LMFDB - Embedded newform 3024.2.k.h.1889.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1889,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, 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("3024.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.k (of order $2$, degree $1$, not 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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.h.1889.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{5} +(1.00000 + 2.44949i) q^{7} +O(q^{10})$	$q+1.73205 q^{5} +(1.00000 + 2.44949i) q^{7} +4.24264i q^{11} +2.44949i q^{13} +1.73205 q^{17} +2.44949i q^{19} +8.48528i q^{23} -2.00000 q^{25} -4.24264i q^{29} -7.34847i q^{31} +(1.73205 + 4.24264i) q^{35} +1.00000 q^{37} -1.73205 q^{41} -7.00000 q^{43} -12.1244 q^{47} +(-5.00000 + 4.89898i) q^{49} +7.34847i q^{55} -8.66025 q^{59} -2.44949i q^{61} +4.24264i q^{65} +10.0000 q^{67} +8.48528i q^{71} -9.79796i q^{73} +(-10.3923 + 4.24264i) q^{77} -5.00000 q^{79} +12.1244 q^{83} +3.00000 q^{85} +10.3923 q^{89} +(-6.00000 + 2.44949i) q^{91} +4.24264i q^{95} +2.44949i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 8 q^{25} + 4 q^{37} - 28 q^{43} - 20 q^{49} + 40 q^{67} - 20 q^{79} + 12 q^{85} - 24 q^{91}+O(q^{100}) $ 4 * q + 4 * q^7 - 8 * q^25 + 4 * q^37 - 28 * q^43 - 20 * q^49 + 40 * q^67 - 20 * q^79 + 12 * q^85 - 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	1.73205			0.774597			0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205			0.774597			0.387298	+	0.921954i	$0.373408\pi$
$6$		0			0
$7$	1.00000	+	2.44949i	0.377964	+	0.925820i
$7$	1.00000	+	2.44949i	0.377964	+	0.925820i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.24264i			1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$			4.24264i			1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$		0			0
$13$			2.44949i			0.679366i	0.940540	+	0.339683i	$0.110320\pi$
$13$			2.44949i			0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.73205			0.420084			0.210042	−	0.977692i	$-0.432640\pi$
$17$	1.73205			0.420084			0.210042	+	0.977692i	$0.432640\pi$
$18$		0			0
$19$			2.44949i			0.561951i	0.959715	+	0.280976i	$0.0906580\pi$
$19$			2.44949i			0.561951i	−0.959715	+	0.280976i	$0.909342\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.48528i			1.76930i	0.466252	+	0.884652i	$0.345604\pi$
$23$			8.48528i			1.76930i	−0.466252	+	0.884652i	$0.654396\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.24264i		−	0.787839i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$		−	4.24264i		−	0.787839i	0.919145	−	0.393919i	$-0.128881\pi$
$30$		0			0
$31$		−	7.34847i		−	1.31982i	−0.751343	−	0.659912i	$-0.770594\pi$
$31$		−	7.34847i		−	1.31982i	0.751343	−	0.659912i	$-0.229406\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.73205	+	4.24264i	0.292770	+	0.717137i
$36$		0			0
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.73205			−0.270501			−0.135250	−	0.990811i	$-0.543184\pi$
$41$	−1.73205			−0.270501			−0.135250	+	0.990811i	$0.543184\pi$
$42$		0			0
$43$	−7.00000			−1.06749			−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000			−1.06749			−0.533745	+	0.845645i	$0.679216\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.1244			−1.76852			−0.884260	−	0.466996i	$-0.845336\pi$
$47$	−12.1244			−1.76852			−0.884260	+	0.466996i	$0.845336\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			7.34847i			0.990867i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.66025			−1.12747			−0.563735	−	0.825956i	$-0.690636\pi$
$59$	−8.66025			−1.12747			−0.563735	+	0.825956i	$0.690636\pi$
$60$		0			0
$61$		−	2.44949i		−	0.313625i	−0.987628	−	0.156813i	$-0.949878\pi$
$61$		−	2.44949i		−	0.313625i	0.987628	−	0.156813i	$-0.0501218\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.24264i			0.526235i
$66$		0			0
$67$	10.0000			1.22169			0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000			1.22169			0.610847	+	0.791748i	$0.290829\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			8.48528i			1.00702i	0.863990	+	0.503509i	$0.167958\pi$
$71$			8.48528i			1.00702i	−0.863990	+	0.503509i	$0.832042\pi$
$72$		0			0
$73$		−	9.79796i		−	1.14676i	−0.819288	−	0.573382i	$-0.805631\pi$
$73$		−	9.79796i		−	1.14676i	0.819288	−	0.573382i	$-0.194369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.3923	+	4.24264i	−1.18431	+	0.483494i
$78$		0			0
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.1244			1.33082			0.665410	−	0.746478i	$-0.268257\pi$
$83$	12.1244			1.33082			0.665410	+	0.746478i	$0.268257\pi$
$84$		0			0
$85$	3.00000			0.325396
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$	−6.00000	+	2.44949i	−0.628971	+	0.256776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			4.24264i			0.435286i
$96$		0			0
$97$			2.44949i			0.248708i	0.992238	+	0.124354i	$0.0396858\pi$
