			0
$883$	−134.367	−	134.367i	−0.152171	−	0.152171i	0.626916	−	0.779087i	$-0.284317\pi$
$883$	−134.367	−	134.367i	−0.152171	−	0.152171i	−0.779087	+	0.626916i	$0.784317\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	414.255			0.467030			0.233515	−	0.972353i	$-0.424977\pi$
$887$	414.255			0.467030			0.233515	+	0.972353i	$0.424977\pi$
$888$		0			0
$889$			11.8903i			0.0133749i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	761.311	−	761.311i	0.852531	−	0.852531i
$894$		0			0
$895$			546.269i			0.610356i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1199.18	−	1199.18i	−1.33390	−	1.33390i
$900$		0			0
$901$	185.678	+	185.678i	0.206080	+	0.206080i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			669.933i			0.740258i
$906$		0			0
$907$	312.683	−	312.683i	0.344744	−	0.344744i	−0.513403	−	0.858148i	$-0.671615\pi$
$907$	312.683	−	312.683i	0.344744	−	0.344744i	0.858148	+	0.513403i	$0.171615\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	391.366i		−	0.429601i	−0.976658	−	0.214800i	$-0.931090\pi$
$911$		−	391.366i		−	0.429601i	0.976658	−	0.214800i	$-0.0689101\pi$
$912$		0			0
$913$	−1921.76			−2.10488
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.55413	+	8.55413i	0.00932838	+	0.00932838i
$918$		0			0
$919$	−294.161			−0.320088			−0.160044	−	0.987110i	$-0.551164\pi$
$919$	−294.161			−0.320088			−0.160044	+	0.987110i	$0.551164\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1347.69	−	1347.69i	1.46011	−	1.46011i
$924$		0			0
$925$	32.7857	−	32.7857i	0.0354440	−	0.0354440i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1015.40			−1.09300			−0.546501	−	0.837458i	$-0.684041\pi$
$929$	−1015.40			−1.09300			−0.546501	+	0.837458i	$0.684041\pi$
$930$		0			0
$931$	526.335	+	526.335i	0.565344	+	0.565344i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	256.520			0.274353
$936$		0			0
$937$			1582.09i			1.68846i	0.535977	+	0.844232i	$0.319943\pi$
$937$			1582.09i			1.68846i	−0.535977	+	0.844232i	$0.680057\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−113.842	+	113.842i	−0.120980	+	0.120980i	−0.765005	−	0.644025i	$-0.777263\pi$
$941$	−113.842	+	113.842i	−0.120980	+	0.120980i	0.644025	+	0.765005i	$0.277263\pi$
$942$		0			0
$943$			361.323i			0.383163i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	388.664	+	388.664i	0.410416	+	0.410416i	0.881884	−	0.471467i	$-0.156275\pi$
$947$	388.664	+	388.664i	0.410416	+	0.410416i	−0.471467	+	0.881884i	$0.656275\pi$
$948$		0			0
$949$	−1248.21	−	1248.21i	−1.31529	−	1.31529i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	211.977i		−	0.222431i	−0.993796	−	0.111215i	$-0.964526\pi$
$953$		−	211.977i		−	0.222431i	0.993796	−	0.111215i	$-0.0354744\pi$
$954$		0			0
$955$	−1290.72	+	1290.72i	−1.35154	+	1.35154i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			30.1976i			0.0314887i
$960$		0			0
$961$	−1174.11			−1.22175
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1075.62	−	1075.62i	−1.11463	−	1.11463i
$966$		0			0
$967$	−516.501			−0.534127			−0.267063	−	0.963679i	$-0.586053\pi$
$967$	−516.501			−0.534127			−0.267063	+	0.963679i	$0.586053\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	795.556	−	795.556i	0.819316	−	0.819316i	−0.166693	−	0.986009i	$-0.553309\pi$
$971$	795.556	−	795.556i	0.819316	−	0.819316i	0.986009	+	0.166693i	$0.0533089\pi$
$972$		0			0