$97$			2.44949i			0.248708i	−0.992238	+	0.124354i	$0.960314\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3923			−1.03407			−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923			−1.03407			−0.517036	+	0.855963i	$0.672965\pi$
$102$		0			0
$103$		−	19.5959i		−	1.93084i	−0.260694	−	0.965422i	$-0.583951\pi$
$103$		−	19.5959i		−	1.93084i	0.260694	−	0.965422i	$-0.416049\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.48528i		−	0.820303i	−0.912017	−	0.410152i	$-0.865476\pi$
$107$		−	8.48528i		−	0.820303i	0.912017	−	0.410152i	$-0.134524\pi$
$108$		0			0
$109$	13.0000			1.24517			0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000			1.24517			0.622587	+	0.782551i	$0.286082\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12.7279i			1.19734i	0.800995	+	0.598671i	$0.204304\pi$
$113$			12.7279i			1.19734i	−0.800995	+	0.598671i	$0.795696\pi$
$114$		0			0
$115$			14.6969i			1.37050i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.73205	+	4.24264i	0.158777	+	0.388922i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.1244			−1.08444
$126$		0			0
$127$	13.0000			1.15356			0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000			1.15356			0.576782	+	0.816898i	$0.304308\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.3923			−0.907980			−0.453990	−	0.891007i	$-0.650000\pi$
$131$	−10.3923			−0.907980			−0.453990	+	0.891007i	$0.650000\pi$
$132$		0			0
$133$	−6.00000	+	2.44949i	−0.520266	+	0.212398i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			21.2132i			1.81237i	0.422885	+	0.906183i	$0.361017\pi$
$137$			21.2132i			1.81237i	−0.422885	+	0.906183i	$0.638983\pi$
$138$		0			0
$139$		−	12.2474i		−	1.03882i	−0.854527	−	0.519408i	$-0.826153\pi$
$139$		−	12.2474i		−	1.03882i	0.854527	−	0.519408i	$-0.173847\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.3923			−0.869048
$144$		0			0
$145$		−	7.34847i		−	0.610257i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			12.7279i			1.04271i	0.853339	+	0.521356i	$0.174574\pi$
$149$			12.7279i			1.04271i	−0.853339	+	0.521356i	$0.825426\pi$
$150$		0			0
$151$	5.00000			0.406894			0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000			0.406894			0.203447	+	0.979086i	$0.434786\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	12.7279i		−	1.02233i
$156$		0			0
$157$			7.34847i			0.586472i	0.956040	+	0.293236i	$0.0947321\pi$
$157$			7.34847i			0.586472i	−0.956040	+	0.293236i	$0.905268\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20.7846	+	8.48528i	−1.63806	+	0.668734i
$162$		0			0
$163$	1.00000			0.0783260			0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000			0.0783260			0.0391630	+	0.999233i	$0.487531\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.0526			1.47433			0.737166	−	0.675712i	$-0.236164\pi$
$167$	19.0526			1.47433			0.737166	+	0.675712i	$0.236164\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$		0			0
$172$		0			0
$173$	20.7846			1.58022			0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846			1.58022			0.790112	+	0.612962i	$0.210022\pi$
$174$		0			0
$175$	−2.00000	−	4.89898i	−0.151186	−	0.370328i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4.24264i		−	0.317110i	−0.987350	−	0.158555i	$-0.949317\pi$
$179$		−	4.24264i		−	0.317110i	0.987350	−	0.158555i	$-0.0506835\pi$
$180$		0			0
$181$		−	4.89898i		−	0.364138i	−0.983286	−	0.182069i	$-0.941721\pi$
$181$		−	4.89898i		−	0.364138i	0.983286	−	0.182069i	$-0.0582795\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.73205			0.127343
$186$		0			0
$187$			7.34847i			0.537373i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.48528i			0.613973i	0.951714	+	0.306987i	$0.0993207\pi$
$191$			8.48528i			0.613973i	−0.951714	+	0.306987i	$0.900679\pi$
$192$		0			0
$193$	7.00000			0.503871			0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000			0.503871			0.251936	+	0.967744i	$0.418933\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.2132i			1.51138i	0.654931	+	0.755689i	$0.272698\pi$
$197$			21.2132i			1.51138i	−0.654931	+	0.755689i	$0.727302\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.3923	−	4.24264i	0.729397	−	0.297775i
$204$		0			0
$205$	−3.00000			−0.209529
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.3923			−0.718851
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.1244			−0.826874
$216$		0			0
$217$	18.0000	−	7.34847i	1.22192	−	0.498847i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.24264i			0.285391i
$222$		0			0
$223$			22.0454i			1.47627i	0.674653	+	0.738135i	$0.264293\pi$
$223$			22.0454i			1.47627i	−0.674653	+	0.738135i	$0.735707\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		−	14.6969i		−	0.971201i	−0.874181	−	0.485601i	$-0.838601\pi$
$229$		−	14.6969i		−	0.971201i	0.874181	−	0.485601i	$-0.161399\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.24264i		−	0.277945i	−0.990296	−	0.138972i	$-0.955620\pi$
$233$		−	4.24264i		−	0.277945i	0.990296	−	0.138972i	$-0.0443799\pi$
$234$		0			0
$235$	−21.0000			−1.36989
$236$		0			0
$237$		0			0
$238$		0			0
$239$			21.2132i			1.37217i	0.727522	+	0.686084i	$0.240672\pi$
$239$			21.2132i			1.37217i	−0.727522	+	0.686084i	$0.759328\pi$
$240$		0			0
$241$			26.9444i			1.73564i	0.496878	+	0.867820i	$0.334480\pi$
$241$			26.9444i			1.73564i	−0.496878	+	0.867820i	$0.665520\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.66025	+	8.48528i	−0.553283	+	0.542105i
$246$		0			0
$247$	−6.00000			−0.381771
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.0526			1.20259			0.601293	−	0.799028i	$-0.294652\pi$
$251$	19.0526			1.20259			0.601293	+	0.799028i	$0.294652\pi$
$252$		0			0
$253$	−36.0000			−2.26330
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.3923			0.648254			0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923			0.648254			0.324127	+	0.946014i	$0.394929\pi$
$258$		0			0
$259$	1.00000	+	2.44949i	0.0621370	+	0.152204i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			21.2132i			1.30806i	0.756468	+	0.654031i	$0.226923\pi$
$263$			21.2132i			1.30806i	−0.756468	+	0.654031i	$0.773077\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.73205			−0.105605			−0.0528025	−	0.998605i	$-0.516815\pi$
$269$	−1.73205			−0.105605			−0.0528025	+	0.998605i	$0.516815\pi$
$270$		0			0
$271$			4.89898i			0.297592i	0.988868	+	0.148796i	$0.0475397\pi$
$271$			4.89898i			0.297592i	−0.988868	+	0.148796i	$0.952460\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	8.48528i		−	0.511682i
$276$		0			0
$277$	−7.00000			−0.420589			−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000			−0.420589			−0.210295	+	0.977638i	$0.567442\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8.48528i		−	0.506189i	−0.967442	−	0.253095i	$-0.918552\pi$
$281$		−	8.48528i		−	0.506189i	0.967442	−	0.253095i	$-0.0814484\pi$
$282$		0			0
$283$			7.34847i			0.436821i	0.975857	+	0.218411i	$0.0700872\pi$
$283$			7.34847i			0.436821i	−0.975857	+	0.218411i	$0.929913\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.73205	−	4.24264i	−0.102240	−	0.250435i
$288$		0			0
$289$	−14.0000			−0.823529
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.66025			0.505937			0.252969	−	0.967474i	$-0.418593\pi$
$293$	8.66025			0.505937			0.252969	+	0.967474i	$0.418593\pi$
$294$		0			0
$295$	−15.0000			−0.873334
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.7846			−1.20201
$300$		0			0
$301$	−7.00000	−	17.1464i	−0.403473	−	0.988304i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4.24264i		−	0.242933i
$306$		0			0
$307$		−	29.3939i		−	1.67760i	−0.544442	−	0.838799i	$-0.683259\pi$
$307$		−	29.3939i		−	1.67760i	0.544442	−	0.838799i	$-0.316741\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−22.5167			−1.27680			−0.638401	−	0.769704i	$-0.720404\pi$
$311$	−22.5167			−1.27680			−0.638401	+	0.769704i	$0.720404\pi$
$312$		0			0
$313$			24.4949i			1.38453i	0.721642	+	0.692267i	$0.243388\pi$
$313$			24.4949i			1.38453i	−0.721642	+	0.692267i	$0.756612\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	18.0000			1.00781
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.24264i			0.236067i
$324$		0			0
$325$		−	4.89898i		−	0.271746i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.1244	−	29.6985i	−0.668437	−	1.63733i
$330$		0			0
$331$	−13.0000			−0.714545			−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000			−0.714545			−0.357272	+	0.934000i	$0.616293\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.3205			0.946320
$336$		0			0
$337$	25.0000			1.36184			0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000			1.36184			0.680918	+	0.732359i	$0.261581\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	31.1769			1.68832
$342$		0			0
$343$	−17.0000	−	7.34847i	−0.917914	−	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	21.2132i		−	1.13878i	−0.822066	−	0.569392i	$-0.807179\pi$
$347$		−	21.2132i		−	1.13878i	0.822066	−	0.569392i	$-0.192821\pi$
$348$		0			0
$349$			19.5959i			1.04895i	0.851427	+	0.524473i	$0.175738\pi$
$349$			19.5959i			1.04895i	−0.851427	+	0.524473i	$0.824262\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.73205			0.0921878			0.0460939	−	0.998937i	$-0.485323\pi$
$353$	1.73205			0.0921878			0.0460939	+	0.998937i	$0.485323\pi$
$354$		0			0
$355$			14.6969i			0.780033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			33.9411i			1.79134i	0.444715	+	0.895672i	$0.353305\pi$
$359$			33.9411i			1.79134i	−0.444715	+	0.895672i	$0.646695\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	16.9706i		−	0.888280i
$366$		0			0
$367$			4.89898i			0.255725i	0.991792	+	0.127862i	$0.0408116\pi$