$973$	32.2120	−	32.2120i	0.0331059	−	0.0331059i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−512.430			−0.524493			−0.262246	−	0.965001i	$-0.584463\pi$
$977$	−512.430			−0.524493			−0.262246	+	0.965001i	$0.584463\pi$
$978$		0			0
$979$	−297.250	−	297.250i	−0.303626	−	0.303626i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−722.200			−0.734690			−0.367345	−	0.930085i	$-0.619733\pi$
$983$	−722.200			−0.734690			−0.367345	+	0.930085i	$0.619733\pi$
$984$		0			0
$985$			758.281i			0.769828i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−803.421	+	803.421i	−0.812357	+	0.812357i
$990$		0			0
$991$			794.715i			0.801932i	0.916093	+	0.400966i	$0.131325\pi$
$991$			794.715i			0.801932i	−0.916093	+	0.400966i	$0.868675\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	497.250	+	497.250i	0.499749	+	0.499749i
$996$		0			0
$997$	1080.18	+	1080.18i	1.08343	+	1.08343i	0.996187	+	0.0872401i	$0.0278047\pi$
$997$	1080.18	+	1080.18i	1.08343	+	1.08343i	0.0872401	+	0.996187i	$0.472195\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.3.m.b.271.2		16
3.2	odd	2	inner	576.3.m.b.271.7		16
4.3	odd	2		144.3.m.b.91.3	yes	16
8.3	odd	2		1152.3.m.e.415.7		16
8.5	even	2		1152.3.m.d.415.7		16
12.11	even	2		144.3.m.b.91.6	yes	16
16.3	odd	4	inner	576.3.m.b.559.2		16
16.5	even	4		1152.3.m.e.991.7		16
16.11	odd	4		1152.3.m.d.991.7		16
16.13	even	4		144.3.m.b.19.3	✓	16
24.5	odd	2		1152.3.m.d.415.2		16
24.11	even	2		1152.3.m.e.415.2		16
48.5	odd	4		1152.3.m.e.991.2		16
48.11	even	4		1152.3.m.d.991.2		16
48.29	odd	4		144.3.m.b.19.6	yes	16
48.35	even	4	inner	576.3.m.b.559.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.m.b.19.3	✓	16	16.13	even	4
144.3.m.b.19.6	yes	16	48.29	odd	4
144.3.m.b.91.3	yes	16	4.3	odd	2
144.3.m.b.91.6	yes	16	12.11	even	2
576.3.m.b.271.2		16	1.1	even	1	trivial
576.3.m.b.271.7		16	3.2	odd	2	inner
576.3.m.b.559.2		16	16.3	odd	4	inner
576.3.m.b.559.7		16	48.35	even	4	inner
1152.3.m.d.415.2		16	24.5	odd	2
1152.3.m.d.415.7		16	8.5	even	2
1152.3.m.d.991.2		16	48.11	even	4
1152.3.m.d.991.7		16	16.11	odd	4
1152.3.m.e.415.2		16	24.11	even	2
1152.3.m.e.415.7		16	8.3	odd	2
1152.3.m.e.991.2		16	48.5	odd	4
1152.3.m.e.991.7		16	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.23991 - 4.23991i) q^{5} +0.262225 q^{7} +(8.60531 - 8.60531i) q^{11} +(-15.9957 + 15.9957i) q^{13} -3.51534 q^{17} +(-10.7566 - 10.7566i) q^{19} +16.4968 q^{23} +10.9536i q^{25} +(-25.9522 + 25.9522i) q^{29} +46.2072i q^{31} +(-1.11181 - 1.11181i) q^{35} +(-2.99313 - 2.99313i) q^{37} +21.9026i q^{41} +(-48.7016 + 48.7016i) q^{43} +70.7760i q^{47} -48.9312 q^{49} +(-52.8193 - 52.8193i) q^{53} -72.9715 q^{55} +(61.7726 - 61.7726i) q^{59} +(-22.9004 + 22.9004i) q^{61} +135.640 q^{65} +(54.9939 + 54.9939i) q^{67} -84.2532 q^{71} +78.0341i q^{73} +(2.25653 - 2.25653i) q^{77} -59.2887i q^{79} +(-111.661 - 111.661i) q^{83} +(14.9047 + 14.9047i) q^{85} -34.5426i q^{89} +(-4.19447 + 4.19447i) q^{91} +91.2142i q^{95} -66.0805 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{19} + 96 q^{37} + 32 q^{43} + 112 q^{49} + 256 q^{55} - 32 q^{61} + 256 q^{67} + 160 q^{85} - 288 q^{91}+O(q^{100}) \) 16 * q + 32 * q^19 + 96 * q^37 + 32 * q^43 + 112 * q^49 + 256 * q^55 - 32 * q^61 + 256 * q^67 + 160 * q^85 - 288 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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