$367$			4.89898i			0.255725i	−0.991792	+	0.127862i	$0.959188\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000			0.569558			0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000			0.569558			0.284779	+	0.958593i	$0.408080\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10.3923			0.535231
$378$		0			0
$379$	35.0000			1.79783			0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000			1.79783			0.898915	+	0.438124i	$0.144357\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−19.0526			−0.973540			−0.486770	−	0.873530i	$-0.661825\pi$
$383$	−19.0526			−0.973540			−0.486770	+	0.873530i	$0.661825\pi$
$384$		0			0
$385$	−18.0000	+	7.34847i	−0.917365	+	0.374513i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	29.6985i		−	1.50577i	−0.658150	−	0.752886i	$-0.728661\pi$
$389$		−	29.6985i		−	1.50577i	0.658150	−	0.752886i	$-0.271339\pi$
$390$		0			0
$391$			14.6969i			0.743256i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.66025			−0.435745
$396$		0			0
$397$			29.3939i			1.47524i	0.675218	+	0.737618i	$0.264050\pi$
$397$			29.3939i			1.47524i	−0.675218	+	0.737618i	$0.735950\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.48528i		−	0.423735i	−0.977298	−	0.211867i	$-0.932046\pi$
$401$		−	8.48528i		−	0.423735i	0.977298	−	0.211867i	$-0.0679545\pi$
$402$		0			0
$403$	18.0000			0.896644
$404$		0			0
$405$		0			0
$406$		0			0
$407$			4.24264i			0.210300i
$408$		0			0
$409$		−	39.1918i		−	1.93791i	−0.247234	−	0.968956i	$-0.579522\pi$
$409$		−	39.1918i		−	1.93791i	0.247234	−	0.968956i	$-0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.66025	−	21.2132i	−0.426143	−	1.04383i
$414$		0			0
$415$	21.0000			1.03085
$416$		0			0
$417$		0			0
$418$		0			0
$419$	29.4449			1.43848			0.719238	−	0.694764i	$-0.244491\pi$
$419$	29.4449			1.43848			0.719238	+	0.694764i	$0.244491\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.46410			−0.168034
$426$		0			0
$427$	6.00000	−	2.44949i	0.290360	−	0.118539i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			16.9706i			0.817443i	0.912659	+	0.408722i	$0.134025\pi$
$431$			16.9706i			0.817443i	−0.912659	+	0.408722i	$0.865975\pi$
$432$		0			0
$433$		−	22.0454i		−	1.05943i	−0.848174	−	0.529717i	$-0.822298\pi$
$433$		−	22.0454i		−	1.05943i	0.848174	−	0.529717i	$-0.177702\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.7846			−0.994263
$438$		0			0
$439$			29.3939i			1.40289i	0.712722	+	0.701447i	$0.247462\pi$
$439$			29.3939i			1.40289i	−0.712722	+	0.701447i	$0.752538\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	12.7279i		−	0.604722i	−0.953194	−	0.302361i	$-0.902225\pi$
$443$		−	12.7279i		−	0.604722i	0.953194	−	0.302361i	$-0.0977748\pi$
$444$		0			0
$445$	18.0000			0.853282
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	21.2132i		−	1.00111i	−0.865704	−	0.500556i	$-0.833129\pi$
$449$		−	21.2132i		−	1.00111i	0.865704	−	0.500556i	$-0.166871\pi$
$450$		0			0
$451$		−	7.34847i		−	0.346026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.3923	+	4.24264i	−0.487199	+	0.198898i
$456$		0			0
$457$	−8.00000			−0.374224			−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000			−0.374224			−0.187112	+	0.982339i	$0.559913\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	29.4449			1.37138			0.685692	−	0.727892i	$-0.259500\pi$
$461$	29.4449			1.37138			0.685692	+	0.727892i	$0.259500\pi$
$462$		0			0
$463$	−29.0000			−1.34774			−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000			−1.34774			−0.673872	+	0.738848i	$0.735370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	10.0000	+	24.4949i	0.461757	+	1.13107i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	29.6985i		−	1.36554i
$474$		0			0
$475$		−	4.89898i		−	0.224781i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−39.8372			−1.82021			−0.910103	−	0.414381i	$-0.863998\pi$
$479$	−39.8372			−1.82021			−0.910103	+	0.414381i	$0.863998\pi$
$480$		0			0
$481$			2.44949i			0.111687i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			4.24264i			0.192648i
$486$		0			0
$487$	−22.0000			−0.996915			−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000			−0.996915			−0.498458	+	0.866914i	$0.666100\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.7279i		−	0.574403i	−0.957870	−	0.287202i	$-0.907275\pi$
$491$		−	12.7279i		−	0.574403i	0.957870	−	0.287202i	$-0.0927249\pi$
$492$		0			0
$493$		−	7.34847i		−	0.330958i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−20.7846	+	8.48528i	−0.932317	+	0.380617i
$498$		0			0
$499$	−19.0000			−0.850557			−0.425278	−	0.905063i	$-0.639824\pi$
$499$	−19.0000			−0.850557			−0.425278	+	0.905063i	$0.639824\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.73205			0.0772283			0.0386142	−	0.999254i	$-0.487706\pi$
$503$	1.73205			0.0772283			0.0386142	+	0.999254i	$0.487706\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$		0			0
$508$		0			0
$509$	19.0526			0.844490			0.422245	−	0.906482i	$-0.361242\pi$
$509$	19.0526			0.844490			0.422245	+	0.906482i	$0.361242\pi$
$510$		0			0
$511$	24.0000	−	9.79796i	1.06170	−	0.433436i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	33.9411i		−	1.49562i
$516$		0			0
$517$		−	51.4393i		−	2.26230i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.73205			−0.0758825			−0.0379413	−	0.999280i	$-0.512080\pi$
$521$	−1.73205			−0.0758825			−0.0379413	+	0.999280i	$0.512080\pi$
$522$		0			0
$523$		−	31.8434i		−	1.39241i	−0.717841	−	0.696207i	$-0.754870\pi$
$523$		−	31.8434i		−	1.39241i	0.717841	−	0.696207i	$-0.245130\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.7279i		−	0.554437i
$528$		0			0
$529$	−49.0000			−2.13043
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	4.24264i		−	0.183769i
$534$		0			0
$535$		−	14.6969i		−	0.635404i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.7846	−	21.2132i	−0.895257	−	0.913717i
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	22.5167			0.964508
$546$		0			0
$547$	−41.0000			−1.75303			−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−41.0000			−1.75303			−0.876517	+	0.481371i	$0.840139\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.3923			0.442727
$552$		0			0
$553$	−5.00000	−	12.2474i	−0.212622	−	0.520814i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8.48528i			0.359533i	0.983709	+	0.179766i	$0.0575342\pi$
$557$			8.48528i			0.359533i	−0.983709	+	0.179766i	$0.942466\pi$
$558$		0			0
$559$		−	17.1464i		−	0.725217i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.7846			0.875967			0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846			0.875967			0.437983	+	0.898983i	$0.355693\pi$
$564$		0			0
$565$			22.0454i			0.927457i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	33.9411i		−	1.42289i	−0.702744	−	0.711443i	$-0.748042\pi$
$569$		−	33.9411i		−	1.42289i	0.702744	−	0.711443i	$-0.251958\pi$
$570$		0			0
$571$	25.0000			1.04622			0.523109	−	0.852266i	$-0.324772\pi$
$571$	25.0000			1.04622			0.523109	+	0.852266i	$0.324772\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	16.9706i		−	0.707721i
$576$		0			0
$577$			19.5959i			0.815789i	0.913029	+	0.407894i	$0.133737\pi$
$577$			19.5959i			0.815789i	−0.913029	+	0.407894i	$0.866263\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.1244	+	29.6985i	0.503003	+	1.23210i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	41.5692			1.71575			0.857873	−	0.513862i	$-0.171786\pi$
$587$	41.5692			1.71575			0.857873	+	0.513862i	$0.171786\pi$
$588$		0			0
$589$	18.0000			0.741677
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.66025			0.355634			0.177817	−	0.984064i	$-0.443096\pi$
$593$	8.66025			0.355634			0.177817	+	0.984064i	$0.443096\pi$
$594$		0			0
$595$	3.00000	+	7.34847i	0.122988	+	0.301258i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			4.24264i			0.173350i	0.996237	+	0.0866748i	$0.0276241\pi$
$599$			4.24264i			0.173350i	−0.996237	+	0.0866748i	$0.972376\pi$
$600$		0			0
$601$			22.0454i			0.899251i	0.893217	+	0.449625i	$0.148443\pi$
$601$			22.0454i			0.899251i	−0.893217	+	0.449625i	$0.851557\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.1244			−0.492925
$606$		0			0
$607$			26.9444i			1.09364i	0.837251	+	0.546819i	$0.184162\pi$
$607$			26.9444i			1.09364i	−0.837251	+	0.546819i	$0.815838\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	29.6985i		−	1.20147i
$612$		0			0
$613$	−4.00000			−0.161558			−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000			−0.161558			−0.0807792	+	0.996732i	$0.525741\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			25.4558i			1.02481i	0.858743	+	0.512407i	$0.171246\pi$
$617$			25.4558i			1.02481i	−0.858743	+	0.512407i	$0.828754\pi$
$618$		0			0
$619$		−	2.44949i		−	0.0984533i	−0.998788	−	0.0492267i	$-0.984324\pi$
$619$		−	2.44949i		−	0.0984533i	0.998788	−	0.0492267i	$-0.0156757\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10.3923	+	25.4558i	0.416359	+	1.01987i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.73205			0.0690614
$630$		0			0
$631$	37.0000			1.47295			0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000			1.47295			0.736473	+	0.676467i	$0.236490\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	22.5167			0.893546
$636$		0			0
$637$	−12.0000	−	12.2474i	−0.475457	−	0.485262i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			21.2132i			0.837871i	0.908016	+	0.418936i	$0.137597\pi$
$641$			21.2132i			0.837871i	−0.908016	+	0.418936i	$0.862403\pi$
$642$		0			0
$643$		−	14.6969i		−	0.579591i	−0.957089	−	0.289795i	$-0.906413\pi$
$643$		−	14.6969i		−	0.579591i	0.957089	−	0.289795i	$-0.0935872\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.7846			0.817127			0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846			0.817127			0.408564	+	0.912730i	$0.366030\pi$
$648$		0			0
$649$		−	36.7423i		−	1.44226i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	29.6985i		−	1.16219i	−0.813835	−	0.581096i	$-0.802624\pi$
$653$		−	29.6985i		−	1.16219i	0.813835	−	0.581096i	$-0.197376\pi$
$654$		0			0
$655$	−18.0000			−0.703318
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	16.9706i		−	0.661079i	−0.943792	−	0.330540i	$-0.892769\pi$
$659$		−	16.9706i		−	0.661079i	0.943792	−	0.330540i	$-0.107231\pi$
$660$		0			0
$661$		−	12.2474i		−	0.476371i	−0.971220	−	0.238185i	$-0.923447\pi$
$661$		−	12.2474i		−	0.476371i	0.971220	−	0.238185i	$-0.0765525\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.3923	+	4.24264i	−0.402996	+	0.164523i
$666$		0			0
$667$	36.0000			1.39393
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.3923			0.401190
$672$		0			0
$673$	16.0000			0.616755			0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000			0.616755			0.308377	+	0.951264i	$0.400214\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−31.1769			−1.19823			−0.599113	−	0.800664i	$-0.704480\pi$
$677$	−31.1769			−1.19823			−0.599113	+	0.800664i	$0.704480\pi$
$678$		0			0
$679$	−6.00000	+	2.44949i	−0.230259	+	0.0940028i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	38.1838i		−	1.46106i	−0.682880	−	0.730531i	$-0.739273\pi$
$683$		−	38.1838i		−	1.46106i	0.682880	−	0.730531i	$-0.260727\pi$
$684$		0			0
$685$			36.7423i			1.40385i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	22.0454i		−	0.838647i	−0.907837	−	0.419323i	$-0.862267\pi$
$691$		−	22.0454i		−	0.838647i	0.907837	−	0.419323i	$-0.137733\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	21.2132i		−	0.804663i
$696$		0			0
$697$	−3.00000			−0.113633
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.7279i		−	0.480727i	−0.970683	−	0.240363i	$-0.922733\pi$
$701$		−	12.7279i		−	0.480727i	0.970683	−	0.240363i	$-0.0772666\pi$
$702$		0			0
$703$			2.44949i			0.0923843i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.3923	−	25.4558i	−0.390843	−	0.957366i
$708$		0			0
$709$	49.0000			1.84023			0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000			1.84023			0.920117	+	0.391644i	$0.128094\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	62.3538			2.33517
$714$		0			0
$715$	−18.0000			−0.673162
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.9090			1.22730			0.613649	−	0.789579i	$-0.289701\pi$
$719$	32.9090			1.22730			0.613649	+	0.789579i	$0.289701\pi$
$720$		0			0
$721$	48.0000	−	19.5959i	1.78761	−	0.729790i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			8.48528i			0.315135i
$726$		0			0
$727$		−	7.34847i		−	0.272540i	−0.990672	−	0.136270i	$-0.956489\pi$
$727$		−	7.34847i		−	0.272540i	0.990672	−	0.136270i	$-0.0435114\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.1244			−0.448435
$732$		0			0
$733$		−	29.3939i		−	1.08569i	−0.839834	−	0.542844i	$-0.817348\pi$
$733$		−	29.3939i		−	1.08569i	0.839834	−	0.542844i	$-0.182652\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			42.4264i			1.56280i
$738$		0			0
$739$	−38.0000			−1.39785			−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000			−1.39785			−0.698926	+	0.715194i	$0.746338\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	21.2132i		−	0.778237i	−0.921188	−	0.389118i	$-0.872780\pi$
$743$		−	21.2132i		−	0.778237i	0.921188	−	0.389118i	$-0.127220\pi$
$744$		0			0
$745$			22.0454i			0.807681i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20.7846	−	8.48528i	0.759453	−	0.310045i
$750$		0			0
$751$	−10.0000			−0.364905			−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000			−0.364905			−0.182453	+	0.983215i	$0.558404\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.66025			0.315179
$756$		0			0
$757$	−53.0000			−1.92632			−0.963159	−	0.268933i	$-0.913329\pi$
$757$	−53.0000			−1.92632			−0.963159	+	0.268933i	$0.913329\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.5167			−0.816228			−0.408114	−	0.912931i	$-0.633813\pi$
$761$	−22.5167			−0.816228			−0.408114	+	0.912931i	$0.633813\pi$
$762$		0			0
$763$	13.0000	+	31.8434i	0.470632	+	1.15281i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	21.2132i		−	0.765964i
$768$		0			0
$769$			4.89898i			0.176662i	0.996091	+	0.0883309i	$0.0281533\pi$
$769$			4.89898i			0.176662i	−0.996091	+	0.0883309i	$0.971847\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−39.8372			−1.43284			−0.716422	−	0.697668i	$-0.754221\pi$
$773$	−39.8372			−1.43284			−0.716422	+	0.697668i	$0.754221\pi$
$774$		0			0
$775$			14.6969i			0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	4.24264i		−	0.152008i
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$			12.7279i			0.454279i
$786$		0			0
$787$			4.89898i			0.174630i	0.996181	+	0.0873149i	$0.0278286\pi$
$787$			4.89898i			0.174630i	−0.996181	+	0.0873149i	$0.972171\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−31.1769	+	12.7279i	−1.10852	+	0.452553i
$792$		0			0
$793$	6.00000			0.213066
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.3923			0.368114			0.184057	−	0.982916i	$-0.441077\pi$
$797$	10.3923			0.368114			0.184057	+	0.982916i	$0.441077\pi$
$798$		0			0
$799$	−21.0000			−0.742927
$800$		0			0
$801$		0			0
$802$		0			0
$803$	41.5692			1.46695
$804$		0			0
$805$	−36.0000	+	14.6969i	−1.26883	+	0.517999i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	25.4558i		−	0.894980i	−0.894289	−	0.447490i	$-0.852318\pi$
$809$		−	25.4558i		−	0.894980i	0.894289	−	0.447490i	$-0.147682\pi$
$810$		0			0
$811$			51.4393i			1.80628i	0.429349	+	0.903139i	$0.358743\pi$
$811$			51.4393i			1.80628i	−0.429349	+	0.903139i	$0.641257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.73205			0.0606711
$816$		0			0
$817$		−	17.1464i		−	0.599878i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			25.4558i			0.888415i	0.895924	+	0.444208i	$0.146515\pi$
$821$			25.4558i			0.888415i	−0.895924	+	0.444208i	$0.853485\pi$
$822$		0			0
$823$	25.0000			0.871445			0.435723	−	0.900081i	$-0.356493\pi$
$823$	25.0000			0.871445			0.435723	+	0.900081i	$0.356493\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	38.1838i		−	1.32778i	−0.747830	−	0.663890i	$-0.768904\pi$
$827$		−	38.1838i		−	1.32778i	0.747830	−	0.663890i	$-0.231096\pi$
$828$		0			0
$829$			31.8434i			1.10597i	0.833193	+	0.552983i	$0.186511\pi$
$829$			31.8434i			1.10597i	−0.833193	+	0.552983i	$0.813489\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.66025	+	8.48528i	−0.300060	+	0.293998i
$834$		0			0
$835$	33.0000			1.14201
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.73205			−0.0597970			−0.0298985	−	0.999553i	$-0.509518\pi$
$839$	−1.73205			−0.0597970			−0.0298985	+	0.999553i	$0.509518\pi$
$840$		0			0
$841$	11.0000			0.379310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.1244			0.417091
$846$		0			0
$847$	−7.00000	−	17.1464i	−0.240523	−	0.589158i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			8.48528i			0.290872i
$852$		0			0
$853$		−	19.5959i		−	0.670951i	−0.942049	−	0.335476i	$-0.891103\pi$
$853$		−	19.5959i		−	0.670951i	0.942049	−	0.335476i	$-0.108897\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.1244			0.414160			0.207080	−	0.978324i	$-0.433604\pi$
$857$	12.1244			0.414160			0.207080	+	0.978324i	$0.433604\pi$
$858$		0			0
$859$		−	39.1918i		−	1.33721i	−0.743619	−	0.668604i	$-0.766892\pi$
$859$		−	39.1918i		−	1.33721i	0.743619	−	0.668604i	$-0.233108\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	16.9706i		−	0.577685i	−0.957377	−	0.288842i	$-0.906730\pi$
$863$		−	16.9706i		−	0.577685i	0.957377	−	0.288842i	$-0.0932703\pi$
$864$		0			0
$865$	36.0000			1.22404
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	21.2132i		−	0.719609i
$870$		0			0
$871$			24.4949i			0.829978i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.1244	−	29.6985i	−0.409878	−	1.00399i
$876$		0			0
$877$	−7.00000			−0.236373			−0.118187	−	0.992991i	$-0.537708\pi$
$877$	−7.00000			−0.236373			−0.118187	+	0.992991i	$0.537708\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.1769			1.05038			0.525188	−	0.850986i	$-0.323995\pi$
$881$	31.1769			1.05038			0.525188	+	0.850986i	$0.323995\pi$
$882$		0			0
$883$	17.0000			0.572096			0.286048	−	0.958215i	$-0.407658\pi$
$883$	17.0000			0.572096			0.286048	+	0.958215i	$0.407658\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.0526			0.639722			0.319861	−	0.947464i	$-0.396364\pi$
$887$	19.0526			0.639722			0.319861	+	0.947464i	$0.396364\pi$
$888$		0			0
$889$	13.0000	+	31.8434i	0.436006	+	1.06799i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	29.6985i		−	0.993822i
$894$		0			0
$895$		−	7.34847i		−	0.245632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31.1769			−1.03981
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	8.48528i		−	0.282060i
$906$		0			0
$907$	−13.0000			−0.431658			−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000			−0.431658			−0.215829	+	0.976431i	$0.569245\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			29.6985i			0.983955i	0.870608	+	0.491977i	$0.163726\pi$
$911$			29.6985i			0.983955i	−0.870608	+	0.491977i	$0.836274\pi$
$912$		0			0
$913$			51.4393i			1.70239i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.3923	−	25.4558i	−0.343184	−	0.840626i
$918$		0			0
$919$	5.00000			0.164935			0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000			0.164935			0.0824674	+	0.996594i	$0.473720\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−20.7846			−0.684134
$924$		0			0
$925$	−2.00000			−0.0657596
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−39.8372			−1.30702			−0.653508	−	0.756920i	$-0.726703\pi$
$929$	−39.8372			−1.30702			−0.653508	+	0.756920i	$0.726703\pi$
$930$		0			0
$931$	−12.0000	−	12.2474i	−0.393284	−	0.401394i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			12.7279i			0.416248i
$936$		0			0
$937$		−	2.44949i		−	0.0800213i	−0.999199	−	0.0400107i	$-0.987261\pi$
$937$		−	2.44949i		−	0.0800213i	0.999199	−	0.0400107i	$-0.0127392\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−53.6936			−1.75036			−0.875180	−	0.483797i	$-0.839257\pi$
$941$	−53.6936			−1.75036			−0.875180	+	0.483797i	$0.839257\pi$
$942$		0			0
$943$		−	14.6969i		−	0.478598i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	16.9706i		−	0.551469i	−0.961234	−	0.275735i	$-0.911079\pi$
$947$		−	16.9706i		−	0.551469i	0.961234	−	0.275735i	$-0.0889211\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	50.9117i		−	1.64919i	−0.565723	−	0.824596i	$-0.691403\pi$
$953$		−	50.9117i		−	1.64919i	0.565723	−	0.824596i	$-0.308597\pi$
$954$		0			0
$955$			14.6969i			0.475582i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−51.9615	+	21.2132i	−1.67793	+	0.685010i
$960$		0			0
$961$	−23.0000			−0.741935
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.1244			0.390297
$966$		0			0
$967$	46.0000			1.47926			0.739630	−	0.673014i	$-0.235000\pi$
$967$	46.0000			1.47926			0.739630	+	0.673014i	$0.235000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32.9090			1.05610			0.528049	−	0.849214i	$-0.322924\pi$
$971$	32.9090			1.05610			0.528049	+	0.849214i	$0.322924\pi$
$972$		0			0
$973$	30.0000	−	12.2474i	0.961756	−	0.392635i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			42.4264i			1.35734i	0.734443	+	0.678671i	$0.237444\pi$
$977$			42.4264i			1.35734i	−0.734443	+	0.678671i	$0.762556\pi$
$978$		0			0
$979$			44.0908i			1.40915i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.73205			0.0552438			0.0276219	−	0.999618i	$-0.491207\pi$
$983$	1.73205			0.0552438			0.0276219	+	0.999618i	$0.491207\pi$
$984$		0			0
$985$			36.7423i			1.17071i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	59.3970i		−	1.88871i
$990$		0			0
$991$	−55.0000			−1.74713			−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000			−1.74713			−0.873566	+	0.486705i	$0.838199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	16.9706i		−	0.538003i
$996$		0			0
$997$		−	36.7423i		−	1.16364i	−0.813317	−	0.581821i	$-0.802340\pi$
$997$		−	36.7423i		−	1.16364i	0.813317	−	0.581821i	$-0.197660\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.h.1889.4		4
3.2	odd	2	inner	3024.2.k.h.1889.2		4
4.3	odd	2		756.2.f.d.377.3	yes	4
7.6	odd	2	inner	3024.2.k.h.1889.1		4
12.11	even	2		756.2.f.d.377.1	✓	4
21.20	even	2	inner	3024.2.k.h.1889.3		4
28.27	even	2		756.2.f.d.377.2	yes	4
36.7	odd	6		2268.2.x.j.377.1		8
36.11	even	6		2268.2.x.j.377.3		8
36.23	even	6		2268.2.x.j.1889.4		8
36.31	odd	6		2268.2.x.j.1889.2		8
84.83	odd	2		756.2.f.d.377.4	yes	4
252.83	odd	6		2268.2.x.j.377.2		8
252.139	even	6		2268.2.x.j.1889.3		8
252.167	odd	6		2268.2.x.j.1889.1		8
252.223	even	6		2268.2.x.j.377.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.f.d.377.1	✓	4	12.11	even	2
756.2.f.d.377.2	yes	4	28.27	even	2
756.2.f.d.377.3	yes	4	4.3	odd	2
756.2.f.d.377.4	yes	4	84.83	odd	2
2268.2.x.j.377.1		8	36.7	odd	6
2268.2.x.j.377.2		8	252.83	odd	6
2268.2.x.j.377.3		8	36.11	even	6
2268.2.x.j.377.4		8	252.223	even	6
2268.2.x.j.1889.1		8	252.167	odd	6
2268.2.x.j.1889.2		8	36.31	odd	6
2268.2.x.j.1889.3		8	252.139	even	6
2268.2.x.j.1889.4		8	36.23	even	6
3024.2.k.h.1889.1		4	7.6	odd	2	inner
3024.2.k.h.1889.2		4	3.2	odd	2	inner
3024.2.k.h.1889.3		4	21.20	even	2	inner
3024.2.k.h.1889.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{5} +(1.00000 + 2.44949i) q^{7} +O(q^{10})\)	\(q+1.73205 q^{5} +(1.00000 + 2.44949i) q^{7} +4.24264i q^{11} +2.44949i q^{13} +1.73205 q^{17} +2.44949i q^{19} +8.48528i q^{23} -2.00000 q^{25} -4.24264i q^{29} -7.34847i q^{31} +(1.73205 + 4.24264i) q^{35} +1.00000 q^{37} -1.73205 q^{41} -7.00000 q^{43} -12.1244 q^{47} +(-5.00000 + 4.89898i) q^{49} +7.34847i q^{55} -8.66025 q^{59} -2.44949i q^{61} +4.24264i q^{65} +10.0000 q^{67} +8.48528i q^{71} -9.79796i q^{73} +(-10.3923 + 4.24264i) q^{77} -5.00000 q^{79} +12.1244 q^{83} +3.00000 q^{85} +10.3923 q^{89} +(-6.00000 + 2.44949i) q^{91} +4.24264i q^{95} +2.44949i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 8 q^{25} + 4 q^{37} - 28 q^{43} - 20 q^{49} + 40 q^{67} - 20 q^{79} + 12 q^{85} - 24 q^{91}+O(q^{100}) \) 4 * q + 4 * q^7 - 8 * q^25 + 4 * q^37 - 28 * q^43 - 20 * q^49 + 40 * q^67 - 20 * q^79 + 12 * q^85 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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